R Estimates and Associated Inferences for Mixed
Models with Covariates in a Multi-Center Clinical Trial
M. Mushfiqur Rashid∗
Center for Drug Evaluation and Research, FDA
Joseph W. McKean
John D. Kloke
Western Michigan University
University of Pittsburgh
Abstract
Robust rank-based methods are proposed for the analysis of data from multi-center
clinical trials using a mixed model (including covariates) in which the treatment effects
are assumed to be fixed and the center effects are assumed to be random. These rankbased methods are developed under the usual mixed model structure but without the
normality assumption of the random components in the model. For this mixed model,
our proposed estimation includes R-estimation of the fixed effects, robust estimation
of the variance components, and studentized residuals. Our accompanying inference
includes estimates of the standard errors of the fixed effects estimators and tests of general linear hypotheses concerning fixed effects. While the development is for general
scores function, the Wilcoxon linear scores are emphasized. A discussion of the relative efficiency results shows that the R-estimates are highly efficient compared to the
1
traditional maximum likelihood (ML) estimates. A small Monte Carlo study confirms
the validity of the analysis and its gain in power over the ML analysis for heavy-tailed
distributions. We further develop a rank-based test for center by treatment interactions. We discuss the results of our analysis for an example of a multi-center clinical
trial which shows the robustness of our procedure.
Key words: Exchangeable; Dispersion function; Nonparametric; Robustness; Variance components; Wilcoxon.
∗
The views expressed in this article are those of the author and not those of the United
States Food and Drug Administration.
1
Introduction
Clinical trials comparing several treatment groups are usually conducted at a number of
centers or sites. This is done mainly to enroll enough patients to test a hypothesis of interest
in a suitable time period. An advantage of conducting multi-center trials is that it can
provide evidence that the trial results are not strongly center dependent. Further, it may
be reasonable to interpret that the results (effectiveness) apply to a broader population of
clinical sites or patient populations. In this article, we assume an experimental design in
which the centers participating in the study are presumed to be a random sample from
the population of centers which might conceivably use the treatments under investigation.
This leads to a two-factor mixed model with treatment groups as one factor (fixed factor)
and centers as a stratification factor (random). Our model, also, allows for any number of
covariates which we treat as fixed effects. See Rashid (2003), Patel (2002), Boos and Brownie
(1992), Chakravarty and Grizzle (1975) and Khatri and Patel (1992) for further discussions
2
about the arguments of analyzing multi-center clinical trials using mixed models.
In Sections 2 and 4.1, we develop rank-based estimates of the fixed effects (including
coefficients corresponding to the covariates). These estimates are invariant to the random
effects. Using the residuals based on this robust fixed effects fit, we develop robust predictors
of the random effects and, hence, robust estimates of the variance components, (Section 5).
We then use these to obtain robust studentized residuals, which, as their counterparts in
regression analysis, standardize residuals for both location in factor space and variation due
to the error distribution. These studentized residuals prove effective as a diagnostic tool in
checking quality of fit and in determining outliers. Our estimates are based on minimizing
a convex function and, hence, are easily computed as discussed in Section 4.3. We have
coded our algorithms in the language R (R Development Core Team 2010) and have made it
available for the interested user. Our development is for general rank scores, which include
the popular Wilcoxon and sign (l1 ) scores; thus, the estimates are efficiently robust and can
be optimized depending on the knowledge of the underlying error distribution. Furthermore,
weighted versions of these rank-based estimates can easily be formulated (similar to the HBR
estimates of Chang et al., 1999) to achieve up to 50% breakdown for designs with messy
covariates.
In Section 4.1, we establish the asymptotic distribution of estimates and provide estimates
of their asymptotic covariance matrix. This allows for the development of a rank-based
analysis which includes confidence intervals for contrasts of the fixed effects and tests of
general linear hypotheses. The analysis has the same robustness and efficiency properties
as the rank-based estimates. The results of the simulation study of Section 7 verify the
adequacy of the analysis based on asymptotic distributions. In the one situation simulated,
there are only eight observations per center. In this case as well as the others, our inference
is empirically valid and possesses robustness of power over the traditional procedure.
3
In a multi-center clinical trial, there is always the possibility of interaction between the
centers (random effect) and the fixed effects. A test of this interaction is recommended by E9
of the International Conference on Harmonization, 1998, which is widely used in practice in
the pharmaceutical industry. In Section 8 we propose a rank-based test for this interaction.
Section 2 is an outline of our proposed analysis. We follow this in Section 3 with a detailed
analysis of an example. The outliers in this data set show the robustness of our rank-based
analysis over the traditional maximum likelihood procedure. In the presence of the largest
outlier in the data set, different practical conclusions are drawn for the two analyses; while,
the conclusions of the analyses are the same when the outlier is omitted.
1.1
Literature Review
For the history of fixed effects model analysis of multi-center trials, see Patel (2002). While
the mixed effects model is widely studied in literature, most of these studies were done under normality assumptions. Among several researchers, Hartley and Rao (1967), Hammerle
and Hartley (1973), Chakravarty and Grizzle (1975) provided likelihood based procedures
for comparing treatment effects for a mixed-effect model. Conventionally, all random components are assumed to be independent normal with zero means. Khatri and Patel (1992)
generalized this model by treating the centers as a random sample from a t-variate normal
population where t is the number of treatments and provided the likelihood ratio test for
comparing the treatment. Gould (1998) provided Bayes and empirical Bayes methods for
multi-center trials.
