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State-Space Models with Exogenous Variables and Missing Data

Permanent Link: http://ufdc.ufl.edu/UFE0021568/00001

Material Information

Title: State-Space Models with Exogenous Variables and Missing Data
Physical Description: 1 online resource (102 p.)
Language: english
Creator: Naranjo, Arlene H
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: algorithm, em, filter, kalman, likelihood, longitudinal, maximum
Statistics -- Dissertations, Academic -- UF
Genre: Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We describe a method to calculate parameter estimates in the state-space model with exogenous variables, with and without missing observations. Exogenous variables are independent of the system and affect the response, but not vice versa. The EM algorithm and the Kalman smoother equations are used in combination to derive maximum likelihood estimates for the model parameters. In the missing data case, two state-space models are proposed to represent the unobserved information that can occur in both the response and exogenous variables. In addition, analytic recursive formulas are derived for calculating parameter estimate standard errors. Simulation studies are performed to determine the effects of varying the number of subjects and time points, differing missing data percentages, and mismatched observations in time. It seems that the exogenous variables are superfluous in the complete case since the previous responses in time appear to be sufficient in predicting future outcomes. However, the exogenous variables add considerable information to the analysis when data are missing and there is strong evidence in favor of including these in the model. The new procedure appears to be relatively robust to moderate percentages of missing data and mismatched observations in time, even with fewer subjects and time points, although several of the variance parameters are being overestimated. The methodology is applied to a data set from an observational study on patients with autoimmune diseases.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Arlene H Naranjo.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Casella, George.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021568:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021568/00001

Material Information

Title: State-Space Models with Exogenous Variables and Missing Data
Physical Description: 1 online resource (102 p.)
Language: english
Creator: Naranjo, Arlene H
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: algorithm, em, filter, kalman, likelihood, longitudinal, maximum
Statistics -- Dissertations, Academic -- UF
Genre: Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We describe a method to calculate parameter estimates in the state-space model with exogenous variables, with and without missing observations. Exogenous variables are independent of the system and affect the response, but not vice versa. The EM algorithm and the Kalman smoother equations are used in combination to derive maximum likelihood estimates for the model parameters. In the missing data case, two state-space models are proposed to represent the unobserved information that can occur in both the response and exogenous variables. In addition, analytic recursive formulas are derived for calculating parameter estimate standard errors. Simulation studies are performed to determine the effects of varying the number of subjects and time points, differing missing data percentages, and mismatched observations in time. It seems that the exogenous variables are superfluous in the complete case since the previous responses in time appear to be sufficient in predicting future outcomes. However, the exogenous variables add considerable information to the analysis when data are missing and there is strong evidence in favor of including these in the model. The new procedure appears to be relatively robust to moderate percentages of missing data and mismatched observations in time, even with fewer subjects and time points, although several of the variance parameters are being overestimated. The methodology is applied to a data set from an observational study on patients with autoimmune diseases.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Arlene H Naranjo.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Casella, George.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021568:00001


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8a93f1739e44994400a2efa99eafdab1
dec4c1cadc71205a628e9d5eba9483ebe01fed67







STATE-SPACE MODELS WITH EXOGENOUS VARIABLES AND MISSING DATA


By
ARLENE HORTENSIA NARANJO



















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007



































S2007 Arlene Hortensia T I1 I.100




































To my late mother, my father, and Amy










ACKENOWLED GMENTS

Writing this dissertation has simultaneously been the most ambitious intellectual

challenge and emotionally demanding experience I have ever undertaken in my life. My

journey has been filled with numerous triumphs but also with many obstacles along the

way. I have been very fortunate to have the support of family and friends to help me up

each time this project has driven me down. Aniv lived through every high and manic low

with me. She was a devoted source of encouragement and I am absolutely convinced that I

would not have made it to the end without her support. I am also forever grateful for the

motivation my father provided me. He ah-- .1-4 believed in me, even when I did not believe

in myself, and this was the only thing that kept me going some d ex-

I was extremely lucky to have had a great coninittee that helped me succeed. In

particular, I would like to thank Dr. Trindade for so selflessly sharing his time. He was

more than willing to guide me when I was stuck and his -II_a-r~!--- 0. ahr-l-.1 led in the right

direction. I am also grateful that Dr. Casella is a statistical encyclopedia and was ahr-7- .

willing to share his expertise in this field with me. My other coninittee nienters, Drs.

Daniels, Ch!~is-I us .1, Sobel, and Richards were also invaluable for improving my thesis with

their insightful ideas. Finally, I would be remiss if I did not thank Dr. Shuster for helping

shape the topic of this project, particularly in the initial stages.











TABLE OF CONTENTS


page

ACK(NOWLEDGMENTS .......... . .. .. 4

LIST OF TABLES ......... ..... .. 6

LIST OF FIGURES ......... .... .. 7

ABSTRACT ......... ...... .. .. 8

CHAPTER

1 INTRODUCTION ......... ... .. 9

2 THE STATE-SPACE APPROACH ........ .. .. 12

2.1 Complete Data Specification ........ .. 12
2.2 Missing Data Adjustments ......... .... 18
2.2.1 The K~alman Recursions ....... .. .. 22
2.2.2 The EM Algorithm ......... ... 27

:3 CALCULATION OF STANDARD ERRORS ..... ... .. :34

:3.1 Complete Data Case ......... . :35
:3.2 Missing Data Case ......... . :37

4 CLARIFICATIONS AND PROOFS . ..... .. 44

4.1 Derivation of Q (8 |8( -0) .... 4
4.2 Estimation of Parameters ...... ... .. 49
4.3 Proof of Exogeneity Theorem ...... .. .. 51
4.4 Proof of Lenina 2.2 ........ .. .. 51
4.5 Proof of Theorem 2.3 ......... .. 5:3

4.6 Q (8,6 O 80 -, OE -0) : Missing Data Case .... ... .. 56
4.7 Parameter Estimation: Missing Data Case .... .. .. 76
4.8 Proof of EC11 Theorem ......... ... .. 81

5 SIMULATION RESITLTS ......... .. .. 8:3

6 DATA ANALYSIS: AN EXAMPLE . ..... .. 91

7 CONCLUSIONS AND FITTIRE RESEARCH .... .... .. 96

REFERENCES ............. ........... 99

BIOGRAPHICAL SK(ETCH ......... .. .. 102











LIST OF TABLES


Table

3-1 Initial Conditions for Complete Data Standard Error Calculations ....

3-2 Initial Conditions for Missingf Data Standard Error Calculations Partial
With Respect to 8 ...

3-3 Initial Conditions for Missingf Data Standard Error Calculations Partial
(of First State-Space Model Functions) With Respect to 0 .. .

3-4 Initial Conditions for Missingf Data Standard Error Calculations Partial


page

. 38

Derivatives
. 42

Derivatives
. 43

Derivatives


(of Second State-Space Model Functions) With Respect to R


5-1 Simulation Subject and Time Point Combinations ...

5-2 True and Initial Parameter Values .....

5-3 Minimum and Maximum Log-Likelihoods: Case with k
time points for all i =1,..., k .....

5-4 Minimum and Maximum Logf-Likelihoods: Case with k
time points for all i =1,..., k .....

5-5 Minimum and Maximum Logf-Likelihoods: Case with k
time points for all i =1,..., k .....

5-6 Minimum and Maximum Logf-Likelihoods: Case with k
time points for all i =1,..., k .....

5-7 Minimum and Maximum Logf-Likelihoods: Mismatched
ni = 15 time points for all i = 1,. ., k .....

6-1 Data Analysis Initial Parameter Values ......

6-2 Data Analysis Results .....


.. 84

. 84

=30 subjects, as = 5
. 86;

= 30 subjects, as = 15
. 86;

= 150 subjects, as = 5
. 87

= 150 subjects, as = 15
. 87

case with k = 150 subjects,
. 90

. 93

. 94










LIST OF FIGURES


Figure


page


6-1 Hs-CRP observations, smoothed values, and forecasts for a typical subject. .. 94










Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

STATE-SPACE MODELS WITH EXOGENOUS VARIABLES AND MISSING DATA

By

Arlene Hortensia ?- I1 I.10

December 2007

Cl. ny~: George Casella
Major: Statistics

We describe a method to calculate parameter estimates in the state-space model

with exogenous variables, with and without missing observations. Exogenous variables

are independent of the system and affect the response, but not vice versa. The EM

algorithm and the K~alman smoother equations are used in combination to derive

nmaxiniun likelihood estimates for the model parameters. In the missing data case, two

state-space models are proposed to represent the unobserved information that can occur

in both the response and exogenous variables. In addition, analytic recursive formulas

are derived for calculating parameter estimate standard errors. Simulation studies are

performed to determine the effects of varying the number of subjects and time points,

differing missing data percentages, and nxisniatched observations in time. It seems that

the exogenous variables are superfluous in the complete case since the previous responses

in time appear to be sufficient in predicting future outcomes. However, the exogenous

variables add considerable information to the analysis when data are missing and there is

strong evidence in favor of including these in the model. The new procedure appears to

be relatively robust to moderate percentages of missing data and nxisniatched observations

in time, even with fewer subjects and time points, although several of the variance

parameters are being overestimated. The methodology is applied to a data set front an

observational study on patients with autoininune diseases.










CHAPTER 1
INTRODUCTION

The integration of state-space models into the field of statistics is attributed in

large part to the development of the K~alman filter. So named because of the significant

paper published by R. E. K~alman in 1960, it is a recursive formula that uses the linear

state-space model with known parameter values for filteringf and forecasting in discrete

time (K~alman, 1960). The advantage of the algorithm is that filtering, prediction, and

smoothing can easily be achieved for any problem cast in state-space form. The goal

of the procedure is to use the observations to estimate the underlying states, which are

unobserved. The K~alman filter was first applied to the problem of projectile tracking for

NASA's Apollo program (K~edem and Fokianos, 2002), and since then has been prevalent

in navigational systems and in the tracking of the trajectory of submarines, aircraft

carriers, and ballistic missiles. K~alman and Bucy (1961) increased the applications of

the state-space model and K~alman filter by expanding the system to the continuous time

domain.

The state-space formulation was thereafter found to be useful in areas other than

guidance systems. Autoregressive integrated moving average (ARIMA) models in time

series can he represented in state-space form, providing a unified approach to this wide

class of models (Brockwell and Davis, 2002). Thus, the flexibility of the state-space

approach allows for analysis of processes that are not necessarily stationary. These

versatile models are widely used beyond the field of statistics, where they are also known

as structural equation models (SEM) or dynamic linear models (DLM). See Shumway

and Stoffer (1982) for an application to economics data, Harvey (1991) for analyses in the

social sciences, or Jones (1984) for an example of the model's use in analyzing biomedical

data.

Further demonstrating the broad applicability of the state-space methodology, it can

also model nonlinear and non-Gaussian processes (West and Harrision, 1997). Analyses










of this kind can be approached from a B li- -i Ia perspective and use some kind of Monte

Carlo scheme, such as the Gibbs sampler, to obtain solutions. Carlin et al. (1992) detail

the procedure in this context. Durbin and K~oopman (2001) advocate the use of a linear

approximating model and importance sampling to treat state-space systems that are

nonlinear and non-Gaussian. In a similar vein, the sophisticated, sequential technique

known as particle filteringf uses simulation to estimate the parameters in models of this

kind (Doucet and Tadic, 2003). If properly designed, both of these methods are likely to

be faster and simpler to execute than the Markov C'I I!1, Monte Carlo approach.

By far, Shumway and Stoffer (2000) had the largest impact on the work done

here. In their text, they present the basic formulation of the state-space model, derive

the K~alman recursions, and explain their power and ease of implementation. They

also show how to compute the likelihood in the state-space construct and provide the

details of the Newton-Raphson algorithm to accomplish parameter estimation. The

most important section is the one relating to the missing data modifications. Here, the

necessary adjustments to the state-space model and K~alman recursions are described. The

technique relies on the EM algorithm for parameter approximation because of the ease

with which the procedure handles missing data. The EM approach to solving missing data

problems cast in state-space form was originally presented in Shumway and Stoffer (1982),

with theoretical results proven in Stoffer (1982).

The contribution of this work is that it incorporates exogenous variables in the

state-space model and allows them to be missing in addition to the responses. In general,

exogenous variables affect the response, but not vice versa. These are independent

variables "determined by factors outside the system under study" (Diggle et al., 2002).

The benefit of this approach is that more of the variability in the response can be

captured instead of attributed to white noise, and it takes advantage of the K~alman

recursions to implicitly fill in any missing values in the covariates. Although Schmid

(1996) also proposes a state-space formulation with covariates, not only must all its a ;liy










of the parameters he known (which we do not require) but the observed explanatory

variables cannot he missing. We suspect the reason for such constraint is that there is no

exogeneity assumption, which is critical for our results to hold. Whenever possible, the

notation used is as in Shumway and Stoffer (2000) and new notation follows the same style

so that extensions made here are clear.

The organizatioon of this paper is as follows. ('!s Ilter 2 deals with model development.

The first part extends the complete data state-space formulation by adding exogenous

variables to the model. It goes on to specify the K~alman recursions, EM algorithm, and

parameter estimates in this case. The remainder of the chapter relates the necessary

corrections to the state-space model, K~alman equations, and estimation procedure in order

to accommodate missing values. It also contains a theorem that provides the theoretical

foundation for the K~alman recursions to continue to hold in the presence of missing data.

This chapter contains the bulk of the new work accomplished.

Standard errors for the parameter estimates derived via the EM algorithm in the

complete and missing data cases are calculated in ('! .pter 3. A procedure is detailed

specifying how to generate the observed information matrix in both situations. Proofs for

all theoretical results appear separately, in ('! .pter 4. This is done so that the main text

flows better. ('!s Ilter 5 describes the results of the simulation study. Several subject and

time point combinations using different initial parameter values are considered in both

the complete data case and under various percent of missing data. In addition, the model

without exogenous variables and the effects on the procedure of mismatched observations

in time are examined. The state-space model with exogenous variables and missing data

is then applied to a real data set in ('! .pter 6. The data analyzed were produced from an

observational study on patients with autoimmune diseases. The derived formulation serves

as a model for the data and using the estimated parameter values, the K~alman recursions

are applied to help predict the behavior of a certain biomarker of interest. Lastly, ('! .pter

7 deals with the implications of this work and -II__- -; areas of further research.










CHAPTER 2
THE STATE-SPACE APPROACH

This chapter is devoted to the development of the state-space model with exogenous

variables in the complete and missing data cases. The first part specifies the K~alman

recursions, EM algorithm, and parameter estimates when there is no missing data. The

necessary modifications to the K~alman equations and estimation procedure in order to

accommodate missing values are presented in the second half of the chapter.

2.1 Complete Data Specification

The state-space model is given by the state equation


xi,t = Oxi,t-1 + Tui,t + wi,t, (2-1)


where


i = 1, ..., k subjects

t = 1,..., n time points

xi~t :p x 1 state vector

# p x p transition matrix

T p x r matrix of coefficients

ui~t : rx 1 vector of exogenous variables

wiat ~ iid N (0, Q) noise vector,


with initial condition xi,o ~ NV (I-o, Co) and the observation equation


Yi,t = Ai,txi,t + Pui,t + vi,t, (2-2)

where


yi~t :q x 1 vector of observed data

Ai~t :q x p observation matrix










S: q x r matrix of coefficients

vi~t ~ iid N (0, R) observational noise vector.


For simplicity, we assume {wi~t} and {vi~t} are uncorrelated.

Let ~,s = (yi,l, yi,s) and Us,s = (usil, .. ,us,s) denote the responses and covariates

(assumed known) at time a for subject i, respectively. If in addition to the observations,

~,,, and Uis,, for i = 1, k, the states Xi,ni = (xi,o, xi,l, xi,n ) were observable, then

the complete data joint density would be


fe (X, Y, U) = fo,,o (iio feT (xis| xI~t-1, us) t fir~n (Ysis| ii, i~) (2-3)
i= 1 t= 1 t= 1



and U = (UI~,,,U~,,, U2,g ** ,n) Since we are assuming normality, the natural logarithm

of the complete data likelihood (2-3), ignoring a constant, can be written as


In L(8 X, vY, U) = k~l In 1 |E '| (x,,o pou)l C, (x,,o pso)
i= 1
+ ~Nln|Q- |


i= 1 t= 1
+ Nln |R- | (2-4)


i= 1 t= 1

where C\, na, = N.

Shumway and Stoffer (1982) detail a procedure to estimate the parameter vector

8 based on the EM algorithm. We can implement the EM algorithm to consecutively

maximize the expectation of the complete data likelihood given the observations to find

the MLEs of 8 based on the observed data, (Y, U) since we do not have the complete

data, (X, Y, U) As will be seen, the EM algorithm achieves state approximation in the

E-step separately from parameter estimation in the M-step. Although other methods









can be used to accomplish parameter identification, such as Newton-Raphson (Shumway

and Stoffer, 2000), the EM algorithm is presented here because of the ease with which

it handles missing values. The popularity of the EM algorithm in statistics is mostly

attributed to the seminal work of Dempster et al. (1977). The particular methodology in

the missing data case is presented in Section 2.2.2.

To execute the EM algorithm, we begin with the expectation (E) step. Define


xi,t


Fi ,ti,


At each iteration j, denote


(2-5)


parametler estimnates. Computing: E [ln L~ (8 | XY, U) Y, U', 8 1)] yields


jk In |,L Eo | -tr Eo ,") rn[+ (x ( po) (x po)1)
i= 1
+ Nln |Q-l |17 rQ- [SII SIo@'I (IS'o +t (ISoo@'I]}

-tr {Q-l [@ISO2T' + TS-'20' S12T' TS'2 + TS22T'] (2-6i)

+11 Nn |R- | tr R-A,nP, nAl I
i= 1 t= 1


i= 1 t= 1


where


i= 1 t= 1


i= 1 t= 1


(Ps~' x x:LX *)


(Pt,t- + x fx "-)


(2-7)


(2-8)


E [(xi,t -x:,t) (xi~t -x~) ,,U~

E [(xi;t -x:;tl) (xi,td t) ~


Qc (8 BC o 3) =E[nL( ,Y )Y ,B ]


Q (8 8 3) >











Soo = )(?,"<'-1 + xy x "_ (2-9)
i= 1 t= 1



SO2 = X _1 ~x~u~, (2-10)
i= 1 t= 1

S2= x :u~u ,t (2-11)
i= 1 t= 1



For more details on how (2-6) was reached, see Section 4.1.

In expressions (2-6)-(2-11), we estimate the mean of the state vector given the

observations, xf,t, and its covariance matrix, Pi), with minor changes to the K~alman

recursions found in Shumway and Stoffer (2000). The K~alman predictor, used when t > s,

and the K~alman filter, applied when t = s, are given by

X -1 = ~X _11 Tui,t (2-13)

Pt- =p- @ + Q, (2-14)


where


xf,t xt- +Ki~. (Yi,L AI,Itx,/ ru1~t) (2-15)



Kci,t = Pi A t (A,tP A~,t + R) ,l (2-17)


for i = 1,. ., k and t = 1,. ., us, with initial conditions x ,o = po and Poo = Co.

The E-step of the EM algorithm is really a smoothing problem, since t < s, but the

recursive nature of the K~alman smoother requires use of both the filter and predictor.

This can be seen in the smoothing equations found in Shumway and Stoffer (2000),


X ,-1 i~- -X -1 (2-18)

Ps ti- ati -11 IDIit- PsiP t-1):t, (2-193)









where


di~t1 = i\-1 Fi ) ,(2-20)

for i =1, ., andl t, = us n 1, 1, initializedr wnith x@i and P" from the K~alman

filter and predictor.

In addition, a lag-one covariance smoother term, Pi t,~,z, is necessary to complete the

E-step. The recursive algorithm also appears in Shumway and Stoffer (2000) as


?.t-,t- 2 s,t-2 -2 i -1 s i,t -1 i 1 t- ,( 1

with initial condition



for i = 1,. ., k and t = us, ni 1, .. ,2, where K4~,,, Ji~t, and Pi e are obtained from the

K~alman filter, predictor, and smoother recursions.

The maximization (j!) step entails maximizing the conditional expectation of the

complete data likelihood given the observations, found in (2-6), at each iteration j. Taking

derivatives with respect to each component of OC ), setting to zero, and solving yields

the updated parameter estimates.

In the past, other authors, including Shumway and Stoffer (2000), have included

exogenous variables in both equations of the state-space model but have failed to provide

M-step estimators for the coefficient matrices T and F. Thus, this is done next. In

addition, the presence of these parameters in the model affects the estimates for 4, Q

and R and so the revised estimates for these are also derived. Consider maximizing with

respect to F. This leads to

crQ (8 le -13) ,6 k
6 ~ 2~ I(~i~ ~!i li:: ii= 1 t= 1

x (yi~ s~x -Fur)










,6 ~~~~i= 1 t= 1~ ~ -y~u~



T -1 HiitU ,t



i=i= 1 t=


crQ (8 | 8 -3)


using matrix differentiation results from Petersen and Pedersen (2007). Setting to zero

and solving for 0 produces the estimator


E~n = it-A~x1 ,s E' (2-
(j ~;i= 1r t=1 1 ~ x,>

As another example, to derive TC ), we must evaluate

crQ (8 81 3) 6
bT ar[tr { Q-1 (OSO2T' + TS 20'- S12T' TS(2 + YaTS22l
--1802 + -1S12 Q-1TS'2 -- 0,


-23)


which produces


Similar calculations yield the following estimators for the remaining parameters:



i= 1




~()= (Sto TC )S(2) Soo

=(Slo S12S221Sr(2) Soo (I 'oSO2SrWS 2Soo)
C23 ~ illS0*b~s1 c Irj)c1- i1i)
Q~~~~~~~ = Sr l"o o+ S2Soo S'o S 2
1-
+ [(,SoSIoo SO2 S12 b )('SoS2 S22) T' ')


(2-24)




(2-25)


(2-26)



(2-27)



(2-28)












i= 1 t= 1


i= 1 t= 1

See Section 4.2 for details on the calculation of (2-25)-(2-29).

In essence, the technique entails alternating between evaluation of the K~alman

recursions given in (2-13)-(2-22) and the maximum likelihood estimators (2-23)-(2-29)

until convergence. In practice, the procedure is stopped when the values of either 8 or

the observed likelihood do not change much from one iteration to the next. Shumway and

Stoffer (2000) -11---- -1 evaluating the innovations form of the observed data likelihood at

each iteration to establish convergence. The innovations are defined as


eilt = Yilt E (yilt | Yl~t-l, Us,~t-1) = yilt As,tx~~ Fuilt. (2-30)


Clearly, E (ei~t) = 0and


Ei,L -- Var (es.t) =Va~r [Aige (x1~L x --) +vi,] = ~PIA/~P-A~ R. (2-31)


The innovations form of the natural logarithm of the observed data likelihood, ignoring a

constant, can be expressed as

In L (8 | Y, U) = i (In |E- | C 1i,t (2-32)

i= 1 t= 1

Assuming independence between i subscripts, i = 1,..., k, this follows since the

innovations sequence es~t, t = 1, .. ., us, is a Gaussian process.

2.2 Missing Data Adjustments

The K~alman recursions require that the observation and exogenous variable vectors

be completely observed. Unfortunately, in real-life applications, missing data occur

frequently. This section details the necessary modifications to the K~alman equations and

EM algorithm in order to accommodate the case of missing values.









Some new notation is necessary when there are missing data. The observation vector

can be partitioned as

Yi,t =(a
Yiet
where yI is the yi~ti x 1 vector of~ observed- values~- and y is the gi~t2 X 1 unObser~ved

component, with qi,ti + qi~t2 = q. Similarly, write the vector of exogenous variables as

u i~t ui~ti~(1)(a
clnoin: he7.~t S1 ""'". r"1,~.,.,., ,,-,.,,,,,,.,, 11., ....I);tldIHi7i~2



denoting ted preti b x 1 ector ofosrvedl exgnu vaibesb f n teTat

may be reexpressed as

xi,t = (x1t, ir c,t 2 2 Wt(233
i~t i~t

Yet iui,ti1(a it wi,t i2-33

(1()i~it xi,t + (234
(2) A (2) (l1) (2) (2V()
Yi,t itit2 i,t2itit

respectively, where, for j = 1, 2,

T :) p x ri,tj portion of (state) coefficient matrix corresponding to u -)



Emt~ : yi~tm x r.i,tj p-ortlion of c~oeffic~ient matrix c~orre~spondling to u ), mn = 1, 2

v )~ : qi~tj xv 1 portion of mneasuremrent noise vector c~orre~spondling to y ~(),

and

R -- var (2-35))()RitllR~tl
vRi,t,21 Ri,t,22 (-5










is the covariance between the observed and unobserved portions of the observational error

vector. Although the general model is given and theoretical results are provided for this

model, in practice, we recommend zeroing out either the T or F terms in (2-33) or (2-34)

to avoid a collinearity problem. Out of the two, it seems more appropriate to remove T

from the state equation.

Note that in expressions (2-33) and (2-34), xi~t and yiet depend on the unobserved

portion of the exogenousi variable vector, u ~. W~e propose circumnventinlg th~is difficulty by

modelling ui~t with a second state-space model. Treating uise as the observation vector for

the underlying state process gi~t, this second model is


gi,t = Agi,t-1 + ci,t (2-36)

ui,t = Bi,tgi~, +hi~t, (2-37)


for i = 1,. ., k and t = 1,. ., us, where


giot : mx 1 state vector

A : mx m transition matrix

ci~t ~ iid NV (0, C) noise vector

Bit : rx m observation matrix

hi~t ~ iid NV (0, H) observational noise vector,


with initial condition gi~o ~ N (pg, Eg) .For simplicity, we assume that {wi,t}, {vi},t)

{ci~t}, and {hi~t} are all mutually uncorrelated.

The observation equation (2-37) can be partitioned into observed and missing parts,

yielding
u1 B(1 h~
i,t i,t i,t
U(2) B(2) h(2)
i,t i,t i,t










where, for j = 1, 2,


B(" : i,tj x mI portion of observation matrix corresponding to u!"

~ jh6 : ri.ty x 1 portion of measurementn noise v~ctlor corresponding to u ,l


and the covariance between the observed and unobserved portions of the observational

error vector, hi,t, is given by the matrix



H = var (2-39)) I H~ti
h a H i,t,21 Hi,t,22(239

In the case of missing data, the likelihood must be modified to include several more

terms to account for the second state-space model. Hence, ignoring a constant, the natural

logarithm of the complete data likelihood is


k
k In | Eg | (gso pi) (Ei) (gso pi)
i= 1


In L (8, O | X, Y, G U)


+Nln |C- |-(ls
i= 1


- gi,t-1)' C- (gi,t Agie,t-1


N 1Vn |H-l | (ui,t-Biti)'H (ut-Biti)
i= 1


k j In |C1 Eo | (xi_,o
i= 1
N~1Vn|IQ- |

- Xii~i t ~Xi~t-1
i= 1 t= 1
Nt n |~ 1IR- |

- ( >:yi,t Ai,txi,s
i= 1 t= 1


po)' Lo (xio po)


(2-40)


STui,t> / -1 Xi,t


- Xi,t-1 Tui,t>




Ai,txi~t Fui,t) ,


where R = (pi, Eg, A, C, H) represents the parameters from the second tt-pc

formulation used to model the exogenous variables. Also, G = (G1,,,,...,Gk~,n) denotes

the state information for the second state-space model, with Gis,, = (gi,o, gi,,,) -










2.2.1 The Kalman Recursions

This section presents the modifications to the K~alman equations in the presence

of missing responses and covariates. Before developing the necessary adjustments, it

should be noted that the exogenous nature of uise in relation to yi,t will be an issue.

