BMC Bioinformatics
O
Bio.l Central
Methodology article
A joint model for nonparametric functional mapping of longitudinal
trajectory and timetoevent
Min Lin1,2 and Rongling Wu*1
Address: 'Department of Statistics, University of Florida, Gainesville, FL 32611, USA and 2Department of Biostatistics and Bioinformatics, Duke
University, Durham, North Carolina 27710, USA
Email: Min Lin annie.lin@duke.edu; Rongling Wu* rwu@stat.ufl.edu
* Corresponding author
Published: 15 March 2006
BMC Bioinformatics 2006, 7:138 doi: 10.1 186/147121057138
Received: 13 July 2005
Accepted: 15 March 2006
This article is available from: http://www.biomedcentral.com/14712105/7/138
2006 Lin and Wu; licensee BioMed Central Ltd.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Background: The characterization of the relationship between a longitudinal response process
and a timetoevent has been a pressing challenge in biostatistical research. This has emerged as an
important issue in genetic studies when one attempts to detect the common genes or quantitative
trait loci (QTL) that govern both a longitudinal trajectory and developmental event.
Results: We present a joint statistical model for functional mapping of dynamic traits in which the
event times and longitudinal traits are taken to depend on a common set of genetic mechanisms.
By fitting the Legendre polynomial of orthogonal properties for the timedependent mean vector,
our model does not rely on any curve, which is different from earlier parametric models of
functional mapping. This newly developed nonparametric model is demonstrated and validated by
an example for a forest tree in which stemwood growth and the time to first flower are jointly
modelled.
Conclusion: Our model allows for the detection of specific QTL that govern both longitudinal
traits and developmental processes through either pleiotropic effects or close linkage, or both. This
model will have great implications for integrating longitudinal and event data to gain better insights
into comprehensive biology and biomedicine.
Background
Although there has been a upsurge of interest in jointly
modelling longitudinal and event data during the last dec
ade [19], no statistical models have been developed to
characterize the shared genetic basis for these two types of
traits. In biomedicine, the identification of specific genetic
variants responsible for an HIV patient's timedependent
CD4 count and for the time to onset of AIDS symptoms
can help to design individualized drugs to control this
patient's progression to AIDS. Similarly, in studies of
prostate cancer, a shared genetic basis between prostate
specific antigen, repeatedly measured for patients follow
ing treatment for prostate cancer, and the time to disease
recurrence can be used to make optimal treatment sched
ules for patients. In plants, knowledge about whether the
genetic loci for reproductive behaviors, such as the time to
first flower and the time to form seeds, also govern growth
rates and sizes of plants helps to understand the etiology
of plant's adaptation to the environment in which they are
grown.
The genetic mapping of quantitative trait loci (QTL) that
are responsible for longitudinal traits has long been a dif
ficult issue because of the dynamic features of these traits.
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More recently, part of this difficulty has been solved by
integrating the statistical analysis of longitudinal data into
a QTL mapping framework, leading to a socalled func
tional mapping strategy [1016]. Statistical models for func
tional mapping were established on the belief that
biological processes can be described by mathematical
functions. One of the most significant examples for this is
the use of Sshaped logistic curves to model growth trajec
tories. West et al. [ 17] indicated from fundamental princi
ples of biophysical processes that logistic forms of growth
are biologically crucial for the maintenance of optimal
metabolic level and, thereby, the best use of available
resources for an organism from birth to adulthood.
Because of the embedment of fundamental biological
principles within the modelling model, functional map
ping provides a quantitative framework for testing biolog
ically relevant hypotheses at the interplay between gene
actions and development.
The concept of functional mapping can be further
extended to jointly mapping a longitudinal variable and a
timetoevent by incorporating statistical theories devel
oped to characterize the relationships between longitudi
nal response and event processes [19]. However, original
functional mapping models for a dynamic trait reply
upon explicit mathematical functions that describe the
development of the trait. In practice, there are also many
situations in which no appropriate curves can be used to
describe a biological process. To model an arbitrary shape
of curves, a different statistical model based on nonpara
metric theory should be formulated. Polynomial analyses
that can be specified by varying orders have power to fit
curves with arbitrary shapes. As shown by Kirkpatrick and
Heckman [18], Legendre polynomials have several favora
ble properties for curve fitting which include: (1) the func
tions are orthogonal, (2) it is flexible to fit sparse data, (3)
higher orders are estimable for high levels of curve com
plexity and (4) computation is fast because of good con
vergence.
