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NONPARAMETRIC COMPARISON OF SLOPES OF REGRESSION LINES RAYMOND RICHARD DALEY 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 1984 To my parents ACKNOWLEDGEMENTS Dr. P.V. Rao suggested the research topic and, as my major professor, provided many hours of consultation with me during the preparation of this dissertation. For this I thank him sincerely. I also thank the other members of my committee, the Department of Statistics, and the staff of the Biostatistics Unit. I am grateful to Dr. Anthony Conti for allowing me flexibility in my work schedule that facilitated completion of this dissertation. I will not forget Alice Martin for her practical advice and heartfelt support during the various qualifying exams. I thank my close friends for supporting my choice of academic goals while showing me the benefits of a balanced life. I especially thank Christopher Kenward for his encouragement and positive attitude. Finally, I appreciate the love and support of my parents in all areas of my life and I am proud to share the attainment of this longawaited goal with them. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS . . . . . . . . . iii A13STRACT . . . . . . . . . . . v CHAPTER ONE INTRODUCTION . . . . . . . 1 1.1 The Basic Problem . . . . . 1 1.2 Literature Review . . . . . 3 1.3 Objectives and overview . . . . 9 TWO COMPARING THE SLOPES OF TWO LINES . . 11 2.1 Introduction . . . . . . 11 2.2 Asymptotic Distributions . . . 16 2.3 Large Sample Inference . . . . 32 2.4 Asymptotic Relative Efficiencies . 35 2.5 Small Sample Inference . . . . 68 2.6 Monte Carlo Results . . . . . 75 THREE COMPARING THE SLOPES OF SEVERAL LINES . 99 3.1 Introduction . . . . 99 3.2 Asymptotic Theory a d a Proposed Test . . . . . . 104 3.3 An Exact Test . o . o 138 3.4 A Competitor to the Proposed Te S t . 141 3.5 Asymptotic Relative Efficiencies . 161 FOUR CONCLUSIONS . . . . . . . . 171 BIBLIOGRAPHY . . . . . . . o . . . 175 BIOGRAPHICAL SKETCH . . . . . . o . . 378 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 NONPARAMETRIC COMPARISON OF SLOPES OF REGRESSION LINES By Raymond Richard Daley December, 1984 Chairman: Dr. Pejaver V. Rao Major Department: Statistics Distributionfree confidence intervals and point estimates are defined for the difference of the slope parameters in a linear regression setting with two lines, assuming common regression constants are used for both lines. A statistic, T, is proposed for this setting that is an application of a statistic discussed by Sievers in his paper titled "Weighted Rank Statistics for Simple Linear Regression," appearing on pages 628631 of The Journal of the American Statistical Association in 1978. The statistic T employs weights, a rs that are chosen by the user. If the regression constants are x 1,. nJ then under the weights a r = XSx r the null distribution of T depends on the regression constants. An. associated exact confidence interval for the slope difference can be obtained by calculation of a permutation distribution, requiring use V of a computer. Under the weights a rs = 1 if X s X r is positive, zero otherwise, the null distribution of T is essentially that of Kendall's tau. In this case, an associated eizact confidence in terval for the slope difference can be calculated using readily available critical values of this distribution. Simulation results indicate the power of a test of parallelism based on T under either of the two sets of weights dominates the power of other available eXact procedures. Pitman asymptotic relative efficiencies under equal spacing of the regression constants also favor a test based on T over other exact tests. Under unequal spacing of the regression constants, the simulation results suggest use of the weights a rs r__ xsx r when feasible. The method applied in the case of two regression lines is generalized to construct a test for use in comparing the slopes of several regression lines with the slope of a standard line. The proposed test statistic is a quadratic Zorm in a set of statistics, each having the structure of T. The asymptotic relative efficiency of the statistic with respect to a potential competitor is examined. CHAPTER ONE INTRODUCTION 0 1.1 The Basic Problem There are many experimental settings where the appropriate statistical analysis involves comparing the slopes of regression lines. For an example, consider a dilution assay in which two different drugs are being compared to get a measure of relative potency. A linear regression of the response on the chosen dosage levels is constructed for each drug. A fundamental assumption of this type of assay is the assumption that the two regression lines are parallel (Finney, 1964, p. 108). Hence a statistical test of parallelism of two regression lines would be desirable in this type of study. In other applications, an estimate of the difference in. slopes might be needed. Consider a study comparing the effectiveness of two different fluoride dentifrices in reducing dental decay, where the participants engaged in varying numbers of supervised brushing sessions. The response measured is the change in the number of decayed and filled surfaces over the duration of the study, called the DFS increment. For each fluoride, the linear regression of DFS increment on number of supervised brushings usually 2 results in a negative slope; as the number of supervised brushings increases, the average DFS increment tends to decrease. In this case the dental researcher might be interested in an estimate of the difference between the rate of decrease for the two fluoride groups, that is, an estimate of the difference of the slopes of the two regression lines. We have just offered two examples of the basic problem discussed in this dissertation, comparing the slopes of two or more regression lines. In this work we deal with linear regression models in the designed experiment, where the levels of the independent variable are determined by the experimenter, such as dose in a dilution assay. The methods of inference we propose assume the levels of the independent variable are chosen to be the same for each regression line, as is frequently the case in a designed experiment. Thus, in the dilution assay example, the chosen dosage levels for both drugs would be the same. Before proceeding further, we might mention that the problem of comparison of slopes does not suffer from lack of attention. As will be clear from the literature review in Section 1.2, there are many methods for comparing the slopes of two or more regression lines. The classical methods based on least squares theory are exact when the underlying distribution is normal and these methods are asymptotically distibutionfree over the class of distributions having finite positive variance. Nonparametric methods exist which 3 are either distributionfree or asymptotically distributionfree over a larger class of underlying error distributions, including heavily tailed ones such as the Cauchy distribution under which the least squares methods perform poorly. The nonparametric, asymptotically distributionfree methods usually have good efficiency properties. Unfortunately, not only is their performance for small sample sizes suspect, but these methods are hard to implement in practice. In this work, we concentrate on the problem of developing better distributionfree methods of inference about slope differences in situations where the experimenter has control over the selection of the levels of the independent variable. 1.2 Literature Review In this section we will review articles in the literature discussing nonparametric analysis of linear regression models. We focus on works that are pertinent to the topic of this dissertation, comparing the slopes of two or more regression lines. Often a nonparametric procedure designed to make inference about the slope of a single line may be adapted to the case of several lines. In Chapter Two we will indicate a simple method of adapting a test designed for the single line setting to the two line setting, when the levels of the independent variable for both lines are chosen to be the same. For these reasons we include a 4 discussion of the techniques applicable in the simple (single line) linear regression setting. To begin the discussion, consider the simple linear regression model, Y.3 a + Ox.i + E.,(.21 I ..., N. The Y's are the observed responses, a and 5 are unknown parameters, the x's are the known regression constants (levels of the independent variable), and the E's are unobservable random errors. In this setting, nonparametric tests based on rank statistics have been examined by many authors (Hoeffding, 1951; Terry, 1952). Hajek (1962) discussed the general properties of tests based on the linear rank statistics, N V= E (x x)cp(R.) (1.2.2) :1=1 J where R. is the rank of Y. among Y 1F Y2' ..., Y N' p is a score function which transforms the ranks to scores in an N appropriate way, and x = Z I.N. Hajek discussed j=1 asymptotic properties of these tests which depend on the chosen score function. Good discussions of properties of the class of linear rank statistics (1.2.2) are found in Hajek and Sidak (1967) and Randles and Wolfe (1979). Tests of the hypothesis a = 0 based on the linear rank statistics (1.2.2) can be used to derive estimates of a in the model (1.2.1) by the application of a general technique developed by Hodges and Lehmaann (1963). Since its appearance in the literature, this technique has been applied to estimation problems in a wide variety of settings. Because HodgesLehmann estimates are used frequently in this dissertation, we now give a brief example describing this estimation technique. Suppose we assume the simple linear model (1.2.1) but with zero intercept, Y.i = Ox + E., (1.2.3) j =1 .,N, and we use the statistic (the numerator of the Pearson product moment correlation) N Q Z (x.X)Y. (1.2.4) j=1 i J to test =0 against the alternative 3 # 0. Suppose further that the errors E., j = 1,1 ...,I N, are symmetrically distributed about zero. Then N QM~ Z (x .x)(Yj ax ) 6 N Z (x.x)E. (1.2.5) j=1 has median zero. Thus a desirable property for an estimate Sof $ is that Q($) be as near as possible to the median of the distribution of Q($), that is, as near as possible to zero. Then the HodgesLehmann estimate of based on Q is the value of such that Q( ) = inf IQMg). Of course this example is in a special setting, but it does provide the basic motivation for a HodgesLehmnann estimate. Adichie (1967) applied the technique of Hodges and Lehmnann to derive estimates of the slope and intercept in the simple linear regression setting (1.2.1) using linear rank statistics of the form (1.2.2). Jureckova (1971) generalized these estimates to the multiple regression setting. An undesirable feature of the methods of Adichie and Jureckova is that calculation of the estimates and confidence intervals requires iterative procedures. Sen (1968) gave a complete development of an estimate of slope in the simple linear regression setting (1.2.1), that had earlier been suggested by Theil (1950). The estimate is the median of the slopes Y Y s r 7 which is easy to calculate and intuitively appealing. The TheilSen estimate can be derived by applying the HodgesLehmann technique to the test statistic ZZ sgn(x 5x r)sgn(Y Y r), (1.2.7) r where sgn(u) = 1, 0, 1 as u is greater than, equal to, or less than zero. The statistic (1.2.7) is the numerator of a statistic known as Kendall's (1955) tau, a measure of association between the pairs (xj,, Y.). Since the null (a=0) distribution of Kendall's tau is distributionfree and has been tabulated, and the statistic (1.2.7) has an easily invertible form, exact smallsample confidence intervals for Share readily constructed using the HodgesLehmann technique applied to the method of Theil and Sen; no iterative procedure is necessary.' Scholz (1977) and