There are varying definitions for an exogenous explanatory variable, and so the statement

from Diggle et al. (2002) defining a covariate process to be exogenous if at time t, it

is "conditionally independent of all preceding response measun in~ !I- will be taken.

Symbolically, this is given as





For our purposes, this definition is more rigorous than necessary. The results we will prove

depend on the weaker, implied condition


E ( U, t t ,, Ui, a 1)= E ( Ui, t|U, I 1) ->


Before presenting the results needed in order to establish the required changes to the

K~alman estimates when there is missing data, some more notation must be given. Let









x~ = E xgt ,

P~lt = E xi,,t xit- x gt i,s U

like ~ the deintin give inScin21 xetta tecvraesaenwrndma el










Note that the definitions of g~,1, F i,t an r o oniinl n .Ti

term drops out because the covariates, ui~t, are exogenous from the responses, as will be

shown below. The assumption of exogeneity is key for the K~alman recursions to hold for

the second state-space model. It is not clear if these equations will follow without it. Note

that the exogeneity assumption is on the missing data mechanism and it only affects how

the missing values are implicitly filled in. Based on this condition, the missing exogenous

variables are generated only from observed covariates using the second state-space model

(2-36),(2-37), and not from any responses. This assumption does not affect the original

model (2-33),(2-34) and the EM algorithm will still produce the MLEs. The K~alman

formulas can be handled separately because of the next result.

Theorem 2.1. Conditional expectation terms relating to the second state-space model

(2-6) (2-87)~ areindependent o,~ f v~)due to ,,,, ,, :i; 7

Proof. See Section 4.3. O

The modified K~alman recursions in the presence of missing values in the observational

vector for the second state-space model in terms of gi~t have been derived in Stoffer (1982).

The routine works when there are either no exogenous variables (which is the case here)

or when the exogenous vector is completely observed. The procedure l~~l entails seting

and B~~ inl (2-38) to 0, substituting 0 for Hi~t,12 anld Hi,t,21 inl (2-39), anld thlen running

the K~alman equations as usual. Thus, expressions (2-18)-(2-20) can be used to generate

smoothed values, g ,,tn by adjusting (2-38) and (2-39) as discussed above. Takling the

different notation into account, these smoother equations become


Fi = t F 1) t1 (- F D ,t)

DLi,t-1) = F AF









for i = 1,. ., k and t = us, ni 1, .. ,1, initialized with g an d rmteKla

predictor and filter equations, which are


(t-1)

i,t


AZF A':) + C',


where


(t)

Li,t


L~i~t)


(t-1)
it + L(it) u(it) -

(I L(it)B(it)) F


(t-1)
B(it)84,t


HII~t)


F B 4,t) B (it-)F B 4t)+


and


U(i~t)



B~i~t)



H~i~t)


0

Hi~t,22


pi and F


Eg. The lag-one covariance smoother,


with initial conditions g ,o)


necessary for parameter estimation, becomes


F "')1l,t-2 L i D it-)+ -lt1 Ltt_z


AF D it-2)


with initial condition


F = I L4,,,)(4,a ) AF


n i, a1 1, .. ,2, whler~e L i,n,) D~i~t),1 and Fij~ are asi above.


for i = 1, .. ,k and t


Hi,t

0~~


B









Since the case of missing exogenous data has not been treated before, the K~alman

recursions for xi,t must be derived. Before proceeding, a preliminary result is necessary. It

is derived using a geometric approach, in which conditional expectations are handled from

the perspective of Hilbert spaces as orthogonal projections.

Lemma 2.2. For7 each i = 1, .. ., k, let N Ft~ be thle Hilbert subspace spanned by thle

elements of i) teU .i~) Thl~en1~Fj ,rr,;l De decomp~osed as theC diectI( suml ofIi~~ @,)1 and Zi,t,

wuheret Zi~t is thle Hilbert s~ubspace sp~anrned by thle elements of zi~t = ,a Ai~ ,xjtt 1)



Proof. See Section 4.4. O

The result detailing the necessary modifications to the K~alman filter and predictor of

xi,t in the presence of missing data can now be specified.
Theorem 2.3. The minimum variance filter and predictor equations and corresponding

covariance matrices for the state-space model given by (2-SS) and (2-84) are

x (- = @x t -1) + (itg1 (2-41)

P Fi5 )' + Q+ T B,~i,tF, B,t + H T', (2-42)








Yii~t
wher0





















0t Ri,t,2







initial conditions xi, o) =ILU po an P=


R(i,tl)







for i = 1,. ., k and t = 1,. ., us, with


Proof. See Section 4.5.


Shumway (1982) demonstrates that smoothed values can be obtained in the case of

missing data from the smoother result given by expressions (2-18)-(2-20). Only a minor

change need be made, and that is to use the filtered values produced by Theorem 2.3. In

other words, in the presence of missing values, the smoother equations become


xt (it-1 x (i

P P ,


x(t l


x ',




J~i~t-1)


(2-46)



(2-48)


for i = 1, and t = usl ni 1, ..,1, initialized with xiT and P~~~ from Th'leor~em 2.3.

It is also proven in the aforementioned work that the same is the case with the lagf-one

covariance smoother term. Namely, (2-21) and (2-22) become


i -,t- 2 i5 ,t-2) (i,t-1) i t~-1 l i,t- 2)


(2-49)


with initial condition


Fi~nin -1 = I-K ,, As,3 P


(2-50)









res pectivelyi, for i = 1,. ., k anld t = usl~ n, 1, .. ,2, wh~er~e Kg,~,,), J~i~t), and P are

obtained from Theorem 2.3 and the smoother equations above.

In summary, the K~alman recursions in the case of missing data are similar to those

when the data set is complete except for several adjustments. First, the filter and

predictor and their corresponding errors must be modified so that missingness in the

exogenous variable vector is taken into account. This is seen by an estimate of ui,, taking

its place in both the filter and predictor recursions. The errors also contain extra terms to

account for the estimation of ui~t. ClI Imy, are also made to handle incompleteness in the

responses, yi~t. This is done by substituting zeroes: (i) in place of the missing values, (ii)

in the row(s) of Aige and F corresponding to the missing data, and (iii) in the off-diagonal

elements of R for subject i at update time t. Finally, the smoother, its error, and lag-one

covariance terms can be obtained as in the complete data case except that they use the

missing data corrected filter and predictor as described in this section.

2.2.2 The EM Algorithm

As is generally the case, the parameter vectors 8 and R are not known and must

be estimated in order to approximate the states, xi,t and gi~t, for i = 1,..., k, and

t = 1, .. ., us. This section presents the adjustments to the maximum likelihood estimation

procedure (via the EM algorithm) given in Section 2.1 when there is missing data.

Throughout this section, the observed responses and covariates are given by Y(1) =

1, '' k and Ut') = U :, .. U respectively, with state information
X =(X,, .. k~ )andu G (G1I,,,,.. ,Gk,n) Proceeding from the complete data

likelihood (2-40), the E-step is, for each iteration j,


Q2 (8, O 8 )l,: n 3) =E[nL(,O|X ,G )Y ,U ,8 1 (2-51)




Sti= 1 (b' [ ,

+ Nln|IC |- tr(c C SQ -So)A' -AS: o+A oo)A









N 1Vn | H- |

k asii


;z i) ~) +


i= 1 t= 1

+kln|E | ,


0

H ~t22


N 1Vn |Q-l | + tr {Q It~I- T (S12)


Sl 'O2)


~tr {Ql(iS(12)


S(10)O' 9S1n)+OSioo) I'])


TS(22)) T'}

0z~z' ] ;


(2-52)


i=1 t=1

N 1Vn | R- |


i= 1 t= 1


X()it)


k C
i= 1


0 0


itr
t= 1


x 8 t i, t


R


-1


i=1 t=1



i= 1 t= 1


B(it)8


uI,t)


2i=1 1 "'[ ( x(i -o xci


- S(02) -


p-1 y(4,,)


A (i,t Xii)x "' o D~,a2Sittu~i~t )
0 0


A(nitx DIt tuit))


0 0


I 0]


0 0

0 i,t,22 1


0 0










where


Sr = tF + g g ~t" (2-53)

i= 1 t= 1

S~ ~ ~~(i (n0) = +g g 2-4
i= 1 t= 1



i= 1 t= 1
(1)
u 4,t) =i (2-56)
B(2) (ni)
i,t gi,t


S 11) = Pi~i + x~) x (257
i= 1 t= 1


S(10) = P 1 +~;' x iiixr, "n (2-58)
i= 1 t= 1


Sioo) = P + x~~ xx ,-t (i~, l2-59)
i= 1 t= 1


S(02) =X.L- 1U i,t) 1it(2-60)
i= 1 t= 1


S(12) =X i) U i,t)Cl~ (2-61)
i= 1 t= 1


S~i=1 t=1i,t

i,t U )Ui,t) gi~ C"utI.O I


i= 1 t= 10


Note that Si~t "is a permutation matrix that reorders the [exogenous] variables at time

t [from subject i] in their original order" (Shumway and Stoffer, 2000) and ~Di~t is the

cor~responlding permutations mratr~ix for thle resiponlses. Here, Hi, 2p iS the previous value

of the variance-covariance matrix of the observational error vctor (h 2) RS~ociated wiith

the unobserved portion of the response (us: in thle second state-space model specified by

62(j-1). Similarly, Ri ,22 IS the corresponding variance-covariance matrix associated with









the unobserved portion of the response y ntemoe pciidb B (.Thsrsl

hinges on the assumptions that the observable and unobservable portions of the exogenous
andobsrvaio vetor hve ncoreate eror, o va h, h~a Hi,t,12 = 0 and

vara vt= ,~ v3 Rit1 0. Fr molre detailsi on (2-52), see Sectio:n 4.6..

Proceeding to the M-step of the algorithm, we must now maximize (2-52) with respect

to each of the parameters in 8 = (I-o, Co, 4, T, Q, P, R) and R = (pg, Eg, A, C, H) .

This will yield the desired parameter updates to be performed at each iteration. In what

follows, we still maintain the assumptions Hi,t,12 = 0 = Ri~t,12-

Similar to the E-step for the portions of Q (8,O: 81 Oi), DC )) relating to the second

state-space model, the M-step estimators for R are also given in Shumway (1982). These

are


( ): gno" (2-63)
i= 1


A =S10 S oo) (2-65)


1~j 1



1i= 1 t=] 2t12 g,


(it) i~ t ,Oai ) ,t (2-67)



The remaining parameters relating to 8 are now presented. The similarities between

the corresponding lines of (2-6) and (2-52) relating to po and Co imply that the estimates

in the case of missing data are, respectively,


p = x "(2-6i8)
i= 1










(2-69)


at each iteration j. The derivation of the remaining parameter estimates,


T3) = (S(j12) Of )S(02)) S-


(2-70)

(2-71)


',


S = S(22) it
i=1 t=1


0
H(j-1)
Hi,t,22


QC ) (S(1) T ) S O2)

=(S(l0) S(12) S- S'O2)) S,(0 I (


)1
S(02) S:- S'O2) S to


(2-72)


1
Q j) =N(S~I1)


S(10) O'") 4(" S;,) 1 ") S~oo) 4 i))


S'O2) O' 3) + (S(12)


0

Hi,t,22


8,< T'


1
+ rf


(2-73)


Y(i;,


SDi~a
i= 1 t= 1


R j) =


Hi, ,2 i,t i i) t2
0


(2-74)


0 0


1 k n
i=1 t=1


1 k n
i=1 t=1


i=1


p~i) x p ,


1
[T 3( 1)


AC ) S(02) -- T(" S ~) S(22)


i=1 t=1


A ,tx oD itui)
.,x;~ 0r:nri~yl i


A~i~t)xZ oD~NS~u4t ,t
00


1 k~ n i I 0 ; i
i= 1 t= 1 0 0


i Git I0
0 0


0R,t,22









k 'H~1t2 +~ B(2)(ni) /(2) 0Oi
vec 0(j I=1 Si,( u (i~t)U44,t) + i(t,22 iB i(ti)



Sv l i= 1 t= 1oo 0 0


and ~ Dieclaiyngrmrs apea 0 (R c~io 4.)-1 7 0D (2-75) ~ rpeetsIeul





mandx j som cltarfingd fremakaperiScto47I (2-75), R represents the un-oeatr



The M-step estimators for F and R at the je" iteration, given by (2-75) and (2-74),

respectively, are complicated functions of each other. Since neither function is easy to

solve in terms of the other to allow substitution, we so-----~ -r using PC -1) instead of FC )

in (2-74), solving for RC ), and then substituting RC ) in (2-75) to obtain Fl ). As will
be shown by the next theorem, this modification to the EM algorithm changes the EM

into the Expectation/Conditional Maximization, or EC'j I algorithm as defined by Meng

and Rubin (1993). Under the EC11L algorithm, the E-step of the EM algorithm remains

the same but the M-step is replaced by a series of conditional maximization steps that

optimnize Q (8, O 81 0(1), DC-1)) in (2-52) with a functlion of (8, R) fixed at its previous
value. The following result shows that our procedure converges.

Theorem 2.4. The l,,;.../.#,I EM~ procedure described above constitutes a converging ECM~~



Proof. See Section 4.8. O

In short, in the missing data case, we alternate between evaluation of the K~alman

recursions in the E-step and the maximum likelihood estimators in the M-step until

convergence. The M-step in this case additionally performs two conditional maximization










steps in order to ensure appropriate results for 0 and R, thus making our procedure an

EC11L algorithm. As in the complete data case, the process is stopped when the values of

the observed likelihood vary little from one iteration to the next. The innovations form of

the natural logarithm of the observed data likelihood, ignoring a constant, is now


I n L ( 8 O Y ', U () = I n | E f 6 ( ,\ (2 -7 6 )

with
(1) (1) -A ()x(t-1) fp(l) B F(2) (2) (BTt-1)
'i,t = i,t i,t iX ii,t t it1 i,t it

and


C, =1 A (1 ft Fi A +~i Ri~t,ll +rl ,)B +1 (a B Fa 0 ft)B + 2 B B









CHAPTER 3
CALCULATION OF STANDARD ERRORS

The EM algorithm greatly eases maximum likelihood estimation in the presence of

missing data but it does not automatically yield parameter estimate standard errors.

Luckily, techniques abound to calculate standard errors when using the EM algorithm. A

method applied to state-space models utilizing the bootstrap is described in Stoffer and

Wall (1991), but the nested looping involved in the algorithm would probably be very

computationally intensive if applied to our situation. Another procedure we considered

applying was the supplemented EM, or SEM, algorithm of Meng and Rubin (1991).

Although this technique requires relatively few modifications to the basic EM algorithm,

the numerical differentiation involved was all in r drawback considering that our

parameters of interest can be matrices. A variety of other methods are given by Givens

and Hoeting (2005). Since under general conditions the MLE is .I-i-mptotically normal

(Shumway and Stoffer, 2000), most of these approaches give some sort of simplification
to direct calculation of the inverse of the observed Fisher information matrix. Robert

and Casella (2004) explain the use of Oakes' identity (Oakes, 1999), a formula that also

provides a shortcut to outright computation of the Hessian matrix of second derivatives.

At first, this relation appeared to be viable for our problem because the observed Fisher

information matrix is written in terms of the complete data likelihood, which we have

already calculated (see (2-4) and (2-40)). However, calculation of the mixed derivative

term proved rather complicated.

Under our complete and missing data state-space models, direct differentiation of the

observed data likelihood is a simpler alternative. The formula for the information matrix

is comprised of recursive calculations using the K~alman equations and parameter estimates

evaluated at the MLEs. Gupta and Mehra (1974) demonstrate that the (j, s)th element of

the information matrix is










zjs = E -,~ Int LY,





where ~ ~ 04i hej omoet o ithe paramte vetor Th s~ itadr eror are thequar






3. Coplt DaaCs


Ahr 8 s show inh Sctiponn t 2.we hr ren isn aa the parameter vector. h tnaderr r h isqu


treat o teah gna element s of each. paaee n8as nthed vaib le ofu a interst. e Thus, ) weclclt

pathe, stanar Herrosia forixcnb prxmtdb rpigteepcain




Note) tha o he o symme d tr ic matixod n yU pa ameer Lo, Qi, and Ronyt hei deisatinct delement

abe ovee and incluingth diagoal carecniee. Thus, vnwa olwte) cwonsisuts ofs 2pip+)+rb

q)mie + s 1 nqe clrparameyi m epter.

Exresin 2-2)giesth oseve dalta lo-lkeiho C ase

As~ ~ ~ I Lhw (8 |eto Y., U)nthr = r Ino | E / 6t,< theit ,aaee etri
O = (Ilo, C, ~, Q, R) Sinc equai=n (1) t= 1 ie ntrso clrprmtrw










where es~t = yi~t Ai,tx ,/ ru1~L and Ei,< stP A, R. Hence, it follows that in

general, for i = 1,..., k and t = 2,..., n,


ti,t = -Aige x Xi 0 us,(32

x, =- x _x Kitli,t-1> ii ii


K i,t-1 + -,t-1 T u s~r>i (3-3)

Ki-1 = A,t Ki,t) ,, (3-4)


[i- KPt1i~-8 ib- A ,t1 -

-~ K6 -18j- 68 i,t-1it1 35

E,t- =~ Age P As +,t- R~~ (3-6

wit th aid ft heKla eusosi 21)(-7.Tefloigcluain r





o8 =8 i(by1 ij6,ba ,.,;b ,,p(38




=c~ liby (,j aba=1..,;b ,..p(-0
= Hj be r ist n= Usb, =1,.,;b 1.. (3-11)






R = J I(0; = T~ab), a =1,...,q;b ,.. (3-13)










where I is the indicator function and Ji, is a matrix of dimension p x r with a one in row

a, column b and zeroes elsewhere. The partial derivative of the transpose of a parameter

matrix with respect to one of its elements is simply the transpose of the corresponding J

matrix given in (3-7)-(3-13).

Since equations (3-2)-(3-6) hold only for t > 2, these derivatives must be evaluated

separately when t = 1. It follows that for i = 1,. ., k,






xp, = @Eo@ '+0~



Egyl = Ai,l (@Eo@' + Q) A~~ + R,


using the K~alman formulas from (2-13)-(2-17). The results of the derivative calculations,

or the initial values for the standard error recursions, appear in Table 3-1. Note that the

partial derivative with respect to each of the parameters in the left-most column appears

along the rows.

3.2 Missing Data Case

When there are missing values in the response and exogenous variable vectors, the

model parameters are 8 = (Po, Co, 4, T, Q, 0, R) and R = (pg, Eg, A, C, H) As in Section

3.1, we will consider each matrix element separately. In addition to vec(8) as described

previously,

vec(0 = p ..9 ,P m ,11, ** ,1m,,, ,T22, .,n,2 m,r:.. aK,mmb1 1vec(A); cll,..., cim,,


C22,..., Cam, ..., Cmm; 11,.** hI, 22a,.**, 2ro***., rr,


are the parameters of interest for the standard error calculations. In the missing data case,

there are 2m(m + 1) + r,/r + 1) additional scalar parameters










Table 3-1. Initial Conditions for Complete Data Standard Error Calculations


Partial Derivative


S i1


xo


E~ C


o" 0
a= 1,...,p


-A # ~J" J"


~O0,ab
1 ,...,p



~ab

1 ,...,p


x Jab i' E


Jaby 0




+ ECoI,): A EC~~


aJb #' Ai,l@ #',~ A


+,Co#'
palxp


+ OCoJ b:) A:~


Ai,] Uppo app


Dub
a= 1,...,p


Ast usab


Jab H,


4ab

1,. ..,p


(I Kg A )
xy AE


~ab


As, Az


0




-Jaub


a= 1,...,q


Tab
a= 1,...,q
b = a,...,p


K Jabq 114C


Fr-om (2-76), the observed data log-likelihood is now


2i= 1 t=] 1 1 -


In L (8, O Y ), U 3)


with


6 t = A t) x [1 ,O = y Ai,t x e-[,]Dtf~B,~>


i@i d,









and


i,t = A, iAt + ip, +0,)i B + 0(2 B(it-1 F 0 ,\B (1(2 B 2>


i,t1


i = 1,..., k and t


= 2,...,4,'68j(t


x (1


iIB'>t#


-
[I,0]Dt SitB 8e


(3-14)


X (t-21) K _(1, -1







K (1b t)( >


Bitg


(3-15)

(3-16)


6
K(1)
68, i~t
6 1)
68,


-it K_


K 1 A t-2


K1 AL i ( (ta


(3-17)


+ 0 t H


=A1,\F A + [I,0]D-,[ ROis+ [Ir,0]D~tfHP'Di[


+[I, 0] ~D~,tFi,tBi,tt> F ~(-)B's,t>,t'i-, t -[ I


NUote that /,sl anld E i,t)l ar~e written inl terms of the permutation mratr~ices so that they:

can be explicitly differentiated with respect to the parameters in 8 and R. Thus, for


-~ [I~,0D


TBi,tg


-(


-t P8f A (1


-


A -, -


/Q>T BiyR,tF B R,t H T' + T H T'

+ T~r ( F B,tT+TBiFBt+H T









C(,) = A,\ At~i + [I, 0] ~D ,t RDi,t o


+ [I, 0] D~,s Io HPD [I, 0] ~D~,tF H O'DPr[IO


+ [I, 0] D~,tfH (3' ~Di,t I (3-18)


+ [I, 0]D~t i tB i,tF B ,fD~


+ [I, 0] ~D~,tFi,tB~ i~t> F'' B'it t' Io


+ [I,O] D~,tF i,tB~ i~t>F (-1 B's,t> ,t 0 Iot

where I and 0 are suitably conformable identity and zero matrices, respectively, and
Kt = P )Aiti ,s These recursive derivatives follow from the missing data

K~alman formulas given by Theorem 2.3, while expressions (3-7)-(3-13) continue to hold.

Equations (3-14)-(3-18) depend on the partial derivatives of K~alman predictors and
variances and parameters in the model for the second state-space formulation. To resolve

these, define the innovations for the model for uise as #(i~t =ut) ) (it-)g ), with

(i~) Vr #(it) =B~i,t)F B 4ig ,t) + H~i~t). The following equantions are necessary for
the evaluation of (3-1) and hold for i = 1,. ., k and t = 2, .. ., as :


~(i,t) = -B(i,t) 8t (3-19)

g A g + A) g (t -21ALia-)(it
668;A Lt 1) (~)g-1) ( i ,t-1 ) +A(,t-) (i~t-1 (3-20

-A L6 t) = F ,)-LiI)~) ,) (3-21)


00j 6 0

+ 8, H i,t) (3-22)

oI 60s0












AL~~(tt-1)B~(t- -1)h F '+FA' .(-3








C =) ,i Job ,, m O ab), a 1,.., ; = ,..,




Ah sbefoe ic h ata derivatives givenw byo expresma rcsions (3 h ecn tt-14)-3-18 ande o

(3-19)-(3-23) farial s toe hod or t = 1,thi cas iom s give n b elo for ivlut = 1, k :



d,\)= 7,) A )@ po A I)TB4 ,zp [I ,0]D(,7041,4

x o ~) =a I(p, +0~a> TB4 z p

P86 26m18 = Xao) + Q +, TB4 (AA + C) B(;T += THT'

E~~ = A~ 1(o ++T,(AA+Ca)B( a T'+THT' a
Ie Ix







ti i = uYi,1) -~ B(4, )A












g ,



L i~


Apgl


(ACg A' + C) B(4,l)i:


Ac~nA +


Table 3-2.


Initial Conditions for Missingf Data Standard Error Calculations Partial
Derivatives With Repc to

Partial Derivative


a=1



b= a,.



b= 1,





qab

b= a,.


l I }


xJa,~. Jo~


++\~o~ I cJ~



I-e~,~


++Ea"ep




r~~B2,a (AE\ A+ C) B:(3T'
+TB,, (AEA'+ C) B:(3Jffl
+.100 77, t


++Ea\).


.1 /B2.1ApL / a~BsliA


[I, 0] D:(3Jor,[H +E ~B


+ [I, 0] ~D F, [H + E 1Bs,,I

(AE A' + C) B'E,l q 3iD,



[I,O]l I /J~~~D~


-[I,0O] ~D:(Jair


a=1,...,q


0] ~D' I~a
z I
|), 0 | )


a=1,.

b= a,.


~(i,1) = B(4,1) (AcgA' + CU~~) B +]-I 4,,HS4, +oo ooo4i .


These equations are then differentiated and the results are summarized in Tables 3-2,


3-3, and 3-4. Because the partial derivatives of I' ,,,g jJ1, L(4,1), F W41,ad ,,)wt

respect to the parameters of 8 are all zero, these are omitted from the tables that follow.













Table 3-3.








a =1,...,m


Initial Conditions for Missing Data Standard Error Calculations Partial
Derivatives (of First State-Space Model Functions) With Respect to R
Partial Derivative


, + [I, 0] ~D~


TBi,1Ahilxt


-[I,0O] D Eri~Bi,1Bi,>A~ab






+ [I, O] JD~, i,1B~il,>
x (Jabxmcn / ox~m)



x B', O] D~~ri~,i~l









-[I, 0] ~D z~ar ,


a,..., m


(111


ri,1 Jasxn


Xab
a= 1,...,m
b=1,..., m


,, ,+ [I, 0] ~D ~


P 1
11)b 11~ -1


Ti,l abbxm O


Ca


a,. m


hab


-1l
i~~~_ i, ~


TBiIJ ,T'


(1
o. i il
-A ,, l jz)


T Jab Ti


Table 3-4. Initial Conditions for

Derivatives (of Second


MVissinlg Data Stand~ardU Er'ror Calculations Partial
State-Space Model Functions) With Respect to R

Partial Derivative


6, 6 I, g )


SF(O


sL )


6~\lr(i~l)


a= 1,...,m
9
"O,ab


b = a,...,m



b-1,...,m

Cab

-~ m


B ,1)~ix,


Jmx, O


aJb / Bi -1
0 xm (,i) (,i)





L,,O( 6 1;1)) I11)


Jb / -
0 -L, ( cb i




0 -Ljo 6b il


abxm '


(i,1) abxm i,1)



(i,1 i,1


(i,1) bxm i,1)

IO IO
[i E, 1b~ [i,
T001~ a r001
0n 0I 0~ 0~Ln


(i,1 J "


mbxm





0










CHAPTER 4
CLARIFICATIONS AND PROOFS

4.1 Derivation of Q (8 81-13-))

Let wi,t = xi~t Oxi~t-1 Tui~t and denote yi,t Ai,txi~t Fui~t by vi~t. Recall

that E (z'Az | W) = p's, wAp w + tr (ACz, w) Also, it is understood that E represents

conditional expectation with respect to OC -1) (and thus dropped after the first line).