The purpose of this article is to develop a joint statistical
model for nonparametric functional mapping of longitu
dinal trajectories based on the Legendre polynomials,
integrated with timetoevents. This joint model is con
structed within the maximum likelihood context, includ
ing simultaneously modelling of the mean vector (based
on nonparametric approaches) and covariance matrix
(based on parametric approaches). By analyzing stem vol
ume growth data in an example of a forest tree, we will
demonstrates the implications of our joint model. Lastly,
the advantages of our model in general biomedical and
biological research and the areas in which the model can
be further refined are discussed.
The Model
The likelihood function
Consider a mapping population of size n for which a
number of molecular markers are genotyped, aimed to
identify QTL for a longitudinal trait and timetoevent.
Every individual of the mapping population is measured
for the longitudinal trait at multiple (say T) time points
(y) and a timetoevent (z). Variable z can be the time to
first flower, the timing of cancer malignance, the time of
mortality, or the events that happen at a time. The infer
ence of unknown QTL genotypes for the phenotypic traits
based on observed marker information (M) can be made
due to cosegregation between the QTL and markers.
Suppose there are two segregating QTL for longitudinal
and event traits in the mapping population, each with
genotypes 2, 1 and 0. These two QTL are assumed to be
linked or associated with and, therefore, can be inferred
from, markers. The joint likelihood function of the two
types of phenotypic data and marker information at the
two underlying QTL is written as
n 2 2
L(2  y,z,M) = n1 [ 4J2,fJ2 (YyZ';U2,. 2
i = l 0 b0
_I, (1)
where 0 is the unknown vector that defines the QTL posi
tions, timedependent QTL effects (UjJ2 and vjj2 ) and
covariance matrix (E), 011. 2 is the mixture proportion
expressed as the conditional probability of a joint geno
type J1J2 for the longitudinal and event QTL given marker
genotypes for individual i and fhJ2 is the (T + 1)dimen
sional multivariate normal distribution function with
mean vector (UjlJ2, vJ2 ) and covariance matrix Y.
Conditional probabilities
There are different descriptions of the conditional proba
bility, depending on the type of the mapping population.
If the mapping population is an experimental cross initi
ated with two contrasting parents, such as the F2 or back
cross, the conditional probability is described in terms of
the recombination fractions between the markers and
QTL [12,19]. If the two QTL are bracketed by different
pairs of markers, the conditional probability of joint QTL
genotypes given the marker intervals can be expressed as
the product of the corresponding conditional probabili
ties for QTL genotypes given a single marker interval. If
the two QTL are located at the same marker interval, the
conditional probabilities should be derived using the
principle of 4point analysis. For a natural population, the
association between the QTL and markers can be
described by the coefficients of linkage disequilibria [17]
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A
2,
K
" 1
Age (year)
4 5 6
Age (year)
Figure I
Plots of stem volume index growth vs. ages for each of the
90 genotypes used to construct linkage maps in poplar
hybrids (Yin et al. 2002). The relationships between growth
and age are displayed for untransformed (A) and logtrans
formed data (B).
Modelling the mean vector
The choice of a mean function for a longitudinal trait is
based on theory or past experience that suggests a certain
mathematical form for the timedependent mean. How
ever, it would be essential to derive a general approach
that can fit any kind of curves. By choosing different
orders of orthogonal polynomials, the Legendre function
has potential to approximate the functional relationships
between trait values and times to any specified degree of
precision. The Legendre polynomials are solutions to a
very important differential equation, the Legendre equa
tion,
2
(1 2) d 2xd + r(r+1)z = 0.
dx2 dx
The polynomials may be denoted by P,(x), called the Leg
endre polynomial of order r. The polynomials are either
even or odd functions of x for even or odd orders r.