Sievers (1978) extended the method of Theil and Sen by using a weighted Kendall's tau. The definition of a statistic equivalent to the one defined by Sievers will be given in Chapter Two. Sievers also showed how to construct confidence intervals and point estimates for the slope $. Note that although Scholz's work appeared prior to Sievers, the Scholz reference is only an abstract of an unpublished paper and hence we hereafter refer to this technique as the SieversScholz approach. Further discussion of the SieversScholz approach will be given in Chapter Two. 8 Having considered nonparametric methods of inference about the slope in the simple linear regression setting, we now consider those methods applicable to the multiple regression setting. Recall the method of Jureckova (1971), which was an extension of Adichie's (1967) results to the multiple regression case. A test of the parallelism of k > 2 lines using Jureckova's method requires the computation of k individual HodgesLehmann estimates, each calculated by an iterative technique. Sen (1969) and Adichie (1974) specify tests of parallelism of k > 2 lines that require only a single HodgesLehmann estimate of the overall slope, arrived at iteratively. The Sen and Adichie method has good efficiency properties. However neither Sen and Adichie's approach, nor the technique of Jureckova, provides exact confidence intervals for the individual slopes. Jaeckel (1972) suggested estimating regression parameters by minimizing a chosen dispersion function of the residuals. He showed that for a certain class of dispersion functions his estimates are asymptotically equivalent to Jureckova's, but easier to compute. However, Jaeckel's estimates are also only asymptotically distributionfree and require iterative computations. Of the methods reviewed, only the TheilSen and SieversScholz statistics, applicable in the simple linear regression setting, have readily invertible forms enabling easy computation of exact distributionfree confidence intervals for the slope parameter. In the two line setting, 9 two exact procedures are found in the literature. The first, due to Hollander (1970), is based on a statistic in the form of the Wilcoxon signed rank statistic. The second exact procedure, due to Rao and Gore (1981), is applicable when the regression constants of the two lines are equally spaced. The statistic of this second procedure takes the form of a WilcoxonMannWhitney twosample statistic. These two procedures are used to test the null hypothesis that the two regression lines are parallel. The null distributions of the two statistics are distributionfree, and exact distributionfree confidence interals for the difference in slopes can be readily computed by applying the HodgesLehmnann technique. More discussion about the Hollander and the RaoGore methods will be given in Chapter Two. 1.3 Objectives and Overview Having reviewed the literature regarding nonparametric comparisons of slopes of regression lines, we now state concisely the two objectives of this dissertation. The first objective is to develop efficient, exact nonparametric methods for comparing the slopes of two regression lines when the researcher has control over the choice of the levels of the independent variable. The methods we propose will enable construction of exact distributionfree confidence intervals for the difference between the two slopes. The exact techniques of Hollander (1970) and Rao and Gore (1981), discussed briefly in the previous section, will be 10 direct competitors of the methods we suggest. A comparison of these three techniques in Chapter Two using their Pitman asymptotic relative efficiencies and a simulation study will establish the superiority of the new methods whenever they are appropriate. The second objective of this work is to generalize the new methods suggested in Chapter Two to the setting where the slopes of several regression lines are being compared. In Chapter Three we will extend these new methods to the multiple line case when the purpose is to compare the slope of one of k lines, considered a standard or control, to the slopes of the kl other lines. A comparison of our proposed test to a modification of Sen's (1969) test will show that our proposed test, inaddition to allowing an exact distributionfree test for small samples not available with Sen's approach, is almost as efficient as the modification of Sen's test when the sample size is large and the error distribution is not too heavily tailed. CHAPTER TWO COMPARING THE SLOPES OF TWO LINES 2.1 Introduction In this chapter we examine the case of two regression Lines, assuming the same regression constants are used for both lines. in this section we first establish the notation for the linear regression model and the statistic used in this chapter. We then motivate the form of the statistic and give some background concerning its development. A special characterization of the HodgesLehmann estimate associated with the statistic is given. We close this section with a brief look at the contents of the rest of Chapter Two. Consider the linear regression model Y ij = ai a i x i + E ij, (2.1.1) i=1,2r j=lf ... N, and x In this model al. 1 < x2 < ... < xN' a 2' al, and a2 are unknown regression parameters, the x's are known regression constants, and the E's are mutually independent, unobservable random errors. We are interested here in making inference about the slope difference, 11 12 61_2 = a1 Since the regression constants are assumed the same for both lines, we suggest the use of the SieversScholz approach introduced in Section 1.2 when discussing techniques appropriate in the simple linear regression setting, now applied to the differences Zj = Y  Y2j' j=I,...,N. A statistic equivalent to the one considered by Sievers and Scholz has the representation T*(b) = (1/N)ZZ a sgn(Z Z bx ), (2.1.2) rsrs s r rs r where Xrs = s Xr, sgn (u) = 1,0,1 as u is less than, equal to, or greater than zero, and ars > 0 are arbitrary weights with ars =0 ifxr x Note that under (2.1.1), the differences Z., j=1,...,N, follow the linear model Z= 12 + S12xj + EI2,j' (2.1.3) where aI2 = a1 a2' 812 = 81 2' and El_2,j = E E2j, j=1,...,N. In computing the differences Z., j=1,...,N, ] we have reduced the two line case to the simple linear regression case, enabling us to apply the approach of Sievers and Scholz to the two line case. Of course the assumption of common regression constants is crucial to this reduction. Let us now motivate the form of T*(b) by first discussing a special case of (2.1.2) due to Theil and Sen. 13 Let Z.(b) = Z. bx. and consider the pairs (x., Z.(b)), j=l,...,N. Under H0: 12 = b, the value of x. has no effect on the value of Z (b) = a12 + E2,j However under H1: 12 > b, a larger x will tend to result in a relatively larger observed Z. (b) = a12 + (8 12b)x + E12,j. Thus a method of constructing a statistic for testing H0: 812 = b is to employ some measure of association between the x. and the Z.(b), j=1,...,N. Theil and Sen used this approach, selecting Kendall's tau as the measure of association: ZZ sgn(x rs)sgn(Z (b)Z (b)) r~s rs s r r [CZ sgn(xtu)ZE sgn(Z w(b)Z (b))] t EZ sgn(x rs)sgn(Z Z bxrs r~s rs s r rs r 2 Here N is the number of positive differences x Xr, 1 < r < s < N (keep in mind x < x < ...< xN), (N) = N(N1)/2, and we assume no ties among the Z's. The statistic UN(b) is Kendall's tau between the x and Z (b), j=1,...,N, for each fixed value of b. Let S =rs (Z Z )/x 1 < r < s < N, denote the slope rs s r rs of the line connecting the observed differences Z and Z. r s Since Z Zr bxrs > 0 if and only if S > b, we see that ic s Zr brs rs U I(b) is a function of these slopes. The numerator of UN (b) is equal to the difference between the number of slopes, 14 Srs, that exceed b and the number of these slopes which are less than b. * Comparing UN(b) given by (2.1.4) with T (b) given by (2.1.2), we see the SieversScholz statistic is an extension of the statistic due to Theil and Sen obtained by replacing sgn(x ) with a general weighting function ar. This allows rs 0rs' the slopes determined from points which are farther apart to be given more weight than the slopes determined from closer points. Under fairly general regularity conditions, Sievers has shown the highest efficacy of a test based on his statistic is attained when the slopes Srs are given weights a rs= xs xr x rs. Rewriting T (b) using the optimal weights, a = x we define weiht, rs rs T(b) = (I/N)ZZ x rsSgn(Z sZ rbx rs). (2.1.5) r In the remainder of this chapter, we explore the appropriateness of T(b) for inference about s We have seen the SieversScholz statistic is a generalization of the statistic due to Theil and Sen. There is a similar relationship between HodgesLehmann estimates of the slope parameter associated with the TheilSen and the SieversScholz statistics. A HodgesLehmann estimate of $ 12 based on the TheilSen statistic can be shown to be equal to the median of the set of slopes {Srs: 1 < r < s < N, xr x s} (Sen, 1968). The corresponding HodgesLehmann estimate associated with the 15 SieversScholz statistic T(b) is a generalization of the TheilSen estimate and can be viewed (Sievers, 1978) as a median of the distribution of a random variable V, where P{V=v} = x rs/x.. if V=Srs, (2.1.6) X.. =tZuxtu,Z 1 < t < u < N, xt 3 xu. Thus the Sieverst Scholz estimate is a weighted median of the slope estimates {S rs: 1 < r < s < N, xr 7 xsi. In the next section we briefly summarize Sievers' results concerning the asymptotic distribution of the SieversScholz test statistic and estimate, now applied to the two regression line setting. Using these results, we describe large sample inference of $12 in Section 2.3. Pitman asymptotic relative efficiencies (PAREs) of the SieversScholz procedure with respect to other nonparametric approaches as well as the least squares procedure are given in Section 2.4. These PAREs are derived assuming equally spaced regression constants. In Section 2.5 we propose two exact tests of H0: a12 = 0 which are easily implemented for small sample sizes. Finally, we close Chapter Two with a Monte Carlo study in Section 2.6. The first part of this study concentrates on comparisons of the SieversScholz asymptotic procedure with others under moderately large samples, while the second deals with exact tests and small samples. Because PARE's are available only when the x's are equally spaced, these Monte Carlo simulations emphasize 16 comparisons of the various test procedures under unequally spaced regression constants. 2.2 Asymptotic Distributions In this section we present some important results concerning the asymptotic distributions of the SieversScholz statistic and estimate. All of these results follow from straightforward modifications of Sievers' (1978) results for the simple linear regression setting. Sievers presents his theorems without proofs, giving reference primarily to the text by Hajek and Sidak (1967). We repeat these results, now applying them to the two line setting. We indicate references to proofs or supply the basic steps, since some of the results in this section will be needed in Chapter Three when considering several regression lines. As mentioned in the previous section, we are assuming the optimal weights, a = x5 ,r r < s, and thus we state rs r the results for T(b) in (2.1.5). Note that under these optimal weights, T(b) can be expressed in terms of the ranks of the differences Z.(b), j=1,...,N, as follows: N* T(b) =(2/N) Z [Rank(Z.(b))x. (N+1)x, (2.2.1) j=1 where Rank (Z.(b)) is the rank of Z.(b) amoncr {Zr (b): N r = 1,.. .,N} and x Z x.IN. Hence, T(b) is a linear rank j=1 j 17 statistic and distributional theory for this class of statistics applies to T(b). For example, assuming some regularity conditions, asymptotic normality of T(O) under H0: 81_2 = 0 is immediate (see Theorem 2.2.1). Basic notation. We now give some notation that will be maintained throughout Chapters Two and Three. Let 2 (xx) 2N, ax E j x )lN (2.2.2) j=1 a2 Z(x 2/3 2Na 2/3 (2.2.3) T,A N si x.