Starting from (2-5),


Q2 t8 8 )i) = [ln L(8 XI Y, ) Y', U, 8l)

= k~ In |C1 Eo C|- [(xi,o po)l Co (xi,o p) Yu,]
i= 1
ik a
N 1Vn|Q-l | wtawsYs
i= 1 t= 1


N1Vn |R- | E i[Fv:Rv,tR v,1 s
i= 1 t= 1



i= 1


-~ tr { Eo Var [(xi.o- o|Ya]
i= 1
ik a
N 1Vn |Q-l | '[a u]Q ws|Ys
i= 1 t= 1

-)) tr {Q- Var [wias | Yu,]} (4-1)
i= 1 t= 1



i= 1 t= 1



i= 1 t= 1

F'or clarity, we will evaluate the individual pieces of Qo (8 8 3-l) separately. Thus,





(4-2)









and


i= 1


itar; FL E(xi,o
i= 1




i= 1


x ) (x ~i,o -x) ,





- x e) (xh /co) ~i




(x6- pCo) ( o




r8 pO) (~x~ o)'] }


i= 1


i= 1


i= 1


i= 1


i= 1


-p~o)'E.


[,? (x


pbo)/ Co (x (


(4-3)


Similarly, we have


E~~ [ws 7.,, Q-1 E [ws ),2,
i= 1 t= 1


i= 1 t= 1


X t1 -T i~t -1X


X -1 ,- Tui,~t


(4-4)


po) Y,2,] }


Po) (xi,o Po)' ~i)


tr (,Eg E(x h










and


i~~lr-1 Var w,,v ]}
i= 1 t= 1


i= 1 t= 1


i= 1 t= 1


i= 1 t= 1


(xils x) (X1




1 x 1) (x:1 x t


i= 1 t= 1


i= 1 t= 1


i= 1 t= 1


s


-Q1E [(xi,


x )~ (xj~t-1


x ) # 1


-1'E [(x l


x ) 1


cIx Tui,t) (xi,s


i= 1 t= 1


cIx Tue,~t


t= 1


-1 x


cIx Tue,t) (xc~t-1


x _


i= 1 t= 1


x ) 11


x I) (xi,t


i= 1 t= 1


i= 1 t= 1


- 9x Tue,t/lPi






X _1 ,- Tui,~t


i= 1 t= 1


XI, _1- Tui,~t -1 (X ~


[wi,tw:,, I Yn])


~X:j _1 T i, t






- Tuiert/ 112


X X X _t- -ia I 4,g


(QE #(xi,t-1










i= 1 t= 1


i t


+(

T -o

i ( ?


-1 i,tt-
i= 1 t= 1



i= 1 t= 1


i= 1 t= 1


i= 1 t= 1


1 X X5;-' t i-



t-1 X _X~t-1) O'


(4-5)


X ~i~ -1- I Ui,tl U ,t

ni ~X _1-,


i= 1 t= 1
X (X:L X:I _1- U4,g


Finally,


i= 1 t= 1


R-1 E [vi,t Yn ]


i= 1 t= 1


Ailtx~ rui~t)' R-l (ysis


Ai~tx~ rui~t)


(4-6)


and


i= 1 t= 1


i= 1 t= 1


Ai,tx )t I ui,t) (yi,s


-1Var [as As] tr Q if itxt
i= 1 t= 1 n i~~~


'Var [vi,t | Yn ]}


i= 1 t= 1


Astx Fust)











i= 1 t= 1


i= 1 t= 1


i= 1 t= 1


Ai,txi(1 us,t) (Ys,s -


- As:Ix:L uZ,L) (xiZs


-~ tr R E

i= 1 t= 1


i= 1 t= 1


Ai~tx(1 rui~t)


Astx( ui~i)l


i= 1 t= 1


i= 1 t= 1


-'AstgAit}


(4-7)


Ai~tx) ( Fui~t)


Combining (4-2)-(4-7) and substituting expressions (2-7)-(2-12) into (4-1) yields


Q2(8 18!1-) = ~kln|~ol|-:

N 1Vn |Q- |


ka p
i r o + xi,


jLo) (x: -po)1


~tr { Q [S 1


- Slo@' OS:o + @Soo ']

- S12T' TS:2 22S~T]

R-1Ais,PZgA:~,)


-tr { Q [@802ol' 0't20'

N1Vn|IR- |- r
i= 1 t= 1


i= 1 t= 1


Ai,tx(1 rui~t) (yet


which is expression (2-6).


1 Var [vi,t I Yni]}


A-txL -U~ ~ Ius i])


A,:e (xilt x )> (xi,, -





- As,txl ( ru1~L) (Yz,s


As r) usi


Ast) us,t)l R-l (ysis


Asi FusI)1) ,









4.2 Estimation of Parameters

Tob derive equationsi (2-25)-(2-29), we simrply maximrrize Q2 (8 81 8i ) in (2-6) with

respect to each parameter. Note that matrix differentiation results from Petersen and

Pedersen (2007) are used throughout, except when the parameter of interest is symmetric

(i.e., the variance-covariance matrices Co, Q, and R). Thus, starting with I-o, we have

sQ t8 81 3) 6k

z=1


i= 1 =(

-~, [k~o1op


0,

which yields, as in (2-25),


i= 1
So long as there are replications (k > 2) po and Co can be simultaneously solved

for. Hence, in this case, it follows from (2-6) that the expression we must maximize with

respect to Eo is


~k n|Eo |I tr Eo P + (xr~ $- o) (x $po

Since Eo is a variance-covariance matrix, it and its inverse are positive semi-definite.

Using a standard result in multivariate analysis from Anderson (1984), this function is

maximzed t Eo = kP J (x: 8 -o) (x 8 iPo) Solvingfor Eo gives
expression (2-26),

~' =~ii=1I~k
i= o 1 ~ i~









Tob obtain ant estimnator for 4, we mrust differentiate Q (8 8 O~-I)) with~ respect to 4,
set to zero, and solve. Therefore,


6Q( Ii1) [t. {Q- (-SIo@'I (IS:o + O'oo'50'))]

b [tr Q-l (OSO2T' + TSO2~))
=Q 1Slo Q-~ Sio Q- TS'2

0,

produces the estimator in (2-27),




If we substitute TC ) above with expression (2-24), we obtain the alternate form of (2-27),




Moving on to the equation for Q, we maximize






in terms of Q. Revisiting the Anderson (1984) result, since Q-l is positive semi-definite,

we obtain the maximum Q-l = NV (S11 Slo@' OS 'o + OSoo + OSO2T' +TS '2

-S12T' TS:2 + TS22 ; -1. This yields (2-28) as the solution for Q,






N[S11 Slo~Soo ISo + T( jS2,Soo So- S:2

+N [(Smt-o~o O 1)T')-T)(dSo O 2

The last expression is obtained by substitution of (2-27) in place of #(j









Finally, thle estimrator for R is: obtained by mnaximrizing Q (8 1 O~-3)) with respect to

R, which amounts to maximizing


N1Vn |R- | -1 tri~ R- l (ysi I.x P ut) ().t Asit I( LU iu~))
St ,i= 1 t= ] n as

i=1 t=a

Again, Anderson (1984) demonstrates that the maximum in terms of R is


i= 1 t= 1

since R-l is positive semi-definite. This is equation (2-29).

4.3 Proof of Exogeneity Theorem

By the exogenous assumption made in Section 2.2.1, E (Uie I,t, Ui,t-1)

E (Uise | Ui,t-1) Thus, the conditional mean of the variables ui,t = / (gi,t) and hence of
the underlying process that generated the observations, gi,t, i = 1,. ., k, t = 1,. ., us,

depend only on previous instances of the covariates and not at all on the responses, yi,t.

TIhus. ,! drops out of any conditional expectations termrr related to gsi~t and ui~t due to
exogeneity.
4.4 Proof of Lemma 2.2
For each i = 1,..., k, we must demonstrate that y,, A~l) xt ) -1 ,1t)B ge l)

-0 Bg I% ,\-1,i.e., that


EA1 y A) ,txl B 0,\) B + 0 ge u =0, 4-8



for j, mb = 1 1. N~ow, from (2-34), we\ have thty =A_ is+E)uf tu

v Also, note that v~l 1 % ,F- and hist I 0,1t)-1. Thus, for each j = 1:, .., t 1, it









follows that


Ey- A ,)tx-l ,\\lB +1 (2 $B get Y~

= EA xi' s x~Lt y ,t exh y~


8


E E xa- x -1 t) (t
E U ,\\Bit i 0 B ot- i


xt-)IV 1 T(1) 1(


1,. ., t 1, we have


using (2-34). Similarly, for each m


E y :


A -1) _r~ ~t1 2 B B2) g!"~) U-1


x e-) u ~]E[ v ,t()1 h~12 h,2 ui


x ~ i 1) u


(t'>u Uii ;())


x 1, U ui~


gt 1 U u(1


=0.


Hence, (4-8) holds.


E ,\\B B g


EA E xis


E 0 \)iB B E got


gt' UT1 yii


E A!: xi,;


E1 (2 \ B2) 0 B gl


E E A x a


E(1 E2 B2 ,\ B B g


E [A ,E (xi,;


.E [ ,\)iB j + B~1R1j E got l










4.5 Proof of Theorem 2.3


Calculation of (2-41) is obtained directly since with (2-33) and (2-38), we have


Ea xa + ,,_ a <1-) DU(
E O i t 1+ tu, it) u t-l, DUt

xt -11) +E ETrl fn Bgi,t + h,\>+i~ \U,t2 B (giot + hit2 U rr_,

@xit -1) + Tui,tgi t


x -l


for i = 1, .. ,k and j


1,. ., us. Thus,


(t-1)
Fit


-1'


x ( xi~t-1 XL-) xt -1 T,ll (g~;


gt'' Thi,t ws i #1 vU 1


Pi O'Q+T B,tF


Bt+H T


is as in (2-42), since


(t-1:)' (1) (1)


(t-1)
E [(xit-1 xit-1 Kit


xLt -l) (<-1)


EE [xi,t-11 !\U >, xi(t-1 < -) DU
E x~tIE g t Ui <- D t -18 (<-1


(4-9)


Now, to obtain the filter term and its covariance, first note that the innovations are


now


6 1t) (1 /, E y t ) T/1 ) i


r1it1 2 2 1 h a DU


=yi~ A i,tx 2 ,\) B 0/B g


E xi(r xi,

E ~xi,t-1


) (X. -(t-l1) (1) T (1)
xt-xi,t i,t-l, i,t--1

xt-1 ) + TBi,tI (g,t g!~) T hi~r + was










using (2-34) and (2-38). The next step is to find the joint conditional distribution of xi,t

and tel ,s iven the observed data, "I) andc U .1 To this end, the conditional mean of the

innovations is




EI ,()it ,\(1 (2, U2 =t E A ft xi -x) 2) V1) I


=~~1 E(2 (,2B) gt 8 e--) 1/(1h,

+0 ~~ ()h1, U 0,


with the aid of (2-34) and (2-38). The conditional variance of the innovations is


E f)xi,t xz~ ,tB 0

x (g ,si -1) v1 (1 ,\h (1)h

xt'j A ,1 t) x1 (1 x h (2)

+ ,\iBt ~ +I B gU-8--,U-

A ,\) Fi -1Ai) R~i, ti r B +1 (a B Fa


x ~it i0 ,)B + B ,) Hi, t >

i~itt1


This result follows, from~~~ ~ n (23)(-3)(-) and sinc P;, ,t and F(-l are the conditional

error covariances for xi~t and gi~t, respectively. Finally, the conditional covariance term is

given by


(t-1l) (1) (t-1

cov xi~t xi~t i,t xi,t x~




iti,tl i>


Var ~/,\i1) T, U l


((1) V(1) (1)
covU xi,t, Ei,t i-1 i,t-1










again using (4-9). Summarizing these results, we have


P l)


P~ A~
(1)
i,t


,t 1 i,t-]1


(4-10)


Hence ,


x ~t) = 4 Ui~

= E i~s 1, U ,0,\ by Lemma 2.2




Using expression (2.201) from Shumway and Stoffer (2000) and (4-10), it follows that

K~ =i~ P~ A ~t.Ths


= 4 ) ( ,t P t) ) R(i, t) + (i, t ) B~i t F ) B's ,ti
(ii~ti) H~t~l> F i


as ive by(2-5).Latly toobtin ,note that we can write


(t t-) (t-) ;(t1) (1)B, a))g~'
xi~ ~~r is i~ i,t 'li,t Aget Tli,

=xi~t xit l)- K A ,tx 1 0 B sl s h ,\)


+ E B g + haIj v l')


AB !t,t )x


(1)
ti,t


()t-1) (-)
=~~ xi,t K ,t) 'l,t) Ai,)xYi,s



as in (2-43), with


K(i,t) = i 4t 0t


A (1) (~t-it1) ()~) g~'
F(i,tl)B 84,tt ,)










xs-xt) = I t xi,t xit ~ K ,\ ift it) + v ft
-i ~ K 0 i,<)1 Fe B 4, -



Thus, with (4-9) and (2-45),


P~ =~ E~ xi~t x x~t- t

=r I-K A~)),( 1 I-K A(1 d~)K 0 B +Iit B F~

x ~ ~ ~ ~ ~ 1 fpl [\B 0/B \ K 0 \,,)1 H (2 t > K ,t\-
i~i, ti

+~i K(a Ri t,1


+ (r~1i,t) +Fi,t(1) B2 2)'i,tF B' I i,tlHi)t K 'l) i,t) 11


= I -Ki,t AL,t)l Fit





in agreement with (2-44).

4.6 Q (8, O 8(1 -2(1),: O -3):issinlg Data Case

Let w~i,t> = xi,t Oxi,t-1 T(i,t)u~i,t> and denote y~i,t> A~i,t>xi,t L(i,t)u by

v~i,t>, where T(i,t) = rr,\ u' = us u ,~)) y'=@,y ,A't> =~) -
A ~ ~ ~ l~ ,l~ A adFit)= 4t 4 .Hr, E rerset cnitiona expcttio with
L(1) ~(2)
i,t2 i,t2

respect to OC -1) and RC -l) and thus both are dropped after the first line. Starting from

(2-51),




= k~In |Eg| -i E Ce(ge,o ; ,i g~ pb L~)I Y ) U
i= 1
+ 1Vn | C- |


i= 1










N 1Vn | H- |


i= 1

~kl In |Eo1|-


- Bi,tgi~,)' H- (ui,t


E ~~ [(xi~o po)'
i= 1


cl


i= 1 t= 1



aE [(g,,o
i= 1


[v'R- v~a,t> I ~I:Clt


pi) (Ei) (gio


p) ),C


N 1Vn | C- |


i= 1
N 1Vn | H- |

-1 E [(uiit
i= 1
+kl In |Eo |

-1 E [(xiio
i= 1


Agi,t-1)l C- (gi,s


Age,t-1) I1 Ct)


Bi,tgi~,)' H- (ui,t


Bitgit) 1 C


(4-11)


Ipo)l I T~), C ~~] Co E [(xi~o


- ) tr { Eo Var [(xl,o
i= 1
+~ N1Vn|Q- |


i= 1 t= 1


-~ tr



i= 1 t= 1


o (xi,o po) I1 L, IC)


N 1Vn |Q- |


tNln |R- |


Q (8,62O 80-) 90-1)) = ~kln| |-


po) 1 C


po) 1 C )] }


rw-,t>l ( CQ E [w~a,t> I I), Ci 1


{ Q-Var [w~i~t> I ,~I C l)] }



[v'-i~ V1 ),U (1 R E [v 1 12!, iin

{R a vi~t ( C)]}










The assumption of exogeneity is pivotal for the EM methodology to work. Without

it, the proposed theory as described fails to hold. The first six terms of (4-11) relate to the

second state-space model, which treats the exogenous variables, ui,t, as responses. These

terms can be handled separately because of Theorem 2.1, which is proven in Section 4.3

The EM algorithm with missing values in the response without exogenous variables

appears in Stoffer (1982). Assuming Hi,t,12 = 0, the E-step for these six terms gives




i= 1
+ Nln|C' |-tr C [~ 1S 0h- St)'A o)~,+ AnSoon')A

+ NIln |H- | (4-12)


i= 1 t= 1


tr H B~~i, t) ,
i=1 t=1 0~n) H ,22)O Ij

wherethe xpresionsfor S10), and S oo,) ar~e given by (2-53)-(2-55).

Each remaining part of Q (8, O2 8 31, O 1) wZill be simplified individually.. Thlus,


E [(i~o- po v Y), U i()] E E(xi~o o) Y ~), U~
i= 1

=~ x oni (no /ox P 4-3
i= 1

and



i= 1

=~ tr { Eo E [(xi,o po)(xi,o po)' Y )
i= 1


f~i=1 1 ~(x~











i= 1


ftr
i= 1


Eo E


[xi,o


x ~nxi,o


i= 1


i= 1




i= 1


[xi.,o


x!uj x !~


o o


k~t


i= 1


Io o ,


(4-14)


p j-), ,A 1


Now, note that given RC )


Ci -'j, Hi )) whichl arec thec


parameter values produced by the previous iteration of the algorithm, we have from (2-38)
that

u ,tit -( ~i,t i]t11ir,1
i~t r it,
u(2) B(2) H~jl H j-)
i~t i~ti,t,21 i,t,22


Henceforth, the superscript (j


1) is dropped from all Hi~t terms to ease the notation.


Thus, it follows that


B )ge,


K)g,+i~ t21l11u


u (2).


(4-15)


Definingf


B g ,


H() H~t2H, 1 u


(4-16)


x (1


), IilI


x po xio


-X~ p1o x-p


xs -~i (Xo


(xii llo .


E u2) )










then we have


EB goti + Hi,t,21, 1 u ,t) B got ni)

us ii,1 ,a 1 u -B


(4-17)


Also, it follows that


Hi,t,22 Hi~,1 11Hiut1


(4-18)


Therefore, using (4-17),


i= 1 t= 1


n i) -r T1 u ( -r T2) E( u 2


-i x -Tu T2) u ,L)


(nx" -T ( u 1 -Tr2) Ut +H


i= 1 t= 1




i= 1 t= 1


(4-19)


t, t- 1,


To e~valuatle t;ii r VLar [wi,t>, Iv1 2!, C })] first note that for j
i= 1 t= 1


(4-20)


E (ut' 2)))


E u ) (a Ua 2(a H -U 2),


E [w' 1 C Q w,"nt In ), C


x"i) -r T1 u 1 T2)2) + H )


E [(x ui x u ,



=Ex [, E( u L (1) ,n C xu) + H(I)1 i,i byeogeneity

x~) (of + H .I)~ x u ( H .I)









using (4-17). Also,


E u u L


Hi~,t,22 Hi,t,21 H, a11 Hi,t,12

+ E Bt glot +t Hi,t,21Hi a 11 us


S(g i;t + Hi,t,21H,

Hilt,22 Hi2t,21H, a 1IHi,t,12


B ~g 'ttn) + H .) ,


+ B it,21H, /11 ,~)) F B,
+ B, ;, )B g )+ H.)


Hi,t,22 Hi~t,21H- 1 Hit,2 ) + Hij) u 2) + H .))

+ Bi Hi,t,21H, /1, F

x Bt Hi,t,21Hi~, /1B


(4-21)


using, the fcts F~n)', = E go

and results (4-15)-(4-18). Thus,


8 git


i= 1 t= 1

= tr{Q- [w<>w' ,t> n 1 }
i= 1 t= 1


i= 1 t= 1


Xi 1 r ,1) U 1) (2) u (2) + H j)


i= 1 t= 1


x x


i= 1 t= 1


EL [xi~t


x xi~t-] x (1) ), C


Bl giot>


us, gt US


g U an g


E got U, ,


- 1 X n )


(X X u) ~X~ -r 11) U It (2) 2a) H~)I


QBF E xrr


-1E [xi,s u T 1













i= 1 t= 1

x( x cn,- ~x cn,-r T1 u [


i= 1 t= 1



i= 1 t= 1


x x,<-1


i= 1 t= 1


xni _r 1) u (1


x (x l


-~ tr Q

i= 1 t= 1



i= 1 t= 1


E, x ,,


xi _r T1 u1


X~j~UL(ii X -Xr )n ) r() 1


i= 1 t= 1

X Xi,t-1 X ri 1'


f~ij ~ -1E xini ~x (ni (1t u u1 T '( C
i= 1 t= 1


E xi,t-1 x xt X t,s x ,


E xit-1


xi~- ni iCl


IE [I xi,t-l -] x u T 1 C


E ~T u~ xin-

E T[r a u c xi,t-1 xi~- ,; )


Ea)uu~)a T Y'- 1 ,


-1E T a u xa ( ,


xi _r 1) u (1 CI











k as)

i= 1 t= 1


(nx -r T1 u 1


i= 1 t= 1





i= 1 t= 1


Xi- 1-riT( uit1 -riT~ u ~


i= 1 t= 1





i= 1 t= 1





i= 1 t= 1



i= 1 t= 1


1~ -1~t X n )Xit 1


1 ( -1 Xni 1X tni))


(8"Ax)9x- )e}


1 2) (Hi,t,22


Hi,t,21H, a 11Hi t,12i


1 2) )


Hi~ll~,t,21, a t1)>B~


- Hiz~t,21H~ e1BT


(4-22)


k ast

i= i i 1 t= 1:


1(1) r (2) U (2)X (ni)
it it it it


(1) r (2) (2)
~x(ni) u u
it it it


i= 1 t= 1


lr (1) (1) uit(1, +r ca, ca,
it it it it


X( X cni ) ni X r 1 Ui> Itl (7


Xni 1-T ( u 1 --T u2 +H(


+H.~)


lr (a (a) Hij) u2 + H~~) Y: T


(ni)
it-























































Var [w~i,t> Y I, U }!]


T 2a) 2a) U3i) l



- T ) (a) x -r1 T 1 {

-~(Ii xit- x-T,t) H T


+~ tr Q



i= 1 t= 1



i= 1 t= 1


x i
i


t 1

T u ,(i

- ~ T ,t }


using (4-20) and (4-21).

If we further suppose that cov h Hi,t,12 = 0, then H .) = 0. The

assumption that the observed and unobserved components are uncorrelated greatly

simplifies calculations and is not unreasonable. This reduces (4-19) to


E w-,t Y ~), U l] Q-lE [w Y~l i, U ]
i= 1 t= 1



i= 1 t= 1


(4-23)


and (4-22) to



i= 1 t= 1


i= 1 t= 1


-~ tr Q

i= 1 t= 1



i= 1 t= 1


-1 i -1 X 1X )


f (n)1 + i x x )>


: (;t~ u H ,>>
















rt )


- ~


i=1 t=1



i=1 t=1


i= 1 t= 1




i= 1 t= 1


1 U(it) X i)


(4-24)


- 1 :[

0Tit


T 4, t)


i= 1 t= 1

- X i)
i= 1 t= 1


't~x;~l-r~b~);(Lb)) -


1~Xi) i


X 1-(it)U(it)


with fi(i~t) as given by (2-56). In (4-24), O and I are suitably conformable zero and

identity matrices, respectively.

N\ow, to simnplify the portions of Q (8, O 81 -1), O -1)) that relate to R, first notice
that given OC -1) and RC -l),


~ i


xi,t +


01)
i,t1
(1)
i,t2


,(j-1)
,


(1)
Yi,t
(2)
Yi,t


A (1
i~ (a


0 (2)
i,t1
(2)
i,t2


(1) (2)


i~t,1 i~,12 The (j 1) superscript is henceforth dropped from the
p(j-1) (j-1)
i,t,21 i,t,22
simplify the notation. Using the same technique and results as before,


with R ]1)

Ri~t terms to


E (y ~


Y ), U ), xit, u,)


nA xi,t + i,1t2 Ut 1) )t2U it


+i.t,21 [t 11 ,1t)


A ft xi,t 0 ,1t)1u t r ,tiutt

(4-25)


Q-1T~~)I~ o1Tit
0Hi,t,22


Q-1Tit 2~~~) r(i) i) Rl~i) 0.Bt
,[iI:~t


X 1


1 X i) X 1 U it) it)


U 1) i,) i,)

u ~t u I.O
















it,21 t 11 ,


A sxi,t C(1 ,\\u ,l ()1 + H .))

(4-26)


and


E y Y U ), u


EE yU \Y ~ Un ),n ,xi,t, Ut Ui
A (2x(i 0 }2 H 1) ~() U2


+Ri~t,21KI [t11 ,1)


A ()x -0,\\1)ut }t2u (2


(4-27)


Lettingf


(. Li,t,21 [it~l 11 ,


(4-28)


then


it,21 t 11


= A(2 xni Ct2lU 1) t()2 g




Analogous to (4-18), it follows that


(4-29)


y((2) 2)


Ri.t,22 F i:t,21 R[tl R1 .t12.


(4-30)


It follows that


E y Y U ), xi,


E E yU Y U ) x~i~t Y ,U)
A xis E,1t(21) +I()i2 H2) + H-j


Aitx -~~ i0t i,\t 1 +H ,


E y Y ,U ))


E Et y, Yt i,U i,x i~ 2 (,>
A x) (i ,t2l U 1) t( )2 t


A~ x -~ 0 ,tu -~ ,~t +i H .))


2)~ 1) ) Xi,t, U ~


E: [y










Hence, with (4-17) and (4-29),


i= 1 t= 1


E [v' 4,t> I ,n' C~l) R-E [v I C ~)]


E (y~i~t 1 C (3 A L,t~x"-
i= 1 t= 1

x R E (yi,t>, 1 1 Lif)) A~ri,tnx "


i= 1 t= 1


x R l


Ax F(it)



- Ax F(it)


(4-31)


tr { RVar [v IVt>1) 1,~~ C ,note that usingr (4-26) andl (4-29)
i= 1 t= 1



E~ xi,tE y Lx, 1 L -" 1,


E xi~ A 2Xi~t 1)2U, 1) (2)2 H 2) +H:1) +R~i,t,21R 11

x y Ax ~t-I 0(1 u -~t u 2)1 2)+ ()


x


To resolve



E [(xi,t -


since E xic~tx:;,t "jl U

(4-27)-(4-29),


A R Rit,12 ,


(4-32)


~I) x x l4n). Also, note that with (4-17),(4-21), and


-111 it,12


- Fi",t)E (u i~l )]


E[utt E y la~1 ), ,u ,


E u A 2~)X a ) (1)2 1) it21 1

x \y ,t A i~x Ii tui


+E (u LlU2)U I~f)r










Hi~tt,21HTit1,Hi,t,12

(4-33)


Another term that will be necessary is


E y )y Y ),U


EE y yi 2) al Xi,t, Ui I nil

Rilt,22 Ci~t,21 sTt 11 i,t,12

+ E A 2)Xi t 1)2 (1) () 2U ) ,1 t1

x y ~sxi,t 0 2U 1) )U

xA x~Ii,+Et)2U1)t )'i2U it,1 t1


x (y :


A ex 4, 0 t1 2 H 1) ()2 H (


Ri~t,22 ,t,21 [ 11i~t,12 (. (.) i~

+ A(2 RCi~t,21 sTtl 11Als Ri,t,21 st 1 it1)

+ 0 2 i,t,2 1,)1 [Hi t,22 Hi,t,21H l1Hi,t,12


+ (B :- Hi;t,21H l1B F~ >')


(4-34)


,


Rilt,21 [lt 11 1 >)


which is obtained with the aid of equations (4-17),(4-20), (4-21),(4-25) and (4-30).