The general form of a Legendre polynomial of order k is
given by the sum,
K = ( (2r 2k)! r2k2)
Pr (X) = I ( 1)rk (r Xr2 (2)
k=0 2k!(r k)! (r 2k)!
where K = r/2 or (r 1)/2 whichever is an integer. This pol
ynomial is defined over the interval [1, 1]. From Eq. 5, we
show the first few polynomials as
Po(x) = 1
P (x)= x
2
P2(x) =2(3x2 1)
P3(x) = 1 (5X3 3x)
2
P4(x)= (35x4 30x2 +3)
8
P5(x) = 1 (63x5 70x3 + 15x)
8
P6(x) = 1 (231x 315x4 + 105x2 5).
16
In this modelling, independent variable x is expressed as
time t, which is adjusted, to rescale the measurement
times to the range of the orthogonal function [1, 1], by
t* = 1+ 2( tmin
tmax tmin
where tmin and tma are respectively the first and last time
points.
Our aim is to model the timedependent genotypic values
for different QTL genotypes jj2, using the orthogonal
Lengedre polynomial with a particular order r. A family of
such polynomials is denoted by
Pr (t*) = [Po(t*), P(t*), ... ,P r(t*)]
and a vector of genotypic values, which is timeindepend
ent, denoted by
wjij2 = (WjiJ2, Wjlj21"' Wjjr)
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a 4 8 A 8 9s 8 8 8 9 8 os 8 8 Q Q
a D N m n 6
Figure 2
The profile of the loglikelihood ratios between the full
(there is a QTL) and reduced (there is no QTL) model that
combines stem volume index growth trajectories and flower
timing across linkage groups in the Populus deltoides parent
map. The genomic positions corresponding to the peaks of
the curve are the MLEs of the QTL localization. The thresh
old values for claiming the existence of QTL are given as the
horizonal solid lines for the genomewide level and broken
lines for the chromosomewide level. Blue color corre
sponds to the unifying model for jointly mapping growth tra
jectories and flower trait, whereas red color corresponds to
a model for mapping growth trajectories only. The positions
of markers on the linkage groups (Yin et al. 2002) are indi
cated at ticks.
The timedependent genotypic values uj j2 (t) can be
described as a linear combination of fwvj2 weighted by
the family of the polynomials, i.e.,
Substituting the mean vector of the likelihood (1) by the
above expression (3), we will need to estimate timeinvar
iant genotypic values for the longitudinal trait, fvj i2 and
the genotypic mean for the event trait, vy2 .
Modelling the covariance matrix
A general form for the covariance matrix among longitu
dinal trajectories and development event in the likelihood
(1) is expressed as
Zy Za2
2i2 
where Y and a 2 are the covariance matrix and variance
for the longitudinal and event traits, respectively, and y
= Iz, is the covariance matrix between these two types of
traits. The structures of Y and z can be empirically mod
elled on the basis of prior knowledge or results. Several
approaches for parametric modelling of the covariance
matrix, reviewed by Zimmerman and NunezAnton [20],
can be utilized.
The most common approach for modelling the covari
ance structure is based on a variancecorrelation specifica
tion, in which functions for the responses' variances and
correlations are specified. In previous QTL mapping [10
14], the covariance structure for longitudinal traits is
modelled by the simplest, most parsimonious and most
flexible firstorder autoregressive (AR(1)) model in which
there are two parameters, stationary variance (y ) and
correlation (p,). Relaxing the stationary variance assump
tion for growth data, Wu et al. [15] adopted a transform
bothsides (TBS) model to obtain an empirically homoge
neous variance. The results from simulation studies sug
gest that the TBSbased mapping model provides more
precise estimates for curve parameters and residual vari
ancecorrelation than the untransformed model.
The TBSbased model displays the potential to relax the
assumption of variance stationarity, but the covariance
stationarity issue remains unsolved. Zimmerman and
NunfiezAnt6n [20] proposed a socalled structured ant
edependence (SAD) model to model the agespecific
change of correlation in the analysis of longitudinal traits.
The SAD model has been employed in several studies and
displays many favorable properties [21,22].
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u (t) = PW)W .
http://www.biomedcentral.com/14712105/7/138
~1
0 1 2 3 4 5 6 7 8 9 10
Age (year)
0 1 2 3 4 5 6 7 8 9 10
Age (year)
Figure 3
Volume growth curves for two different QTL genotypes for the QTL detected on linkage group 2 by the Legendre polynomial
based model. Left panel: logtransformed curves; Right panel: antetransformed curves. Growth trajectories for all the individ
uals studied are indicated in yellow background. The effect of the detected QTL on the time to first flower is indicated.