= (X X.) (2.2.4) s j=1 S N X = E (xxr) and so (2.2.5) jr+l x.. x.. = N(xjx) and (2.2.6) 2 2 N 2 ZE x = EE (XsXr) N Z (x x) (2.2.7) r by simple algebra. For ease in notation, we assume the underlying error distributions of the two lines are the same, with cumulative distribution function (cdf) F and probability density function (pdf) f. Let E., j=1,2,..., 18 designate independent and identically distributed random variables with cdf F and pdf f; G and g will denote, respectively, the cdf and pdf of E1 E The cdf and pdf of E1 E2 E3 + E4 will be denoted by H and h, respectively. For general pdf q, let I(q) = fq2(x)dx, provided the integral is finite. The normal distribution 2 with mean p and variance a will be designated by the notation N(pa,o 2). We adopt the O(.), o(.), O (.), o (); and u notations as in Serfling (1980, p. 1, P p 89). Let P indicate convergence in probability and d indicate convergence in distribution. Finally, let a(bc) be notation for the interval (JabI acj, jabl + lacl). Conditions. The following conditions are used in the statements of the theorems of this section: N 2 2 I. Z (x _X)2 Na2 as N + j=1 x 2 2 II. max (x.x) max (xjx) I jN I 1j. N = < 3 1 Z(xjx) x j=1 III. G is continuous. IV. G has a square integrable absolutely continuous density g. Condition II, the familiar Noether (1949) condition for repression constants, ensures that no individual regression constant dominates the others in size as N . 19 Theorem 2.2.1: Under H0: 12 = b and condition III, 2 the exact variance of T(b) is [(1/3)+(1/N)]Nax If x H0 12 = b and conditions I, II, and III hold, then T(b)/a d N(0,1) (2.2.8) T,A + as N + =. Proof: From the definition of the Z.'s (2.1.3), J Z Zr bxrs = E 2,s E12,r + (01_2b)Xrs*. (2.2.9) Clearly then, the distribution of T(b) (2.1.5) when $12 = b is the same as the distribution of T(0) when 812 = 0. So assuming 812 = 0, we derive the variance of T(0). Var[T(0)] = Var[(i/N) rZ x sgn (E2,s E 2,r)] 2 1N){~x Var [sgn (2~ E12 r)1 (I/2){77 rsrsI, 1 2, , rr r + 2 ZE x x Cov[sgn(E12E 12), sgn (E 12sE 2,)] + 2 ZEE x x Cov[sgn(E 2sE 2), sgn(E 2 E 2 )] r r rs su 12,s 1s2,r 12,u 10) =(1/N 2) {c 1 + c 2 + c 3 + c 41, (2.2.10) 20 where cl, c2, c3, and c4 denote the 4 terms in this variance expression. Since E12,s E12,r is symmetric about zero, it follows that E[sgn(E12,s E12,r)] = 0. Also since E12,s is continuous (condition III) and consequently E12,s E12,r is continuous, it follows that sgn2 (E12,s E 12,r) = 1 with probability one. Thus 2 c = ZE x2 Var[sgn (E E r)] 1 r = zEE x2 E[sgn2 (E E r)] r Z= X rs (2.2.11) r To simplify c2 we first note Coy [sgn (E12,sE 12,r), sgn (E12,uE12,r)] = E[sgn (E12,sE12,r) sgn (E12,uE12,r)] = [P {E12,s E12,r > 0, E12,u E12,r > 0} + P {E12,s E12,r < 0, E 2,u E2 < 0}] [P {E12,s E12,r > 0, E12,u E12,r < 01 + P {E12,s E12,r < 0, Ei2,u E12,r > 0] = [ f[1F(e)] 2dF(e) + IF2 (e)dF(e)] [ I [1F(e)]F(e)dF(e) + IF (e)[1F(e) ] dF (e)] = [(1/3) + (1/3)] [2(1/6)1 21 = 1/3, (2.2.12) and so it follows that c2 = 2 ZEE x rsx Cov[sgn(E 2,sE 2,r),sgn(E 2,uE 2,r)] 2 ~ urs ru 12,s 12,r~ 12,ul12,r r = (1/3) (2 EZ xxrs ru) r = (1/3) (Ex2 ) (using (2.2.5)). (2.2.13) r. r Similar basic manipulations yield the following simplifications of c3 and c4: 2 c = (1/3) ( x ), (2.2.14) r c = (1/3) (2 E Xr. x ). (2.2.15) 4 rr. r r Substituting these simplifications into (2.2.10) we have Var [T(0)] = (1/N2) { z x2 rs r 2 2 + (1/3) Z (x + x 2 2xx )} r. .r r. .r r 2 2 2 = (1/N ) {zz x + (1/3) Z (x x ) }. (2.2.16) rs r. .r r Using (2.2.6) and (2.2.7) in (2.2.16) yields 22 2 N 22 Var[T(0)] (1/N ){N E (x.i) + (113)N E (xx) j=1 ( j=1 x N (xjx)2 {(1/N) + (1/3)} j=l = [(1/3) + (1/N)] Na2, (2.2.17) verifying the expression for the exact variance of T(b) under H0: a12 = b. If we take k=2 in Theorem 3.2.3 then this theorem states that under H0: 0 12 = 0, assuming conditions I, II, and III T(O)/a d N(0,1) (2.2.18) T0/T,A as N + From the remarks at the beginning of this proof, T(b)/oT,A has the same limiting distribution under H0: 12 = b, and so the proof is complete. Note that a2,A is the asymptotic variance of T(b) under H0: a12 = b. Theorem 2.2.2: Assume a sequence of alternatives HN: a12 = a1_2(N) = /(N a x) to the null hypothesis H0: a12 = 0, where w is a constant. Then under conditions IIV, T() u d N(2 (3 2)WI(g) ,i) (2.2.19) T 0/T,A + as N 23 Proof: A detailed argument is given, since results obtained here are used in the proof of Theorem 3.2.4 in Chapter Three. Let w N = m/(N o x). From (2.2.9) and the definition of T(b) (2.1.5), it is clear that the distribution of T(0) under 812 = N is the same as the distribution of T( N) under 812 = 0. Therefore the proof is complete if we assume 812 = 0 and show that T( N) has the desired limiting distribution. We first state and prove two lemmas. Lemma 2.2.3: Assume 812 = 0 and conditions IIV. Then E[T(wN)/oTA] + 2(32)wI(g) (2.2.20) as N + . Proof of lemma: E[T( N)] = (1/N)E x rsE[sgn(Z Z + N x rs)] N rsrs s r N rs r Now E[sgn(Z sZr +WN_)] = 1 2fG(Z.NXrs)dG(z). (2.2.21) 24 Conditions III and IV allow us to write a Taylor's series expansion of G(zwNLxrs) : G(zw Nx rs) = G(z)Nxrsg(z rs(z)), where 0 < 0 rs (Z) < Nx rs. Since max Ix.xl INxrs = l(Iw(xsXr)/(Nox)l < 2 j l N N o x it follows from condition II that 0 rs(z) + 0 uniformly in r and s as N + m. That is, for all E > 0, there exists N(e) such that N > N(s) => IwNxrsi < e for all r < s. (2.2.22) Then by absolute continuity of g (condition IV), for all 6 > 0, there exists N (6) such that N > N (6) => g(z rs(z)) E (g(z)6), (2.2.23) for all z. Since g(.) and 6 are nonnegative and g(*) is square integrable (condition IV), multiplying by g(z) and integrating we see N > N (6) => fg(ze0 rs(z))g(z)dz (I(g)6). (2.2.24) Substituting the Taylor's series expansion into (2.2.21) we have E[sgn(Z sZr +wx )] = 2N x rsg(z0rs (z))g(z)dz, 25 and so E[T(wmN)/T,A] = (2w N/(NTA ))Z X2rsg(zrs (z))g(z)dz. r Assume w > 0. Using (2.2.24) in the above representation we see N > N (6) => ~ 2 E[T(wN)/CT,A] (2N/(NT,A))ZE x2rs (I()6) r N S(2w N/(NcT,A))(N Z (x jx) )(I(g)) (from (2.2.7)) j=1 2(3h)m(I(g)+6), and so, N > N (6/(2(3 ))) => E[T(wN)/cT,A] (2(32 )WI(g)6), and (2.2.20) is proved for w > 0. The proof is similar for for w < 0. Lemma 2.2.4: Assume B12 = 0 and conditions I, II, and III. Then T(N) [T(0)+E(T(uN p 0 (2.2.25) aT ,A 26 asN + . Proof of lemma: Since under 612 = 0, E[T(wN) T(0)] = E[T(w ), we prove (2.2.25) by showing Var[(T(wN)T(0))/aT,A + 0 (2.2.26) 1 as N + m. First, T(wN) T(0) = EE x H (w N N rs rs N r where H (w ) = sgn(Z Z + x ) sgn(ZsZ r). rs N s r N rs) ssg(Z r Assuming w > 0, E 2 Nx < Z Z <0 Hrs(WN) wNXrs s r 0 otherwise. We will suppress the argument wN of Hrs(W N) Then, since IwNXrsI + 0 uniformly in r and s (see (2.2.22)) and the cdf of Z Z is continuous (condition III), it follows that s r 2 E[H ] = 4P{w x N + . Similarly, E[H H ,] + 0 uniformly in r, s, r', rs r s s' as N . So Var(Hrs), Cov(Hrs,Hru), and other terms of that form are all uniformly o(1) as N + =. Now Var[ (T(wN)T(0))/cT,A] 2 2 1 = (N2 2) Var[ZZ x H ] TA r s rs 27 (N 2 a 2 1{z 2 Var H T,A X~srs rs r~r~s 2 ZE x rsx sCov (H ,sH s r + 2 ZZE sxs X Cv(H sHu r =(N 2Ta) (c +c +c3 +c4) (2.2.27) where c .. c2' c3 and c4 denote the 4 terms in this variance expression. Since r < s < u implies 0 < x rs< xru' it follows that Icl< 2 ZEE xrs x Ju Cov(H Hr) c21 r < ~ r CVHrs ,H) M ~ (ur)x 2Cov(HH~j r Now (ur1) < N for all 1 < r < u < N, and so cj< 2NEEx 2ICov(H iHru)1 28 The two similar terms c3 and c4 in (2.2.27) can also be bounded in this way. Combining these bounds with the fact that all variance and covariance terms in (2.2.27) are uniformly o(l) we find Var[((TwN)T(0))/oT,A] < BN, 2 2 2 where BN = (o(1)/N a2 )(6N+1)EZ x T,A r 3 2 2 2 = (30(1)/N ox)(6N+I)N ax = o(1), as N + , and hence (2.2.26) is proved for w > 0. The proof is similar for w < 0. Proof of Theorem 2.2.2 (continued): We see by Lemma 2.2.3, Theorem 2.2.1, and Slutzky's theorem (Randles and Wolfe, 1979, p. 424) that under H0: 12 = 0, T(0) + E((T(mN)) d N(2(3)WI(g),1) 0 +N(2 (3 )wlI(g) ,1) aT,A as N + =. Using this result, Lemma 2.2.4 and another application of Slutzky's theorem show that under H0: 812 = 0, 29 T(wN)/TA d N(2(3 )cI(g) ,1) N (N/T,A + as N + =. Then by our remarks at the beginning of the proof of Theorem 2.2.2, we are done. Consider the HodgesLehmann estimate of 812, say 812' associated with the statistic T(b). If we let ^U ^L 812 = sup{8: T(8) > 0} and 812 = inf{8: T(8) < 0}, then ^ ^U ^L we may define 812 = (81 2+81_2)/2. We now give a theorem concerning the asymptotic distribution of 812. Theorem 2.2.5: Under conditions IIV, No(i28 12 N(0,(12I (g))) (2.2.28) x(1212) + as N + . Proof: Let N = w/(Nk x) where w is a constant. Let P {*} denote the probability calculated when 812 = 8, and let 0 designate the standard normal cdf. From Theorem 2.2.2 it follows that lim P {T(0)<0} N *m N = lim P {(T(0)/a ) 2(3 )w I(g) < 2(3 )wI(g)} N N  30 = t (2(3 )owl(g)), or equivalently, lim P {T(0)<0} = D(2(3 )WI(g)). (2.2.29) N+= N Byth dfniio o L ^U By the definition of 1_2 and 12 and using the fact that T(b) is nonincreasing in b, we have ^L ^L 12 > b => T(b) > 0 => 812 > b, (2.2.30) U_ < b => T(b) < 0 => U < b. (2.2.31) 12 12 Since by condition III the underlying distribution is continuous, it can be shown (using a proof similar to that of Theorem 6.1 of Hodges and Lehmann (1963)) that the ^ L ^ U distributions of 012 and S12 are continuous. Thus (2.2.30) and (2.2.31) imply that ^L P { 12 12. 12 U Pp _1 12 ^ ^L ^U ^L <^U ,i olw Since 812 = (1/2) (12 + 12) and 2' it follows that P {1U 12 12 12 12 12 ~ 12 12 31 and then substituting (2.2.32) and (2.2.33) into this result we have P {T(b)<} < P {812 8 8 12 P12{Tb<} Because the distribution of T(a) when 812 = 8 is the same as the distribution of T(0) when 812 = 0 (see proof of Theorem 2.2.1) it can be shown using the definition of 812 that the distribution of (81212) under 812 is the same A as the distribution of 812 when %12 = 0. Since the distribution of 812 is continuous and the limiting distribution in (2.2.29) is continuous, we apply (2.2.34) to obtain lim P {N a (8 ) < W} N2 x 1212 N+ 12 = lim P0 12 N} 0 x11N+= = lim P0{812 < w/ (N )} N+. = lim P0{T(w/(N x )) < 0} (from (2.2.34)) N+@ = lim P 0{T(w N) < 0} N+o 32 = lim P {T(0) < 0} N+w wN = D (2(32) wI(g)) (from (2.2.29)), (2.2.35) and the result (2.2.28) follows immediately. 2.3 Large Sample Inference Using the results of the previous section, we now proceed to construct tests and confidence intervals for a1_2 These tests and confidence intervals follow from those presented by Sievers, now applied to the two line setting. Consider a test of H0:aI2 = 0 against H 1 12 >0 based on the statistic T(0). Of course to test for parallelism of the regression lines we take 0 = 0. Large values of T(%0) indicate the alternative H1 holds. For a given 0 < a < 1, let z denote the 100(1a) percentile of the N(0,1) distribution. Then from Theorem 2.2.1, the test which rejects H0 if T( 0 )/,T,A > z is an approximate level a test. A twosided test of H0: a12 = 0 is derived by making the usual modifications. Since T(b) (2.1.5) is a nonincreasing step function of b with jumps at the slopes Srs (Z z )/xri 1 < r < s < N, rs S. r s a confidence interval for 12 can be constructed by 33 inverting the twosided hypothesis test. This interval has endpoints that are properly chosen values from the set of slopes {S : 1 < r < s < N}. rs Let J(s) be the cumulative distribution function of a discrete random variable that takes the value rs with rs probability x rs/x.., where x.. = Z x rs. Then for each b rs rsrs r such that b # Srs, 1 < r < s < N, J(b) = (1/2)[l(NT(b)/x..)]. (2.3.1) If the distribution of the Z's is continuous, then P{Srs = b for some 1 1 a = P{ (l(Nt2/x..)) < J(B) < (l(Ntl/x..))} = P{J 1( (1(Nt2/x..))) < < J ((1(Ntl/x..)))