There has been some work on mixed model analysis under nonparametric settings. In the
absence of covariates and treatment by center interactions, Van Elteren’s (1960) weighted
Wilcoxon-Mann-Whitney test can be used when there are only two treatments. Boos and
Brownie (1992) proposed a nonparametric method where the test statistic is the standardized
4
average of Wilcoxon-Mann-Whitney type estimators associated with centers. However, there
are two problems with this method: first, their procedure does not incorporate covariates
and second, it considers only two treatments. In the absence of covariates, Rashid (2003)
developed a rank-based procedure for testing a non-inferiority hypothesis for multi-center
trials using a mixed model.
2
MR Analysis
Consider a multi-center clinical trial conducted over c centers, involving a primary efficacy
variable Y . For center j, j = 1, . . . , c, suppose we have a sample of nj patients and we let
Y j denote the nj × 1 vector of responses for this efficacy variable. Consider the mixed model
Y j = µ1nj + W j θ + 1nj bj + ej ,
(2.1)
where 1nj is a vector of nj ones, W j is the design matrix, θ is a k × 1 vector of fixed
effects parameters, bj is the random effect for center j, and ej is an nj × 1 vector of random
errors. Assume that bj and ej are independent and that the components of ej are iid with
the marginal probability density function f (t) and distribution function F (t). We assume
that the random effect is a continuous random variable. Other than this assumption, no
additional assumptions are needed for the distribution of the random effects. Because we
can reparameterize, we assume without loss of generality that W j has full column rank. We
further assume that observations between centers are independent of one another.
For j = 1, . . . , c, let ǫj = 1nj bj + ej . Then by Model (2.1) the distribution of ǫj is
exchangeable; i.e., L(ǫj1 , . . . , ǫjnj ) = L(ǫjα1 , . . . , ǫjαnj ),
i = 1, . . . , nj , where L denotes
distribution and α1 , . . . , αnj is a permutation of the integers 1, . . . nj .
Suppose additionally that the variance-covariance matrix of ǫj exists. Let σb2 and σe2
5
denote the variances of bj and eji , respectively. The variance-covariance matrix of ǫj satisfies
the property of compound symmetry given by
Var(ǫj ) = σ 2 Anj (ρ),
(2.2)
where Anj (ρ) = (1 − ρ)I nj + ρJ nj , σ 2 = σe2 /(1 − ρ), and ρ = σb2 /(σe2 + σb2 ). The parameter
ρ is often called the intraclass correlation coefficient. The analysis we propose, though, does
not assume the existence of the variance-covariance matrix of ǫj . The parameters σ 2 , σe2 , σb2 ,
and ρ are called the variance components of the model.
2.1
MR Estimates
For center j, consider a set of rank scores, aj (i) = ϕj [i/(nj + 1)], i = 1, . . . , nj , where ϕj (u)
is a specified nondecreasing, square-integrable function defined on the interval (0, 1) which,
R1
R1
without loss of generality, is standardized so that 0 ϕj (u) du = 0 and 0 ϕ2j (u) du = 1. In
this paper, we use the same score function for each center which we denote as ϕ(u). The
√
most popular score function is the Wilcoxon with ϕ(u) = 12[u − (1/2)]. This is the score
function we use in our examples and Monte Carlo study. The sign scores, generated by
ϕ(u) = sign[u − (1/2)], are also frequently used. For regression, the fit based on sign scores
is equivalent to an l1 fit; see Hettmansperger and McKean (2011).
For each center j, consider Jaeckel’s (1972) dispersion function given by
Dj (θ) = kY j − W j θkj ,
6
(2.3)
where k · kj is the pseudo-norm
kvkj =
nj
X
aj [Rj (vi )]vi ,
i=1
v ∈ R nj ,
(2.4)
and Rj (vi ) denotes the rank of vi among v1 , . . . , vnj , i.e., intra-center rankings. Our objective
function is the dispersion function
D(θ) =
c
X
Dj (θ).
(2.5)
j=1
Because the functions Dj are convex, their sum is also. Our estimator of the fixed effects θ
is given by the value that minimizes D(θ), i.e.,
bM R = Argmin D(θ).
θ
(2.6)
We call this estimator the MR estimate, where the M stands for multiple or many rankings.
Because D(θ) is convex, as discussed in Section 4.3 there are simple algorithms for the
bM R . As discussed in Section 4.3, the authors have programmed a Newton
computation of θ
procedure in R, (R Development Core Team, 2010), Section 4.3 which they have made
available R, (R Development Core Team, 2010) to the interested user.
The invariance of the MR estimate to the random effects is easy to see. Because the
rankings are invariant to a constant shift, we have for center j that
Rj (Yij − µ − bj − w′ij θ) = Rj (Yij − w′ij θ).