The emergence of a developmental event (z) at time t* can
be correlated with the longitudinal trait. For example,
larger tumor sizes may be likely to lead to earlier malig
nance of cancer than smaller tumor sizes. An AIDS patient
would die when his/her HIV load accumulates to a partic
ularly high level. In plants, first flowering only appears
after some investment of vegetative growth. All such com
mon knowledge suggests that the correlation between the
event trait at time t* and longitudinal trait measured at
time t (before t*) decays with time difference (t* t). In
fact, a similar pattern of correlation should also hold for t
> t* because of the autocorrelation nature. With all this
consideration, the correlation between the event and lon
gitudinal traits can be modelled by the power equation,
expressed as
.. =o for t* > t,
corr(y(t),z(t*))= '1, A for
S, 1, = 0,
ll' , f=o r, *
where 0 7 < 1. Equation (5) suggests that the event is
correlated with the longitudinal trait, to the same extent,
before and after its emergence. The event trait should be
individualspecific when it is the timing of development,
such as the time to first flower. In this case, Equation (5)
and, therefore, the covariance matrix (4), expressed as i,
should be individualspecific. If LY is modelled by the
AR(1) model, one can derive the explicit expressions of
the determinant and inverse of ,i specified by (ar, p,
2
a 7, ).
Computational algorithms
The unknown parameters (0) contained with the mixture
model (1) include three types, QTLmarker recombina
tion fractions for a pedigree or QTLmarker linkage dise
quilibria for a natural population reflected in the
conditional probabilities 01,,, ,, the curve parameters
( i 2 VjJ2 ) that model the mean vector, and the param
eters (ay p, yz 27, 2) that model the structure of the
covariance matrix. We derived the EM algorithm to esti
mate these parameters. Using the (prior) conditional
probability and the likelihood, we define the posterior
probability for individual i to bear on a QTL genotype jj2
as
.liJ lJ2 i 2 f 2
I2 1i 0 2= OZj2 O l21ifl2(Y Z'; l2' ,2 12
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0 1 2 3
4 5 6 7 8 9 10
Age (year)
0 1 2 3
4 5 6 7 8 9 10
Age (year)
Figure 4
Volume growth curves for two different QTL genotypes for the QTL detected on linkage group 5 by the Legendre polynomial
based model. See Figure 3 for all the explanations.
The posterior probabilities are then used to derive a
closedform maximum likelihood estimates of the QTL
locations, expressed as the ratio of recombination frac
tions, for linkage analysis or QTLmarker haplotype fre
quencies for linkage disequilibrium analysis [17]. For
functional mapping, in which the mean vectors and cov
ariance matrix are modelled by mathematical parameters
based on nonlinear equations, it is impossible to derive
the closed forms for these parameters, the simplex algo
rithm, widely used in operations research, is found to pro
vide a fast and precise estimation of the curve parameters
and the parameters that model the residual covariances
[23]. Thus, we implement the simplex algorithm in the
maximization process of the EM algorithm.
For linkage analysis based on an experimental cross,
oJ l2l, 's are expressed in the recombination fraction
between the QTL and two flanking markers. In practical
computations, the QTL position parameter can be viewed
as a fixed parameter because a putative QTL can be
searched at every 1 or 2 cM on a map interval bracketed by
two markers throughout the entire genome. The amount
of support for a QTL at a particular map position is often
displayed graphically through the use of likelihood maps
or profiles, which plot the likelihood ratio test statistic as
a function of map position of the putative QTL. The peak
of the profile corresponds to the position of the QTL over
the genome.
For linkage disequilibrium analysis of a natural popula
tion, we have derived a closed form for the EM algorithm
to estimate QTLmarker haplotype frequencies. From the
estimated haplotype frequencies, the allele frequencies of
QTL and QTLmarker linkage disequilibria can be esti
mated. How the markers are associated with the underly
ing QTL in the population can be tested for the
significance of QTLmarker linkage disequilibria.