}, (2.3.2) 1 1 where J (u) = inf {s: J(s)>u} and J_ (u) = inf {s: J(s)>u} are inverses of J defined appropriately for our purpose. Thus [J1 ( (1(Nt2/x..))), J ( (1(Nt /x..)))) (2.3.3) 34 is a 100(1a) percent confidence interval for 812* If we use the asymptotic normality of T() in Theorem 2.2.1 to determine t1 and t2 we find that 1x)) 1 [jl ( z 2(NT/2x..)), J ( +z /2(NoTA/2x..))) + a/2 T,A + a/2 T,A (2.3.4) is an approximate 100(1a) percent confidence interval for 812. We can write this interval more explicitly in terms of the slopes Srs as [SL,SU), where L S = min{S : J(S rs)> z /2(NaTA /2x..), 1 (2.3.5) and SU = min{S rs J(S )+z (NaT /2x..), l (2.3.6) An alternative confidence interval follows directly from the asymptotic normality of 812 (Theorem 2.2.5). An approximate 100(1a) percent confidence interval for 812 is given by 812 z /2(2(3 N )a i(g)) , where I(g) is a consistent estimate of I(g). Such estimates have been proposed by Lehmann (1963) and Sen (1966). 35 2.4 Asymptotic Relative Efficiencies In thi4S section we compare the SieversScholz approach for testing H0: 12 =_ b with the procedure due to Hollander and the procedure due to Rao and Gore. As noted in Chapter One, the Hollander and RaoGore procedures include exact confidence intervals for the slope difference a 12* We will see in the next section that our proposed application of the SieversScholz approach also includes exact confidence intervals for a 12 Hence these three exact, nonparametric procedures are the primary focus of the asymptotic relative efficiency comparisons presented here. Comparisons of the SieversScholz test with the classical ttest based on least squares theory and Sen's (1969) test are also made. First we show how to construct each of the competing nonparametric. test statistics, along with a brief illustration of the rationale behind each one. We then describe a sequence of sample sizes tending to infinity for the purpose of computing the Pitman asymptotic relative efficiencies (PAREs). Finally', we compute these efficiencies and compare their values assuming several different underlyin g error distributions. Since all three exact, nonparametric procedures can be expressed in terms of basic slope estimates for each line, we first define S.is (Y. iY.i )/x rs, x r< x S, (2.4.1) 36 the estimate of the slope of line i (i=1,2) resulting from the responses at xr and x s. Recall that in the previous sections we used a similar notation, Srs = (ZsZr)/xrs, to designate slope (difference) estimates computed from the differences Z., j=1,...,N. Of course S rs rs' and the additional subscript indicating line 1 or 2 in the estimates in (2.4.1) should help to avoid any confusion. There will be (4) = N(N1)/2 of these estimates associated with each line. The slope estimates of line 1 are naturally independent of those of line 2, but the () slope estimates of a single line are not mutually independent. One way of motivating and comparing the three exact, nonparametric procedures is to examine how they utilize these basic slope estimates in forming their test statistics. The SieversScholz Statistic In the two line setting, the SieversScholz statistic is appropriate only when the lines have common regression constants. As before, let x 1 x2< ... xN denote the regression constants. We can write the SieversScholz statistic T(O) in terms of the slope estimates: 1 sgn(S ) N0= rs irs 2rs. r 37 Examining this representation we see that each line 1 slope estimate iS compared with the line 2 slope estimate resulting from the observations at the same regression constants. Thus all ( N) slope estimates of each line are used, but an estimate from line 1 is compared only with the corresponding estimate from line 2. This results in(N comparisons across lines. Each comparison is weighted by the distance between the x's used in its construction. The Hollander Statistic Unlike the SieversScholz statistic, the Hollander statistic is applicable even when the lines do not share common regression constants. However, use of the Hollander statistic requires a grouping scheme designating N/2 pairs of regression constants for each line. Assume N = 2k. When the x's are equally spaced, that is, when the regression constants for line i are x. ~ L + mc.,l m = 0, 1, ...1 2ki, (2.4.2) i = 1, 2, for some constants L1. L 21 cl, and c2 Hollander's grouping scheme pairs xrm with xjm+kl m = 1, ..., k, (2.4.3) i = 1, 2. The first step in Hollander's procedure is to utilize the observations at each pair of xvalues to 38 construct N/2 independent slope estimates of the form (2.4.1) for each line. Hollander notes that under equal. spacing (2.4.2) his grouping scheme (2.4.3) minimizes the variance of the slope estimates used among all grouping schemes that produce identically distributed slope estimates. Under unequal spacing of the regression constants, some ambiguity exists as to the choice of a grouping scheme. Hollander suggests devising a scheme that will yield pairs of xvalues situated approximately a constant distance apart. Having computed N/2 = k independent estimates of the slope of line 1 and k independent estimates of the slope of line 2, the next step in Hollander's procedure is to randomly pair the slope estimates of line 1 with those of line 2. Let (Silrs' S 2tu ) designate one of the k pairs. For each pair the difference of the form Sir 5t is calculated. If we label these differences dl,..., d then Hollander's statistic is computed as k W= Z XR 6(d), (2.4.4) m=1 l where R m is the rank of Id M I among {1d mf: m=1,...,k}, and 6(a) = 1 if a > 0, 0 otherwise; W is the Wilcoxon signed rank statistic computed using the k slope difference estimates as the observations. Writing a slope difference 39 estimate d min terms of the underlying parameters, we see d m has the form. l a2+ i s i r E2u xE2t, where x irs xis x. i,i 1, 2. Since Eir and E ls are independent and identically distributed, it follows that the disribtio ofEis, Eir issymmetric about zero. Clearly then, i s i r E 2u, 2t x lrsX 2tu is symmetrically distributed about zero. Thus Hollander' s approach does not require the same regression constants or equal spacing since the Wilcoxon distribution will apply under $12 = 0 regardless of the spacing or choice of regression constants. However, if the regression constants are the same and equally spaced, the asymptotic relative efficiency (ARE) to be presented in this section as Result 2.4.2 indicates superiority of the SieversScholz approach. Note that Hollander' s approach does not use all available ( basic slope estimates from each line. Instead a subset of N/2 independent line 1 slope estimates is selected from the ( N) possible. A similar set is selected from the line 2 slope estimates. Each member of the first 40 set is compared with only one randomly selected member of the second set, resulting in only N12 comparisons of slope estimates across lines. This is a smaller number of comparisons across lines than the ( N) comparisons of the SieversScholz method. Although there are dependencies among the SieversScholz comparisons, the greater number of them would lead us to expect the SieversScholz approach to be superior to Hollander's method when both are applicable. Again, the ARE Result 2.4.2 will confirm this. The RaoGore Statistic Let A(B) designate the set of N/2 slope estimates for line 1(2) used in constructing Hollander's statistic. Note that under equal spacing (2.4.2) with c 1 = c, = c and using Hollander' s recommended grouping scheme (2.4.3), the distribution of the slope estimates in A differ from those in B only by a shift in location of S12 2a a S +E 1S E 1 51rs 1 kc S a+ E2u E2t S2tu= 2 + kc Here xi.~ x. =' kc is the common distance between pairs of x's used in forming the slope estimates. Thus, under equal spacing, Rao and Gore proposed the MannWhitneyWilcoxon statistic of the form 41 U = S Z A S ZrB} Y (SlrsIS2tu)' (2.4.5) 1rs' 2tu where y(ab) = 1 if a > b, 0 otherwise. It is clear that the RaoGore procedure would apply under any spacing and grouping scheme resulting in the same distance between all pairs of x's used to form slope estimates. Under such spacing, the RaoGore procedure eliminates the extraneous randomization needed by Hollander's procedure to pair the slope estimates of the two lines. We see the RaoGore procedure uses the same two sets of slope estimates as Hollander's approach. However all possible (N/2) 2 comparisons across lines are made. This leads us to expect that the RaoGore procedure will compete favorably with Hollander's method. Comparing the RaoGore technique with the SieversScholz approach in the previous intuitive way is not as revealing. The RaoGore technique makes all possible comparisons across lines among two relatively small sets of independent slope estimates. The SieversScholz approach makes only pairwise comparisons across lines, but uses all possible slope estimates. The sets of slope comparisons used in the two procedures are such that neither set is a subset of the other. We will see, however, that the ARE Result 2.4.3 indicates superiority of the SieversScholz approach. 42 The Sen (1969) Statistic Sen (1969) proposed a statistic for testing the parallelism of several regression lines. We introduce his statistic here in the two line setting. Note that Sen's statistic does not require common regression constants for the two lines as required by the SieversScholz statistic. However we will assume common regression constants to ease the notation. Modifications needed to construct Sen's statistic in the more general case of different x's for the two lines will be obvious. Thus we assume the basic linear model (2.1.1). Let q (u) be an absolutely continuous and nondecreasing function of u: 0 < u < 1, and assume that Oiu) is square integrable over (0,1). Let U(1 < U(2) < '''N be the order statistics of a sample of size N from a uniform [0,1] distribution. Then define the following scores: E E[OIU (j)) (2.4.6) or E. = (j/N+1), (2.4.7) j=1, ..,N. Define 1 =f (u)du (2.4.8) 0 and 43 A 2 2 (u) du 2 4* (2.4.9) 0 and consider the statistics (. j x) ]/(AN a X), (2.4.10) 1j=1 Jj i=1, 2, where R jis the rank of Y among Y il Yi~ YiN' the observations of the ith line. Statistics such as (2.4.10) are used in the single line setting to test hypotheses about the slope. The function Oiu) is called a score function. It is a function applied to the observed ranks of the Y's. The choice of 0 can be made to achieve desirable power properties of a test based on (2.4.10). These properties depend in part on the underlying error distribution. A general discussion of score functions is beyond the scope of this work. We will discuss the score function here only to the extent needed to clearly present Sen's statistic. Assume F, the underlying error cdf of the assumed model (2.1.1), is absolutely continuous and IF) = f [(x)]I2dF (x) < ~.(2.4.11) fx) 44 Then we reserve the symbol T (u) for the following score function: T(u) = f' (F1 (u)) 0 < u < 1 (2.4.12) f(FI (u)) 1. It can be shown, 1 f '(u)du = 0, and 0 12 T 2 (u)du = I*(F). 0 The score function T+(u) has been shown to have certain desirable properties when applied to the twosample location problem (Randles and Wolfe, 1979, p. 299). We also define 12 p(Tf) = [f T(u) (u)duj/[A 2I*(F)] (2.4.13) 0 which can be regarded as a measure of the correlation between the chosen score function c and the optimal one (T) for the error distribution being considered. The expressions p(',f) (2.4.10) and I*(F) (2.4.11) will appear in the development of the statistic based on Sen's (1969) work. 45 We now define Sen's statistic. Let V. (Y.