(i + HI.) () + R(.)I [Hi,t,22-
+ B Hi,t,2H 1B F

x B ) Hi~t21HT/1B 02


E us YP En ), U


n 1) ):~


Hit,21Hzil11B





u(s Y ,
Y U )


Hence, we have


i= 1 t= 1

= tr(R- [v I~tv Y~) U )ni }
i= 1 t= 1


-Aii,tzx i


x


-Axir~l


n,
Str R
St=1

(1)
i,t
(2)
i,t



i= 1 t= 1


(1)
-1 t
Ea
Iy ~


(4-35)


A x )



-Ax ~i


F(i~t)


F(i~t)


x R-


Calculation of the first term in (4-35) is done in pieces. The first expression is


A ,t xi,t F ,1t) u ,) -a (au


A!s xi~t I-,tl u,t -~ i,t u )


x (y ,s


A,t x~ F ,\)ut ,t x -F[\


- A ,ex :" ) (1 ,< U ,a x4 ,1(2 x A
-A~itx -F,\)u\ u i0t Y ,


t~~r RI Yet
i=l t 1y~ +R)


U(1)
I (i~t) i,t
ui,t + H~)


(1)
Hi,t
( i,t) +H~) 1


y+ R i,t 1


(1)x~tI (1) U(1) (2) U(2)
aXit i,t1 i,t i,t1 i,t

Alit xi,t ,t2 t )2U t


(1)(1) (1) (2) (2)

A(2)(1) U(1) (2) U(2)
Axi~t X it i,t2 i,t i,t2 i,t


E y ,t


E. [y -

-. E y ,t

-E (y~t









xni y,(1 A x ) ,< u '

xi \ x,t -~ x ~ A~ t Ui,

x u 0 ,
1n)-A(C~, x ni -~ ) 1 u ,l)


E 0 u 2) U ,~)1


+ E lr(a u 2a) (Xi~t X i) A~ ), C

A(1 xi,t ( u -1 (a u a


A xi,t 0 ,t u -1 u 2)


x (y t


x (y l


(4-36)


+t F I (Hi,t,22
+I~~~ ~()fa
1~t \ i,


Hi,t,21H e 1H,t,12) 0 + A[ I 'Ia A~


Hi,t,2iH1H B (1))BI,[ 02


Hi,~t21H, ,,,it 1 B ~, t F B :


using (4-17), (4-20), and (4-21). Simplification of the second term yields


E y,1< -Ai xi,s 0,tu ift -~t iu

x y 2) A 2)Xit1)2 (1) (2)2 (2)


A ( x -ni (1lu y C1


E y 't

-E y


(1)
r
it


E [y :


A6~'x' 1 -0 u xet-

Axlni -Fu u1 Ui(, I


2U1


-[ EA( xi,t x y 1


+E i,t xi~t- A


-. EAj: xi~s -

+. EAj: xi~s -

+EA: xi;,-

-E [r%' u2)


u;l 1, i['


E y 't


),(1


A xi -Fl~,tu, 2+


A t) x1 1 Ec -0 2) H )


A x~ni -F u A1 2)X i)(1 t21) ) )


xi> A'2 1 1,
X ,t li~


()


,L










+EAt xit-" i0~n [r-E l0t lu y1
+ E [A0 (xu~ 2) ; A 2)I4 12U 1 ) ~u~tca

+E [Ta lu2) Xi,t-X ,n) ( A~(1, )u )


EE yT(a -A) xixt-E u It> -F2)1U ~r(2 2)'2)~

x ~ ~ ~ 1 y 2)-A2X ( 1) 2U ) 2)2 )


= yt -A x"1 -tu -F)2)1 U 2) Hi
x y2)+ -A )(n) 1U )_ 2 U )+
Hi~,22 i,t,21 i e 1 i,t12 t 1 ~ t1

( B Hi)t,1H U (1)(B F ) B(2)- it,1H 1B
x ~ i0 R i,t,1 +~ A~ t I A R it,2



E A x~- ,u u y )Ait~t ,1)U1 22 2


E y A2)X(i,t -,12U 1 )it2U,

x~ t y A 2)iit 12U ) 2)2U2)~; )itl )13

= E y fa Ht,L1 H-E y2) X(i) t-Xn) Ait2 ,L] 1)


2)~t1 X't,,] a ) 1)2 H(i 1 ) ); 0 1 ,

+l~t E A 2)X~ i,t n)A ,U



+ E A 2) i,t X~1 n) ~) A 2)X n ) 1 ()2 (1) () ~2
+ ~li EA 2)t Xi X~t n~t )




E A (2) Xi,t a g) (1)2 U, 2 12 )":










+~ E.~ A ,t)2U 1) Xit X.t n ~ ) Yf)
+ E Aani x @ )2 U1) ni Aex @, t 2 1) ):~

+E 4:A ni ex" +Ft21)) U! n) () 1'1 l


+i~ E~ i0 t)2 u~t A~ xi /2 1


En y~t -At xi2)1(2 0,tU 1) )2U
x y A iX,t 0 U (1) )2; U~~a I
= y + .) x" -0 t) H1) )2 uX~ Hn dla)
xr2 +2 R(.) -V1, A tU1)], )211 t +H(.
+ R~,1 t 11 t1 H-i~t2 it21 1Hi,t12s11 it2
+E Ri i,2 t 11i~ X, )12 I(1 Hit,1H /1B 4-8

x~ ~~~~~~it B it,1 1B 0 R 11it1

+ Rit,22 i,t ,2 t 11 it,1 i,2 [t1A,)? A R 11 it1

wit the aid of (4-1)x (4- 20),(4-21) ,2 (4-9,a d(-2)(43)


r {R Var [vt i,t> Y i, 2U
=~ tr R~



x2 (n' -1 A \x ,\u,) 2 (, 2 i)+H(
(( i~tXt ~ U~~t R))










i,t1


Hi~t,21 Hzl/11Hi;,t,12


3 1i,t,1 2


Hi,~t21HT,,B/t1)>B F


Hit,21Hzl

0
-
0


(4-39)


A ,\) Fi A


i= 1 t= 1


-A

- F(i~t)


X R-1


Ahssuming ~o; (vt l, v j2)


Ri,t,12 = 0 in addition to Hi,t,12


H .), thus reducing (4-31) to


E~r Iv Y ,nl) U l)] R-I E [v Y ),! U ri()]
i= 1 t= 1


fYi (i, t) -
i= 1 t= 1
X -1 yiir)


A (i, t) x F (i, t i) u (i, t)

- A(it)x F(iti)u(it)


(4-40)


i= 1 t= 1
~i~(
x [I.%; iI: 1


Hil~,t,22


r R




i=1 t=1


/1 B (1 .0 la Re 1 Ri~t,12



(Ri~,22 i~t,1 [ 11ilt,12 I


i= 1 t= 1


[I 1lRi,t,121


0 implies R .)









and (4-39) to




i=i- 1 t=


= t R y~~t)- ~ it~x" Fi,t uL,t )r

x ~ it)- (i,t)x -F~~tiul)t
k~~ ~ ~ t=] Hi,t,22 B ,'2) tii4) B (2) 0



fi tri~ R- 1i (4-41)
i=1 =1 0 Ri,t,22


k~~~ ~ n~ A P1 "'n) A 0l)O

i=1 t=1

(ilY!~ t)-Aitx itiui)



with? PFi,tI) given? by l'lit
00O
Notice that T(i,t) is simply a reordering of the columns of T, determined by what is

observed and what is missing in ui,t. Thus, post-multiplication of T by Si~t yields T(i,t),

i~e., T(i,t) = T~i~t. Also, note from the definition that F(i~t) is a reordering of the columns
of 0 with respect to the exogenous variables, ui,t, and a permutation of the rows of 0

determined by the observed and unobserved portions of the responses, yi,t. Thus, pre-

and post-multiplication of 0 by ~DI,s and Si~t, respectively, reconstitutes F. In other words,

F(i~t) = Iqfit.G Herncel by he forml of.ri)givn above we see thatl F.ir) = I
00
F(i~t), where I represents the qi,ti x qi,ti identity matrix.








If we substitute (4-12)-(4-14), (4-23),(4-24), (4-40), and (4-41) into (4-11) and use

expressions (2-57)-(2-62), we obtain

Q (8,1 O el 1), (1 13) = ~k In |E|

i= 1


S10~2 o)A'-AS o)+A-~0oo1)A


N~1Vn|IC |


N 1Vn | H |


B~i,t)g~~n) u((~t) -B i),t)8


0
H(j-1)
Hi,t,22


i=1 t=1
+kl In | |

k~ l


N1Vn |Q-l | t~r {Q- IT~ In (S12)


Soa 'O2)


-t(1 (S(12)


i= 1 t= 1


0
0


-1


y(;,)


tIr CC [S ,


1 8; (ni)g~ ) +


S(10)O' 9S1n)+ OSioo) I'])

'ISl(02) TS(22)) /


N 1Vn | R- |


i= 1 t= 1

Y(i,t)


A i,tx "' l D aS,t ui)
0 0


A~i~t) x -D, Suit









k I0 Hi,2 ,ii 2
i=1 t=1 0 0 0 0


x 6 i,[ t0 I
0 0



i= 1 t= 1 Rt 2l

k n1 A (i) P 1 "' A 0
=1 t=t 1 0~ 0~


which is expression (2-52).

4.7 Parameter Estimation: Missing Data Case

Calculation of the parameter update formulas for 8 = (I-o, Co, 4, T, Q, 0, R) when

there are missing: values e~ntails mnaximnizing Q (8, O2 81 1), DC 1)) inI (2-52) wzith retspectl
to each parameter. Again, matrix differentiation results from Petersen and Pedersen

(2007) are used throughout, except in the case of the symmetric variance-covariance
matrices Q and R.

Starting with the estimator for T, we have

6Q(OR ("l):R~l)) I b[tr { Q- T (S;12) S'O2)(I )


[tr { Q ((12 -

i=1 t=1

Q- (S(12) gS(02) TSi


ka 0 ~l


(~t22)


C1
C~it


OS(02) -sa, TS(22


0Ti~ Hi: ,t,22O ,,)









Solvingf this equation produces


-1


i= 1 t= 1


0

i 2-l


(S(12) 4") S(02)) S-


which is expression (2-70).
To derive an estimator for 4, we proceed in similar fashion. Therefore,


crQ (8, O2 81 -,~1) 01)


S6
2 [tr {Q- (-S(10)@' ~o)+Soo

[tr{ Q (- S'O)@'- S(02)T )


Q S(10) Q- OS'oo)


Q- TS'(O2)


Setting to zero and solving produces estimator (2-72),

QC ) = (S(10) TC )S'IO2)) S ~:.

If we substitute TC ) above with expression (2-70) and simplify, we obtain the alternate

form of (2-72),


S(02 S- S'O2 ,)] S


where S is given by (2-71).
To obtain the estimator for Q, we maximize


~tr {Q- (S~try


S(10o)@' O5';o) + @S~oo) '


T (S12) S'O2)@') (S(12) OS(02)


TS(22) r


0

Hi,t,22


(S(12) 4") S(02))


O = [S(10) S(12) S- S'O2) ] S I


N1Vn|IQ- |


i=1 t=1










in terms of Q. Appealing to the multivariate analysis result in Anderson (1984), we see

that this function is maximized with Q as in (2-73),











cr1 k8 n' 0(l,~3) 0
6~~i= t=1 0 H ~ii=, ,22 ((I) .~I)xn





(j-1 (j-1 k as ~ ~ ii it))
iY~i~-1 y(4 At)Xtittox


I St i t)
00 =]0






x Hi 2 i, ii) ()0 gt' It 0
0 0 00



i= 1 t= 1 0 0

(i~t) A ,tx



i= 1 t= 1 0 0



00










-1 Y(i,t) A;1,t!3x "







I 0 ,t tio


i= 1 t= 1


X u i, t) ~, t i



rii= 1 t=l 1


t
2(


Hi,2 ,
0Lt2 i~


x


2(


TQ (8, O 81


1) 02(1 )


A~i~t u 4,t) C4,t


k n i

~~iit


IO
0 0


0 0


product result (for matrices) from Harville (1997) stating that C5= AjXBy


C is


equivalent to C= (B


vec C, we obtain


(j) I


t=1


Gi,t


H ( + B(2) F(ni) /(2)j 0
i,t,22 i,t i,t i
0 0


u 4,t) u~i,t) +


(R )) y(it)


k:~l n I 0I
i o ,)


IOC
-1 Uo:~iU i~ t) U i,t) ,~t
0 0

IO t~,
-1 o


x ,,2 Si, t,


In order to solve this equation set to zero, we have to isolate U. Using the K~ronecker


Al)] vec X


-1 I 0 ,t


~Di,t I 0
0 0


x vec Diizi- s o 0
i= 1 t= 1 0


A (it)x u 4,t)4,










This is precisely expression (2-75). Since vec PC ) is the qr x 1 vector created by stacking

the columns of 0 j) one below the other, we simply undo the vec function to obtain the

q x r matrix FC ). Note that the equation for FC ) uses the unordered estimator of R at

iteration j, R) given in (4-42) (see below). This adjustment is made because the formula

for FC ) already performs any necessary reordering.

Finally, thle estimrator for R is obtained by mnaximrizing Q (8, 2 81 O~- ), DC 3) with

respect to R. Thus, to maximize


N1Vn |R- | -1 tr R- y(i,t) A~i,t)x l Dtff ntiit



x y(i,t) A~i,t)xtt "' o t tui)


k I0Hi 2j1 i(2 (i) (2) 0 lo

i~~aTi,t i,t it



i= 1 t= 1 0 R =1t

we refer to Anderson (1984), which gives the maximum for R as


R =y(i,t) A~i,t)xZ~ o Do tuot
i= 1 t= 10



x y(i,t) A~i,t)xni -: lo Dsuo ~


I 0 ,N ,t i, ,2 it i) 2)
i=1 t=1 0 0



i=1 t=1 0Ri~t,22 i=1 t=1 0


(4-42)


0 ,t' DO
0 0 0









Now, this must be rearranged to its original order to find the true estimator for R. Hence,

pre- and post-multiplying by ~Di~t and ~D~,, respectively, gives

RC f) = Di, t= jY(i, t) A (i, t) x o Dt)S, u(,t

x ~it)- ~~tx l Dt i,u ,) s




1 k~~ ~~ n= [I 0o~ Hi,-1 2 O2 ii) B(t2) 0O






1 k n 0 0 k n P ") A' 0l)
1 t= i,t,22 1i=1 t=1 0 0~


which is equation (2-74).
4.8 Proof of ECM Theorem

We must first show that our adjustments indeed constitute an EC11L algorithm.




version of Q (8,n O 81 -), D 1 ) is mnaximized with respects to R w~ith (2-74) except

that PC ) takes the place of PC ). If we denote this estimate as R(j-1+1/2), then clearly

Q (J-31)\R( 3 ) R(j-1 1/2) 1(j-1031(j1) 2(j-1) > (j-31), ~(j-31 ~(j-1)(-)) This
constitutes the first of two steps in the C11l cycle. Next, set the constraints R = 01 j-l) and



(2-75) wvith R j-1+1/2) in plaC6e Of R~i maximnizes Q (8,n O 1 0(1), DC -1)) subjects to these
constraints. Note that R j-+1/2) represents the .Unordere~d versions of R(j-1+1/2) and the

formula is given by (4-42). If we express this estimate by F(j-1+2/2), the update ensures









Q (8 ) \0 ~-1+22) (-1)(j-1)(j-31))





These tw~o steps together result in (BC), Of 3i) ,T),Q ,j122
R(j-1 1/2) ~(j)) an~d the Q functlion is guaranteed to increase. Thlus, by Definition 1 of

Meng and Rubin (1993), our procedure represents an EC'jLl algorithm.

Next, we show that our EC'jL algorithm converges. Theorem 1 in Meng and Rubin

(1993) shows that any EC'jLl algorithm is also a Generalized Expectation Maximization, or

GEM, algorithm. Briefly, Dempster et al. (1977) define a GEM algorithm as one in which
the M-step iterates increase the Q function. Note that the conditional maximizations

given by R(j-1+1/2) and F(j-1+2/2) are unique. The proof of Theorem 2 in Meng and Rubin

(1993) demonstrates that any EC'j l with unique C'j l steps satisfy Theorem 1 of Wu

(1983), which shows that the limit points of any GEM sequence are stationary points and
the log-likelihood converges monotonically. Thus, the desired result follows.










CHAPTER 5
SIMULATION RESULTS

In this chapter, the results of the simulation analyses are discussed. The ultimate goal

is to assess the performance of the EM algorithm in both the complete data case and the

missing data case. The procedure is evaluated under differing number of subjects, time

points, initial parameter values, and percent missing. Results with and without exogenous

variables in both the complete and missing data cases, as well mismatched time points, are

also examined.

The simulation methodology is now presented. The state, observation, and exogenous

vectors were set to dimension p = q = r = 1 and the parameters were given arbitrary

values. The procedure was replicated on 3 different data sets for each subject/time

point combination. The arrangements considered are di-phi-- 4~1 in Table 5-1. For each

subject/time point combination examined, several starting values were considered for

the parameters to ensure that the algorithm was converging to the global (and not

a local) maximum. See Table 5-2 for the true initial and the combination of starting

parameter values examined. Various percentages of missingness (CI' 5' .~ and 211' .) in

the response and exogenous variables were evaluated. In addition, the model without

exogenous variables and the effect on the procedure of mismatched observations in time

were examined. On each run, the simulated data were alternatingly processed with the

K~alman recursions and EM algorithm described in Section 2.1 or 2.2, depending on

whether any data were missing, until convergence. The estimation procedure was deemed

to have converged when the difference in the likelihood from one iteration to the next was

smaller than 0.01 in absolute value. The tolerance was increased from 0.001 to ensure that

the simulations converged within a reasonable length of time. All analyses were performed

using R statistical software (R Development Core Team, 2007).

Tables 5-3 to 5-6 summarize the results of the simulation an~ llh-- a performed in

the complete and missing data cases using different parameter values to initialize









Table 5-1. Simulation Subject and Time Point Combinations
Number of Subjects Number of Time Points
k = 30 us = 5
k =30 us 15
k = 150 us = 5
k =150 us 15

Table 5-2. True and Initial Parameter Values
Parameter True Value Initial Values

I-0 0 -5, 0, 5
Co 1 1, 5,10
0 0.22 -0.9, 0.22, 0.9
T 1 -5, 1, 5
Q 2 2, 5, 10
0 0.9 -5, 0.9, 5
R 3 1, 3, 10
p 0 -5, 0, 5
E~ 1.5 1.5, 5, 10
A 0.42 -0.9, 0.42, 0.9
C 2.5 2.5, 5, 10
H 3 1, 3, 10


the algorithm and various percentages of missingnesss. The procedure was also run

without exogenous variables in the model for comparison. For each subject/time

point combination, the minimum and maximum log-likelihood values obtained over

all parameter value combinations are given for each data set. In addition, the relative

error is calculated as the absolute value of [(Estimate True Value) /True Value] The

maximum over all parameter estimates (except po, Co, pI-, and Eg) of the median and

75t" percentile of the relative errors are reported as the maximum relative errors (jl1RE).

This gives some indication of how much the parameter estimates vary in each data

set. The MRE is calculated separately for the mean (4 and if applicable, T, P, and A)

and variance (Q, R, and if applicable, C, H) parameters. The initial state mean and

covariance parameter estimates are not included in the MRE calculations because these










are not calculated using any observed data. Hence, we do not expect the estimates in

these cases to be very reliable. In fact, these parameter estimates are calculated by

extrapolating/smoothing backwards, to a baseline time point for which no data are

available. Note that in the case where no exogenous variables are present, F and T do

not appear in the model. Similarly, the second set of parameter values in Table 5-2 are

only used in the second state-space model when there are missing data. In this case, not

all possible parameter combination starting values are examined because this amounts to

running 312 Simulations until convergence for each data set and would take a considerable

amount of time. Instead, all initial parameter combinations are considered for pi, Eg, A, C,

and H but random values are generated for I-o, Co, 4, T, Q, F, and R from their respective

ranges given in Table 5-2.

Some general comments are in order before discussing particular results. The median

and 75" percentile of the relative errors are used instead of the maximum over all

estimates because several histograms were examined and the distributions had long

tails. Thus it seems that a few estimates are really different from the true parameter value

and hence the maximum of the relative error is misleading. In addition, the MRE may not

be the best measure of how good the estimates are. For instance, a pure guess of 0 as an

estimate for a parameter value leads to an MRE of 1.00. So an arbitrary value is better

than the MLE in some cases if the MRE is used to determine the discrepancy between a

parameter and its estimator.

To generate the cases without exogenous variables, the error term from the corresponding

data set was kept the same and the responses were then generated from the model without

covariates. There are less terms in the calculation of the likelihood in the cases without

exogenous variables and hence these should be smaller. However, the responses are

different and hence it is not a fair comparison between the two cases. The fact that the

likelihoods are bigger when the exogenous data are dropped from the model is probably

a consequence of how the data were generated. Overall, it seems that the new procedure





















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cl cp cl cp ~ cp
w ~3 W W ~3 ~J Z ~ ~j 8 8
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d ~z d ~z ~z ~z
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iBi iBi iBi iBi
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fi $ 4 ~
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~Z ~Z ~Z ~Z ~Z ~Z
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converges to the right maximum regardless of starting value, since the largest difference

between the minimum and maximum likelihood across the tables is about :31.34 points

(there are much larger differences, but these occur in cases without exogenous variables).

Several points can he made about the complete case simulation results. Clearly,

the outcomes for the cases without exogenous variables are much better than those

with the covariates. The likelihood converge within a much tighter range of values and

the AIREs are lower, across all four tables. The problem across all subject/time point

combinations considered is with F, the coefficient associated with the exogenous variables

in the response model. The AIREs are considerably large for F, with a median of around

2.6 in all tables for all three data sets, so that half of the estimates are about :3.6 times as

different as the true F value. Although not listed, the AIREs for T, the coefficient for the

exogenous variables in the state equation, are almost as large as those for F. This -II__- -R

that including exogenous variables in the model for the complete data case may be less

important, regardless of sample size, because the estimators produced for the coefficient

parameters are not very reliable. It may be that when there are no missing values, the

previous responses in time are sufficient in predicting future outcomes and the exogenous

variables are superfluous.

The missing data simulations yielded several interesting results. First, the mean

parameter estimates are relatively close to the true values while those for the variance

parameters have deteriorated in comparison to the complete data case, judging by

the AIREs. The mean parameter estimates are about as good as in the complete case

without exogenous variables, where the aforementioned problems with F and T are not

an issue, across all subject/time point combinations. It is not surprising that the variance

parameter estimates have worsened in the presence of missing data, since there is less

information. In addition, the difference between the maximum and minimum likelihood

when there are missing data is generally smaller than the respective complete case across

all four tables. Also, within each data set and missing percentage grouping, for most










sample sizes, the likelihood discrepancies are larger when the exogenous variables are

excluded. These points are strong evidence in favor of including exogenous variables in the

model when data are missing.

One drawback apparent from the missing data simulations is that the variance

parameters in the first state-space model (Q and R) are being overestimated. Those from

the second state-space formulation (C and H) modelling the exogenous variables are being

estimated fairly well. The differences between the minimum and maximum likelihood do

appear to systematically increase as the total sample size increases, or as we move from

Table 5-3 to Table 5-6. But this may be a consequence of calculating the missing data as

a percent of the total number of observations. Hence, with more data there are also more

missing values and thus more uncertainty.

The results of the simulation study to analyze the effects of mismatched observations

in time appear in Table 5-7. To create a time disparity, using the complete data sets (0' .

missing) with k = 150 subjects and as = 15 time points from Table 5-6, each subject

had 10 responses and corresponding exogenous variables randomly set to missing. This

in essence creates a time mismatched grid in which each subject has 5 observations and

the 10 other possible observation times in the group are artificially missing so that all

subjects have the same set of 15 possible time points. These results appear in Table 5-7.

As before, the minimum and maximum log-likelihood values and the MREs obtained over

all parameter value combinations for the mean and variance parameters are given for each

of the 3 data sets. Again, all initial parameter combinations are considered for pi, Eg, A, C,

and H but random values are generated for I-o, Co, 4, T, Q, 0, and R from Table 5-2.

The mismatched time points simulation yielded results similar to those observed

in Table 5-6 with 211' missing data in both the response and exogenous variables. The

difference between the maximum and minimum likelihood are comparable in both

situations. Also, for the first 2 data sets, the MREs for the means are approximately the

same in both tables and the MREs for the variance parameters are in fact smaller in the










Table 5-7. Minimum and Maximum Log-Likelihoods: Mismatched case with k = 150
subjects, as = 15 time points for all i = 1, .. ,k
Data Set min like max like MRE means- MRE vars-
median median


1 -1653.48 -1636.53 108()12(R
1.08 1.28
1.16 (0) 1.13 (R)
2 -1670.84 -1644.17
1.17 1.13
0.64 (T) 6.48 (Q)
3 -1639.93 -1632.03
0.68 7.50


mismatched time points case. However, while the MREs for the mean parameters remain

relatively similar in both situations for the third data set, those for the variance parameter

Q are disproportionately larger in the mismatched case. One possible explanation for

this is that the random selection of missing responses resulted in observations that were

highly variable. The combination of this with the large percentage of missing values

created by the mismatched time points may be the reason why the variance term of the

state equation in the first state-space model is being overestimated. The other variance

parameters, R, C, and H, are being estimated relatively well in the mismatched case for

the third data set. Thus, the only problem is with the estimates for Q, and this may only

be a chance occurrence since it only happens in one data set and not the other two.

In short, the new procedure appears to be fairly robust to moderate percentages of

missing data, even in the analyses with fewer subjects and time points. When data are

missing, the inclusion of exogenous variables seems to add valuable information to the

modelling procedure. In addition, the proposed technique appears to produce acceptable

parameter estimates in the presence of mismatched time points. These are all favorable

results, since these elements are likely to be present in the analysis of real data.