After the point estimates of parameters are obtained by
the EM algorithm, the approximate variancecovariance
matrix and the sampling errors of the estimates (Q0) can
be estimated. The techniques for so doing involve calcula
tion of the incompletedata information matrix which is
the negative secondorder derivative of the incomplete
data loglikelihood. The incompletedata information can
be calculated by extracting the information for the miss
ing data from the information for the complete data [24].
Order selection
For a QTL to be detected, we need to determine the opti
mal order for the Legendre polynomial that fits the data.
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We propose using the AIC information criterion to select
the best model. The AIC value at a particular order, r, is
calculated by
AIC = 2 In L( I r) + 2 dimension(Q r), (7)
where (2 r) is the the MLE of parameters for the Legen
dre polynomial of order r and dimension (0r) represents
the number of independent parameters under order r.
Also, Bayesian Information Criterion (BIC) [25] is used to
determine the optimal order of the Legendre function,
which is calculated by
BIC = 2 In L(2 r) + 2 dimension( r) In (nT). (8)
As compared to AIC, BIC adjusts the effects of sample size
and the number of time points measured.
Hypothesis tests
Our model allows for a number of hypothesis tests to
examine the genetic control of growth processes [14]. All
these tests are helpful to address biological questions
related to the genetic control mechanisms of growth. Test
ing whether specific QTL exist to affect the longitudinal
and event processes is a first step toward the understand
ing of the detailed genetic architecture of complex pheno
types. This can be tested by formulating the following
hypotheses,
H0 : ujl2 = u and vj2 = v vs. HI: Not all qualities in Ho
hold. (9)
The Ho states that there is no QTL affecting longitudinal
and event processes (the reduced model), whereas the H1
proposes that such a QTL does exist (the full model). The
test statistic for testing the hypotheses is calculated as the
loglikelihood ratio of the reduced to the full model:
LR = 2[ln Lo(Q y, z) In L, (2 y, z, M)], (10)
where C and 2 denote the MLEs of the unknown
parameters under Ho and H1, respectively. Because the LR
calculated by equation (9) may not be asymptotically X2
distributed with eleven degrees of freedom due to viola
tion of regularity conditions, an empirical approach for
determining the critical threshold based on permutation
tests is used. By repeatedly shuffling the relationships
between marker genotypes and phenotypes, a series of the
maximum loglikelihood ratios are calculated, from the
distribution of which the critical threshold is determined.
After the QTL are detected to be significant for both longi
tudinal and event traits, we need to test whether the
detected QTL are significant separately for each trait. We
assume two different genetic settings:
(1) Longitudinal and event traits are under control of the
same QTL;
(2) Each process is controlled by different QTL that are
linked on the same chromosomal region.
For the first setting, only after it is significant in two sepa
rate tests for longitudinal and event traits can the tested
QTL be thought to be pleiotropic in affecting both types
of traits. For the second setting, we assume that two tested
QTL are located in the same interval bracketed by two
markers. The comparison of the first and second setting
can examine how the detected QTL jointly affect the dif
ferentiation in longitudinal and event traits. First, we can
test how two genetic mechanisms, pleiotropy or close
linkage, contribute to the correlation between these two
types of traits. If the two QTL are detected to be significant
for both, we then test whether such a correlation is due to
pleiotropy or close linkage. Second, when two QTL exist,
we can test how they epistatically interact to affect longi
tudinal trajectories and developmental events. Wu et al.
[14] formulated a procedure for testing the epistatic effects
on developmental trajectories.
Results
The proposed joint model is used to analyze growth tra
jectories and flowering behavior in a forest tree. The study
material used was derived from the interspecific hybridi
zation of Populus (poplar), P. deltoides and P. eummericana.
This hybrid population was planted at a spacing of 4 x 5
m in the complete randomized design in a field trial near
Xuzhou City, Jiangsu Province, China. The total stem
heights and diameters measured at the end of each of the
first 11 growing seasons are used to calculate stem volume
indices (y) for QTL analysis. Because the vegetative and
reproductive growth processes are generally correlated in
plants [26], the ages to first flower (z) was predicted by a
regression equation for each of these hybrids. Two genetic
linkage maps each based on a different parent were con
structed for a subset of hybrids (90) with different types of
molecular markers that are segregating in a pattern of
pseudotest backcross [271. Our analysis here will be
based on P. deltoides (D)specific linkage map.