+bx) (2.4.14) 1 1J denote the value of Vi (2.4.10) based on Yi + bxl, Yi2 + bx2' ". YiN + bxN. Define V = (V1 + V2)/2. (2.4.15) Assuming H0: 812 = 0, let 6* denote the HodgesLehmann estimate of the common slope of the two lines based on V. Define V. = Vi (Yi8*x), (2.4.16) 1 . i = 1, 2. Then Sen proposed the statistic ~ 2 L= V2 (2.4.17) i=1 1 to test H0: 12 = 0 against H1: 812 # 0. The statistic L is a quadratic form in the Vi, i = 1, 2. We now give an intuitive motivation for the form of L. The statistic V. is the value of V. based on the 1 1 observations Y 8* 8*x ... *x . il 1 i2 2 YiN N 46 Under H 0: a12 =_ 0, the 2 lines are parallel and is an appropriate estimator of the corrarnon slope. In that case, the transformed observations behave essentially as random errors fluctuating about zero. Then L, being the sum of squared random errors with mean zero, has an asymptotic central chisquared distribution, and a test of H 0 may be based on L using this asymptotic null distribution. Under H 1: a 12 A 0, the estimate is not appropriate since the two slopes are not the same. The transformed observations will not, in general, have mean zero. Hence the value of L will be larger than expected under the null, and the use of the null, central chisquared critical values will tend to lead to rejection of H 0. PARE Specifics: Alternatives, Regression Constants,, Sequence of Sample Sizes In computing the PAREs, we assume a sequence of alternatives to the null hypothesis H 0 12 = 0 specified by HN:~ = (1) = w/hc as in Theorem 2.2.2. However, following Hollander (1970), we consider only equally spaced common regression constants resulting from setting c 1 = c 2 = c, L 1= L 2= L in (2.4.2), with n observations per line at each of the 2k xvalues. Clearly then, N = 2kn, a 2= c 2(4k _1)/12, and x H N: 12(N) 12 (2kn) = [w! (2kn) ]23_/(k2 ) (2.4 .18) 47 The PAREs of the test procedures will be derived under two different schemes for allowing the sample size N = 2kn to tend to infinity: Case 1: Let C be a positive finite constant and let c = ck such that kck C as k Consider n fixed, k ).c Case 2: Let c be constant. Consider k fixed, n , resulting in a PARE in terms of k. Then let k in this expression. Case 1 essentially allows the number of distinct regression constants (2k) to tend to infinity over a fixed region of experimentation. Case 2 considers the number of distinct regression constants fixed while the number of replicates (n) tends to infinity. The efficiency expressions derived under case 1 and case 2 will be seen to be identical. In view of (2.4.18), the rate at which the sequence of alternatives converges to the null is inversely proportional to the square root of the sample size, N = 2kn, for both case 1 and case 2. Before proceeding to derive the various PARE's, we state a theorem due to Noether (1955) as presented by Randles and Wolfe (1979, p. 147). Theorem 2.4.1 (Noether's Theorem). Let {S n(i)} and {Tn (i)l be two sequences of tests, with associated sequences of numbers {pS(n(i)) (e)}, {uIT(n, (i)) (e)}, 48 2 2 {2(n(i))()}, and {2 T(n(i))(), and satisfying the S (n (i)) {T((i)) following Assumptions AlA6: S s (e.) T (6.) Al. n(i) S(n(i)) 1and n'(i) "T(n'(i)) 1 aS(n(i)) (6 i T(n'l(i)) ( 0 have the same continuous limiting (i + ) distribution with cdf H(.) and interval support when e. is the true value of e. A2. Same assumption as in Al but with 8i replaced by e0 throughout. lia S(n(i)) (i) lim T(n'(i)) (ei) A3. 1.I 1+* 0(6 ) +. o(6 ) i aS(n(i))() 0 aT(n'(i)) 0) d' A4. d [ (6)] = (n) (6) and de S(n) S(n) d de ["T(n') (6)] = T(n') (6) are assumed to exist and be continuous in some closed interval about e = e0, with u'S(n)e0) and T(n) 0both S(n) 0o an (n')(eO) bt nonzero. S' n (i) ) i'T(n' (i)) ei) lim S(n(i)) lim T(n(i)) () A5. (6) =1 S(n(i)) 0 T(n' (i)) (O) A6. lim S(n) (0) = K and n m 2 ( s (nos(n) 0)) 49 (1 T 2l(e)) KT where K S and K T are positive constants, called the efficacies of the tests based on S n and Tn respectively.. Then the PARE of S relative to T is K2 PARE (S,T). K (2,4 .19) KT Proof: See Randles and Wolfe (1979, p. 147). Note that assuming the equally spaced regression constants described just prior to (2.4.18) it follows that conditions I and II hold under either case 1 or case 2. Hence we need only explicitly assume conditions III and IV to apply Theorem 2.2.2 which establishes the asymnpototic distribution of the SieversScholz statistic under H N* We now present the PAREs. PARE (SieversScholz, Hollander) Result 2.4.2: Assume the sequence of alternatives {H N1 (2.4.18) and the equally spaced regression constants described just prior to (2.4.18). Also assume conditions IIIV, and I(h)<. Then the PARE of the SieversScholz statistic T(O) with respect to Hollanider's W under case 1 or case 2 is 50 4 I2 (g) PARE(SieversScholz, Hollander) = [ 2 ]. (2.4.20) 2 (h) Verification of Result 2.4.2: We apply Noether's theorem. Assumptions AlA6 must hold for T(0) and W. For 2 T(0)/c2, Theorem 2.2.2 establishes Al and A2 with the standard normal limiting distribution and standardizing constants (suppressing the various subscripts and taking =612) (0) = 4knI(g)6 and 22 2 S2(6) = 8kn/[c2 (4k21)]. Then A3, A4, and A5 follow immediately from the form of these standardizing constants. Assumption A6 holds and efficacy (T(0)) = lim (0)/[(2kn) 0(0)] N+. N*co 29I(g) case 1, (2.4.21) c(4k21) 2I(g) case 2. We note that W is the Wilcoxon signed rank statistic computed using nk independent random variables with the distribution of E E E +E 12 kc 51 As in Randles and Wolfe (1979, p. 165166), the assumptions AIA6 can be validated using the equivalent statistic nk W/(2 ), in which case p(6) = (2/(nk1)) [l1H(kc(e))] + 1 fH(kc(t28))dH(kct), 2 a (8) = 1/(3kn), the limiting distribution in Al and A2 is the standard normal, and efficacy (W) = lim Ii' (0)/[(2kn) 2a(0)] N+. 6 CI(h) case 1, = (2.4.22) 6 kcI(h) case 2. Then (2.4.20) follows immediately from (2.4.19) of Noether's theorem. The first four rows of Table 1 show the value of the PARE (2.4.20) when the error distribution F is uniform, normal, double exponential, and Cauchy. The distributions are listed in order of the increasing heaviness of their tails, using the measure 1 1 F (.95) F (.50) (24.23) (2.4.23) 1 1 F (.75) F (.50) 52 Table 1. PARE(SieversScholz, Hollander) and Heaviness of Tails for Selected Error Distributions. Distribution PARE Heaviness of Tails Uniform 1.29 1.80 Normal 1.33 2.44 Double Exponential 1.48 3.32 Cauchy 2.67 6.31 CN(e,a ,5,.9) 1.71 2.82 CN(,a 2,5,.8) 1.94 4.10 CN(,a 2,5,.7) 1.98 5.16 CN(6,a 2,5,.6) 1.86 5.34 CN(e,a2 5,.5) 1.70 5.02 CN(6, 2,10,.9) 2.13 3.07 CN ( ,a ,50,.9) 2.83 3.68 CN(6,o2,100,.9) 2.96 3.92 CN(6, 2,500,.9) 3.07 4.51 CN(,a 2,1000,.9) 3.08 4.78 1 1 FI(.95) F (.50) 1) Heaviness of Tails = (.95) F (.50) 1 1 F (.75) F (.50) 2) CN(6, 2,K,T) represents the mixture of two independent normals. The first, chosen with probability T, has mean e and variance a The second, chosen with probability 2 2 1T, has mean 6 and variance K . 53 defined by Crow and Siddiqui (1967), where F [F(t)] = t. It is clear that for these four distributions, the PARE of the SieversScholz statistic with respect to Hollander's statistic increases with increasing heaviness of tails. To see whether this behavior persists for other distributions, we examined the PARE (2.4.20) for a variety of contaminated normal distributions defined as follows. Let Z = X with probability T and Z = Y with probability a2 2 2 1T, where X is N(e,a ) and Y is independently N(,K2 a ). Then we say Z is distributed as a scale contaminated compound normal which we designate by CN(O,o2,2K,T). When F is 2 the CN(e,a K,T) distribution, straightforward computations yield the following formulas for I(g) and I(h): 4 4i i 1 4 (i)T (z I(g) 1 a 4 {2, (2.4.24) i=0c [2(i(K 2_1)4)2 i=0 8 T8i(l_T)i I_ 8 o (2.4.25) T a i=0 [2 (i(K21)+8)] In addition to the four common error distributions, Table 1 shows the value of the PARE (2.4.20) for several contaminated normal error distributions along with the heaviness of the tails of these distributions. The formulas (2.4.24) and (2.4.25) were used to compute these PAREs while iterative techniques provided the heaviness of tails of the 54 various contaminated normal error distributions. Figure 1 shows a plot of the PARE (2.4.20) against the heaviness of tails for all the error distributions in Table 1. The main purpose of Table 1 and Figure 1 is to show the PARE (2.4.20) of the SieversScholz statistic T(O) to Hollander's W is greater than one over a wide range of underlying distributions. Secondarily, we notice in Figure 1 that although no exact relationship exists, distributions with heavier tails tend to show higher PAREs. Under case 2, using the efficacies in (2.4.21) and (2.4.22), it follows that the PARE of the SieversScholz statistic to Hollander's statistic before allowing k to tend to infinity is 4k 2_ 1 3k 21 2 (h) Thus we see that the PARE is increasing in k, with a minimum value of 1 2 (g)/21 2 (h) for k=1. This is likely due to the fact that Hollander's procedure uses a decreasing proportion of the available slope estimates as k . We have chosen. to present our results after allowing k for ease in interpretation and to avoid the need for a separate discussion of case 1 and case 2. PARE(SieversScholz, RaoGore) Result 2.4.3: Assume the sequence of alternatives {H NI (2.4.18) and the equally spaced regression constants Figure 1. Plot of PARE(SieversScholz, Hollander) versus heaviness of tails for selected error distributions. 1 1 FI(.95) FI(.50) 1) Heaviness of Tails = (.95) F (.50) 1 1 F (.75) F (.50) 2) U = Uniform N = Normal E = Double Exponential C = Cauchy 3) CN(,o 2,K,T) represents the mixture of 2 independent normals. The first, chosen with probability T, has mean e and variance 02. The second, chosen with probability 2 2 1T, has mean 6 and variance K . 2 2 T1 = CN(6,o ,5,.9) T2 = CN(e,2 ,5,.8) <2 = CN(e, ,50,.9) T3 = CN(6,o ,25,.7) <3 = CN(6, 2'100'.9) 2 =C (,2,0 .9 T4 = CN(O,a ,5,.6) <4 = CN(e,a ,500,.9) T5 = CN(e,a2,5,.5) <5 = CN(6,a2'1000,.9) 5 6 PARE K 3 K 2 2.5 13 14 IT TS 1.5 + E u N 0.5 + +        1 2 3 4 5 6 7 Heaviness of Tpils 57 described just prior to (2.4.18). Also assume conditions III and IV. Then the PARE of the SieversScholz statistic T(0) with respect to the RaoGore statistic U under case 1 or case 2 is PARE(SieversScholz, RaoGore) = 4/3. (2.4.26) Verification of Result 2.4.3: As in Randles and Wolfe (1979, p. 170171), the assumptions AlA6 of Noether's theorem can be validated for the statistic U + (nk(nk+1)/2), equivalent to U, in which case p(e) = (nk)2 fG(kc(t+6))dG(kct) + (nk(nk+1)/2), a 2(8) = n3k3/6, the limiting distribution in Al and A2 is the standard normal, and efficacy (U) = lim (0)/[(2kn) 2(0)] N m 3 I(g) case 1, = (2.4.27) 3 kcI(g) case 2. We showed AlA6 hold for T(0) in the previous verification and gave the efficacies of T(0) in (2.4.21). Thus (2.4.26) follows immediately from (2.4.19) of Noether's theorem. 58 PARE(SieversScholz, Classical Least Squares) Result 2.4.4: Consider the classical least squares theory ttest, as specified by Hollander (1970). Assume the sequence of alternatives {H N} (2.4.18) and the equally spaced regression constants described just prior to (2.4.18). Also assume conditions III, IV, and 2 = Var(E1)<. Then the PARE of the SieversScholz statistic T(0) with respect to the classical ttest statistic under case 1 or case 2 is PARE(SieversScholz, classical) = 24a 2[I 2(g)]. (2.4.28) Verification of Result 2.4.4: Let 1 and a2 denote the least squares estimates of a1 and 2' respectively, and let 2 s be the residual mean square error (see (3.1) of Hollander (1970, p. 389)). Then the form of the ttest statistic used to test H0: 812 = 0 under the specified equally spaced regression constants is t 2 2 (2.4.29) [ 12s2 ] nkc2 (4k 1) Usina the fact that s2 is a consistent estimate of a2 under HN and a discussion similar to that in Randles and Wolfe (1979, p. 164165), it follows that assumptions AlA6 of Noether's theorem hold for the statistic t with 59 IM 2 and (2.4.29) 12a 2 nkc2 (4k21) 2 a (e) = 1. The limiting distribution in Al and A2 is the standard normal and efficacy (t) = lim j' (0)/[(2kn) 2(0)1 N o S C21 case 1, 62 (2.4.30) 22 , c (4k2 1) case 2. 