CHAPTER 6
DATA ANALYSIS: AN EXAMPLE

The motivation for selecting this topic for research was the desire to predict when

certain bioniarkers will change for a cohort of patients with the autoininune disease

systemic lupus erythentatosus (SLE). Tracking such changes is important because they

are indicative of an increase in disease activity. If physicians can forecast a flare before it

occurs, they can intervene earlier and thus reduce its impact on the patient.

The analysis provided here is by no means exhaustive or complete, but rather a

sample application of the derived formulation. As an illustration, we will model high

sensitivity C-reactive protein (hs-CRP) using weight as an exogenous covariate. Hs-CRP

has been used as a gauge of disease activity in patients with rheumatic conditions such

as SLE, and its levels are known to increase in response to infection (Barnes et al.,

2005). Detailed demographic, clinical, and medication information has been collected

front this group of patients. Several laboratory variables have been found to correlate

cross-sectionally with hs-CRP (e.g., serum albumin, C3, erythrocyte sedimentation rate

(ESR), weight, hypertension, and apolipoprotein A-I) and these were all candidates for a

predictor (Barnes et al., 2005). Since a variable must he exogenous to hs-CRP in order to

be included in the state-space model as a covariate, weight seemed a reasonable choice.

The available information on SLE patients front this data set will be analyzed using

the K~alman recursions and EM algorithm for the missing data case described in section

2.2. The missing data stent front the fact that some variables were simply not collected at

certain visits, but mostly because patients were observed on different dates. To handle this

difficulty, "Ins!--!xas values were inserted so that all patients had the same number of total

visits with the same time index. Each patient's baseline visit was considered their first

time point and subsequent visits were assigned the number of whole months since the first

visit as the time point. Subjects were included in the analysis if they had at least 5 visits

in a span of 20 months to try to reduce the artificial missingness. In all, there were 68










patients analyzed with 20 time points each. Subjects were missing anywhere from 45' to

T'.of hs-CRP and weight values because of the forced time-griding of observations. The

simulation study in the mismatched time points case with 150 subjects, 15 time points

each and I.T' missing data (Table 5-7) produced reasonable results, so the analysis of this

data set should be feasible.

Although the simulations showed that the starting values had little effect on the

convergence of the procedure, some attempt was made to arrive at rough estimates.

These guesses where then used to generate a range of preliminary values with which to

initialize the EM algorithm. The starting values for the initial state condition parameters

(I-o, Co, pI-, and Eg) were arrived at by taking the mean and variance of baseline

hs-CRP and weight measurements across subjects. Several weighted regression models

were fit, with weights equal to the inverse of the number of visits, to approximate the

remaining initial values. To get at the starting values for the first state-space formulation

parameters, hs-CRP was modelled using weight and the previous value of hs-CRP in

time as covariates. The regression coefficients gave a rough idea of what to use to start

4, T, and F with, and the mean squared error (ilrmi) an approximation for Q and

R's starting points. Similarly, a weighted regression was fit for weight to initialize the

second-state space model parameters. In this case, the coefficient for the previous value

in time of weight was used as a preliminary value for A while the MSE was applied as

initial guesses for C and H. Table 6-1 lists the initial parameter values derived from the

preliminary rough estimates. For the first state-space model, random points were drawn

from the range of values given while all initial parameter combinations were considered for

pI-, Eg, A, C, and H.

Convergence of the estimation procedure was established with a tolerance of 0.01.

The R statistical software function fdHess from the alme package developed by Pinheiro

et al. (2007) was used to calculate parameter standard errors at the MLEs once the

algorithm converged (R Development Core Team, 2007). The fdHess procedure evaluates










Table 6-1. Data Analysis Initial Parameter Values
Parameter Initial Values

I-lo 0 to 100
Co 100 to 200
0 -0.9 to 0.9
T -5 to 5

Qc 10 to 100
0 -5 to 5
R 10 to 100

pi 100, 150, 300
Eg 1000, 4400, 7500
A -0.9, 0.5, 0.9
C 5, 15, 50
H 5, 15, 50


the approximate Hessian matrix of the observed data log-likelihood using the numerical

technique of finite differences. The final model selected was the one yielding the largest

log-likelihood value. Forecasting the future behavior of hs-CRP and the errors associated

with these predictions was accomplished using the K~alman recursions given by Theorem

2.3.

The iteration with the largest likelihood produced the parameter estimates given in

Table 6-2. Front the 1\LEs and standard errors for T and F, it appears that weight is a

poor predictor of hs-CRP. Even though the model may not he very informative, as further

illustration, an approximate trajectory for the response and its prediction interval were

generated. Specifically, smoothed values and forecasts for hs-CRP were obtained front the

fitted model. A graph of the actual hs-CRP observations, smoothers, and 6-nionth ahead

predictors is given by Figure 6-1 for an average subject.

A more thorough analysis of this data should be performed. Hs-CRP should be

modelled hivariately with cystatin C, which is considered an excellent indicator of renal

function (No oI !~ et al., 2007). Simultaneous increases in hs-CRP and decreases in cystatin

C would -II__- -r the potential for a flare in the near future. There are various possible










covariates to include in the model as alluded to earlier, and these should be examined

niultivariately if they are deemed to be exogenous to hs-CRP and cystatin C. An an~ ll-h-

of this kind will likely be very computer intensive, since between February 2000 and June

2005 (the scope of the data considered here), there were 231 SLE patients with between

1 and 18 visits to the clinic. Because of the nxisniatch in visit dates, the number of time

points needed to create a grid could be enormous.









CHAPTER 7
CONCLUSIONS AND FUTURE RESEARCH

We proposed a procedure to calculate parameter estimates in the state-space model

with exogenous variables, with and without missing observations. Exogenous variables are

independent of the system and affect the response, but not vice versa. The EM algorithm

and the K~alman smoother equations were applied in conjunction to derive maximum

likelihood estimates for the model parameters.

In the complete case, we included exogenous variables in the state and observation

equations of the state-space model and derived estimates for T and F, the respective

coefficient parameters. Simulations demonstrated that the inclusion of exogenous

covariates when there were no missing data may be superfluous. It seemed that the

previous responses in time were sufficient in predicting future outcomes.

The new formulation for missing data proposed two state-space models to represent

the unobserved information that can occur in both the response and exogenous variables.

The EC11L algorithm, in which the E-step of the EM algorithm remains the same but

the M-step is replaced by a series of conditional maximization steps, was applied to

derive maximum likelihood estimates of the parameters. The assumption of exogeneity

was pivotal for the EM methodology to work since without it, the proposed theory as

described failed to hold. The simulation studies showed that the exogenous variables

supplied valuable information to the analysis when data were missing, thus providing

persuasive evidence in favor of including these in the model. Overall, the new procedure

appeared to be relatively robust to moderate percentages of missing data and mismatched

observations in time, even with fewer subjects and time points, although several of the

variance parameters were being overestimated.

In addition, analytic recursive formulas were derived for calculating parameter

estimate standard errors. Although the EM algorithm greatly eases maximum likelihood

estimation in the presence of missing data, it does not automatically yield parameter










estimate standard errors. We generated these hv differentiating the observed data

likelihood in both the complete and missing data cases to obtain the information matrix.

As an example of the new technique, the procedure was applied to a real data

set. Since subjects were observed on different dates, a time grid with artificial missing

values was created so that everyone had the same number of total visits and time points.

Parameter standard errors were calculated numerically and smoothed values and forecasts

for the response were obtained using the K~alman recursions for illustrative purposes.

Since the data analysis was only an example demonstrating the use of the newly

developed procedure, a more in-depth study should be performed. It is strongly

recommended that the model consist of a hivariate response and a multivariate exogenous

variable vector, which would take advantage of the full potential of the state-space model

with exogenous variables and missing data that was developed here. This type of analysis

will likely require considerable processing time though, because of the large number of

time points that will be required to create a grid. The idea for this work arose from

the need for physicians caring for this particular set of patients to predict when certain

biomarkers will change. A more comprehensive analysis may allow rheumatologists to

accurately forecast a flare before it occurs and intervene earlier on the patient's behalf.

The predictive capabilities of the model should also be further explored through simulation

to determine how accurate the technique is in forecasting future outcomes.

Although the motivation for developing the state-space model with exogenous

covariates and missing data was in the medical field, examples abound in other areas in

which the procedure would be readily applicable. The nature of the human body makes

it difficult to ascertain which covariates are really exogenous to the outcomes of interest

in the autoimmune diseases data set. It would be interesting to apply the technique in

a case where the independent variables are clearly external forces on the response. One

such example is investigating the effect of environmental factors, such air pollution, on

morbidity, and several data sets can he found in Latini and Passerini (2004).










There are several other potential areas for continued research. The effect of including

endogenous variables in the state-space model can he studied through simulation. For

instance, a multivariate Normal distribution can he placed on the stacked vector of

responses and covariates so that these are correlated. This might give an indication

of how crucial the exogeneity assumption is. A related topic would be to modify the

model to allow endogenous covariates. Although these can he studied using the current

model by stacking the outcomes and endogenous variables in the response vector, it

would still be interesting to incorporate them as dependent variables, calculate their

coefficient parameters, and determine their effect on the outcomes of interest. We suspect

that it will be challenging to derive nmaxiniun likelihood estimates in this case and

that some unrealistic assumptions may have to be made in order for theoretical results

to be developed. More investigation is also necessary to determine how to correct the

overestimated variance parameters. Furthermore, the state-space procedure using the

K~alman recursions and the EM algorithm implicitly assumes that the missing data are

missing at random (jl AR), or that the nmissingness is explained by the observed responses

and covariates (Little and Rubin, 1987). It would be worthwhile to modify the procedure

to handle non-ignorable missing data.










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DEMPSTER, A. P., LAIRD, N. M. and RUBIN, D. B. (1977). 1\aximum likelihood from
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DIGGLE, P. J., HEAGERTY, P., LIANG, K(. and ZEGER, S. L. (2002). A,...el;;-.: of
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DOUCET, A. and TADIC, V. B. (200:3). Parameter estimation in general state-space
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DIJRBIN, J. and KtOOPMAN, S. J. (2001). Time Series A,:tale;,.: by State-Space M~ethods.
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GUPTA, N. K(. and MEHRA, R. K(. (1974). Computational aspects of maximum
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HARVEY, A. C. (1991). Forecasting. Structural Time Series M~odels and the Kalmarn
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BIOGRAPHICAL SKETCH

Arlene Hortensia T I1 I.100 was born in Hialeah, Florida, the third child and only

daughter of Magaly and Claudio No Is ~io. She earned a B.S. in mathematical sciences in

2000 and a M.S. in statistics in 2002 from Florida International University in Miami,

Florida. In 2007, she obtained a Ph.D. in statistics from the University of Florida

under the direction Dr. George Casella. Her interests include longitudinal data analysis,

statistical applications in research, and methods for handling missing data.





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WritingthisdissertationhassimultaneouslybeenthemostambitiousintellectualchallengeandemotionallydemandingexperienceIhaveeverundertakeninmylife.Myjourneyhasbeenlledwithnumeroustriumphsbutalsowithmanyobstaclesalongtheway.Ihavebeenveryfortunatetohavethesupportoffamilyandfriendstohelpmeupeachtimethisprojecthasdrivenmedown.Amylivedthrougheveryhighandmaniclowwithme.ShewasadevotedsourceofencouragementandIamabsolutelyconvincedthatIwouldnothavemadeittotheendwithouthersupport.Iamalsoforevergratefulforthemotivationmyfatherprovidedme.Healwaysbelievedinme,evenwhenIdidnotbelieveinmyself,andthiswastheonlythingthatkeptmegoingsomedays.Iwasextremelyluckytohavehadagreatcommitteethathelpedmesucceed.Inparticular,IwouldliketothankDr.Trindadeforsoselesslysharinghistime.HewasmorethanwillingtoguidemewhenIwasstuckandhissuggestionsalwaysledintherightdirection.IamalsogratefulthatDr.Casellaisastatisticalencyclopediaandwasalwayswillingtosharehisexpertiseinthiseldwithme.Myothercommitteemembers,Drs.Daniels,Christman,Sobel,andRichardswerealsoinvaluableforimprovingmythesiswiththeirinsightfulideas.Finally,IwouldberemissifIdidnotthankDr.Shusterforhelpingshapethetopicofthisproject,particularlyintheinitialstages. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 6 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 8 CHAPTER 1INTRODUCTION .................................. 9 2THESTATE-SPACEAPPROACH ......................... 12 2.1CompleteDataSpecication .......................... 12 2.2MissingDataAdjustments ........................... 18 2.2.1TheKalmanRecursions ......................... 22 2.2.2TheEMAlgorithm ........................... 27 3CALCULATIONOFSTANDARDERRORS .................... 34 3.1CompleteDataCase .............................. 35 3.2MissingDataCase ............................... 37 4CLARIFICATIONSANDPROOFS ........................ 44 4.1DerivationofQ (j1) 44 4.2EstimationofParameters ........................... 49 4.3ProofofExogeneityTheorem ......................... 51 4.4ProofofLemma 2.2 ............................... 51 4.5ProofofTheorem 2.3 .............................. 53 4.6Q; (j1);(j1):MissingDataCase ................. 56 4.7ParameterEstimation:MissingDataCase .................. 76 4.8ProofofECMTheorem ............................ 81 5SIMULATIONRESULTS .............................. 83 6DATAANALYSIS:ANEXAMPLE ......................... 91 7CONCLUSIONSANDFUTURERESEARCH ................... 96 REFERENCES ....................................... 99 BIOGRAPHICALSKETCH ................................ 102 5

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Table page 3-1InitialConditionsforCompleteDataStandardErrorCalculations ........ 38 3-2InitialConditionsforMissingDataStandardErrorCalculations-PartialDerivativesWithRespectto .................................. 42 3-3InitialConditionsforMissingDataStandardErrorCalculations-PartialDerivatives(ofFirstState-SpaceModelFunctions)WithRespectto ............ 43 3-4InitialConditionsforMissingDataStandardErrorCalculations-PartialDerivatives(ofSecondState-SpaceModelFunctions)WithRespectto .......... 43 5-1SimulationSubjectandTimePointCombinations ................. 84 5-2TrueandInitialParameterValues .......................... 84 5-3MinimumandMaximumLog-Likelihoods:Casewithk=30subjects,ni=5timepointsforalli=1;:::;k 86 5-4MinimumandMaximumLog-Likelihoods:Casewithk=30subjects,ni=15timepointsforalli=1;:::;k 86 5-5MinimumandMaximumLog-Likelihoods:Casewithk=150subjects,ni=5timepointsforalli=1;:::;k 87 5-6MinimumandMaximumLog-Likelihoods:Casewithk=150subjects,ni=15timepointsforalli=1;:::;k 87 5-7MinimumandMaximumLog-Likelihoods:Mismatchedcasewithk=150subjects,ni=15timepointsforalli=1;:::;k 90 6-1DataAnalysisInitialParameterValues ....................... 93 6-2DataAnalysisResults ................................ 94 6

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Figure page 6-1Hs-CRPobservations,smoothedvalues,andforecastsforatypicalsubject. ... 94 7

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Wedescribeamethodtocalculateparameterestimatesinthestate-spacemodelwithexogenousvariables,withandwithoutmissingobservations.Exogenousvariablesareindependentofthesystemandaecttheresponse,butnotviceversa.TheEMalgorithmandtheKalmansmootherequationsareusedincombinationtoderivemaximumlikelihoodestimatesforthemodelparameters.Inthemissingdatacase,twostate-spacemodelsareproposedtorepresenttheunobservedinformationthatcanoccurinboththeresponseandexogenousvariables.Inaddition,analyticrecursiveformulasarederivedforcalculatingparameterestimatestandarderrors.Simulationstudiesareperformedtodeterminetheeectsofvaryingthenumberofsubjectsandtimepoints,dieringmissingdatapercentages,andmismatchedobservationsintime.Itseemsthattheexogenousvariablesaresuperuousinthecompletecasesincethepreviousresponsesintimeappeartobesucientinpredictingfutureoutcomes.However,theexogenousvariablesaddconsiderableinformationtotheanalysiswhendataaremissingandthereisstrongevidenceinfavorofincludingtheseinthemodel.Thenewprocedureappearstoberelativelyrobusttomoderatepercentagesofmissingdataandmismatchedobservationsintime,evenwithfewersubjectsandtimepoints,althoughseveralofthevarianceparametersarebeingoverestimated.Themethodologyisappliedtoadatasetfromanobservationalstudyonpatientswithautoimmunediseases. 8

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Theintegrationofstate-spacemodelsintotheeldofstatisticsisattributedinlargeparttothedevelopmentoftheKalmanlter.SonamedbecauseofthesignicantpaperpublishedbyR.E. Kalman in1960,itisarecursiveformulathatusesthelinearstate-spacemodelwithknownparametervaluesforlteringandforecastingindiscretetime( Kalman 1960 ).Theadvantageofthealgorithmisthatltering,prediction,andsmoothingcaneasilybeachievedforanyproblemcastinstate-spaceform.Thegoaloftheprocedureistousetheobservationstoestimatetheunderlyingstates,whichareunobserved.TheKalmanlterwasrstappliedtotheproblemofprojectiletrackingforNASA'sApolloprogram( KedemandFokianos 2002 ),andsincethenhasbeenprevalentinnavigationalsystemsandinthetrackingofthetrajectoryofsubmarines,aircraftcarriers,andballisticmissiles. KalmanandBucy ( 1961 )increasedtheapplicationsofthestate-spacemodelandKalmanlterbyexpandingthesystemtothecontinuoustimedomain. Thestate-spaceformulationwasthereafterfoundtobeusefulinareasotherthanguidancesystems.Autoregressiveintegratedmovingaverage(ARIMA)modelsintimeseriescanberepresentedinstate-spaceform,providingauniedapproachtothiswideclassofmodels( BrockwellandDavis 2002 ).Thus,theexibilityofthestate-spaceapproachallowsforanalysisofprocessesthatarenotnecessarilystationary.Theseversatilemodelsarewidelyusedbeyondtheeldofstatistics,wheretheyarealsoknownasstructuralequationmodels(SEM)ordynamiclinearmodels(DLM).See ShumwayandStoer ( 1982 )foranapplicationtoeconomicsdata, Harvey ( 1991 )foranalysesinthesocialsciences,or Jones ( 1984 )foranexampleofthemodel'suseinanalyzingbiomedicaldata. Furtherdemonstratingthebroadapplicabilityofthestate-spacemethodology,itcanalsomodelnonlinearandnon-Gaussianprocesses( WestandHarrision 1997 ).Analyses 9

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Carlinetal. ( 1992 )detailtheprocedureinthiscontext. DurbinandKoopman ( 2001 )advocatetheuseofalinearapproximatingmodelandimportancesamplingtotreatstate-spacesystemsthatarenonlinearandnon-Gaussian.Inasimilarvein,thesophisticated,sequentialtechniqueknownasparticlelteringusessimulationtoestimatetheparametersinmodelsofthiskind( DoucetandTadic 2003 ).Ifproperlydesigned,bothofthesemethodsarelikelytobefasterandsimplertoexecutethantheMarkovChainMonteCarloapproach. Byfar, ShumwayandStoer ( 2000 )hadthelargestimpactontheworkdonehere.Intheirtext,theypresentthebasicformulationofthestate-spacemodel,derivetheKalmanrecursions,andexplaintheirpowerandeaseofimplementation.Theyalsoshowhowtocomputethelikelihoodinthestate-spaceconstructandprovidethedetailsoftheNewton-Raphsonalgorithmtoaccomplishparameterestimation.Themostimportantsectionistheonerelatingtothemissingdatamodications.Here,thenecessaryadjustmentstothestate-spacemodelandKalmanrecursionsaredescribed.ThetechniquereliesontheEMalgorithmforparameterapproximationbecauseoftheeasewithwhichtheprocedurehandlesmissingdata.TheEMapproachtosolvingmissingdataproblemscastinstate-spaceformwasoriginallypresentedin ShumwayandStoer ( 1982 ),withtheoreticalresultsprovenin Stoer ( 1982 ). Thecontributionofthisworkisthatitincorporatesexogenousvariablesinthestate-spacemodelandallowsthemtobemissinginadditiontotheresponses.Ingeneral,exogenousvariablesaecttheresponse,butnotviceversa.Theseareindependentvariables\determinedbyfactorsoutsidethesystemunderstudy"( Diggleetal. 2002 ).Thebenetofthisapproachisthatmoreofthevariabilityintheresponsecanbecapturedinsteadofattributedtowhitenoise,andittakesadvantageoftheKalmanrecursionstoimplicitlyllinanymissingvaluesinthecovariates.Although Schmid ( 1996 )alsoproposesastate-spaceformulationwithcovariates,notonlymustamajority 10

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ShumwayandStoer ( 2000 )andnewnotationfollowsthesamestylesothatextensionsmadehereareclear. Theorganizationofthispaperisasfollows.Chapter 2 dealswithmodeldevelopment.Therstpartextendsthecompletedatastate-spaceformulationbyaddingexogenousvariablestothemodel.ItgoesontospecifytheKalmanrecursions,EMalgorithm,andparameterestimatesinthiscase.Theremainderofthechapterrelatesthenecessarycorrectionstothestate-spacemodel,Kalmanequations,andestimationprocedureinordertoaccommodatemissingvalues.ItalsocontainsatheoremthatprovidesthetheoreticalfoundationfortheKalmanrecursionstocontinuetoholdinthepresenceofmissingdata.Thischaptercontainsthebulkofthenewworkaccomplished. StandarderrorsfortheparameterestimatesderivedviatheEMalgorithminthecompleteandmissingdatacasesarecalculatedinChapter 3 .Aprocedureisdetailedspecifyinghowtogeneratetheobservedinformationmatrixinbothsituations.Proofsforalltheoreticalresultsappearseparately,inChapter 4 .Thisisdonesothatthemaintextowsbetter.Chapter 5 describestheresultsofthesimulationstudy.Severalsubjectandtimepointcombinationsusingdierentinitialparametervaluesareconsideredinboththecompletedatacaseandundervariouspercentsofmissingdata.Inaddition,themodelwithoutexogenousvariablesandtheeectsontheprocedureofmismatchedobservationsintimeareexamined.Thestate-spacemodelwithexogenousvariablesandmissingdataisthenappliedtoarealdatasetinChapter 6 .Thedataanalyzedwereproducedfromanobservationalstudyonpatientswithautoimmunediseases.Thederivedformulationservesasamodelforthedataandusingtheestimatedparametervalues,theKalmanrecursionsareappliedtohelppredictthebehaviorofacertainbiomarkerofinterest.Lastly,Chapter 7 dealswiththeimplicationsofthisworkandsuggestsareasoffurtherresearch. 11

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Thischapterisdevotedtothedevelopmentofthestate-spacemodelwithexogenousvariablesinthecompleteandmissingdatacases.TherstpartspeciestheKalmanrecursions,EMalgorithm,andparameterestimateswhenthereisnomissingdata.ThenecessarymodicationstotheKalmanequationsandestimationprocedureinordertoaccommodatemissingvaluesarepresentedinthesecondhalfofthechapter. wherei=1;:::;ksubjectst=1;:::;nitimepointsxi;t:px1statevector:pxptransitionmatrix:pxrmatrixofcoecientsui;t:rx1vectorofexogenousvariableswi;tiidN(0;Q)noisevector; whereyi;t:qx1vectorofobserveddataAi;t:qxpobservationmatrix 12

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LetYi;s=(yi;1;:::;yi;s)andUi;s=(ui;1;:::;ui;s)denotetheresponsesandcovariates(assumedknown)attimesforsubjecti;respectively.Ifinadditiontotheobservations,Yi;niandUi;nifori=1;:::;k;thestatesXi;ni=(xi;0;xi;1;:::;xi;ni)wereobservable,thenthecompletedatajointdensitywouldbe with=(0;0;;;Q;;R);X=(X1;n1;X2;n2;:::;Xk;nk);Y=(Y1;n1;Y2;n2;:::;Yk;nk);andU=(U1;n1;U2;n2;:::;Uk;nk):Sinceweareassumingnormality,thenaturallogarithmofthecompletedatalikelihood( 2-3 ),ignoringaconstant,canbewrittenas lnL( 2klnj10j1 2kXi=1(xi;00)010(xi;00)+1 2NlnjQ1j1 2kXi=1niXt=1(xi;txi;t1ui;t)0Q1(xi;txi;t1ui;t)+1 2NlnjR1j 2kXi=1niXt=1(yi;tAi;txi;tui;t)0R1(yi;tAi;txi;tui;t); ( 1982 )detailaproceduretoestimatetheparametervectorbasedontheEMalgorithm.WecanimplementtheEMalgorithmtoconsecutivelymaximizetheexpectationofthecompletedatalikelihoodgiventheobservationstondtheMLEsofbasedontheobserveddata,(Y;U);sincewedonothavethecompletedata,(X;Y;U):Aswillbeseen,theEMalgorithmachievesstateapproximationintheE-stepseparatelyfromparameterestimationintheM-step.Althoughothermethods 13

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ShumwayandStoer 2000 ),theEMalgorithmispresentedherebecauseoftheeasewithwhichithandlesmissingvalues.ThepopularityoftheEMalgorithminstatisticsismostlyattributedtotheseminalworkof Dempsteretal. ( 1977 ).TheparticularmethodologyinthemissingdatacaseispresentedinSection 2.2.2 ToexecutetheEMalgorithm,webeginwiththeexpectation(E)step.Denexsi;t=E(xi;t (j1)=ElnL( with(j1)=(j1)0;(j1)0;(j1);(j1);Q(j1);(j1);R(j1)thepreviousiterationparameterestimates.ComputingElnL( (j1)=1 2klnj10j1 2kXi=1trn10hPnii;0+xnii;00xnii;000io+1 2NlnjQ1j1 2trQ1[S11S100S010+S000]1 2trQ1[S020+S0020S120S012+S220] +1 2NlnjR1j1 2kXi=1niXt=1trR1Ai;tPnii;tA0i;t;1 2kXi=1niXt=1trnR1yi;tAi;txnii;tui;tyi;tAi;txnii;tui;t0owhere 14

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Formoredetailsonhow( 2-6 )wasreached,seeSection 4.1 Inexpressions( 2-6 ){( 2-11 ),weestimatethemeanofthestatevectorgiventheobservations,xsi;t;anditscovariancematrix,Psi;t;withminorchangestotheKalmanrecursionsfoundin ShumwayandStoer ( 2000 ).TheKalmanpredictor,usedwhent>s;andtheKalmanlter,appliedwhent=s;aregivenby where fori=1;:::;kandt=1;:::;ni;withinitialconditionsx0i;0=0andP0i;0=0: ShumwayandStoer ( 2000 ), 15