Although stem height and diameter for each tree follows
a logistic curve [10], the stem volume index derived from
these two traits cannot be fit by the growth equation
mainly because stem volume has not yet reached its
asymptotic growth during this measurement period (Fig.
1A). As shown by Figure 1, the variance of the stem vol
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Table I: Coefficients of the first five Legendre polynomials for adjusted time points (t*) used in the poplar growth study.
t 0 1 2 3 4 5 6 7 8 9 10
t*
Po(t*)
P,(t*)
P2(t*)
P3(t*)
P4(t*)
Ps(t*)
P6(t*)
P7(t*)
4/5
I
4/5
23/50
2/25
24/103
167/418
172/439
I 10/459
3/5
I
3/5
1/25
9/25
51/125
107/70 I
132/767
10/31
2/5
I
2/5
13/50
I 1/25
93/823
59/218
163/557
13/891
I/5
I/5
1 1/25
7/25
29/125
274/891
56/695
231/787
1/5
I
1/5
1 1/25
7/25
29/125
274/89 I
56/695
231/787
2/5
I
2/5
13/50
I 1/25
93/823
59/218
163/557
13/891
3/5
I
3/5
1/25
9/25
51/125
107/701
132/767
10/31
4/5
I
4/5
23/50
2/25
24/103
167/418
172/439
 I 10/459
ume index increases markedly with age, but the logtrans
formation of these indices leads to much parallel curves
(Fig. 2B), suggesting that the variance stationarity assump
tion may be met after the transformation.
We implemented the Legendre function to model the QTL
genotypic mean vector of growth trajectories and the TBS
based AR(1) model to approximate the structure of the
covariance matrix. The joint model also allows the estima
tion of the genotypic means and residual variance for the
age to first flower, as well as the correlation of this trait
with stem wood growth trajectories. Equation (5) pro
vides a general equation for modelling the correlation
between the event and longitudinal traits measured at dif
ferent ages. In this example, it was observed that there
were significant correlations between the age to first
flower and volume growth at all different ages (0.29 
0.82). Thus, for simplicity of computation, we assume
that such correlations are consistent across the ages of vol
ume index, denoted as 77. Using the adjusted ages, we cal
culated the coefficients of the Legendre polynomials for
the first seven orders (Table 1). These coefficients are used
to estimate timedependent genotypic values. The AIC
and BIC values calculated consistently suggested an opti
mal order of 5 to fit stem volume growth (Table 2).
While our joint model was derived to detect two QTL at a
time, it was reduced to a oneQTL model because of a lim
ited sample size for sufficient estimates oftwoQTL model
parameters. For the pseudotest backcross there are two
genotypes, Qq (j = 1) and qq ( = 0), at each QTL. Figure 2
illustrates the profile of the loglikelihood ratio (LR) val
ues for testing the existence of QTL that control either
overall growth curves of stem volume indices from age 0
to 10 years or the ages to first flower, or both, across all of
the 19 Dspecific linkage groups. We performed 100 per
mutation tests to determine critical threshold values for
declaring the existence of QTL. By comparing the peaks of
the LR profile with the thresholds, three significant QTL
were detected, one on linkage group 2 at the 5% genome
wide testing level and two on linkage groups 5 and 12 at
the 5% chromosomewide testing level (Fig. 2; Table 3).
We indicated the positions of these QTL on linkage
groups, which correspond to the peaks of the LR profile. If
only the stem volume growth is analyzed using traditional
functional mapping [10], only the QTL on linkage group
2 is detected, suggesting that the joint model displays bet
ter power than a singletrait analysis.
Each of the three QTL was tested for their pleiotropic effect
on both vegetative growth and reproduction by formulat
ing two independent null hypotheses, one being that the
QTL does not affect stem growth and the second being
that the QTL does not affect flowering age. The rejection
of both the null hypotheses implies that a QTL has a plei
otropic effect on growth and reproduction. As indicated
by Table 3, all the detected QTL on linkage groups 2, 5 and
12 only trigger a significant effect on stem volume growth,
but neither has an effect on both growth and reproduc
tion.
Table 2: The AIC and BIC values used to determine the optimal order for the Lengendre polynomials.