2402 AlA6 hold for T(0) as shown in the verification of Result 2.4.2 with the efficacies given by (2.4.21). Thus (2.4.28) follows immediately from (2.4.19) of Noether's theorem. The efficiency (2.4.28) is identical to the familiar PARE of the twosample WilcoxonMannWhitney test with respect to the twosample normal theory ttest when the cdf of the underlying error distribution is G. Hence, the PARE(SieversScholz, classical) > 0.864 for all G (see Hollander (1970) for a proof that in this case the inequality is strict). Also, this PARE equals 0.955 when G is normal and is greater than one for many nonnormal G. 60 PARE (SieversScholz, Sen) Result 2.4.5: Assume the sequence of alternatives {H } (2.4.18) and the equally spaced regression constants described just prior to (2.4.18). Also assume the following: 1) Conditions III and IV. 2) 1* (F) < as in (2.4.11), where F is the underlying error distribution. 3) The score function flu) is an absolutely continuous and nondecreasiag function of u, 0 < u < 1, that is square integrable over (0,1). Let p(!,c) be defined as in (2.4.13). Then the PARE of the SieversScholz statistic T(0) with respect to Sen's (1969) statistic L under case 1 or 2 is PARE(SieversScholz,Sen) = 24 1(g) (2.4.31) p ( )I*(F) Verification of Result 2.4.5: Sen's statistic L has an asymptotic (N+) chisquared distribution under HN (Sen, 1969, p. 1676). Using results from Sen's work we define a statistic, V that has an asymptotic normal distribution under H N and whose square (multiplied by a constant) is asymptotically equivalent to L under HN. Since the square of V (multiplied by a constant) has the same asymptotic distribution under H N as L, we use V in applying Noether's theorem to derive efficacy 61 expressions for Sen's test and the PARE(SieversScholz, Sen). * Let denote the HodgesLehmann estimate of i based i * on V. (2.4.10), i=1,2, and let 8 = 1/2(81 + 2). From 11 2 equations (3.10) and (3.20) in Sen (1969, p. 1673, 1675) it follows that N8 * a = N2 (i ) = O (1) and b = N 2ox (Si i) = 0 (1) 1 1 p as N+ under HN, i=1,2. Using these definitions of a and b and the notation (2.4.14) we apply Lemma 3.2 in Sen (1969, p. 1674), which is given in Chapter Three of this work as Lemma 3.4.3, yielding Vi (Y i8 x) V.i(Y i8ix) = N a )p(,)[I (F) + 0 p(1) as N+ under HN. Equation (3.22) in Sen (1969, p. 1675) states IVi(YiSx)I = o (1) 1 i 1 p 62 and so, applying this to the previous result, V. = V.i(Yi x) 1 1i = Nxi p as N+ under HN. Applying equation (3.19) in Sen (1969, p. 1675), N (8 ) = o (1), x p we have V. = N o (5.8 )p(Y,4)[I(F)] + o (1) 1 x i p as N* under H.N* It follows that 63 2 2 * 2 L p (Y,)I (F)No (i8 ) + o (1) x p i=1 2 2 * 2 x 1 2 p = ('I,p)I (F)Na (SIB2 pi as N under HN. We define the statistic * 8 * V = (N/2) x 12) (2.4.32) and note that the asymptotic distribution of 2 * 2 p (T,4)I (F)(V ) under HN is the same as that of L. From Lemma 3.4 in Sen (1969, p. 1676), * d p(T,O)[I (F)] [V (N/2) x(8162)] N(0,1) d as N+ under HN. Thus assumptions AlA6 of Noether's theorem hold for the statistic V with p(e) = [c(4k21) /(2(3 ))]a, and 2 2 ( *,)I(F) 1. 0(9) = [o (Y,6)I (F)1 . 64 The limiting distribution in Al and A2 is the standard normal and efficacy (V ) = lim (0)/[(2kn) 2 (0)] N+o [Cp(T,)(I (F)) ]/6 case 1, = (2.4.33) [c(4k21)h p(,O)(I* (F)) ]/[2(6h )] case 2. The assumptions AlA6 hold for T(0) as shown in the verification of Result 2.4.2 with the efficacies given by (2.4.21). Hence (2.4.31) follows immediately from (2.4.19) of Noether's theorem. To evaluate the expression (2.4.31), we first note the following, which results from the definition of A2 in (2.4.9) and p(T,O) in (2.4.13): 2 122 p (T,O)I*(F) = [ 'Y(u)o(u)dul 2/A 0 = [ f 1(u)o(u)du]2/[ f [(u)4*] 2du]. 0 0 (2.4.34) Suppose we assume O(u) = u, 0 < u < 1. Scores resulting from this choice of score function are called Wilcoxon 65 scores. Direct computations show that in this case 1 f [#(u) **] du = 1/12. (2.4.35) 0 Using a derivation similar to one in Randles and Wolfe (1979, p. 308) we show that when #(u) = u, the numerator of the righthand side of (2.4.34) equals 12 (f): 1 1 1 1 1 1 I Y(u)4(u)du = I u[f'(F (u))/f(F (u))]du. 0 0 (2.4.36) 1 Let t = F 1(u), resulting in 1 ((1) I '(u)o(u)du = I F(t)f'(t)dt, (2.4.37) 0 ((0) where F(E(p)) = p. Now let u = F(t), dv = f'(t)dt and apply integration by parts to (2.4.37): 1((1) f I 1 (u) (u)du = {[F(t)f(t)] E(1) f (t)dt} 0(0 0 ((0) E (1)2 = [F(((0))f(E(0))] [F(((1))f(E(1))] + I 2(t)dt. (0) (2.4.38) If we assume the support of F is [a,b] where a < b and f(x) 0 as x + b or x + a, then we take ((0) = a, E(1) = b in (2.4.38) and 66 f 1 T u) qdu) du =f bf2 (t) dt = (f). (2.4.39) 0 a As Randles and Wolfe (1979, p. 313) state, this same form (2.4.39) can be obtained under more general assumptions. Hence substituting (2.4.35) and (2.4.39) into (2.4.34) it follows from (2.4.31) that under 41u) = u, H N, and assumed regularity conditions, PARE(SieversScholz, Sen) =21 (g) (2.4.40) 12W Table 2 gives values of the PARE (2.4.40) for four common error distributions. We see that when compared to the Sen statistic using Wilcoxon scores, the SieversScholz statistic achieves a PARE close to (or equalling) one for error distributions having light to moderately heavy tails (uniform, normal, double exponential). However the SieversScholz statistic has poor PARE under the Cauchy distribution which has very heavy tails. That the asymptotic performance of Sen's test is better than other tests is not surprising since Sen's test is the rank test that maximizes the efficiency relative to the likelihood ratio test. However, Sen's test not only requires iterative calculations, but is distributionfree only asymptotically. More discussion of the relative merits of Sen's test and the SieversScholz testd will be given in Section 2.6. 67 Table 2. PARE(SieversScholz, Sen) for Selected Error Distributions Assuming (u) = u. Distribution PARE (SieversScholz, Sen) Uniform 0.89 Normal 1.00 Double Exponential 0.78 Cauchy 0.50 Summary of PARE Results In conclusion, PAREs derived under equally spaced regression constants favor the SieversScholz approach over the other two exact, nonparametric competitors due to Hollander and Rao and Gore. Specifically, the PARE of the SieversScholz statistic with respect to Hollander's statistic is greater than one over a wide range of underlying error distributions. Indeed this efficiency frequently exceeds two. Even more interesting is the fact that the PARE of the SieversScholz statistic with respect to the RaoGore statistic is 4/3 for all error distributions (subject to certain regularity conditions required to derive the PARE). The SieversScholz statistic achieves the familiar PARE (2.4.28) when compared with the classical least squares theory ttest. Although the PARE of the SieversScholz statistic with respect to Sen's statistic is 68 less favorable under the heavily tailed Cauchy distribution, we noted that Sen's test is distributionfree only asymptotically. We discuss these two methods further in Section 2.6. Note that under equal spacing, the PARE of the SieversScholz statistic to the TheilSen statistic (2.1.4) discussed in Section 2.1 is one (Sievers, 1978). Thus all of the previous ARE comparisons also apply when using the (zeroone weighted) TheilSen statistic in place of the SieversScholz statistic. The advantage of the SieversScholz approach over that of Theil and Sen will appear in the Monte Carlo comparisons under unequal spacing of the regression constants discussed in Section 2.6. 2.5 Small Sample Inference Since the test of parallelism and the confidence intervals for a 12 presented in Section 2.3 depend on asymptotic theory, they are generally only applicable with moderately large samples.' In this section we discuss exact, distributionfree tests of H 0: 12 =_ 0 and corresponding exact confidence intervals for the slope difference, 12* These tests continue our basic approach of applying the method of Sievers and Scholz to the two line setting, again assuming common regression constants. Specifically, we discuss two related techniques. The first utilizes the exact distributio4 fteSeesShl statistic T(O) with the optimum weights, a rs= x rs, under 69 the null hypothesis H 0: a 12 _' 0. This null distribution depends on the chosen regression constants, and hence must be recalculated for each design. The second technique is a straightforward application of the TheilSen approach to the two line setting, as discussed in Section 2.1. Since the required null distribution is essentially that of Kendall's tau, tabled critical values are readily available for small sample sizes. Thus the second technique has favor under sample sizes small enough to discourage the use of asymptotic results, yet too large to allow the computation of the null distribution of T(0) required by the first technique. Let us now discuss in detail the two smallsample techniques we have proposed. First, recall the basic linear model (2.1.1) we have assumed, the resulting distribution of the differences Z. given by (2.1.3), and the representation of T(0) in terms of the ranks of the Z's given by setting b = 0 in (2.2.1): N T(0) = (2/N) Z [Rank(Z.i)x. (N+1)X. (2.5.1) j=1JJ Suppose that for line i (i=1,2) the underlying errors,E j j~l...Nare independent and identically distributed (i.i.d.). The distribution of the line 1 and line 2 errors need not be the same, but assume these two distributions are continuous. Let 4*N denote the set of N! permutations of 70 (1,2,...,N). Then under H0: _12 = 0, the Z's are i.i.d. and hence the vector of ranks of the Z's is uniformly distributed over *N (Randles and Wolfe, 1979, p. 37). Consequently, in view of the representation (2.5.1), the null (a = 0) distribution of T(0) is uniformly distributed over the N! values of T(0) obtained by permuting the ranks of the Z's among themselves. For example, if for r= (r,.. ) , N tr = [(2/N) Z (r ] (N+l)x j=l then the distribution of T(0) under H0: 812 = 0 is given by P0 {T(0)=tr} = 1/N!, rE (2.5.2) where, as before, Pb{.1 denotes the probability calculated assuming a12 = b. Having tabulated this null distribution, a test of H0 is conducted as follows. If t is a constant determined from (2.5.2) such that P0 {T(0)>t } = a, then the test of H0: 812 = 0 against H0: a12 > 0 which rejects H0 if T(0) > t (2.5.3) is an exact level a test. Rather than simply state whether a test has accepted or rejected a null hypothesis at a 71 particular level, one might wish to report the attained significance level, the lowest significance level at which the null hypothesis can be rejected with the observed data. If we observe T(0) = t(0), then the exact test (2.5.3) has an attained significance level of p = [(number of r in P)/N!], where P = {rEs : t(0) Let t /2 be a constant defined analogously to t: PO {t a/2 Applying the argument and notations used to derive (2.3.3), it is easily seen that 12 1 [J l (b(1(Nt /2/x..))), J ( (l+(Nt /x..)))) + a/2 + a/2 (2.5.4) is an exact 100(1a) percent confidence interval for 812' In terms of the slopes Srs = (Z SZ r)/xrs, 1 < r < s < N, we can write this interval as [SeL, sU), where we ea S L= min{S: J(S )> (1(Nt /x..)), 1 and 72 S U=min{S J(S )> (1(Nt /x..)) 1 (2.5.5) Computation of a null distribution based on permutations of the observations (or ranks) such as the one in (2.5.2) has become possible under small sample sizes with the speed of today's computers. At the University of Florida we found that the necessary calculations were feasible when there were N=8 or fewer regression constants per line. In this case the entire null distribution of T(0) could be tabulated at a cost of about $2.00 to the user. We were accessing a system operating an IBM 3081 with MVS/XA and an IBM 3033 with OS MVS/SP JES2 Release 3. With more than 8 regression constants per line, the cost of tabulating the null distribution becomes prohibitive (at least $30.00 when N=10, for