Because the scores sum to 0, for each center, it follows that the objective function D, and
thus the MR estimator, are invariant to the random effects. Notice that it is invariant to the
7
intercept parameter µ also. The intercept is easily estimated by a location estimator based
on the residuals, as discussed in Section 4.4. Besides being invariant to the random effects,
this estimator satisfies the usual regression equivariances. For instance, if T = dY + W γ
bM R (T ) = dθ
bM R (Y ) + γ.
then θ
Note that the negative of the gradient of D(θ) is
S(θ) =
c
X
S j (θ),
(2.7)
j=1
where
S j (θ) =
nj
X
i=1
w ij aj [Rj (yij − w′ij θ)],
and w ′ij is the ith row of W j . As discussed above, Rj denotes the intra-center rankings.
b solves S(θ)
b = 0.
Hence, equivalently, θ
b is discussed in Section 4.1, along with
Asymptotic normality of the MR estimates, θ,
their standard errors. Asymptotic χ2 , Wald type tests of linear hypotheses of the form
H0 : Hθ = 0, for a specified matrix H, are easily constructed based on the asymptotic
b Also, a simple robust test of interaction between the fixed effects and
distribution of H θ.
the centers is discussed in Section 8. Together, these analyses give a complete inference for
the fixed effects based on the MR estimates.
Suppose further that ǫj is compound symmetric, i.e., has covariance given by (2.2). In
Section 5 we present robust estimates of the variance components. We then use these to
obtain studentized residuals for the MR residuals.
Before giving further details of the MR analysis, we illustrate its use with an in depth
discussion of an example in the next section. We used the software discussed above for the
MR fit and the R function lme (see Chapter 1 of Pinheiro and Bates, 2000) to compute the
ML fit for this example.
8
3
Example
Several years ago, one of the authors consulted on a multi-centered clinical trial of lipidlowering agents. Generally, robust analyses appeared to be more powerful than the traditional analysis for these data. The data set, though, is confidential, so for an example we
generated data which are similar to part of the study. We present next the results of the
analyses on the generated data but we discuss it as though these were the results on the
original data.
We will assume that the data are drawn from a randomized, placebo controlled multicenter study which was conducted to compare the relative efficacies of four different active drug
compounds on several lipid levels. We refer to the placebo group as T1 and the four active
drug compounds as T2-T5. The study was performed at two centers, C1 and C2. Center
C1 had 48 patients complete the study, while C2 had 49 patients complete it. Within each
center, patients who passed a screen for high cholesterol were randomly assigned to one of
the five treatments T1-T5. The sample sizes for treatments T1-T5 are respectively 20, 18,
20, 20 and 19. Lipid levels for the patients were measured at specified times.
As our response of interest, we consider the drop in triglyceride levels between the Baseline
Visit and Week 4 of the study. We considered Model (2.1) with the design matrix modeling
the treatment effects. For i = 2, . . . , 5, let θi denote the mean change in the triglyceride level
in the ith treatment group minus the mean change in the placebo group. The main effect
hypotheses of interest are H0 : θ2 = · · · θ5 = 0 versus not all treatment effects are 0. Table 1
summarizes the results of the traditional maximum likelihood (ML) tests and the rank-based
(MR) tests. Both analyses agree that there is no interaction between treatments and centers.
For the main effect hypotheses H0 , however, the analyses lead to dramatically different
conclusions. Based on the MR test, we reject H0 and conclude that there is a treatment
effect with respect to triglyceride levels, while, with the ML results we conclude that the
9
changes in triglyceride levels are the same across all treatments including the placebo.
Table 1: Summary of the Wilcoxon and ML ANOVAS for Trigylceride Data
MR Fit
ML Fit
Test Stat. p-Value Test Stat. p-Value
Treat.
21.67
0.0002
0.678
0.6089
T×C
1.20
0.3258
0.011
0.9731
Table 2: Summary of the Wilcoxon and ML Estimates Trigylceride Data
MR Fit
ML Fit
Est.
SE
Ratio
Est.
SE
Ratio
θ2 −0.86 10.44 −0.08
3.20
25.72 0.12
θ3 −3.40 10.44 −0.33 −12.06 25.72 −0.47
θ4 32.76 10.30 3.18
1.58
25.38 0.06
θ5 26.37 10.43 2.53
29.28 25.72 1.14
Based on the MR results, we further consider the estimates of the contrasts to see where
the treatment differences occur. Table 2 displays a summary of the estimates and their
standard errors. Based on the MR results, Treatments 4 and 5 differ significantly from
placebo; hence, at Week 4, the triglyceride levels of patients under these treatments increased
significantly over the placebo levels between the Baseline Visit and Week 4 of the study. On
the other hand, based on ML’s nonsignificant test of the main effects, this investigation of
contrasts would generally not be made. This is beside the point, though, since for the ML
analysis, none of the contrasts with placebo is significant.
An empirical measure of relative precision or efficiency between competing estimates is the
ratio of the square of their standard errors. For this nearly balanced data set, the standard
errors of the contrasts for an estimator are nearly the same. Hence, choosing the contrast θ5 ,
the empirical relative efficiency between the MR and ML analyses is (25.72/10.43)2 = 6.08.
So, for this data set, the MR analysis is 608% more efficient than the ML analysis.