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fori=1;:::;kandt=ni;ni1;:::;1;initializedwithxnii;niandPnii;nifromtheKalmanlterandpredictor. Inaddition,alag-onecovariancesmootherterm,Psi;t1;t2;isnecessarytocompletetheE-step.Therecursivealgorithmalsoappearsin ShumwayandStoer ( 2000 )as withinitialcondition fori=1;:::;kandt=ni;ni1;:::;2;whereKi;ni;Ji;t;andPti;tareobtainedfromtheKalmanlter,predictor,andsmootherrecursions. Themaximization(M)stepentailsmaximizingtheconditionalexpectationofthecompletedatalikelihoodgiventheobservations,foundin( 2-6 ),ateachiterationj:Takingderivativeswithrespecttoeachcomponentof(j1);settingtozero,andsolvingyieldstheupdatedparameterestimates. Inthepast,otherauthors,including ShumwayandStoer ( 2000 ),haveincludedexogenousvariablesinbothequationsofthestate-spacemodelbuthavefailedtoprovideM-stepestimatorsforthecoecientmatricesand:Thus,thisisdonenext.Inaddition,thepresenceoftheseparametersinthemodelaectstheestimatesfor;QandRandsotherevisedestimatesforthesearealsoderived.Considermaximizingwithrespectto:Thisleadsto (j1) 2 "tr(R1kXi=1niXt=1yi;tAi;txnii;tui;tyi;tAi;txnii;tui;t0oi

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(j1) 2 "tr(R1kXi=1niXt=1Ai;txnii;tu0i;tyi;tu0i;t0)#+1 2 "tr(R1kXi=1niXt=1ui;ty0i;tui;tx0nii;tA0i;t)#1 2 "tr(R1kXi=1niXt=1ui;tu0i;t!0)#=R1kXi=1niXt=1yi;tu0i;tAi;txnii;tu0i;tR1S22;usingmatrixdierentiationresultsfrom PetersenandPedersen ( 2007 ).Settingtozeroandsolvingforproducestheestimator (j)="kXi=1niXt=1yi;tAi;txnii;tu0i;t#S122:(2-23) Asanotherexample,toderive(j);wemustevaluateQ (j1) 2 trQ1(S020+S0020S120S012+S220)=Q1S02+Q1S12Q1S0220;whichproduces (j)=S12(j)S02S122:(2-24) Similarcalculationsyieldthefollowingestimatorsfortheremainingparameters: (j)0=1 (j)=S10(j)S002S100=S10S12S122S002S100IS02S122S002S1001 17

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SeeSection 4.2 fordetailsonthecalculationof( 2-25 ){( 2-29 ). Inessence,thetechniqueentailsalternatingbetweenevaluationoftheKalmanrecursionsgivenin( 2-13 ){( 2-22 )andthemaximumlikelihoodestimators( 2-23 ){( 2-29 )untilconvergence.Inpractice,theprocedureisstoppedwhenthevaluesofeitherortheobservedlikelihooddonotchangemuchfromoneiterationtothenext. ShumwayandStoer ( 2000 )suggestevaluatingtheinnovationsformoftheobserveddatalikelihoodateachiterationtoestablishconvergence.Theinnovationsaredenedas Clearly,E(i;t)=0and i;tVar(i;t)=VarAi;txi;txt1i;t+vi;t=Ai;tPt1i;tA0i;t+R:(2-31) Theinnovationsformofthenaturallogarithmoftheobserveddatalikelihood,ignoringaconstant,canbeexpressedas lnL( 2kXi=1niXt=1lnj1i;tj0i;t1i;ti;t:(2-32) Assumingindependencebetweenisubscripts,i=1;:::;k;thisfollowssincetheinnovationssequencei;t;t=1;:::;ni;isaGaussianprocess. 18

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respectively,where,forj=1;2;(j)i;t:pxri;tjportionof(state)coecientmatrixcorrespondingtou(j)i;tA(j)i;t:qi;tjxpportionofobservationmatrixcorrespondingtoy(j)i;t(j)i;tm:qi;tmxri;tjportionofcoecientmatrixcorrespondingtou(j)i;t;m=1;2v(j)i;t:qi;tjx1portionofmeasurementnoisevectorcorrespondingtoy(j)i;t;and 19

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2-33 )or( 2-34 )toavoidacollinearityproblem.Outofthetwo,itseemsmoreappropriatetoremovefromthestateequation. Notethatinexpressions( 2-33 )and( 2-34 ),xi;tandyi;tdependontheunobservedportionoftheexogenousvariablevector,u(2)i;t:Weproposecircumventingthisdicultybymodellingui;twithasecondstate-spacemodel.Treatingui;tastheobservationvectorfortheunderlyingstateprocessgi;t,thissecondmodelis fori=1;:::;kandt=1;:::;ni,wheregi;t:mx1statevector:mxmtransitionmatrixci;tiidN(0;C)noisevectorBi;t:rxmobservationmatrixhi;tiidN(0;H)observationalnoisevector; Theobservationequation( 2-37 )canbepartitionedintoobservedandmissingparts,yielding 20

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Inthecaseofmissingdata,thelikelihoodmustbemodiedtoincludeseveralmoretermstoaccountforthesecondstate-spacemodel.Hence,ignoringaconstant,thenaturallogarithmofthecompletedatalikelihoodis lnL(; 2klnjg0j11 2kXi=1(gi;0g0)0(g0)1(gi;0g0)+1 2NlnjC1j1 2kXi=1(gi;tgi;t1)0C1(gi;tgi;t1)+1 2NlnjH1j1 2kXi=1(ui;tBi;tgi;t)0H1(ui;tBi;tgi;t)+1 2klnj10j1 2kXi=1(xi;00)010(xi;00) (2-40) +1 2NlnjQ1j1 2kXi=1niXt=1(xi;txi;t1ui;t)0Q1(xi;txi;t1ui;t)+1 2NlnjR1j1 2kXi=1niXt=1(yi;tAi;txi;tui;t)0R1(yi;tAi;txi;tui;t);

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Diggleetal. ( 2002 )deningacovariateprocesstobeexogenousifattimet;itis\conditionallyindependentofallprecedingresponsemeasurements"willbetaken.Symbolically,thisisgivenasf(Ui;t 2.1 exceptthatthecovariatesarenowrandomaswell. 22

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2-36 ),( 2-37 ),andnotfromanyresponses.Thisassumptiondoesnotaecttheoriginalmodel( 2-33 ),( 2-34 )andtheEMalgorithmwillstillproducetheMLEs.TheKalmanformulascanbehandledseparatelybecauseofthenextresult. 2-36 ),( 2-37 )areindependentofY(1)i;sduetoexogeneity. Proof. 4.3 ThemodiedKalmanrecursionsinthepresenceofmissingvaluesintheobservationalvectorforthesecondstate-spacemodelintermsofgi;thavebeenderivedin Stoer ( 1982 ).Theroutineworkswhenthereareeithernoexogenousvariables(whichisthecasehere)orwhentheexogenousvectoriscompletelyobserved.Theprocedureentailssettingu(2)i;tandB(2)i;tin( 2-38 )to0;substituting0forHi;t;12andHi;t;21in( 2-39 ),andthenrunningtheKalmanequationsasusual.Thus,expressions( 2-18 ){( 2-20 )canbeusedtogeneratesmoothedvalues,g(ni)i;t;byadjusting( 2-38 )and( 2-39 )asdiscussedabove.Takingthedierentnotationintoaccount,thesesmootherequationsbecomeg(ni)i;t1=g(t1)i;t1+D(i;t1)g(ni)i;tg(t1)i;tF(ni)i;t1=F(t1)i;t1+D(i;t1)F(ni)i;tF(t1)i;tD0(i;t1);D(i;t1)=F(t1)i;t10F(t1)i;t1;

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24

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4.4 TheresultdetailingthenecessarymodicationstotheKalmanlterandpredictorofxi;tinthepresenceofmissingdatacannowbespecied. 2-33 )and( 2-34 )are

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4.5 Shumway(1982)demonstratesthatsmoothedvaluescanbeobtainedinthecaseofmissingdatafromthesmootherresultgivenbyexpressions( 2-18 ){( 2-20 ).Onlyaminorchangeneedbemade,andthatistousethelteredvaluesproducedbyTheorem 2.3 .Inotherwords,inthepresenceofmissingvalues,thesmootherequationsbecome fori=1;:::;kandt=ni;ni1;:::;1;initializedwithx(ni)i;niandP(ni)i;nifromTheorem 2.3 .Itisalsoprovenintheaforementionedworkthatthesameisthecasewiththelag-onecovariancesmootherterm.Namely,( 2-21 )and( 2-22 )become withinitialcondition 26

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2.3 andthesmootherequationsabove. Insummary,theKalmanrecursionsinthecaseofmissingdataaresimilartothosewhenthedatasetiscompleteexceptforseveraladjustments.First,thelterandpredictorandtheircorrespondingerrorsmustbemodiedsothatmissingnessintheexogenousvariablevectoristakenintoaccount.Thisisseenbyanestimateofui;ttakingitsplaceinboththelterandpredictorrecursions.Theerrorsalsocontainextratermstoaccountfortheestimationofui;t:Changesarealsomadetohandleincompletenessintheresponses,yi;t:Thisisdonebysubstitutingzeroes:(i)inplaceofthemissingvalues,(ii)intherow(s)ofAi;tandcorrespondingtothemissingdata,and(iii)intheo-diagonalelementsofRforsubjectiatupdatetimet:Finally,thesmoother,itserror,andlag-onecovariancetermscanbeobtainedasinthecompletedatacaseexceptthattheyusethemissingdatacorrectedlterandpredictorasdescribedinthissection. 2.1 whenthereismissingdata. Throughoutthissection,theobservedresponsesandcovariatesaregivenbyY(1)=Y(1)1;n1;:::;Y(1)k;nkandU(1)=U(1)1;n1;:::;U(1)k;nk;respectively,withstateinformationX=(X1;n1;:::;Xk;nk)andG=(G1;n1;:::;Gk;nk):Proceedingfromthecompletedatalikelihood( 2-40 ),theE-stepis,foreachiterationj; Q; (j1);(j1)=ElnL(; =1 2klnjg0j11 2kXi=1tr(g0)1F(ni)i;0+g(ni)i;0g0g(ni)i;0g00+1 2NlnjC1j1 2trnC1hSg(11)Sg(10)0Sg0(10)+Sg(00)0io

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2NlnjH1j1 2kXi=1niXt=1trH1u(i;t)B(i;t)g(ni)i;tu(i;t)B(i;t)g(ni)i;t01 2kXi=1niXt=1tr8><>:H10B@B(i;t)F(ni)i;tB0(i;t)+264000H(j1)i;t;223751CA9>=>;+1 2klnj10j1 2kXi=1tr10P(ni)i;0+x(ni)i;00x(ni)i;000+1 2NlnjQ1j+1 2trQ1S0(12)S0(02)01 2trQ1S(11)S(10)0S0(10)+S(00)0+1 2trQ1S(12)S(02)S(22)0 2kXi=1niXt=1tr8><>:Q1Ei;t264000H(j1)i;t;22375E0i;t09>=>;+1 2NlnjR1j1 2kXi=1niXt=1tr8<:R10@y(i;t)A(i;t)x(ni)i;t24I00035D0i;tEi;tbu(i;t)1A0@y(i;t)A(i;t)x(ni)i;t24I00035D0i;tEi;tbu(i;t)1A09=;1 2kXi=1niXt=1tr8><>:R124I00035D0i;tEi;t264H(j1)i;t;22+B(2)i;tF(ni)i;tB0(2)i;t000375E0i;t0Di;t24I000359=;1 2kXi=1niXt=1tr8><>:R1264000R(j1)i;t;223759>=>;1 2kXi=1niXt=1tr8><>:R1264A(1)i;tP(ni)i;tA0(1)i;t0003759>=>;;

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+kXi=1niXt=1Ei;t8<:264I0375u(1)i;tbu0(i;t)+2640I375B(2)i;tg(ni)i;tu0(1)i;tI...09=;E0i;t:NotethatEi;t\isapermutationmatrixthatreordersthe[exogenous]variablesattimet[fromsubjecti]intheiroriginalorder"( ShumwayandStoer 2000 )andDi;tisthecorrespondingpermutationmatrixfortheresponses.Here,H(j1)i;t;22isthepreviousvalueofthevariance-covariancematrixoftheobservationalerrorvectorh(2)i;tassociatedwiththeunobservedportionoftheresponseu(2)i;tinthesecondstate-spacemodelspeciedby(j1):Similarly,R(j1)i;t;22isthecorrespondingvariance-covariancematrixassociatedwith 29

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2-52 ),seeSection 4.6 ProceedingtotheM-stepofthealgorithm,wemustnowmaximize( 2-52 )withrespecttoeachoftheparametersin=(0;0;;;Q;;R)and=(g0;g0;;C;H):Thiswillyieldthedesiredparameterupdatestobeperformedateachiteration.Inwhatfollows,westillmaintaintheassumptionsHi;t;12=0=Ri;t;12: (j1);(j1)relatingtothesecondstate-spacemodel,theM-stepestimatorsforarealsogiveninShumway(1982).Theseare g(j)0=1 (j)=Sg(10)Sg(00)1 Theremainingparametersrelatingtoarenowpresented.Thesimilaritiesbetweenthecorrespondinglinesof( 2-6 )and( 2-52 )relatingto0and0implythattheestimatesinthecaseofmissingdataare,respectively, 30

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ateachiterationj:Thederivationoftheremainingparameterestimates, (j)=S(12)(j)S(02)S1 (j)=S(10)(j)S0(02)S1(00)=S(10)S(12)S1S0(02)S1(00)IS(02)S1S0(02)S1(00)1

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4.7 .In( 2-75 ),R(j)representstheun-orderedestimatorforRatiterationj(see( 4-42 )fortheexplicitexpression),sincetheexpressionfor(j)automaticallyhandlesanynecessaryreordering.Finally,theqrmatrix(j)isobtainedfrom( 2-75 )byreversingthevecoperator. TheM-stepestimatorsforandRatthejthiteration,givenby( 2-75 )and( 2-74 ),respectively,arecomplicatedfunctionsofeachother.Sinceneitherfunctioniseasytosolveintermsoftheothertoallowsubstitution,wesuggestusing(j1)insteadof(j)in( 2-74 ),solvingforR(j);andthensubstitutingR(j)in( 2-75 )toobtain(j):Aswillbeshownbythenexttheorem,thismodicationtotheEMalgorithmchangestheEMintotheExpectation/ConditionalMaximization,orECM,algorithmasdenedby MengandRubin ( 1993 ).UndertheECMalgorithm,theE-stepoftheEMalgorithmremainsthesamebuttheM-stepisreplacedbyaseriesofconditionalmaximizationstepsthatoptimizeQ; (j1);(j1)in( 2-52 )withafunctionof(;)xedatitspreviousvalue.Thefollowingresultshowsthatourprocedureconverges. Proof. 4.8 Inshort,inthemissingdatacase,wealternatebetweenevaluationoftheKalmanrecursionsintheE-stepandthemaximumlikelihoodestimatorsintheM-stepuntilconvergence.TheM-stepinthiscaseadditionallyperformstwoconditionalmaximization 32

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lnL; 2kXi=1niXt=1lnj(1)i;tj10(1)i;t(1)i;t1(1)i;t;(2-76) with(1)i;t=y(1)i;tA(1)i;tx(t1)i;t(1)i;t1B(1)i;t+(2)i;t1B(2)i;tg(t1)i;t

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TheEMalgorithmgreatlyeasesmaximumlikelihoodestimationinthepresenceofmissingdatabutitdoesnotautomaticallyyieldparameterestimatestandarderrors.Luckily,techniquesaboundtocalculatestandarderrorswhenusingtheEMalgorithm.Amethodappliedtostate-spacemodelsutilizingthebootstrapisdescribedin StoerandWall ( 1991 ),butthenestedloopinginvolvedinthealgorithmwouldprobablybeverycomputationallyintensiveifappliedtooursituation.AnotherprocedureweconsideredapplyingwasthesupplementedEM,orSEM,algorithmof MengandRubin ( 1991 ).AlthoughthistechniquerequiresrelativelyfewmodicationstothebasicEMalgorithm,thenumericaldierentiationinvolvedwasamajordrawbackconsideringthatourparametersofinterestcanbematrices.Avarietyofothermethodsaregivenby GivensandHoeting ( 2005 ).SinceundergeneralconditionstheMLEisasymptoticallynormal( ShumwayandStoer 2000 ),mostoftheseapproachesgivesomesortofsimplicationtodirectcalculationoftheinverseoftheobservedFisherinformationmatrix. RobertandCasella ( 2004 )explaintheuseofOakes'identity( Oakes 1999 ),aformulathatalsoprovidesashortcuttooutrightcomputationoftheHessianmatrixofsecondderivatives.Atrst,thisrelationappearedtobeviableforourproblembecausetheobservedFisherinformationmatrixiswrittenintermsofthecompletedatalikelihood,whichwehavealreadycalculated(see( 2-4 )and( 2-40 )).However,calculationofthemixedderivativetermprovedrathercomplicated. Underourcompleteandmissingdatastate-spacemodels,directdierentiationoftheobserveddatalikelihoodisasimpleralternative.TheformulafortheinformationmatrixiscomprisedofrecursivecalculationsusingtheKalmanequationsandparameterestimatesevaluatedattheMLEs. GuptaandMehra ( 1974 )demonstratethatthe(j;s)thelementoftheinformationmatrixis 34

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ji;t01i;t si;t+1 2tr1i;t ji;t1i;t si;t+1 4tr1i;t ji;ttr1i;t si;t; wherejisthejthcomponentoftheparametervector.Thestandarderrorsarethesquarerootsofthediagonalelementsof1 ShumwayandStoer ( 2000 ),inpractice,theHessianmatrixcanbeapproximatedbydroppingtheexpectationfrom( 3-1 ).Theobserveddatalikelihood,lnLY;U;andi;t;i;t;andtheirderivativesdierinthecompleteandmissingdatacases.Thus,inwhatfollows,thetwosituationswillbeexaminedseparatelyinmoredepth. 2.1 ,whentherearenomissingdata,theparametervectoris=(0;0;;;Q;;R):Sinceequation( 3-1 )isgivenintermsofscalarparameters,wetreateachelementofeachparameterinasthevariableofinterest.Thus,wecalculatethestandarderrorsforvec()=[01;:::;0p;0;11;:::;0;1p;0;22;:::;0;2p;:::;0;pp;vec();vec();q11;:::;q1p;q22;:::;q2p;:::;qpp;vec();r11;:::;r1q;r22;:::;r2q;:::;rqq]: 2q(q+1)uniquescalarparameters. Expression( 2-32 )givestheobserveddatalog-likelihoodaslnL( 2kXi=1niXt=1lnj1i;tj0i;t1i;ti;t;

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ji;t=Ai;t jxt1i;t jui;t jxt1i;t= jxt2i;t1+ jxt2i;t1+ jKi;t1i;t1+ jKi;t1i;t1+Ki;t1 ji;t1+ jui;t jKi;t= jPt1i;tA0i;tKi;t ji;t1i;t jPt1i;t= jPt2i;t10+ jPt2i;t10+Pt2i;t1 j0+ jQ jKi;t1Ai;t1Pt2i;t10 jKi;t1Ai;t1Pt2i;t10Ki;t1Ai;t1 jPt2i;t10Ki;t1Ai;t1Pt2i;t1 j0 ji;t=Ai;t jPt1i;tA0i;t+ jR; withtheaidoftheKalmanrecursionsin( 2-13 ){( 2-17 ).Thefollowingcalculationsarealsonecessarytoevaluateexpressions( 3-2 ){( 3-6 )andarebasedonamatrixderivativeresultfrom Graham ( 1981 ): j0=Ja1p1I(j=0a);a=1;:::;p j0=JabppI(j=0;ab);a=1;:::;p;b=a;:::;p j=JabqrI(j=ab);a=1;:::;q;b=1;:::;r j=JabppI(j=ab);a=1;:::;p;b=1;:::;p j=JabprI(j=ab);a=1;:::;p;b=1;:::;r jQ=JabppI(j=qab);a=1;:::;p;b=a;:::;p jR=JabqqI(j=rab);a=1;:::;q;b=a;:::;q; 36

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3-7 ){( 3-13 ). Sinceequations( 3-2 ){( 3-6 )holdonlyfort2;thesederivativesmustbeevaluatedseparatelywhent=1:Itfollowsthatfori=1;:::;k;i;1=yi;1Ai;10Ai;1ui;1ui;1x0i;1=0+ui;1Ki;1=(00+Q)A0i;11i;1P0i;1=00+Qi;1=Ai;1(00+Q)A0i;1+R; 2-13 ){( 2-17 ).Theresultsofthederivativecalculations,ortheinitialvaluesforthestandarderrorrecursions,appearinTable 3-1 .Notethatthepartialderivativewithrespecttoeachoftheparametersintheleft-mostcolumnappearsalongtherows. 3.1 ,wewillconsidereachmatrixelementseparately.Inadditiontovec()asdescribedpreviously,vec()=g01;:::;g0m;g0;11;:::;g0;1m;g0;22;:::;g0;2m;:::;g0;mm;vec();c11;:::;c1m;c22;:::;c2m;:::;cmm;h11;:::;h1r;h22;:::;h2r;:::;hrr] aretheparametersofinterestforthestandarderrorcalculations.Inthemissingdatacase,thereare2m(m+1)+1 2r(r+1)additionalscalarparameters. 37

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InitialConditionsforCompleteDataStandardErrorCalculations PartialDerivativej ji;1 jx0i;1 jKi;1 jP0i;1 ji;1 2-76 ),theobserveddatalog-likelihoodisnowlnL; 2kXi=1niXt=1lnj(1)i;tj10(1)i;t(1)i;t1(1)i;t;

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j(1)i;t=A(1)i;t jx(t1)i;t[I;0]D0i;t jEi;tBg(t1)i;t[I;0]D0i;tEi;tB jg(t1)i;t jx(t1)i;t= jx(t2)i;t1+ jx(t2)i;t1+ jK(1)i;t1(1)i;t1+ jK(1)i;t1(1)i;t1+K(1)i;t1 j(1)i;t1+ jBi;tg(t1)i;t+Bi;t jg(t1)i;t jK(1)i;t= jP(t1)i;tA0(1)i;tK(1)i;t j(1)i;t(1)i;t1 jP(t1)i;t= jP(t2)i;t10+ jP(t2)i;t10+P(t2)i;t1 j0 jK(1)i;t1A(1)i;t1P(t2)i;t10 jK(1)i;t1A(1)i;t1P(t2)i;t10K(1)i;t1A(1)i;t1 jP(t2)i;t10K(1)i;t1A(1)i;t1P(t2)i;t1 j0 + jQ+ jBi;tF(t1)i;tB0i;t+H0+ jH0+Bi;t jF(t1)i;tB0i;t0+Bi;tF(t1)i;tB0i;t+H j0

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j(1)i;t=A(1)i;t jP(t1)i;tA0(1)i;t+[I;0]D0i;t jRDi;t24I035+[I;0]D0i;t jH0Di;t24I035+[I;0]D0i;t jH0Di;t24I035+[I;0]D0i;tH j0Di;t24I035 +[I;0]D0i;t jEi;tBF(t1)i;tB0E0i;t0Di;t24I035+[I;0]D0i;tEi;tB jF(t1)i;tB0E0i;t0Di;t24I035+[I;0]D0i;tEi;tBF(t1)i;tB0E0i;t j0Di;t24I035; 2.3 ,whileexpressions( 3-7 ){( 3-13 )continuetohold. Equations( 3-14 ){( 3-18 )dependonthepartialderivativesofKalmanpredictorsandvariancesandparametersinthemodelforthesecondstate-spaceformulation.Toresolvethese,denetheinnovationsforthemodelforui;tas(i;t)=u(i;t)B(i;t)g(t1)i;t;with(i;t)Var(i;t)=B(i;t)F(t1)i;tB0(i;t)+H(i;t):Thefollowingequationsarenecessaryfortheevaluationof( 3-1 )andholdfori=1;:::;kandt=2;:::;ni: j(i;t)=B(i;t) jg(t1)i;t jg(t1)i;t= jg(t2)i;t1+ jg(t2)i;t1+ jL(i;t1)(i;t1)+ jL(i;t1)(i;t1)+L(i;t1) j(i;t1) jL(i;t)= jF(t1)i;tB0(i;t)L(i;t) j(i;t)1(i;t) j(i;t)=B(i;t) jF(t1)i;tB0(i;t)+24I00035E0i;t jHEi;t24I00035+24000I35E0i;t jHEi;t24000I35 40

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jF(t1)i;t= jF(t2)i;t10+ jF(t2)i;t10+F(t2)i;t1 j0+ jC jL(i;t1)B(i;t1)F(t2)i;t10 jL(i;t1)B(i;t1)F(t2)i;t10L(i;t1)B(i;t1) jF(t2)i;t10+F(t2)i;t1 j0: ThesederivativesfollowfromtheKalmanrecursionsforthesecondstate-spacemodelfortheexogenousvariables.Thenextsetofformulasareneededtoevaluate( 3-19 ){( 3-23 )andfollowfromamatrixderivativeresultfrom Graham ( 1981 ): jg0=Ja1m1I(j=g0a);a=1;:::;m jg0=JabmmI(j=g0;ab);a=1;:::;m;b=a;:::;m j=JabmmI(j=ab);a=1;:::;m;b=1;:::;m jC=JabmmI(j=cab);a=1;:::;m;b=a;:::;m jH=JabrrI(j=hab);a=1;:::;r;b=a;:::;r; 3-14 ){( 3-18 )and( 3-19 ){( 3-23 )failtoholdfort=1;thiscaseisgivenbelowfori=1;:::;k:(1)i;1=y(1)i;1A(1)i;10A(1)i;1Bi;1g0[I;0]D0i;1Ei;1Bg0x(0)i;1=0+Bi;1g0K(1)i;1=00+Q+Bi;1(g00+C)B0i;10+H0A0(1)i;1(1)i;11P(0)i;1=00+Q+Bi;1(g00+C)B0i;10+H0(1)i;1=A(1)i;100+Q+Bi;1(g00+C)B0i;10+H0A0(1)i;1+[I;0]D0i;1RDi;124I035+[I;0]D0i;1H0Di;124I035+[I;0]D0i;1Ei;1B(g00+C)B0E0i;10Di;124I035(i;1)=u(i;1)B(i;1)g0

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3-2 3-3 ,and 3-4 .Becausethepartialderivativesof(i;1);g(0)i;1;L(i;1);F(0)i;1;(i;1);andH(i;1)withrespecttotheparametersofareallzero,theseareomittedfromthetablesthatfollow. Table3-2. InitialConditionsforMissingDataStandardErrorCalculations-PartialDerivativesWithRespectto PartialDerivativej j(1)i;1 jx(0)i;1 jK(1)i;1 jP(0)i;1 j(1)i;1 abP(0)i;1A0(1)i;1(1)i;11K(1)i;1 ab(1)i;1(1)i;11JabprBi;1(g00+C)B0i;10+Bi;1(g00+C)B0i;1Jbarp+JabprH0+HJbarpA(1)i;1 abP(0)i;1A0(1)i;1 ab(1)i;1(1)i;110[I;0]D0i;1Jabqr[H+Ei;1B(g00+C)B0E0i;10Di;124I035+[I;0]D0i;1[H+Ei;1B(g00+C)B0E0i;1JbarqDi;124I035