Order
739.2
222.8
764.6
917.6
936.2
933.8
931.9
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71 1.0
255.7
802.2
959.9
983.2
985.5
1010.3
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Table 3: The MLEs and their sampling errors (SE, in the parentheses) of the QTL position, timeinvariant QTL effects on growth
curves (expressed in the Legendre polynomials), QTL effect on the time to first flower, residual variance and residual correlation
under the logtransformed model for the interspecific poplar hybrid mapping population.
Linkage group 2
Linkage group 5
Linkage group 12
1.60 (0.0750) 1.83 (0.0731) 1.62 (0.0727) 1.84 (0.0869) 1.85 (0.0664) 1.54 (0.0707)
3.04 (0.0610) 3.15 (0.0593) 3.09 (0.0622) 3.16 (0.0734) 3.19 (0.0591) 3.02 (0.0626)
1.68 (0.0475) 1.94 (0.0470) 1.71 (0.0467) 1.94 (0.0557) 1.92 (0.0457) 1.69 (0.0492)
0.60 (0.0403) 0.84 (0.0402) 0.61 (0.0401) 0.85 (0.0475) 0.81 (0.0390) 0.62 (0.0422)
0.17 (0.0295) 0.45 (0.0279) 0.18 (0.0305) 0.48 (0.0335) 0.44 (0.0283) 0.16 (0.0306)
0.00 (0.0284) 0.21 (0.0270) 0.03 (0.0286) 0.23 (0.0315) 0.21 (0.0274) 0.01 (0.0292)
0.28 (0.0339)
0.88 (0.0154)
6.6 (0.2015)
0.28 (0.0375)
0.87 (0.0176)
6.8 (0.1951)
7.1 (0.2200)
1.42 (0.2394)
0.50 (0.0686)
0.25 (0.0274)
0.86 (0.0168)
7.1 (0.1974)
7.1 (0.2275)
1.41 (0.2352)
0.51 (0.0580)
6.7 (0.1813)
1.28 (0.1841)
0.48 (0.0503)
The LR, LR and LRz values are the test statistics for testing the existence of a QTL for both growth and the time to first flower, the existence of a
QTL for growth but not for the time to first flower, and the existence of a QTL for the time to first flower but not for growth. The locations of the
detected QTL are described by the genetic distance (in cM) from the first marker of a linkage group.
The MLEs of growth parameters for stem volume indices,
covariancestructuring parameters and the parameters
dealing with reproductive behaviors, as well as their
standard errors estimated from the Fisher information
matrix, were tabulated in Table 3. It can be seen that the
estimates of all the parameters from our joint model pro
vide reasonable precision, using the estimates of growth
curves, we draw two different curves each corresponding
to a genotype at each of the detected QTL (Fig. 3). Note
that growth curves were first drawn from the estimates of
the Legendre parameters (the left panel of Fig. 3) and then
transformed back to the normal scale (the right panel of
Fig. 3). In general. these QTL are switched on to affect the
overall stem growth process after age 45 years at which
strong intertree competition sets in the stand due to can
opy closure. Figure 3 also displays genotypic differences in
the age to first flower at each of the growth QTL. But as
tested, only QTL on linkage group 12 has a significant
impact on the age to first flower (Table 3). At this QTL, the
slowergrowing genotype flowers about 0.7 year earlier
than the fastergrowing genotype. Through this QTL, the
fastgrowing attribute and the capacity to efficiently
occupy growth resources can be transmitted to the next
generation.
Discussion
A theoretical framework has been constructed for func
tional mapping of quantitative trait loci (QTL) underlying
longitudinal growth [1015]. Functional mapping was
grounded on biological reality that every organism fol
lows universal growth laws that can be derived from fun
damental principles for the allocation of metabolic energy
between maintenance of existing tissue and the produc
tion of new biomass [17]. In a couple with linkage dise
quilibrium mapping, functional mapping has been
extended to map host QTL for HIV dynamics for a natural
human population [16].