The reason for such a difference in the MR and ML analyses can readily be seen from
10
the boxplot and q−q plots in Figure 1. The boxplot of responses versus treatments verifies
the significant differences found by the MR analysis between Treatments 4 and 5 and the
placebo. It also shows an outlier in the T4 group which impaired the ML analysis. Even
more so, the outlier had a dramatic effect on ML’s estimate of θ4 . Instead of being the
largest estimated effect over placebo, as it is for the MR estimates, the ML estimate of θ4 is
next to the smallest. The normal q−q plot of the Wilcoxon studentized residuals is overlayed
with usual ±2 benchmarks. It clearly shows the large outlier, but it also depicts six other
potential outliers.
Note: the original data had an outlier of the same size and also several moderate size
−5
−20
−600
−15
−10
Studentized Wilcoxon residuals
−200
−400
Response
0
0
5
outliers. The data did not appear to be normally distributed.
T1
T2
T3
T4
T5
−2
Treatments
−1
0
1
2
Normal quantiles
Figure 1: Plots for Triglyceride Data
As a final note, the robust estimates of the variance components are σ
bM R,b = 5.47 and
σ
bM R,e = 32.59. Their ML counterparts are σ
bM L,b = 0.84 and σ
bM L,e = 78.19.
To show the robustness of MR analysis, we reran the analyses on the data set without the
11
large outlier. Tables 3 and 4 summarize the results of the MR and ML analyses. For these
data (with the outlier deleted), the ML analysis essentially agrees with the MR analysis.
Now the ML analysis shows a significant treatment effect and that treatments 4 and 5 are
significantly different from the placebo. The MR analysis is still more efficient with the
empirical efficiency at (13.75/10.12)2 = 1.84, due to the remaining moderate outliers in the
data set. Note that the MR analysis for this data set differs little from its analysis for the
data set containing the outlier. This illustrates the robustness of the MR analysis. This
robustness is exhibited in the MR estimates of the variance components, also. For the data
without the outlier, the MR estimates of the variance components are σ
bM R,b = 4.12 and
σ
bM R,e = 33.44; while, the ML estimates are σ
bM L,b = 0.26 and σ
bM L,e = 41.80. The ML
estimates are more similar to the MR estimates for this data.
Table 3: Summary of the Wilcoxon and ML ANOVAS for Trigylceride Data
MR Fit
ML Fit
Test Stat. p-Value Test Stat. p-Value
Treat.
24.23
0.0000
4.21
0.0036
T×C
1.24
0.3076
0.0012
0.9720
Table 4: Summary of the Wilcoxon and ML Estimates Trigylceride Data
MR Fit
ML Fit
Est.
SE
Ratio
Est.
SE
Ratio
θ2 −0.83 10.12 −0.08
3.20
13.75 0.23
θ3 −4.07 10.12 −0.40 −12.06 13.75 −0.88
θ4 35.09 10.12 3.46
35.04 13.75 2.55
θ5 25.46 10.12 2.52
29.28 13.75 2.12
4
Inference
In this and the next section, we complete our discussion of the inference of the MR analysis
(see Model 2.1). Let ϕ(u) be a specified score function. For the examples in this paper, we
12
use the Wilcoxon scores given by ϕ(u) =
√
bM R denote the MR estimator
12[u − (1/2)]. Let θ
as defined in (2.6). We assume that nj > p. Let W mj = (I nj − n−1
j J nj )W j , where the m
denotes that the column means are 0. Next stack these matrices in the n × p matrix W ∗m ;
i.e., W ∗m = [W ′m1 · · · W ′mc ]′ and define the matrix
∗
V n = n−1 W ∗′
mW m.
(4.1)
As discussed around (2.1), let f (t) and F (t) denote the pdf and cdf of the random errors eij .
In order to define a necessary scale parameter, defined the function ϕf (u) as
ϕf (u) = −
f ′ [F −1 (u)]
,
f [F −1 (u)]
0 < u < 1.
(4.2)
Next, define the scale parameter τ by
τ=
Z
1
ϕ(u)ϕf (u) du
0
−1
.
(4.3)
The parameter τ is a score parameter; for instance, if the errors have a N(0, σe2 ), then
p
τ = π/3 σe .
4.1
bM R
Asymptotic Distribution of θ
For asymptotic theory, we assume that nj → ∞, for each j so that nj /n → λj , where
Pc
j=1 λj = 1. While this assumption is for theory, we do note that the one Monte Carlo
study of Section 7 shows the validity of our inference for situations where the center sample
size is only eight. The asymptotic theory for these MR estimators follows similarly to the
theory for fixed effect linear models and it is briefly sketched in the appendix. There it is
13
shown that
bM R is asymptotically Np [θ, τ 2 (W ∗′ W ∗ )− ]
θ
m
m
(4.4)
where the scale parameter τ is given in (4.3) and the notation for generalized inverse of a
matrix A is A− where A− = A− AA− . Note for the compound symmetry model (2.2) that
b
τ is a function of σe2 . It follows as in the case of MR estimators for fixed effects that the θ
has bounded influence in the response-space but not in the factor-space. The MR estimator,
however, can easily be generalized to a weighted HBR estimator which has bounded influence
in both spaces and can achieve a 50% breakdown point; see Chang et al. (1999).