PAGE 43

InitialConditionsforMissingDataStandardErrorCalculations-PartialDerivatives(ofFirstState-SpaceModelFunctions)WithRespectto PartialDerivativej j(1)i;1 jx(0)i;1 jK(1)i;1 jP(0)i;1 j(1)i;1 g0;abP(0)i;1A0(1)i;1(1)i;11K(1)i;1 g0;ab(1)i;1(1)i;11Bi;1Jabmm0B0i;10Ai;1 g0;abP(0)i;1A0(1)i;1+[I;0]D0i;1Ei;1BJabmm0B0E0i;10Di;124I035 abP(0)i;1A0(1)i;1(1)i;11K(1)i;1 ab(1)i;1(1)i;11Bi;1Jabmmg00+0g0JbammB0i;10Ai;1 abP(0)i;1A0(1)i;1+[I;0]D0i;1Ei;1BJabmmg00+g0JbammB0E0i;10Di;124I035 cabP(0)i;1A0(1)i;1(1)i;11K(1)i;1 cab(1)i;1(1)i;11Bi;1JabmmB0i;10Ai;1 cabP(0)i;1A0(1)i;1+[I;0]D0i;1Ei;1BJabmmB0E0i;10Di;124I035 hab(1)i;1(1)i;11Jabrr0Ai;1Jabrr0A0(1)i;1+[I;0]D0i;1Jabrr0Di;124I035 InitialConditionsforMissingDataStandardErrorCalculations-PartialDerivatives(ofSecondState-SpaceModelFunctions)WithRespectto PartialDerivativej j(i;1) jg(0)i;1 jL(i;1) jF(0)i;1 j(i;1) g0;ab(i;1)1(i;1)Jabmm0B(i;1)Jabmm0B0(i;1) abF(0)i;1B0(i;1)1(i;1)L(i;1) ab(i;1)1(i;1)Jabmmg00+g0JbammB(i;1) abF(0)i;1B0(i;1) cab(i;1)1(i;1)JabmmB(i;1)JabmmB0(i;1) hab(i;1)1(i;1)024I00035E0i;1JabrrEi;124I00035+24000I35E0i;1JbarrEi;124000I35

PAGE 44

(j1) W)=0z wAz w+tr(Az w):Also,itisunderstoodthatErepresentsconditionalexpectationwithrespectto(j1)(andthusdroppedaftertherstline).Startingfrom( 2-5 ), (j1)=ElnL( 2klnj10j1 2kXi=1E(xi;00)010(xi;00) 2NlnjQ1j1 2kXi=1niXt=1Ew0i;tQ1wi;t 2NlnjR1j1 2kXi=1niXt=1Ev0i;tR1vi;t 2klnj10j1 2kXi=1E0[(xi;00) 2kXi=1tr10Var[(xi;00) 2NlnjQ1j1 2kXi=1niXt=1E0[wi;t 2kXi=1niXt=1trQ1Var[wi;t +1 2NlnjR1j1 2kXi=1niXt=1E0[vi;t 2kXi=1niXt=1trR1Var[vi;t (j1)separately.Thus, 44

PAGE 45

Similarly,wehave 45

PAGE 47

and

PAGE 48

4-2 ){( 4-7 )andsubstitutingexpressions( 2-7 ){( 2-12 )into( 4-1 )yieldsQ (j1)=1 2klnj10j1 2kXi=1trn10hPnii;0+xnii;00xnii;000io+1 2NlnjQ1j1 2trQ1[S11S100S010+S000]1 2trQ1[S020+S0020S120S012+S220]+1 2NlnjR1j1 2kXi=1niXt=1trR1Ai;tPnii;tA0i;t1 2kXi=1niXt=1trnR1yi;tAi;txnii;tui;tyi;tAi;txnii;tui;t0o; 2-6 ). 48

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2-25 ){( 2-29 ),wesimplymaximizeQ (j1)in( 2-6 )withrespecttoeachparameter.Notethatmatrixdierentiationresultsfrom PetersenandPedersen ( 2007 )areusedthroughout,exceptwhentheparameterofinterestissymmetric(i.e.,thevariance-covariancematrices0;Q;andR).Thus,startingwith0;wehaveQ (j1) 2 0"kXi=1trn10hPnii;0+xnii;00xnii;000io#=1 2 0"tr(10kXi=1xnii;0!00+100kXi=1x0nii;0!)#1 2 0k10000=10kXi=1xnii;0k0!0; 2-25 ),(j)0=1 2-6 )thattheexpressionwemustmaximizewithrespectto0is 2klnj10j1 2tr(10kXi=1hPnii;0+xnii;00xnii;000i): Anderson ( 1984 ),thisfunctionismaximizedat10=kPki=1hPnii;0+xnii;00xnii;000i1:Solvingfor0givesexpression( 2-26 ),(j)0=1

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(j1)withrespectto;settozero,andsolve.Therefore,Q (j1) 2 trQ1(S100S010+S000)1 2 trQ1(S020+S020)=Q1S10Q1S000Q1S0020; 2-27 ),(j)=S10(j)S002S100: 2-24 ),weobtainthealternateformof( 2-27 ),(j)=S10S12S122S002S100IS02S122S002S1001: 2NlnjQ1j1 2trQ1(S11S100S010+S000)1 2trQ1(S020+S0020S120S012+S220) Anderson ( 1984 )result,sinceQ1ispositivesemi-denite,weobtainthemaximumQ1=N(S11S100S010+S000+S020+S0020S120S012+S220)1:Thisyields( 2-28 )asthesolutionforQ;Q(j)=1 2-27 )inplaceof(j):

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(j1)withrespecttoR;whichamountstomaximizing1 2NlnjR1j1 2tr(R1kXi=1niXt=1yi;tAi;txnii;t(j)ui;tyi;tAi;txnii;t(j)ui;t0)1 2tr(R1kXi=1niXt=1Ai;tPnii;tA0i;t):Again, Anderson ( 1984 )demonstratesthatthemaximumintermsofRisR(j)=1 2-29 ). 2.2.1 ,E(Ui;t 2.2 forj;m=1;:::;t1:Now,from( 2-34 ),wehavethaty(1)i;t=A(1)i;txi;t+(1)i;t1u(1)i;t+(2)i;t1u(2)i;t+v(1)i;t:Also,notethatv(1)i;t?H(1)i;t1andhi;t?H(1)i;t1:Thus,foreachj=1;:::;t1;it 51

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2-34 ).Similarly,foreachm=1;:::;t1;wehaveEhy(1)i;tA(1)i;tx(t1)i;t(1)i;t1B(1)i;t+(2)i;t1B(2)i;tg(t1)i;tu0(1)i;mi=EhA(1)i;txi;tx(t1)i;tu0(1)i;mi+Ehv(1)i;t+(1)i;t1h(1)i;t+(2)i;t1h(2)i;tu0(1)i;mi+Eh(1)i;t1B(1)i;t+(2)i;t1B(2)i;tgi;tg(t1)i;tu0(1)i;mi=EhEA(1)i;txi;tx(t1)i;tu0(1)i;m 4-8 )holds. 52

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2.3 2-41 )isobtaineddirectlysincewith( 2-33 )and( 2-38 ),wehavex(t1)i;t=Exi;t 2-42 ),since Now,toobtaintheltertermanditscovariance,rstnotethattheinnovationsarenow(1)i;t=y(1)i;tEy(1)i;t

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2-34 )and( 2-38 ).Thenextstepistondthejointconditionaldistributionofxi;tand(1)i;tgiventheobserveddata,Y(1)i;t1andU(1)i;t1:Tothisend,theconditionalmeanoftheinnovationsisE(1)i;t 2-34 )and( 2-38 ).TheconditionalvarianceoftheinnovationsisVar(1)i;t 2-35 ),( 2-39 ),( 4-9 ),andsinceP(t1)i;tandF(t1)i;taretheconditionalerrorcovariancesforxi;tandgi;t;respectively.Finally,theconditionalcovariancetermisgivenbycovxi;t;(1)i;t

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4-9 ).Summarizingtheseresults,wehave Hence,x(t)i;t=Exi;t 2.2 =x(t1)i;t+K(1)i;t(1)i;t: ShumwayandStoer ( 2000 )and( 4-10 ),itfollowsthatK(1)i;t=P(t1)i;tA0(1)i;t(1)i;t1:Thus,x(t)i;t=x(t1)i;t+P(t1)i;tA0(1)i;t(1)i;t1hy(1)i;tA(1)i;tx(t1)i;t(1)i;t1B(1)i;t+(2)i;t1B(2)i;tg(t1)i;ti=x(t1)i;t+K(i;t)y(i;t)A(i;t)x(t1)i;t(i;t1)Bg(t1)i;t; 2-43 ),withK(i;t)=P(t1)i;tA0(i;t)1(i;t)=P(t1)i;tA0(i;t)A(i;t)P(t1)i;tA0(i;t)+R(i;t)+(i;t1)BF(t1)i;tB00(i;t1)+(i;t1)H0(i;t1)1 2-45 ).Lastly,toobtainP(t)i;t;notethatwecanwritexi;tx(t)i;t=xi;tx(t1)i;tP(t1)i;tA0(1)i;t(1)i;t1hy(1)i;tA(1)i;tx(t1)i;t(1)i;t1B(1)i;t+(2)i;t1B(2)i;tg(t1)i;ti=xi;tx(t1)i;tK(1)i;thA(1)i;txi;t+(1)i;t1B(1)i;tgi;t+h(1)i;t+(2)i;t1B(2)i;tgi;t+h(2)i;t+v(1)i;tA(1)i;tx(t1)i;t(1)i;t1B(1)i;t+(2)i;t1B(2)i;tg(t1)i;ti

PAGE 56

4-9 )and( 2-45 ),P(t)i;t=Exi;tx(t)i;txi;tx(t)i;t0 2-44 ). (j1);(j1):MissingDataCase 2-51 ), (j1);(j1)=ElnL(; 2klnjg0j11 2kXi=1Eh(gi;0g0)0(g0)1(gi;0g0) 2NlnjC1j1 2kXi=1E(gi;tgi;t1)0C1(gi;tgi;t1)

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2NlnjH1j1 2kXi=1E(ui;tBi;tgi;t)0H1(ui;tBi;tgi;t) 2klnj10j1 2kXi=1E(xi;00)010(xi;00) 2NlnjQ1j1 2kXi=1niXt=1Ew0Q1w 2NlnjR1j1 2kXi=1niXt=1Ev0R1v (j1);(j1)=1 2klnjg0j11 2kXi=1Eh(gi;0g0)0(g0)1(gi;0g0) 2NlnjC1j1 2kXi=1E(gi;tgi;t1)0C1(gi;tgi;t1) 2NlnjH1j1 2kXi=1E(ui;tBi;tgi;t)0H1(ui;tBi;tgi;t) 2klnj10j 2kXi=1E(xi;00)0 2kXi=1tr10Var(xi;00) 2NlnjQ1j1 2kXi=1niXt=1Ew0 2kXi=1niXt=1trQ1Varw 2NlnjR1j1 2kXi=1niXt=1Ev0 2kXi=1niXt=1trR1Varv

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4-11 )relatetothesecondstate-spacemodel,whichtreatstheexogenousvariables,ui;t;asresponses.ThesetermscanbehandledseparatelybecauseofTheorem 2.1 ,whichisproveninSection 4.3 TheEMalgorithmwithmissingvaluesintheresponsewithoutexogenousvariablesappearsin Stoer ( 1982 ).AssumingHi;t;12=0;theE-stepforthesesixtermsgives 2klnjg0j11 2kXi=1tr(g0)1F(ni)i;0+g(ni)i;0g0g(ni)i;0g00+1 2NlnjC1j1 2trnC1hSg(11)Sg(10)0Sg0(10)+Sg(00)0io+1 2NlnjH1j 2kXi=1niXt=1trH1u(i;t)B(i;t)g(ni)i;tu(i;t)B(i;t)g(ni)i;t01 2kXi=1niXt=1tr8><>:H10B@B(i;t)F(ni)i;tB0(i;t)+264000H(j1)i;t;223751CA9>=>;;wheretheexpressionsforSg(11);Sg(10);andSg(00)aregivenby( 2-53 ){( 2-55 ). EachremainingpartofQ; (j1);(j1)willbesimpliedindividually.Thus, and

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Now,notethatgiven(j1)=g(j1)0;g(j1)0;(j1);C(j1);H(j1);whicharetheparametervaluesproducedbythepreviousiterationofthealgorithm,wehavefrom( 2-38 )that Dening 59

PAGE 60

Also,itfollowsthat Therefore,using( 4-17 ), ToevaluatekXi=1niXt=1trQ1Varw E[(xi;jx(ni)i;ju0(2)i;t 60

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4-17 ).Also, 4-15 ){( 4-18 ).Thus,

PAGE 64

4-20 )and( 4-21 ). Ifwefurthersupposethatcovh0(1)i;t;h0(2)i;t0=Hi;t;12=0;thenH(:)=0:Theassumptionthattheobservedandunobservedcomponentsareuncorrelatedgreatlysimpliescalculationsandisnotunreasonable.Thisreduces( 4-19 )to and( 4-22 )to

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2-56 ).In( 4-24 ),0andIaresuitablyconformablezeroandidentitymatrices,respectively. Now,tosimplifytheportionsofQ; (j1);(j1)thatrelatetoR;rstnoticethatgiven(j1)and(j1); 65

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and Letting then Analogousto( 4-18 ),itfollowsthat 66

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4-17 )and( 4-29 ), ToresolvekXi=1niXt=1trR1Varv 4-26 )and( 4-29 ), sinceExi;tx0i;t 4-17 ),( 4-21 ),and( 4-27 ){( 4-29 ),

PAGE 68

4-17 ),( 4-20 ),( 4-21 ),( 4-25 )and( 4-30 ). 68

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4-35 )isdoneinpieces.Therstexpressionis

PAGE 70

+(2)i;t1Hi;t;22Hi;t;21H1i;t;11Hi;t;120(2)i;t1+A(1)i;tP(ni)i;tA0(1)i;t+(2)i;t1B(2)i;tHi;t;21H1i;t;11B(1)i;tF(ni)i;tB(2)i;tHi;t;21H1i;t;11B(1)i;t00(2)i;t1;using( 4-17 ),( 4-20 ),and( 4-21 ).Simplicationofthesecondtermyields

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+(2)i;t1B(2)i;tHi;t;21H1i;t;11B(1)i;tF(ni)i;tB(2)i;tHi;t;21H1i;t;11B(1)i;t00(2)i;t1R1i;t;11Ri;t;12i+A(1)i;tP(ni)i;tA0(1)i;tR1i;t;11Ri;t;12;using( 4-17 ),( 4-20 ),( 4-21 ),( 4-29 ),( 4-32 ),and( 4-33 ).Now,thethirdterm,Ey(1)i;tA(1)i;txi;t(1)i;t1u(1)i;t(2)i;t1u(2)i;ty(2)i;tA(2)i;txi;t(1)i;t2u(1)i;t(2)i;t2u(2)i;t0 4-37 ).Thus,thelastexpressionis

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4-17 ),( 4-20 ),( 4-21 ),( 4-29 ),and( 4-32 ){( 4-34 ). Substitutionofexpressions( 4-36 ){( 4-38 )into( 4-35 )andsimplicationgive

PAGE 73

+kXi=1niXt=1tr8><>:R1264000Ri;t;22Ri;t;21R1i;t;11Ri;t;123759>=>;+kXi=1niXt=1tr8><>:R1264IRi;t;21R1i;t;11375A(1)i;tP(ni)i;tA0(1)i;tI...R1i;t;11Ri;t;129>=>;kXi=1niXt=12640B@y(1)i;tby(2)i;t+R(:)1CAAx(ni)i;t(i;t)0B@u(1)i;tbu(2)i;t+H(:)1CA3750R12640B@y(1)i;tby(2)i;t+R(:)1CAAx(ni)i;t(i;t)0B@u(1)i;tbu(2)i;t+H(:)1CA375: 4-31 )to 73

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4-39 )to +kXi=1niXt=1tr8><>:R1264A(1)i;tP(ni)i;tA0(1)i;t0003759>=>;kXi=1niXt=1y(i;t)A(i;t)x(ni)i;t(i;t1)bu(i;t)0R1y(i;t)A(i;t)x(ni)i;t(i;t1)bu(i;t);with(i;t1)givenby264(1)i;t1(2)i;t100375. Noticethat(i;t)issimplyareorderingofthecolumnsof;determinedbywhatisobservedandwhatismissinginui;t:Thus,post-multiplicationofbyEi;tyields(i;t);i.e.,(i;t)=Ei;t:Also,notefromthedenitionthat(i;t)isareorderingofthecolumnsofwithrespecttotheexogenousvariables,ui;t;andapermutationoftherowsofdeterminedbytheobservedandunobservedportionsoftheresponses,yi;t:Thus,pre-andpost-multiplicationofbyD0i;tandEi;t;respectively,reconstitutes:Inotherwords,(i;t)=D0i;tEi;t:Hence,bytheformof(i;t1)givenabove,weseethat(i;t1)=24I00035(i;t);whereIrepresentstheqi;t1qi;t1identitymatrix. 74

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4-12 ){( 4-14 ),( 4-23 ),( 4-24 ),( 4-40 ),and( 4-41 )into( 4-11 )anduseexpressions( 2-57 ){( 2-62 ),weobtainQ; (j1);(j1)=1 2klnjg0j11 2kXi=1tr(g0)1F(ni)i;0+g(ni)i;0g0g(ni)i;0g00+1 2NlnjC1j1 2trnC1hSg(11)Sg(10)0Sg0(10)+Sg(00)0io+1 2NlnjH1j1 2kXi=1niXt=1trH1u(i;t)B(i;t)g(ni)i;tu(i;t)B(i;t)g(ni)i;t01 2kXi=1niXt=1tr8><>:H10B@B(i;t)F(ni)i;tB0(i;t)+264000H(j1)i;t;223751CA9>=>;+1 2klnj10j1 2kXi=1tr10P(ni)i;0+x(ni)i;00x(ni)i;000+1 2NlnjQ1j+1 2trQ1S0(12)S0(02)01 2trQ1S(11)S(10)0S0(10)+S(00)0+1 2trQ1S(12)S(02)S(22)01 2kXi=1niXt=1tr8><>:Q1Ei;t264000H(j1)i;t;22375E0i;t09>=>;+1 2NlnjR1j1 2kXi=1niXt=1tr8<:R10@y(i;t)A(i;t)x(ni)i;t24I00035D0i;tEi;tbu(i;t)1A0@y(i;t)A(i;t)x(ni)i;t24I00035D0i;tEi;tbu(i;t)1A09=;

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2kXi=1niXt=1tr8><>:R124I00035D0i;tEi;t264H(j1)i;t;22+B(2)i;tF(ni)i;tB0(2)i;t000375E0i;t0Di;t24I000359=;1 2kXi=1niXt=1tr8><>:R1264000R(j1)i;t;223759>=>;1 2kXi=1niXt=1tr8><>:R1264A(1)i;tP(ni)i;tA0(1)i;t0003759>=>;;whichisexpression( 2-52 ). (j1);(j1)in( 2-52 )withrespecttoeachparameter.Again,matrixdierentiationresultsfrom PetersenandPedersen ( 2007 )areusedthroughout,exceptinthecaseofthesymmetricvariance-covariancematricesQandR: (j1);(j1) 2 trQ1S0(12)S0(02)0+1 2 trQ1S(12)S(02)S(22)01 2 264kXi=1niXt=1tr8><>:Q1Ei;t264000H(j1)i;t;22375E0i;t09>=>;375=Q1S(12)S(02)S(22)Q1kXi=1niXt=1Ei;t264000H(j1)i;t;22375E0i;t0:

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2-70 ). Toderiveanestimatorfor;weproceedinsimilarfashion.Therefore,Q; (j1);(j1) 2 trQ1S(10)0S0(10)+S(00)0+1 2 trQ1S0(02)0S(02)0=Q1S(10)Q1S0(00)Q1S0(02): 2-72 ),(j)=S(10)(j)S0(02)S1(00): 2-70 )andsimplify,weobtainthealternateformof( 2-72 ),(j)=S(10)S(12)S1S0(02)S1(00)hIS(02)S1S0(02)S1(00)i1; 2-71 ). ToobtaintheestimatorforQ;wemaximize1 2NlnjQ1j1 2trQ1S(11)S(10)0S0(10)+S(00)0S0(12)S0(02)0S(12)S(02)S(22)0+0B@kXi=1niXt=1Ei;t264000H(j1)i;t;22375E0i;t1CA01CA9>=>;

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Anderson ( 1984 ),weseethatthisfunctionismaximizedwithQasin( 2-73 ),Q(j)=1 (j1);(j1) 2 "kXi=1niXt=1trnR1y(i;t)A(i;t)x(ni)i;t24I00035D0i;tEi;tbu(i;t)1A0@y(i;t)A(i;t)x(ni)i;t24I00035D0i;tEi;tbu(i;t)1A09=;351 2 24kXi=1niXt=1tr8<:R124I00035D0i;tEi;t264H(j1)i;t;22+B(2)i;tF(ni)i;tB0(2)i;t000375E0i;t0Di;t24I000359>=>;375=1 2 24kXi=1niXt=1tr8<:R124I00035D0i;tEi;tbu(i;t)y(i;t)A(i;t)x(ni)i;t01 2 24kXi=1niXt=1tr8<:R124I00035D0i;tEi;tbu(i;t)bu0(i;t)E0i;t0Di;t24I000359=;35

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2 "kXi=1niXt=1trnR1y(i;t)A(i;t)x(ni)i;tbu0(i;t)E0i;t0Di;t24I000359=;351 2 24kXi=1niXt=1tr8<:R124I00035D0i;tEi;t264H(j1)i;t;22+B(2)i;tF(ni)i;tB0(2)i;t000375E0i;t0Di;t24I000359>=>;375Q; (j1);(j1) Harville ( 1997 )statingthatPrj=1AjXBj=CisequivalenttohPrj=1B0jAjivecX=vecC;weobtainvec(j)=24kXi=1niXt=10@Ei;t8<:bu(i;t)bu0(i;t)+264H(j1)i;t;22+B(2)i;tF(ni)i;tB0(2)i;t0003759=;E0i;tDi;t24I00035R(j)124I00035D0i;t1A351vec24kXi=1niXt=1Di;t24I00035R(j)1y(i;t)A(i;t)x(ni)i;tbu0(i;t)E0i;t35:

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2-75 ).Sincevec(j)istheqr1vectorcreatedbystackingthecolumnsof(j)onebelowtheother,wesimplyundothevecfunctiontoobtaintheqrmatrix(j):Notethattheequationfor(j)usestheunorderedestimatorofRatiterationj;R(j);givenin( 4-42 )(seebelow).Thisadjustmentismadebecausetheformulafor(j)alreadyperformsanynecessaryreordering. Finally,theestimatorforRisobtainedbymaximizingQ; (j1);(j1)withrespecttoR:Thus,tomaximize1 2NlnjR1j1 2tr8<:R10@kXi=1niXt=10@y(i;t)A(i;t)x(ni)i;t24I00035D0i;tEi;tbu(i;t)1A0@y(i;t)A(i;t)x(ni)i;t24I00035D0i;tEi;tbu(i;t)1A0+kXi=1niXt=124I00035D0i;tEi;t264H(j1)i;t;22+B(2)i;tF(ni)i;tB0(2)i;t000375E0i;t0Di;t24I00035+kXi=1niXt=1264000R(j1)i;t;22375+kXi=1niXt=1264A(1)i;tP(ni)i;tA0(1)i;t0003751CA9>=>;;wereferto Anderson ( 1984 ),whichgivesthemaximumforRas +1

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2-74 ). (j1);(j1)ismaximizedwithrespecttoRwith( 2-74 )exceptthat(j1)takestheplaceof(j):IfwedenotethisestimateasR(j1+1=2);thenclearlyQ(j1)nR(j1);R(j1+1=2);(j1) 2-75 )withR(j1+1=2)inplaceofR(j)maximizesQ; (j1);(j1)subjecttotheseconstraints.NotethatR(j1+1=2)representstheunorderedversionofR(j1+1=2)andtheformulaisgivenby( 4-42 ).Ifweexpressthisestimateby(j1+2=2);theupdateensures 81

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MengandRubin ( 1993 ),ourprocedurerepresentsanECMalgorithm. Next,weshowthatourECMalgorithmconverges.Theorem1in MengandRubin ( 1993 )showsthatanyECMalgorithmisalsoaGeneralizedExpectationMaximization,orGEM,algorithm.Briey, Dempsteretal. ( 1977 )deneaGEMalgorithmasoneinwhichtheM-stepiteratesincreasetheQfunction.NotethattheconditionalmaximizationsgivenbyR(j1+1=2)and(j1+2=2)areunique.TheproofofTheorem2in MengandRubin ( 1993 )demonstratesthatanyECMwithuniqueCMstepssatisfyTheorem1of Wu ( 1983 ),whichshowsthatthelimitpointsofanyGEMsequencearestationarypointsandthelog-likelihoodconvergesmonotonically.Thus,thedesiredresultfollows. 82

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Inthischapter,theresultsofthesimulationanalysesarediscussed.TheultimategoalistoassesstheperformanceoftheEMalgorithminboththecompletedatacaseandthemissingdatacase.Theprocedureisevaluatedunderdieringnumberofsubjects,timepoints,initialparametervalues,andpercentmissing.Resultswithandwithoutexogenousvariablesinboththecompleteandmissingdatacases,aswellmismatchedtimepoints,arealsoexamined. Thesimulationmethodologyisnowpresented.Thestate,observation,andexogenousvectorsweresettodimensionp=q=r=1andtheparametersweregivenarbitraryvalues.Theprocedurewasreplicatedon3dierentdatasetsforeachsubject/timepointcombination.ThearrangementsconsideredaredisplayedinTable 5-1 .Foreachsubject/timepointcombinationexamined,severalstartingvalueswereconsideredfortheparameterstoensurethatthealgorithmwasconvergingtotheglobal(andnotalocal)maximum.SeeTable 5-2 forthetrueinitialandthecombinationofstartingparametervaluesexamined.Variouspercentagesofmissingness(0%,5%,and20%)intheresponseandexogenousvariableswereevaluated.Inaddition,themodelwithoutexogenousvariablesandtheeectontheprocedureofmismatchedobservationsintimewereexamined.Oneachrun,thesimulateddatawerealternatinglyprocessedwiththeKalmanrecursionsandEMalgorithmdescribedinSection 2.1 or 2.2 ,dependingonwhetheranydataweremissing,untilconvergence.Theestimationprocedurewasdeemedtohaveconvergedwhenthedierenceinthelikelihoodfromoneiterationtothenextwassmallerthan0.01inabsolutevalue.Thetolerancewasincreasedfrom0.001toensurethatthesimulationsconvergedwithinareasonablelengthoftime.AllanalyseswereperformedusingRstatisticalsoftware( RDevelopmentCoreTeam 2007 ). Tables 5-3 to 5-6 summarizetheresultsofthesimulationanalysesperformedinthecompleteandmissingdatacasesusingdierentparametervaluestoinitialize 83