example). However, in dealing with similar problems, Pagano and Tritchler (1983) and Mehta and Patel (1983) give algorithms that greatly reduce the amount of computation involved. Although their results do not apply directly to this problem, we anticipate the exact test (2.5.3) and computation of the confidence interval (2.5.4) will soon become feasible for larger sample sizes due to the development of similar efficient algorithms and the steadily increasing speed of computer hardware. At the present time we could resort to estimation of the null distribution (2.5.2) of T(0) based on a random sample of permutations of the observed rank vector. Boyett 73 and Shuster (1977) and Ireson (1983) report good approximations resulting from the use of such a sampling of permutations to approximate a permutation distribution. However, there is a possibility that approximate procedures such as these suffer a loss of power due to the restricted sample space (Dwass, 1957). Also, there is the problem of specification of the number of permutations that must be sampled to achieve adequate approximations of the null distribution. Clearly; more study is needed before this technique can be recommended without reservation. Another method of overcoming the computational problem associated with the exact test under larger sample sizes is to replace the SieversScholz statistic by the TheilSen statistic based on the differencesZ.,, as described in Section 2.1. Recall that this consists of using the weights ars = sgn(x rs) in the expression for T (b) given by (2.1.2). The null distribution is essentially that of Kendall's tau, which has been tabulated for many values of N. These tabled critical values and a precise specification of the TheilSen test and confidence interval are given in Hollander and Wolfe (1973). This second approach is appropriate when an exact, distributionfree technique is desired but the number of regression constants per line is too large (N>8, at our facility) to allow complete enumeration of the null distribution (2.5.1) of T(O). If the regression constants are highly unequally spaced, the method based on SieversScholz procedure probably has greater power, but the results of a 74 simulation study discussed in the next section indicate that in many situations the difference in power of the two techniques is slight. One detail that has not been mentioned in this section is how to deal appropriately with ties in the data. If ties occur among the Z's when applying the first technique, simply use all permutations of the vector of midranks in the computation of the null distribution of T(O). The null distribution computed is the conditional null distribution of T(O), given the observed midranks, and the exact, distributionfree properties of the test and interval are retained. We assume the TheilSen approach is only being applied when the sample size prohibits computation of the exact null distribution of T(O). In this case, modifications of the TheilSen approach based on Kendall's tau in the presence of ties are referenced by Hollander and Wolfe (1973, p. 192), but they do not retain the exact nature of the test and confidence interval. As discussed in Section 2.4, the Hollander and RaoGore methods also provide exact, distributionfree tests of H0: S12 = 0 and confidence intervals for the slope difference, 12. The null distributions required to use their methods have been tabulated and are readily available for several sample sizes. Hence these methods are competitors to the exact, small sample techniques proposed in this section. Monte Carlo comparisons of the power of these tests are given in the next section. 75 2.6 Monte Carlo Results To compare the powers of the test statistics discussed in the previous sections, we conducted a Monte Carlo study. The study concentrated on unequally spaced regression constants for the following reasons. The asymptotic relative efficiency results in Section 2.3 indicate superiority of the SieversScholz procedure in a wide variety of cases when the regression constants are equally spaced. Since the structure of the SieversScholz statistic T(b) (2.1.5) utilizes information about the spacing of the regression constants, one would expect that its relative performance improves when unequal spacing is used. Hence, we were particularly interested in comparisons of the test statistics under unequal spacing of the regression constants. We begin our discussion with a description of the sample sizes, regression constants, error distributions, and parameter values used in our simulation study. Choice of Regression Constants Recall from Section 2.4 that Hollander's technique requires a scheme for pairing the regression constants on each line to form slope estimates. Although Hollander clearly specifies the pairing scheme (2.4.3) under equal spacing (2.4.2) of the regression constants, there exists some ambiguity in the choice of such a scheme when the regression constants are unequally spaced. To avoid this 76 ambiguity, we use what we call mirrored spacing, which we now describe. Consider a smooth nondecreasing function t(.) defined over the interval [0,1/2]. Suppose t(.) maps this interval onto itself with t(0) = 0 and t(1/2) = 1/2. Then we define 2k regression constants, xI, x2, ..., X2k, over the interval [0,1] by the relations x tm m tk+ Xm+k =xm + 1/2, for m = I, ..., k. Multiplication by a scale factor can be used to make these constants fall over a more natural range. We use the term mirrored spacing since, by definition, Xm+k xm 1/2 for m = 1, ..., k, and so the arrangement of the set of x's {X k+, ..., X2k} over [ ,1] is identical to the arrangement of the set {xl, ..., xk} over [0, ]. When constructing Hollander's statistic we pair responses at xm with those at X m+k, thus conforming to Hollander's recommendation to choose a pairing scheme approximating that used with equal spacing. Mirrored spacing also allows the use of the RaoGore statistic, which is not generally applicable with unequally spaced regression constants. i We experimented with t(.) of the form t(u) = au where a is chosen to satisfy t(1/2) = 1/2 and i is a positive integer. Note that i = 1 results in two groups of equally 77 spaced constants over the intervals (0,1/2) and (1/2,1). As i increases, the regression constants tend to group closely just below 1/2 and 1. For our simulation study we selected i = 3, t(u) = (1/3)hu 3. This choice of i results in regression constants that depart sufficiently from equal spacing without the excessive clumping observed under larger values of i. Selection of Error Distributions and Parameter Values To generate simulated random variates we used the Fortran subroutines of the International Mathematical and Statistical Library (IMSL). We selected four error distributions: uniform, normal, double exponential, and Cauchy. In terms of the heaviness of tails, these distributions cover a broad range. They are listed above in order of increasing heaviness of tails, from the uniform distribution which has very light tails to the Cauchy distribution which has very heavy tails. The standard normal distribution (N(0,1)) was used, and scale factors of the other distributions were selected such that the probability between 1 and I was the same for all four distributions. The values of a 12 at which the power was estimated were determined by selecting multiples of the estimated standard deviation of the difference of the least squares estimates of 1 and in each case. These multipleswere chosen to achieve a wide range of power. 78 Designs and Computational Details There are basically two parts to the Monte Carlo study presented here. In the first part we used moderately large samples and applied approximate tests based on asymptotic theory. The second part of the simulation used small sample sizes and exact nonparametric tests. The form of the classical ttest was the same in both parts of the Monte Carlo simulation. For the first part of the Monte Carlo study, dealing with moderately large samples, we chose two designs which we call design A and design B. In presenting the sample sizes used in each design, we give the number of regression 'constants per line. Since we are assuming two lines, the total sample size is twice of what we give below. Design A consists of 3 replicates at each of 20 distinct regression constants, resulting in 60 observations per line. The 20 distinct regression constants were selected by using mirrored spacing as described and multiplying by a scale factor of 20. The resulting regression constants are listed in Table 3. Design B uses 30 distinct regression constants, with one observation per line taken at each of these constants. The constants for design B were selected by generating 30 random numbers between zero and one, and multiplying them by a scale factor of 30. We wanted one design chosen to allow power comparisons of the tests under regression constants whose spacing followed no structured pattern, and 79 Table 3. Regression Constants Used for Designs A and B. Design A x = 0.0075 Xll = 10.0075 x2 = 0.0601 x12 = 10.0601 x = 0.2029 x13 = 10.2029 x4 = 0.4808 x14 = 10.4808 x = 0.9391 x15 = 10.9391 x = 1.6228 x16 = 11.6228 x = 2.5770 x17= 12.5770 x8 3.8467 x = 13.8467 x9 = 5.4771 x19= 15.4771 X = 7.5131 x20= 17.5131 Design B x = 0.65 x16= 17.59 x2= 1.16 x17 18.81 x3= 5.46 x18= 18.83 x4= 6.09 x19= 19.53 x = 8.34 x20= 19.99 x6= 8.52 x21 = 20.13 x7= 9.5.0 X22 = 20.78 x = 10.83 x23 21.01 x = 13.61 x = 22.76 Xl0 = 13.84 x25 = 23.78 X = 13.93 x26= 25.86 x12 = 15.20 x27 26.37 x13 = 15.42 x28= 26.59 x14 = 15.58 x = 28.06 x15 = 16.51 x30= 28.92 1) For Design A, 3 responses per line were observed at each of the 20 regression constants. 2) For Design B, 1 response per line was observed at each of the 30 regression constants. 80 hence this accounts for our departure from mirrored spacing in this case. The regression constants for design B are listed in Table 3. To compute Hollander's statistic under design B, we used the usual grouping scheme, pairing the response at x m with that at x m+151 where m = 0, 1, 1 15. Note that given the arbitrary spacing of the regression constants, the RaoGore statistic is not applicable for design B. All tests applied in the simulations under design A and B used a nominal level of a = .05, and in each case the null hypothesis H 0 12 0 was being tested against the onesided alternative H 1 al2 > 0. The tests used were the SieversScholz test described in Section 2.3, the TheilSen test (based on T*(b) in (2.1.2) with a rs = sgn(x Sx r )), the Hollander test based on the statistic W defined in Section 2.4, the RaoGore test (design A) based on U defined in Section 2.4, and the ttest based on the difference of the least square estimates of a 1 and 2* Each of these tests were employed at an approximate a = .05 level utilizing their respective asymptotic distributions. For comparisons of the various procedures under small samples we selected three designs, which we call design C, design D, and design E. All three designs consist of one response per line at each of the regression constants. Sample sizes per line of 6, 8, and 12 were used in .designs C, D, and E, respectively. mirrored spacing was 81 used for all three designs. The design points were multiplied by scale factors of 10, 10, and 15 for designs C, D, and E, respectively. The resulting regression constants for these three designs are given in Table 4. when using designs C and D, the exact tests of H0: a12 =2 0 against H 1: 512 > 0 associated with each of the four nonparametric procedures were used. The exact SieversScholz and TheilSen tests as discussed in Section 2.5 were applied. Thus the exact distribution of the SieversScholz statistic T(0) was first computed for designs C and D to determine appropriate critical values. Anexample of a portion of this distribution for design C is given in Table 5. Exact versions of the Hollander and RaoGore tests rely on the exact null distributions of the Wilcoxon signed rank and WilcoxonMannWhitney statistics, respectively, as tabulated and discussed in Hollander and Wolfe (1973). Randomization was used to bring these exact procedures to the same level. The natural alevels of the tests were compared to select a level for each design at which the amount of randomization needed was minimal. A nominal level of a = .125 was used for design C while a.057 was selected for design D. With 12 observations per line, design E does not allow the feasibility of the exact SieversScholz test. Instead, we replace. this with an approximate procedure, randomly selecting 10,000 permutations of the rank vector and computing the proportion of these permutations resulting in a 82 Table 4. Regression Constants Used for Designs C, D, and E. Design C x = 0.078 x4 = 5.078 x2 = 0.625 x5 = 5.625 x3 = 2.109 x6 = 7.109 Design D x = 0.040 x5 = 5.040 x = 0.320 x6 = 5.320 x = 1.080 x7 = 6.080 x = 2.560 x8 = 7.560 Design E xI = 0.022 x = 7.522 x2 = 0.175 x = 7.675 x3 = 0.590 x = 8.090 x4 = 1.399 X10 = 8.899 x5 = 2.733 X = 10.233 x6 = 4.723 x12 = 12.223 For Design C, D, and E, one response per line was observed at each design point. 