In the absence of covariates, Rashid and Bagchi (1997) developed the asymptotic distribution of MR-estimates of the fixed effects based on the Wilcoxon scores and assuming
equal cell size (fixed) for a large number of centers. Hence, their procedure is applicable for
clinical trials (with rare disease) where the number of centers are large to ensure adequate
patient recruitment.
A consistent estimator of the scale parameter τ can be obtained as follows. For jth center,
bM R . For center j, denote by τbj the estimator of
form the vector of residuals b
rj = Y j − W j θ
τ proposed by Koul et al. (1987). This is based on the differences of the residuals, rbij − rbi′ j ,
b is invariant to the random effect, these differences are, also. Thus
over center j. Because θ
the estimator τbj is invariant to the random effect. Furthermore, it is a consistent estimator
of τ . As our estimator of τ , we take the average of these estimators, i.e.,
c
1X
τbj ,
τb =
c j=1
which is consistent for τ .
14
(4.5)
4.2
Tests of General Linear Hypotheses
Let γ = h′ θ be a contrast of interest. It follows from (4.4) that an asymptotic confidence
interval for γ is
q
∗ −
γb = h θ ± zα/2 τb h′ (W ∗′
m W m ) h,
′b
(4.6)
where zα/2 = Φ−1 (1 − α/2) and Φ(x) is the cdf of a standard normal random variable. Next
consider the general hypothesis
H0 : Hθ = O versus HA : Hθ 6= O,
(4.7)
where H is a q × k contrast matrix with full row rank. Based on the asymptotic distribution
b the Wald type test statistic is
of θ,
b ′ [H(W ∗′ W ∗ )− H ′ ]− H θ/b
b τ 2.
T = (H θ)
m
m
(4.8)
An asymptotic level α decision rule based on T is to reject H0 provided T > χ2α (q).
Another test statistic is based on the reduction of dispersion in passing from the reduced
bH ) and D(θ)
b denote the minimum of the dispersion function under
to the full model. Let D(θ
the constraint H0 : Hθ = O, (reduced model), and the minimum under the full model,
respectively. Denote the reduction in dispersion by
bH ) − D(θ).
b
RD = D(θ
(4.9)
Large values of RD are indicative of a lack of agreement between the collected data and the
15
null hypothesis. It is shown in the Appendix that under H0 , (4.7),
D∗ =
RD
τb/2
converges in distribution to the χ2 (q) distribution
(4.10)
as n → ∞. A nominal α decision rule is to reject H0 in favor of HA , if D ∗ > χ2α (q).
Rashid, Aubuchon, and Bagchi (1993) obtained a similar result for balanced incomplete
block designs without covariates. In the absence of covariates, Rashid (2003) developed
R-estimates for testing non-inferiority hypothesis in multimember trials (assuming the cell
size large ) using a sum of Jaeckel’s dispersion function based on Wilcoxon scores. See, also,
Hettmansperger and McKean (2011) for a related result concerning the fixed effects model.
The drop-in-dispersion test for general linear hypotheses is analogous to −2logΛ in maximum likelihood procedure and has similar interpretation. The use of a measure of dispersion
to assess the effectiveness of a model fit to a set of data is common in regression analysis.
As shown in the appendix, RD is based on a finite sum of reductions over the centers. The
influence function of each of these reductions has bounded influence in the response-space.
Hence, because there are only a finite number of centers, the test statistic D ∗ has bounded
influence in the response-space. While it does not have bounded influence in the covariatespace, it can easily be generalized to a high break down (Chang, et. al. 1999) analysis which
has bounded influence in both response and factor spaces.
4.3
bM R
Computation of θ
bM R . The function D(θ)
Since D(θ) is convex, there is software available to compute θ
can be minimized using the Nelder-Mead algorithm (see Olsson 1974) or a quasi Newton
algorithm. Both algorithms do not require second derivatives. In SAS, we can use subroutines
NLPNMS (for Nelder-Mead algorithm) and NLPQN (for quasi-Newton algorithm) of PROC NLP to
16
minimize the dispersion function. In the statistical software R, the non-linear minimization
function lme of the library nlm can be used. See McKean, Terpstra and Kloke (2009) for a
recent review article on computation of rank-based fitting procedures.
It is also easy to write software for the following k-step (Newton-type) algorithm. Let
b
θ
(0)
and τb(0) be initial estimators of θ and τ , respectively. Using the expression (A.1.3) of
the appendix, it follows that the first Newton step is
(0) (1)
(0)
(0) −1 −
b
b
b
θ = θ + τb n V n S θ
,
(4.11)
where S(θ) is defined by (2.7). This can be iterated for a k-step procedure. This is easy
to code in the R or S-Plus language. A shrinkage factor can be used to modify the step
(0)
b
size, τb(0) n−1 V −
n S(θ ), to obtain a fully iterated estimate. This algorithm converges in a
probabilistic sense; see Abebe and McKean (2007). Code in the R language is available from
one of the author’s website at http://www.pitt.edu/~jdk61/.
4.4
Estimation of the Intercept
Since rank is location invariant, we are unable to estimate µ using the dispersion function.
b denote the
However, µ can be estimated by the median of the residuals. Let b
r = Y − W ∗m θ
vector of residuals. For the purpose of practical interpretation with covariates in the model,
we use the centered design matrix W ∗m . Then our estimator of µ is
µ
b = mediani,j {b
ri,j }.