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SimulationSubjectandTimePointCombinations NumberofSubjectsNumberofTimePoints Table5-2. TrueandInitialParameterValues ParameterTrueValueInitialValues thealgorithmandvariouspercentagesofmissingness.Theprocedurewasalsorunwithoutexogenousvariablesinthemodelforcomparison.Foreachsubject/timepointcombination,theminimumandmaximumlog-likelihoodvaluesobtainedoverallparametervaluecombinationsaregivenforeachdataset.Inaddition,therelativeerroriscalculatedastheabsolutevalueof[(Estimate-TrueValue)=TrueValue]:Themaximumoverallparameterestimates(except0;0;g0;andg0)ofthemedianand75thpercentileoftherelativeerrorsarereportedasthemaximumrelativeerrors(MRE).Thisgivessomeindicationofhowmuchtheparameterestimatesvaryineachdataset.TheMREiscalculatedseparatelyforthemean(andifapplicable,;;and)andvariance(Q;R;andifapplicable,C;H)parameters.TheinitialstatemeanandcovarianceparameterestimatesarenotincludedintheMREcalculationsbecausethese 84

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5-2 areonlyusedinthesecondstate-spacemodelwhentherearemissingdata.Inthiscase,notallpossibleparametercombinationstartingvaluesareexaminedbecausethisamountstorunning312simulationsuntilconvergenceforeachdatasetandwouldtakeaconsiderableamountoftime.Instead,allinitialparametercombinationsareconsideredforg0;g0;;C;andHbutrandomvaluesaregeneratedfor0;0;;;Q;;andRfromtheirrespectiverangesgiveninTable 5-2 Somegeneralcommentsareinorderbeforediscussingparticularresults.Themedianand75thpercentileoftherelativeerrorsareusedinsteadofthemaximumoverallestimatesbecauseseveralhistogramswereexaminedandthedistributionshadlongtails.Thusitseemsthatafewestimatesarereallydierentfromthetrueparametervalueandhencethemaximumoftherelativeerrorismisleading.Inaddition,theMREmaynotbethebestmeasureofhowgoodtheestimatesare.Forinstance,apureguessof0asanestimateforaparametervalueleadstoanMREof1.00.SoanarbitraryvalueisbetterthantheMLEinsomecasesiftheMREisusedtodeterminethediscrepancybetweenaparameteranditsestimator. Togeneratethecaseswithoutexogenousvariables,theerrortermfromthecorrespondingdatasetwaskeptthesameandtheresponseswerethengeneratedfromthemodelwithoutcovariates.Therearelesstermsinthecalculationofthelikelihoodsinthecaseswithoutexogenousvariablesandhencetheseshouldbesmaller.However,theresponsesaredierentandhenceitisnotafaircomparisonbetweenthetwocases.Thefactthatthelikelihoodsarebiggerwhentheexogenousdataaredroppedfromthemodelisprobablyaconsequenceofhowthedataweregenerated.Overall,itseemsthatthenewprocedure 85

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MinimumandMaximumLog-Likelihoods:Casewithk=30subjects,ni=5timepointsforalli=1;:::;k DataSetminlikemaxlikeMREmeans-median75%-tileMREvars-median75%-tileminlikemaxlikeMREmeans-median75%-tileMREvars-median75%-tileminlikemaxlikeMREmeans-median75%-tileMREvars-median75%-tileminlikemaxlikeMREmeans-median75%-tileMREvars-median75%-tileminlikemaxlikeMREmeans-median75%-tileMREvars-median75%-tile 1-194.49-187.172.76()4.360.45(Q)0.70-299.64-295.080.81()1.07()1.81(R)1.90-296.29-292.920.77()1.05()2.13(R)2.23-244.63-241.920.96()1.05()2.12(R)2.16-250.78-249.530.95()1.05()2.64(R)2.71noexogenous-189.90-187.491.07()2.010.73(Q)0.76-188.64-181.450.38()1.070.39(Q)0.65||||-180.16-145.730.47()0.750.38(R)0.79(Q)|||| 2-204.09-201.462.62()4.420.33(Q)0.54-306.06-302.520.82()0.902.38(R)2.50(Q)-297.87-295.430.80()0.942.02(R)2.14-263.87-261.340.93()1.36()2.66(R)2.78-266.87-263.730.93()1.33()2.82(R)2.90noexogenous-202.33-201.820.65()1.080.47(Q)0.98-194.82-189.170.71()1.030.51(Q)0.90||||-179.58-136.870.49()1.251.31(Q)1.32|||| 3-189.59-182.542.62()4.390.59(Q)0.70-308.61-306.420.89()1.013.66(Q)3.73-300.48-299.170.77()0.903.72(Q)3.83-270.66-269.450.89()1.034.93(Q)5.23-261.36-258.110.91()0.965.34(Q)5.48noexogenous-184.72-182.810.22()0.450.51(R)0.71-179.79-170.410.11()0.180.63(R)0.75||||-161.56-143.300.83()1.140.34(R)0.46|||| MinimumandMaximumLog-Likelihoods:Casewithk=30subjects,ni=15timepointsforalli=1;:::;k PercentMissing0%5%response,5%exogenous5%response,20%exogenous20%response,5%exogenous20%response,20%exogenous DataSetminlikemaxlikeMREmeans-median75%-tileMREvars-median75%-tileminlikemaxlikeMREmeans-median75%-tileMREvars-median75%-tileminlikemaxlikeMREmeans-median75%-tileMREvars-median75%-tileminlikemaxlikeMREmeans-median75%-tileMREvars-median75%-tileminlikemaxlikeMREmeans-median75%-tileMREvars-median75%-tile 1-607.28-594.472.58()4.390.26(Q)0.48-921.00-910.310.76()0.842.30(R)2.39-875.65-866.670.76()0.842.15(R)2.33-804.56-793.830.87()0.922.74(R)2.82-767.56-755.780.89()0.942.89(R)2.97noexogenous-595.34-594.710.74()0.900.41(Q)0.87-579.46-561.510.52()0.880.40(Q)0.83||||-545.01-482.841.25()1.330.42(Q)0.86|||| 2-619.51-603.662.60()4.440.31(Q)0.56-932.50-914.720.83()0.922.49(R)2.54-890.42-885.490.82()0.912.57(R)2.73-793.19-786.120.93()0.982.54(R)2.77-769.44-751.730.97()0.982.96(R)3.00noexogenous-604.78-603.370.50()0.630.41(Q)0.96-594.46-577.320.41()0.540.56(Q)1.00||||-539.10-474.330.70()1.060.42(Q)0.78|||| 3-619.04-591.552.61()4.390.29(Q)0.52-906.93-902.500.34()0.473.49(Q)3.51-886.37-878.900.33()0.46()3.46(Q)3.70-773.66-769.690.33()0.42()3.65(Q)3.91-760.01-751.930.54()0.654.63(Q)4.83noexogenous-597.35-596.280.61()0.750.41(Q)0.89-583.75-566.050.59()0.690.41(Q)0.88||||-542.26-477.990.67()0.800.43(Q)0.86||||

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MinimumandMaximumLog-Likelihoods:Casewithk=150subjects,ni=5timepointsforalli=1;:::;k PercentMissing0%5%response,5%exogenous5%response,20%exogenous20%response,5%exogenous20%response,20%exogenous DataSetminlikemaxlikeMREmeans-median75%-tileMREvars-median75%-tileminlikemaxlikeMREmeans-median75%-tileMREvars-median75%-tileminlikemaxlikeMREmeans-median75%-tileMREvars-median75%-tileminlikemaxlikeMREmeans-median75%-tileMREvars-median75%-tileminlikemaxlikeMREmeans-median75%-tileMREvars-median75%-tile 1-995.13-964.702.73()4.370.86(Q)0.93-1489.93-1477.830.91()0.962.13(R)2.19-1454.77-1444.940.88()0.962.34(R)2.47-1282.26-1274.160.96()0.982.09(R)2.16-1263.39-1256.650.96()0.982.56(R)2.59noexogenous-970.91-965.670.26()0.340.45(Q)0.93-942.92-905.910.32()0.400.44(Q)0.94||||-737.61-598.820.53()0.690.61(Q)0.88|||| 2-988.73-968.032.68()4.460.59(Q)0.78-1516.91-1488.860.95()0.972.15(R)2.18-1459.72-1438.510.92()0.972.45(R)2.55-1329.95-1302.880.95()0.992.26(R)2.39-1290.66-1267.100.96()0.982.81(R)2.83noexogenous-972.48-968.020.44()0.840.40(Q)0.69-951.84-905.370.32()0.560.45(Q)0.82||||-784.67-602.510.78()1.320.95(Q)1.12|||| 3-979.19-968.742.63()4.400.75(Q)0.91-1472.63-1465.440.62()0.672.79(Q)3.00-1424.41-1418.000.54()0.603.08(Q)3.30-1285.11-1278.210.64()0.683.94(Q)4.35-1245.26-1240.400.67()0.784.39(Q)4.55noexogenous-978.43-972.940.19()0.340.52(Q)0.95-957.30-910.140.47()0.471.31(Q)1.31||||-759.14-579.890.92()1.190.88(Q)0.97|||| MinimumandMaximumLog-Likelihoods:Casewithk=150subjects,ni=15timepointsforalli=1;:::;k PercentMissing0%5%response,5%exogenous5%response,20%exogenous20%response,5%exogenous20%response,20%exogenous DataSetminlikemaxlikeMREmeans-median75%-tileMREvars-median75%-tileminlikemaxlikeMREmeans-median75%-tileMREvars-median75%-tileminlikemaxlikeMREmeans-median75%-tileMREvars-median75%-tileminlikemaxlikeMREmeans-median75%-tileMREvars-median75%-tileminlikemaxlikeMREmeans-median75%-tileMREvars-median75%-tile 1-2981.09-2905.812.86()4.340.71(Q)0.95-4513.77-4490.130.78()0.892.21(R)2.31-4392.04-4372.540.82()0.912.55(R)2.69-3915.26-3903.650.99()0.992.48(R)2.49-3808.65-3777.310.96()0.972.71(R)2.74noexogenous-2993.44-2993.140.21()0.330.52(Q)0.97-2919.37-2827.140.13()0.290.54(Q)0.98||||-2788.13-2381.880.40()0.430.29(Q)0.80|||| 2-2965.14-2900.852.66()4.460.62(Q)0.89-4555.91-4536.790.79()0.872.33(R)2.40-4402.69-4384.140.80()0.872.38(R)2.53-3965.73-3949.410.98()0.992.48(R)2.50-3849.73-3822.150.93()0.972.83(R)2.86noexogenous-2939.71-2939.130.41()0.530.45(Q)0.88-2875.64-2784.590.42()0.560.45(Q)0.87||||-2708.61-2316.220.60()1.850.74(Q)0.92|||| 3-2952.33-2893.502.64()4.380.81(Q)0.94-4554.35-4535.910.50()0.523.59(Q)3.62-4422.67-4408.170.51()0.533.42(Q)3.70-3896.18-3888.390.53()0.644.87(Q)4.99-3852.75-3840.560.54()0.563.80(Q)4.28noexogenous-2955.55-2955.090.33()0.450.42(Q)0.88-2898.05-2796.200.37()0.500.46(Q)0.87||||-2743.51-2342.360.57()0.890.69(Q)0.95||||

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Severalpointscanbemadeaboutthecompletecasesimulationresults.Clearly,theoutcomesforthecaseswithoutexogenousvariablesaremuchbetterthanthosewiththecovariates.ThelikelihoodsconvergewithinamuchtighterrangeofvaluesandtheMREsarelower,acrossallfourtables.Theproblemacrossallsubject/timepointcombinationsconsiderediswith;thecoecientassociatedwiththeexogenousvariablesintheresponsemodel.TheMREsareconsiderablylargefor;withamedianofaround2.6inalltablesforallthreedatasets,sothathalfoftheestimatesareabout3.6timesasdierentasthetruevalue.Althoughnotlisted,theMREsfor;thecoecientfortheexogenousvariablesinthestateequation,arealmostaslargeasthosefor:Thissuggeststhatincludingexogenousvariablesinthemodelforthecompletedatacasemaybelessimportant,regardlessofsamplesize,becausetheestimatorsproducedforthecoecientparametersarenotveryreliable.Itmaybethatwhentherearenomissingvalues,thepreviousresponsesintimearesucientinpredictingfutureoutcomesandtheexogenousvariablesaresuperuous. Themissingdatasimulationsyieldedseveralinterestingresults.First,themeanparameterestimatesarerelativelyclosetothetruevalueswhilethoseforthevarianceparametershavedeterioratedincomparisontothecompletedatacase,judgingbytheMREs.Themeanparameterestimatesareaboutasgoodasinthecompletecasewithoutexogenousvariables,wheretheaforementionedproblemswithandarenotanissue,acrossallsubject/timepointcombinations.Itisnotsurprisingthatthevarianceparameterestimateshaveworsenedinthepresenceofmissingdata,sincethereislessinformation.Inaddition,thedierencebetweenthemaximumandminimumlikelihoodswhentherearemissingdataisgenerallysmallerthantherespectivecompletecaseacrossallfourtables.Also,withineachdatasetandmissingpercentagegrouping,formost 88

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Onedrawbackapparentfromthemissingdatasimulationsisthatthevarianceparametersintherststate-spacemodel(QandR)arebeingoverestimated.Thosefromthesecondstate-spaceformulation(CandH)modellingtheexogenousvariablesarebeingestimatedfairlywell.Thedierencesbetweentheminimumandmaximumlikelihoodsdoappeartosystematicallyincreaseasthetotalsamplesizeincreases,oraswemovefromTable 5-3 toTable 5-6 .Butthismaybeaconsequenceofcalculatingthemissingdataasapercentofthetotalnumberofobservations.Hence,withmoredatatherearealsomoremissingvaluesandthusmoreuncertainty. TheresultsofthesimulationstudytoanalyzetheeectsofmismatchedobservationsintimeappearinTable 5-7 .Tocreateatimedisparity,usingthecompletedatasets(0%missing)withk=150subjectsandni=15timepointsfromTable 5-6 ,eachsubjecthad10responsesandcorrespondingexogenousvariablesrandomlysettomissing.Thisinessencecreatesatimemismatchedgridinwhicheachsubjecthas5observationsandthe10otherpossibleobservationtimesinthegrouparearticiallymissingsothatallsubjectshavethesamesetof15possibletimepoints.TheseresultsappearinTable 5-7 .Asbefore,theminimumandmaximumlog-likelihoodvaluesandtheMREsobtainedoverallparametervaluecombinationsforthemeanandvarianceparametersaregivenforeachofthe3datasets.Again,allinitialparametercombinationsareconsideredforg0;g0;;C;andHbutrandomvaluesaregeneratedfor0;0;;;Q;;andRfromTable 5-2 ThemismatchedtimepointssimulationyieldedresultssimilartothoseobservedinTable 5-6 with20%missingdatainboththeresponseandexogenousvariables.Thedierencebetweenthemaximumandminimumlikelihoodsarecomparableinbothsituations.Also,fortherst2datasets,theMREsforthemeansareapproximatelythesameinbothtablesandtheMREsforthevarianceparametersareinfactsmallerinthe 89

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MinimumandMaximumLog-Likelihoods:Mismatchedcasewithk=150subjects,ni=15timepointsforalli=1;:::;k minlikemaxlike MREmeans-median75%-tile MREvars-median75%-tile 1-1653.48-1636.531.08()1.081.27(R)1.28 2-1670.84-1644.171.16()1.171.13(R)1.13 3-1639.93-1632.030.64()0.686.48(Q)7.50 mismatchedtimepointscase.However,whiletheMREsforthemeanparametersremainrelativelysimilarinbothsituationsforthethirddataset,thoseforthevarianceparameterQaredisproportionatelylargerinthemismatchedcase.Onepossibleexplanationforthisisthattherandomselectionofmissingresponsesresultedinobservationsthatwerehighlyvariable.Thecombinationofthiswiththelargepercentageofmissingvaluescreatedbythemismatchedtimepointsmaybethereasonwhythevariancetermofthestateequationintherststate-spacemodelisbeingoverestimated.Theothervarianceparameters,R;C;andH;arebeingestimatedrelativelywellinthemismatchedcaseforthethirddataset.Thus,theonlyproblemiswiththeestimatesforQ;andthismayonlybeachanceoccurrencesinceitonlyhappensinonedatasetandnottheothertwo. Inshort,thenewprocedureappearstobefairlyrobusttomoderatepercentagesofmissingdata,evenintheanalyseswithfewersubjectsandtimepoints.Whendataaremissing,theinclusionofexogenousvariablesseemstoaddvaluableinformationtothemodellingprocedure.Inaddition,theproposedtechniqueappearstoproduceacceptableparameterestimatesinthepresenceofmismatchedtimepoints.Theseareallfavorableresults,sincetheseelementsarelikelytobepresentintheanalysisofrealdata. 90

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Themotivationforselectingthistopicforresearchwasthedesiretopredictwhencertainbiomarkerswillchangeforacohortofpatientswiththeautoimmunediseasesystemiclupuserythematosus(SLE).Trackingsuchchangesisimportantbecausetheyareindicativeofanincreaseindiseaseactivity.Ifphysicianscanforecastaarebeforeitoccurs,theycaninterveneearlierandthusreduceitsimpactonthepatient. Theanalysisprovidedhereisbynomeansexhaustiveorcomplete,butratherasampleapplicationofthederivedformulation.Asanillustration,wewillmodelhighsensitivityC-reactiveprotein(hs-CRP)usingweightasanexogenouscovariate.Hs-CRPhasbeenusedasagaugeofdiseaseactivityinpatientswithrheumaticconditionssuchasSLE,anditslevelsareknowntoincreaseinresponsetoinfection( Barnesetal. 2005 ).Detaileddemographic,clinical,andmedicationinformationhasbeencollectedfromthisgroupofpatients.Severallaboratoryvariableshavebeenfoundtocorrelatecross-sectionallywithhs-CRP(e.g.,serumalbumin,C3,erythrocytesedimentationrate(ESR),weight,hypertension,andapolipoproteinA-I)andthesewereallcandidatesforapredictor( Barnesetal. 2005 ).Sinceavariablemustbeexogenoustohs-CRPinordertobeincludedinthestate-spacemodelasacovariate,weightseemedareasonablechoice. TheavailableinformationonSLEpatientsfromthisdatasetwillbeanalyzedusingtheKalmanrecursionsandEMalgorithmforthemissingdatacasedescribedinsection 2:2 91

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5-7 )producedreasonableresults,sotheanalysisofthisdatasetshouldbefeasible. Althoughthesimulationsshowedthatthestartingvalueshadlittleeectontheconvergenceoftheprocedure,someattemptwasmadetoarriveatroughestimates.TheseguesseswherethenusedtogeneratearangeofpreliminaryvalueswithwhichtoinitializetheEMalgorithm.Thestartingvaluesfortheinitialstateconditionparameters(0;0;g0;andg0)werearrivedatbytakingthemeanandvarianceofbaselinehs-CRPandweightmeasurementsacrosssubjects.Severalweightedregressionmodelsweret,withweightsequaltotheinverseofthenumberofvisits,toapproximatetheremaininginitialvalues.Togetatthestartingvaluesfortherststate-spaceformulationparameters,hs-CRPwasmodelledusingweightandthepreviousvalueofhs-CRPintimeascovariates.Theregressioncoecientsgavearoughideaofwhattousetostart;;andwith,andthemeansquarederror(MSE)anapproximationforQandR'sstartingpoints.Similarly,aweightedregressionwastforweighttoinitializethesecond-statespacemodelparameters.Inthiscase,thecoecientforthepreviousvalueintimeofweightwasusedasapreliminaryvalueforwhiletheMSEwasappliedasinitialguessesforCandH:Table 6-1 liststheinitialparametervaluesderivedfromthepreliminaryroughestimates.Fortherststate-spacemodel,randompointsweredrawnfromtherangeofvaluesgivenwhileallinitialparametercombinationswereconsideredforg0;g0;;C;andH: Pinheiroetal. ( 2007 )wasusedtocalculateparameterstandarderrorsattheMLEsoncethealgorithmconverged( RDevelopmentCoreTeam 2007 ).ThefdHessprocedureevaluates 92

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DataAnalysisInitialParameterValues ParameterInitialValues theapproximateHessianmatrixoftheobserveddatalog-likelihoodusingthenumericaltechniqueofnitedierences.Thenalmodelselectedwastheoneyieldingthelargestlog-likelihoodvalue.Forecastingthefuturebehaviorofhs-CRPandtheerrorsassociatedwiththesepredictionswasaccomplishedusingtheKalmanrecursionsgivenbyTheorem 2.3 TheiterationwiththelargestlikelihoodproducedtheparameterestimatesgiveninTable 6-2 .FromtheMLEsandstandarderrorsforand;itappearsthatweightisapoorpredictorofhs-CRP.Eventhoughthemodelmaynotbeveryinformative,asfurtherillustration,anapproximatetrajectoryfortheresponseanditspredictionintervalweregenerated.Specically,smoothedvaluesandforecastsforhs-CRPwereobtainedfromthettedmodel.Agraphoftheactualhs-CRPobservations,smoothers,and6-monthaheadpredictorsisgivenbyFigure 6 -1foranaveragesubject. Amorethoroughanalysisofthisdatashouldbeperformed.Hs-CRPshouldbemodelledbivariatelywithcystatinC,whichisconsideredanexcellentindicatorofrenalfunction( Narainetal. 2007 ).Simultaneousincreasesinhs-CRPanddecreasesincystatinCwouldsuggestthepotentialforaareinthenearfuture.Therearevariouspossible 93

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DataAnalysisResults ParameterMLEStandardError Hs-CRPobservations,smoothedvalues,andforecastsforatypicalsubject. 94

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Weproposedaproceduretocalculateparameterestimatesinthestate-spacemodelwithexogenousvariables,withandwithoutmissingobservations.Exogenousvariablesareindependentofthesystemandaecttheresponse,butnotviceversa.TheEMalgorithmandtheKalmansmootherequationswereappliedinconjunctiontoderivemaximumlikelihoodestimatesforthemodelparameters. Inthecompletecase,weincludedexogenousvariablesinthestateandobservationequationsofthestate-spacemodelandderivedestimatesforand;therespectivecoecientparameters.Simulationsdemonstratedthattheinclusionofexogenouscovariateswhentherewerenomissingdatamaybesuperuous.Itseemedthatthepreviousresponsesintimeweresucientinpredictingfutureoutcomes. Thenewformulationformissingdataproposedtwostate-spacemodelstorepresenttheunobservedinformationthatcanoccurinboththeresponseandexogenousvariables.TheECMalgorithm,inwhichtheE-stepoftheEMalgorithmremainsthesamebuttheM-stepisreplacedbyaseriesofconditionalmaximizationsteps,wasappliedtoderivemaximumlikelihoodestimatesoftheparameters.TheassumptionofexogeneitywaspivotalfortheEMmethodologytoworksincewithoutit,theproposedtheoryasdescribedfailedtohold.Thesimulationstudiesshowedthattheexogenousvariablessuppliedvaluableinformationtotheanalysiswhendataweremissing,thusprovidingpersuasiveevidenceinfavorofincludingtheseinthemodel.Overall,thenewprocedureappearedtoberelativelyrobusttomoderatepercentagesofmissingdataandmismatchedobservationsintime,evenwithfewersubjectsandtimepoints,althoughseveralofthevarianceparameterswerebeingoverestimated. Inaddition,analyticrecursiveformulaswerederivedforcalculatingparameterestimatestandarderrors.AlthoughtheEMalgorithmgreatlyeasesmaximumlikelihoodestimationinthepresenceofmissingdata,itdoesnotautomaticallyyieldparameter 96

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Asanexampleofthenewtechnique,theprocedurewasappliedtoarealdataset.Sincesubjectswereobservedondierentdates,atimegridwitharticialmissingvalueswascreatedsothateveryonehadthesamenumberoftotalvisitsandtimepoints.ParameterstandarderrorswerecalculatednumericallyandsmoothedvaluesandforecastsfortheresponsewereobtainedusingtheKalmanrecursionsforillustrativepurposes. Sincethedataanalysiswasonlyanexampledemonstratingtheuseofthenewlydevelopedprocedure,amorein-depthstudyshouldbeperformed.Itisstronglyrecommendedthatthemodelconsistofabivariateresponseandamultivariateexogenousvariablevector,whichwouldtakeadvantageofthefullpotentialofthestate-spacemodelwithexogenousvariablesandmissingdatathatwasdevelopedhere.Thistypeofanalysiswilllikelyrequireconsiderableprocessingtimethough,becauseofthelargenumberoftimepointsthatwillberequiredtocreateagrid.Theideaforthisworkarosefromtheneedforphysicianscaringforthisparticularsetofpatientstopredictwhencertainbiomarkerswillchange.Amorecomprehensiveanalysismayallowrheumatologiststoaccuratelyforecastaarebeforeitoccursandinterveneearlieronthepatient'sbehalf.Thepredictivecapabilitiesofthemodelshouldalsobefurtherexploredthroughsimulationtodeterminehowaccuratethetechniqueisinforecastingfutureoutcomes. Althoughthemotivationfordevelopingthestate-spacemodelwithexogenouscovariatesandmissingdatawasinthemedicaleld,examplesaboundinotherareasinwhichtheprocedurewouldbereadilyapplicable.Thenatureofthehumanbodymakesitdiculttoascertainwhichcovariatesarereallyexogenoustotheoutcomesofinterestintheautoimmunediseasesdataset.Itwouldbeinterestingtoapplythetechniqueinacasewheretheindependentvariablesareclearlyexternalforcesontheresponse.Onesuchexampleisinvestigatingtheeectofenvironmentalfactors,suchairpollution,onmorbidity,andseveraldatasetscanbefoundin LatiniandPasserini ( 2004 ). 97

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LittleandRubin 1987 ).Itwouldbeworthwhiletomodifytheproceduretohandlenon-ignorablemissingdata. 98

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ArleneHortensiaNaranjowasborninHialeah,Florida,thethirdchildandonlydaughterofMagalyandClaudioNaranjo.SheearnedaB.S.inmathematicalsciencesin2000andaM.S.instatisticsin2002fromFloridaInternationalUniversityinMiami,Florida.In2007,sheobtainedaPh.D.instatisticsfromtheUniversityofFloridaunderthedirectionDr.GeorgeCasella.Herinterestsincludelongitudinaldataanalysis,statisticalapplicationsinresearch,andmethodsforhandlingmissingdata. 102