Although functional mapping has proven to be both bio
logically and statistically advantageous in terms of the
estimates of the QTL positions and effects, its practical
applications may be limited for two reasons. First, a lon
gitudinal variable, such as HIV dynamics, tumor growth
or plant vegetative growth, may be related to timeto
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Test/Parameter
Location
UtJl
2
Sj3
Sj
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lU
0 1 2 3
4 5 6 7 8 9 10
Age (year)
0 1 2 3
.1
4 5 6 7 8 9 10
Age (year)
Figure 5
Volume growth curves for two different QTL genotypes for the QTL detected on linkage group 12 by the Legendre polyno
mialbased model. See Figure 3 for all the explanations.
events, like time to onset of AIDS symptoms, time to first
malignancy or time to first flower, through a common set
of QTL [28,291. Second, not all longitudinal data meas
ured at a series of discrete time points can be fit by a math
ematical function with biological means.
In this article, we have proposed a joint model for func
tional mapping of longitudinal trajectories and timeto
events with the nonparametric context. Several statistical
models have been proposed to jointly analyze longitudi
nal and event processes [19]. Different from those tradi
tional models, our joint model has been constructed
within the mixture model framework, with each mixture
component assigned by biological rationale. We incorpo
rated Legendre polynomials to characterize an arbitrary
form of growth curves. In a real example for a forest tree,
the model has detected a few QTL that affect growth proc
esses and the age to first flower. The detection of the com
mon genetic basis for vegetative and reproductive growth
supports the views that any developmental event is not
isolated from the growth process [28,291. Our model pro
vides a complete genetic analysis of growth courses for
various organisms at different organization levels. From a
statistical perspective, it increases the power of QTL detec
tion and the precision of parameter estimation because
the information about growth and development is jointly
utilized. Meanwhile, our model allows for the test of sev
eral important hypotheses regarding the genetic control of
developmental events occurring from fertilized ovum to
reproductive maturity.
Our model is based on nonparametric Legendre orthogo
nal polynomial approaches for growth and development
processes. Orthogonal polynomials (including Legendre)
have been extensively used in random regression analyses
for longitudinal traits with repeated records [18,3032].
There are several favorable properties for Legendre poly
nomials to be utilized in curve fitting, i.e., (1) the func
tions are orthogonal, (2) it is flexible to fit sparse data, (3)
higher orders are estimable for high levels of curve com
plexity and (4) computation is fast because of good con
vergence. Nonparametric regression methods for
modelling the mean structure of longitudinal data have
been based on more commonly used Bspline basis func
tions [33]. Brown et al. [9] extended the Bspline basis to
model multiple longitudinal variables. As compared to
the Bspline approach that constructs curves from pieces
of lower degree polynomials smoothed at selected
pointed (knots), Legendre polynomials are simpler in
which only fewer regression coefficients are needed to
model the curve. However, polynomials often overem
phasize the observations at the extremes and may be prob
lematic for high orders of fit due to oscillations at the
extremes of the curve [34]. It is therefore worthwhile
implementing more flexible Bspline basis functions into
the nonparametric functional mapping model.
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In our joint model, we assumed that the timetoevent is
multivariate normally distributed together with longitudi
nal data (see also [35]). An alternative to model the distri
bution of longitudinal and event data is to take the
product of the normal distribution function of longitudi
nal trajectories and the distribution of the event trait and
sensoring indicator given the trajectory function [9,36]. In
addition, our model should be extended to consider mul
tiple longitudinal variables based on a framework by Lin
et al. [5], multiple timetoevents [8] and structured covar
iance matrices among unbalanced repeatedmeasures
[37]. In order to unravel the genetic architecture of com
plex phenotypes that are characterized by a network of
biological processes, such extensions will be essential.
With appropriate improvements, our joint model will
have great power to unlock the genetic secrets hidden in
various complicated and biologically realistic life proc
esses.
Conclusion
We have developed a joint statistical model that can detect
specific QTL governing both longitudinal traits and devel
opmental processes through either pleiotropic effects or
close linkage, or both. This model was integrated by non
parametric approaches that do not rely on mathematical
equations to model growth curves. The model will have
great implications for integrating longitudinal and event
data to gain better insights into comprehensive biology
and biomedicine.
Authors' contributions
ML derived the models, programmed the method and per
formed data analyses. RW conceived the idea and drafted
the manuscript.
Acknowledgements
We thank the two anonymous referees for their constructive comments
on this manuscript. The preparation of this manuscript has been partially
supported by NSF grant (0540745) and NIH grant (ROI NS041670) to R.
W. The publication of this manuscript is approved as journal series No.
10579 by the Florida Agricultural Experimental Station.
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