Note that unlike the MR estimator of θ, µ
b is not invariant to the random effect.
(4.12)
To illustrate a use of this estimate, consider a design with a one-way fixed effect factor
17
which we will label Treatment. Suppose Treatment has t levels. Assume at first there is no
covariate. We then select a full column rank design by using treatment contrasts. That is,
for i = 2, . . . , t, let xi be the indicator vector for Treatment i and let T = [x2 · · · xt ]. For
i = 2, . . . , t, let ∆i be the shift in location from Treatment 1 to Treatment i. Then the model
is
Y = µ1n + T ∆ + Zb + e,
b denote the MR estimate of ∆ and let µ
where ∆ = (∆1 , . . . , ∆t )′ . Let ∆
b denote the median
of the residuals from this fit. Then µ
b is the estimate of the effect of Treatment 1 and, for
b i is the estimate of the effect of Treatment i. If the model also contains
i = 2, . . . , t, µ
b+∆
covariates then these would be the estimates adjusted for the covariates.
5
Variance Component Estimators and Studentized Residuals
Recall that the model in the jth center is given by Y j = µ1j + W j θ + bj 1j + ej . As in
Section 2, let ǫj = bj 1j + ej . For this section, assume that the covariance matrix of the
vector of errors, ǫj , exists. Then, as in (2.2), Var(ǫj ) = σ 2 Anj (ρ) = σ 2 [(1 − ρ)I nj + ρJ nj ],
where σ 2 = σe2 /(1 − ρ) and ρ = σb2 /(σe2 + σb2 ). In this section, we obtain estimates of the
variance components σe2 , σb2 and ρ and then use them to obtain studentized residuals.
b denote the MR residuals. Rewriting the model
In the jth center, let b
ǫj = Y j − µ
b1j −W j θ
as Y j − µ1j − W j θ = bj 1j + ej , we see that b
ǫj estimates the left side of this formulation of
the model. In the literature, these residuals are often called the marginal residuals. These
are generally used for model checking; see McKean and Sheather (2009) for discussion.
Because the median of eij is zero, a natural prediction of bj is bbj = medi {b
ǫij }; see
18
Rashid and Nandram (1998) or Kloke at al. (2009). Thus, a robust estimate of the variance
component σb2 is the median of absolute deviations (MAD); i.e.,
σ
bb2 = (MADj {bbj })2 ,
(5.1)
where MADj {bbj } = 1.483 med {bbj − medl {bbl }}. The usual tuning constant 1.483 ensures
consistency of the estimate at the normal distribution.
Next, define the residuals of the independent errors ej as b
ej = b
ǫj − bbj 1j . These residuals
are called the conditional residuals. They are used to check the normality and constant
variance conditions; see McKean and Sheather (2009). As an estimate of the variance component σe2 , consider
eij })2 ,
σ
be2 = (MADij {b
(5.2)
Thus estimates of the total variance and intraclass correlation coefficient ρ are respectively
σ2.
be2 and ρb = σ
bb2 /b
σ
b2 = σ
bb2 + σ
b denote the
We next turn our attention to Studentizing the residuals. Let Yb M R = W θ
fit based on the MR estimates. Denote the residuals by
b
ǫM R = Y − Yb M R .
(5.3)
Since the model contains an intercept parameter, we assume without loss of generality that
W is centered.
One result of the theory provided in the appendix is the asymptotic representation of the
MR estimator, which is given by
b = θ + τ (W ∗′ W ∗ )−
θ
m
m
c
X
√
W ′m ϕ[F (ǫj )] + op (1/ n),
j=1
19
(5.4)
where ϕ[F (ǫj )] is the vector with components ϕ[F (ǫij )]. Based on this representation it
follows similar to the independent error case, (McKean et al., 1990), that a first-order approximation to variance-covariance matrix of ǫ is
−
−
∗
′
∗
∗′
∗
′
CMR =
˙ σ 2 A(ρ) + τ 2 W (W ∗′
m W m ) W − τ E[ǫϕ[(ǫ)]]W m (W m W m ) W
−
∗
∗′
−τ E[ǫϕ[(ǫ)]]W (W ∗′
mW m) W m,
(5.5)
where A(ρ) is the block diagonal matrix diag {An1 (ρ), . . . , Anc (ρ)} and Anj (ρ) is defined in
expression (2.2).
In order to compute the standardized residuals, estimates of the parameters in expression
(5.5) are needed. Expressions (5.1) and (5.2) present estimators of σ 2 A(ρ) while expression
(4.5) gives our estimator of τ . A simple moment estimator of the E{ǫϕ[F (ǫ)]} is given by
m
1 X
\
bM R ).
Dj ( θ
E{ǫϕ[F (ǫ)]} =
N j=1
(5.6)
Substituting these estimators for their corresponding parameters in C M R , we have our estib M R . Hence, the vector of studentized residuals based on the MR fit is given by
mator C
b
ǫ∗ = D −1/2b
ǫ,
(5.7)
b M R . In practice, we correct these residuals for the
where D is the main diagonal of C
median before Studentizing. The typical benchmark for studentized residuals is two, that is,
cases whose studentized residuals exceed two in absolute value are often flagged as potential
outliers.