83 Table 5. Upper Portion of the Exact Null Distribution of T(0) when using Design C. t (0) P 0 {T(0) > t(0)} 6.64062 0.0583 6.69269 0.0542 6.82289 0.0528 6.82292 0.0514 6.87499 0.0486 7.00519 0.0472 7.13542 0.0444 7.18749 0.0431 7.31769 0.0389 7.36979 0.0347 7.49999 0.0333 7.68229 0.0236 7.81249 0.0194 7.86459 0.0167 7.99479 0.0139 8.17709 0.0111 84 value of T(0) larger than the one calculated from the observed data. Rejection of H 0: 12 20 in favor of Hi1: 6 12 > 0 occurred when this proportion was less than or equal to the nominal level, which was set at a = .047. Recall that this type of approximate procedure was discussed in Section 2.5. The implementation of this approximate procedure was facilitated by an algorithm due to Knuth (1973), which presents a onetoone association between the integers 1 ..N! and the N! permutations of (1,...,N). Because of the cost of this approximate SieversScholz procedure, the number of simulations for design E was set at 500, and only two error distributions were selected (normal and Cauchy). Discussion of Results Empirical levels and powers of the SieversScholz, TheilSen, Hollander, RaoGore, and classical tests under design A are presented in Tables 6, 7, 8, and 9, for the uniform, normal, double exponential, and Cauchy distributions, respectively. Empirical levels vary but generally remain within two standard errors of the nominal .05 level. It is seen that in terms of their power, the SieversScholz and TheilSen tests uniformly dominate the RaoGore and Hollander tests. The power of the classical ttest is highest under uniform and normal errors, but falls below the powers of SieversScholz and the TheilSen tests for double exponential errors. The ttest has the lowestL power of the 85 Table 6. Empirical Power Times 1000 Under Design A for the Uniform Distribution (a=.05). 12 0 .024 .064 .096 SieversScholz 058 192 675 934 TheilSen 059 2 18 14 Hollander 063 4 57 35 RaoGore 061 15 58 36 Classical 061 17 71 31 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding SieversScholz result, with a negative value indicating a lower power than SieversScholz. Table 7. Empirical Power Times 1000 Under Design A for the Normal Distribution (a=.05). 0 .032 .064 .096 12 SieversScholz 041 217 561 861 TheilSen 036 5 21 15 Hollander 046 10 70 51 RaoGore 041 25 41 53 Classical 042 33 27 26 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding SieversScholz result, with a negative value indicating a lower power than SieversScholz. 86 Table 8. Empirical Power Times 1000 Under Design A for the Double Exponential Distribution (a=.05). 12 0 .032 .064 .096 SieversScholz 052 234 476 808 TheilSen 043 16 6 14 Hollander 057 46 69 116 RaoGore 055 29 53 67 Classical 049 16 32 22 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding SieversScholz result, with a negative value indicating a lower power than SieversScholz. Table 9. Empirical Power Times 1000 Under Design A for the Cauchy Distribution (a=.05). 0 12 0 .064 .096 .160 SieversScholz 052 394 590 896 TheilSen 052 13 1 1 Hollander 053 140 245 275 RaoGore 062 41 44 31 Classical 043 289 451 667 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding SieversScholz result, with a negative value indicating a lower power than SieversScholz. 87 five tests under the heavily tailed Cauchy distribution. The SieversScholz test tends to have higher power than the TheilSen test, but the difference is not pronounced. Comparing Hollander's test with that of Rao and Gore, we see that the power of the RaoGore test dominates the power of Hollander's test for the two heavily tailed distributions (double exponential and Cauchy) but they perform about equally under normal and uniform errors. A summary of all the Monte Carlo results will be given at the conclusion of this section. Tables 10, 11, 12, and 13 present empirical levels and powers of the SieversScholz, TheilSen, Hollander, and classical tests under design B for the uniform, normal, double exponential, and Cauchy distributions, respectively. Empirical levels of the tests appear somewhat depressed, but generally fall within two standard errors of the nominal .05 level. For the uniform, normal, and double exponential distributions, the order of the tests in decreasing power is classical, SieversScholz, TheilSen, and Hollander. The dominance of the classical approach lessens with increasing heaviness of tails of theerror distribution. For Cauchy errors the classical test again falls into last place, with the SieversScholz and the TheilSen tests exhibiting highest powers. For design C, Tables 14, 15, 16, and 17 present empirical levels and powers under uniform, normal, double exponential, and Cauchy distributions, respectively. 88 Table 10. Empirical Power Times 1000 Under Design B for the Uniform Distribution (a=.05). $ 12 0 .034 .068 .102 SieversScholz 039 253 651 934 TheilSen 036 16 18 12 Hollander 037 62 161 89 Classical 043 37 76 31 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding SieversScholz result, with a negative value indicating a lower power than SieversScholz. Table 11. Empirical Power Times 1000 Under Design B for the Normal Distribution (a=.05). a 12 0 .034 .068 .102 SieversScholz 042 209 569 878 TheilSen 038 20 28 18 Hollander 055 24 128 154 Classical 046 25 48 26 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding SieversScholz result, with a negative value indicating a lower power than SieversScholz. 89 Table 12. Empirical Power Times 1000 Under Design B for the Double Exponential Distribution (a=.05). 12 0 .034 .068 .102 SieversScholz 047 201 495 753 TheilSen 039 4 24 23 Hollander 045 21 137 171 Classical 053 18 2 0 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding SieversScholz result, with a negative value indicating a lower power than SieversScholz. Table 13. Empirical Power Times 1000 Under Design B for the Cauchy Distribution (a=.05). 12 0 .051 .102 .153 SieversScholz 044 225 577 770 TheilSen 047 6 5 7 Hollander 041 90 250 324 Classical 043 133 377 455 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding SieversScholz result, with a negative value indicating a lower power than SieversScholz. 90 Table 14. Empirical Power Times 1000 Under Design C for the Uniform Distribution (a=.125). $12 0 .218 .436 .654 SieversScholz 121 424 824 977 TheilSen 125 27 61 37 Hollander 113 53 101 50 RaoGore 122 26 57 7 Classical 128 55 51 17 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding SieversScholz result, with a negative value indicating a lower power than SieversScholz. Table 15. Empirical Power Times 1000 Under Design C for the Normal Distribution (a=.125). $12 0 .218 .436 .654 SieversScholz 112 371 687 918 TheilSen 117 17 48 35 Hollander 110 52 88 91 RaoGore 103 30 49 44 Classical 117 6 58 36 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding SieversScholz result, with a negative value indicating a. lower power than SieversScholz. 91 Table 16. Empirical Power Times 1000 Under Design C for the Double Exponential Distribution (la=.125). S12 0 .218 .436 .654 SieversScholz 110 354 612 854 Theii.Sen 117 16 20 47 Hollander 123 46 41 65 RaoGore 113 25 26 34 Classical il1 23 61 37 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding SieversScholz result, with a negative value indicating a lower power than SieversScholz. Table 17. Empirical Power Times 1000 Under Design Cafor the Cauchy Distribution (a=.125). 012 0 .218 .436 .654 SieversScholz 131 320 465 574 TheilSen 122 13 9 1 Hollander 132 57 50 41 RaoGore 125 42 25 11 Classical 141 23 2 4 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding SieversScholz result, with a negative value indicating a lower power than SieversScholz. 92 Tables 18, 19, 20, and 21 present thesesame results for design D. Recall that designs C and D consisted of samples of size 6 and 8 per line, respectively, and the exact nonparametric tests were used under these designs. The relative performance of the tests using these small sample sizes is similar to that of the approximate tests under larger samples, except that the dominance of the classical ttest is more dramatic under the smaller sample sizes. Only for Cauchy errors under design D does the power of the SieversScholz test clearly dominate the classical test. The SieversScholz and TheilSen tests generally exhibited greater powers than the Hollander and RaoGore tests. The one exception to this occurred under uniform errors, where the observed power of the RaoGore test was slightly higher than that of the TheilSen test, although we note that in these cases the SieversScholz test had highest power among these three tests. For design E, Tables 22 and 23 present empirical levels and powers for the normal and Cauchy distributions, respectively. When compared with the TheilSen test, the approximate SieversScholz test had higher power under normal errors and lower power under Cauchy errors. Hence these results do not indicate a clear choice between these two methods. However, the SieversScholz and TheilSen tests once again exhibit greater powers than the Hollander and RaoGore tests (except in one instance where the power of the RaoGore test marginally exceeded that of the 93 Table 18. Empirical Power Times 1000 Under Design D for the Uniform Distribution (c,.=.057). $12 0 .186 .372 SieversScholz 059 300 661 TheilSen 064 26 36 Hollander 056 68 160 RaoGore 064 16 25 Classical 057 31 81 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding SieversScholz result, with a negative value indicating a lower power than SieversScholz. Table 19. Empirical Power Times 1000 Under Design D for the Normal Distribution (a=.057). al2 0 .186 .372 SieversScholz 059 223 551 TheilSen 056 9 41 Hollander 050 41 120 RaoGore 054 28 47 Classical 058 34 94 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding SieversScholz result, with a negative value indicating a lower power than SieversScholz. 94 Table 20. Empirical Power Times 1000 Under Design D for the Double Exponential Distribution (a=.057). 1l2 0 .372 .558 SieversScholz 059 451 684 TheilSen 054 11 23 Hollander 053 92 135 RaoGore 059 22 29 Classical 063 60 90 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding SieversScholz result, with a negative value indicating a lower power than SieversScholz. *Table 21. Empirical Power Times 1000 Under Design D for the Cauchy Distribution (a=.057). al2 0 .372 .930 SieversScholz 058 342 645 TheilSen 057 17 17 Hollander 064 71 112 RaoGore 057 28 63 Classical 062 65 44 The first row and first column give the empirical power times 1000. Entries in the remainder of the table are expressed as the difference from the corresponding SieversScholz result, with a negative value indicating a lower power than SieversScholz. 