20
6
Asymptotic Relative Efficiency
Assuming Model 2.1, we discuss the asymptotic relative efficiency (ARE) of the MR estimator
bM R and the maximum likelihood estimator (under normality of the random errors and
θ
bM L . For this discussion, consider an estimable function Hθ where H is a
random effects), θ
√
bM L is given by
r × k matrix of full row rank. The asymptotic covariance matrix of nH θ
σ2H
" c
X
(1/n)W ′j B −1
j Wj
j=1
#−1
H ′,
′
2
2
2
−1
−1
where B −1
j = (1 − ρ) [I nj − ρ{1 + (nj − 1)ρ} 1nj 1nj ] and σ = Var(bj + eij ) = σa + σe ,
and ρ = σa2 /(σa2 + σe2 ).
Based on (4.4), it follows that the efficiency of
√
bM R relative to
nH θ
√
bM L is the
nH θ
(1/r)th root of the ratio of the asymptotic generalized variances, (recall that the generalized variance is the determinant of the covariance matrix). Hence, the asymptotic relative
efficiency (ARE) is

1/r
#−1
" c


X
−1
′
′
(1/n)W j B j W j
ARE(MR, ML) = lim σ 2 H
H /{|τ 2 H(V ∗n )− H ′ |}1/r }.
n→∞ 

j=1
(6.1)
When W j 1nj = 0, it follows from (6.1) that the ARE(MR, ML) = σe2 /τ 2 , which is
similar to the ARE between R and LS estimators in a fixed effects linear model. In this
case, the ARE’s corresponding to exchangeable multivariate normal, multivariate t (with η
degrees of freedom), multivariate logistic with standard form and multivariate exponential
distributions, respectively, are given by
3/π, [6/{π(η − 2)}]{Γ(η/2 + .5)}2 , π 2 /8, and (1 − ρ)(3 + 4ρ)2 /3.
21
Note that for most distributions, the ARE’s are independent of ρ and thus achieve full
Wilcoxon efficiency. Thus, there are gains in precision for the MR estimators over the
normal theory estimators for the heavier tailed distributions. Further, at the multivariate
normal distribution, the loss in efficiency of the MR estimator with respect to the normal
theory estimator is less than five percent. Hence, we loose very little efficiency relative to the
optimal estimate in this case. In addition, there are gains in precision of the MR estimators
with respect to the normal theory test for the skewed distributions. In general, the ARE’s
are substantially larger than unity for heavy-tailed distributions. Similar comments can be
made on the efficiencies of the MR and normal theory test statistics.
7
Simulation Study
To assess the benefits that can be achieved using the rank based methods of this article, a
small scale simulation study is performed. We consider a design which has two levels for the
treatment, two centers, but with no covariate. The design is balanced within centers. Let
nij denote the number of observations under Treatment i within center j; i, j = 1, 2. We
considered two cases of sample sizes, namely nij = 4 and nij = 10, so that overall sample
sizes are 16 and 40, respectively. Thus the center sample size for the first situation is only 8.
Let δ = θ1 − θ2 denote the effect of interest. For the random effect and error distributions,
we chose the multivariate t distribution (with 3 degrees of freedom) with scale parameter
σ 2 = 1 and the intraclass correlation coefficient ρ varied over the set {0.1, 0.5, 0.7, 0.9}.
We decided to measure the performance of the procedures based on their tests of H0 :
δ = 0. For procedures, we selected the drop in dispersion test based on the statistic D ∗ with
asymptotic critical values based on its asymptotic null distribution, (4.10); the Van Elteren
(1960) test based on the asymptotic distribution of the gradient S(0) given in (A.1.1) of
22
Appendix A; and the normal theory F -test, (based on the normal theory mixed model). We
chose the nominal α level to be 0.05. We simulated the null model, δ = 0, and alternative
models with δ ranging from 0.25 to 2.00 in increments of 0.25. For each situation, we ran
10,000 simulations.
7.1
Empirical Levels
Table 5 reports the empirical levels of the three procedures. As a check we used the usual
confidence interval for proportions based on a proportional value of 0.05 and a sample size
p
of 10,000. This interval is (0.0456, 0.0543); i.e., 0.05 ± 2 0.05 × 0.95/10000. It can be seen
from Table 5 that the estimated sizes for all the tests were within this interval except for
the one situation for the test based on D ∗ when nij = 4 and ρ = .9.
Table 5: Empirical Levels for a Nominal
Level of 0.05a
(ρ, nij )
Drop
Scores Test
F test
(0.1, 4)
.0520
.0513
.0525
(0.5, 4)
.0539
.0511
.0510
(0.7, 4)
.0509
.0477
.0483
(0.9, 4)
.0548
.0504
.0504
(0.1, 10)
.0503
.0506
.0496
(0.5, 10)
.0520
.0515
.0494
(0.7, 10)
.0482
.0467
.0534
(0.9, 10)
.0534
.0501
.0499
a
Based on 10,000 Simulations, when σ = 1
and the data were generated from a T
distribution with 3 degrees of freedom.
23
7.2
Empirical Powers
The empirical powers of the drop-in-dispersion test, the Van Elteren test, and the F -test are
given respectively in Tables 6-8. Note that the empirical powers increase as either nij , ρ, or
δ increases for all three procedures.
Tables 6-8 show that the drop-in-dispersion test has slightly higher empirical power than
the scores test although both tests have the same asymptotic relative efficiency. However,
both drop-in-dispersion and scores tests have much higher empirical powers than those of
the F test. In general, the drop-in-dispersion test has the highest empirical power among
the three tests.
Table 6: Power of Drop-in-Disersion Testa
(ρ,m) | δ
.25
.5
.75
1
1.25
1.5
1.75
2
(.1, 4)
.0707
.1577
.2851
.4205
.5576
.6583
.7323
.7984
(.5, 4)
.1014
.2423
.4355
.5944
.7156
.7986
.8645
.8871
(.7, 4)
.1366
.3481
.5766
.7242
.8203
.8864
.9186
.9439
(.9, 4)
.2826
.6574
.8399
.9215
.9551
.9718
.9808
.9887
(.1, 10)
.1295
.3331
.5559
.7149
.8173
.8805
.9161
.9379
(.5, 10)
.1887
.4929
.7159
.8407
.8072
.9407
.9606
.9741
(.7, 10)
.2712
.6419
.8313
.9108
.9722
.9798
.9863
.9881
(.9, 10)
.5519
.8869
.9555
.9793
.9899
.9913
.9963
.9972
a
Based on 10,000 simulations with a level of 0.05, δ = θ1 − θ2 , σ = 1, and the
data were generated from a T distribution with 3 degrees of freedom
24
Table 7: Power of Scores Testa
(ρ, m) | δ
.25
.5
.75
1
1.25
1.5
1.75
2
(.1, 4)
.0769
.1499
.2785
.4094
.5432
.6436
.7194
.7918
(.5, 4)
.0967
.2333
.4213
.5803
.7066
.7883
.8602
.8833
(.7, 4)
.1295
.3405
.5635
.7136
.8101
.8809
.9135
.9430
(.9, 4)
.2714
.6384
.8334
.9162
.9526
.9602
.9794
.9886
(.1, 10)
.1266
.3261
.5488
.7120
.8136
.8792
.9119
.9376
(.5, 10)
.1821
.4862
.7109
.8356
.9042
.9367
.9587
.9735
(.7, 10)
.2676
.6330
.8270
.9079
.9720
.9796
.9852
.9868
(.9, 10)
.5411
.8860
.9554
.9798
.9897
.9912
.9960
.9971
a
8
Based on 10,000 simulations with a level of 0.05, δ = θ1 − θ2 , σ = 1, and the data
were generated from a T distribution with 3 degrees of freedom
A Robust Test of Interaction between Model and
Centers
The theory of Sections 2 and 4 was developed under the assumption that Model (2.2) holds
for each center; that is, there is no interaction between the model and the centers. The E9
document of the International Conference on Harmonization, (ICH), (1998) suggests testing
for interaction after fitting a model with only main effects for treatment and center first.
There are various views in the literature on the level of this test. Bancroft and Han (1980)
suggest the significance level of 0.25, while Fleiss (1986) suggests the level of 0.10; see Anello,
O’Neill and Dubey (2005) for further discussion about the choice of level and its impact.
In this section, we offer a robust test based on MR estimators for each center. If a
treatment by center interaction is found, then the results of the study need to be interpreted
with care and efforts need to be directed toward explaining this finding. We recommend
25
Table 8: Power of F Testa
(ρ, m)| δ
.25
.5
.75
1
1.25
1.5
1.75
2
(.1, 4)
.0540
.0714
.0991
.1271
.1514
.1814
.2073
.2440
(.5, 4)
.0650
.0902
.1239
.1611
.1988
.2375
.2895
.3132
(.7, 4)
.0654
.1085
.1556
.2145
.2578
.3000
.3547
.3959
(.9, 4)
.0954
.1759
.2716
.3540
.4196
.4835
.5509
.6032
(.1, 10)
.0662
.1058
.1435
.1906
.2341
.2818
.3243
.3539
(.5, 10)
.0773
.1334
.1830
.8545
.9138
.9459
.9665
.9764
(.7, 10)
.0946
.1694
.2409
.3165
.4589
.4979
.5681
.5750
(.9, 10)
.1472
.2824
.3951
.5124
.6037
.6703
.7300
.7819
a
Power based on 10,000 simulations with a level 0.05, δ = θ1 − θ2 , σ = 1, and the
data were generated from a T distribution with 3 degrees of freedom
that if a treatment by center interaction is found, then the investigator should report the
center-specific treatment effects and an overall estimate of the treatment benefit using the
average of the center specific treatment benefits. The MR estimates discussed in Section 8.1
can be used to estimate these center-specific treatment effects.
8.1
Test for Interaction
Our concern is that the interaction affects the estimators of the fixed effects, i.e., the components of θ in Model (2.1). Hence, our test is a goodness-of-fit test based on center-wise
b
estimators of θ. For each center j = 1, . . . , c, let θ
(j)
denote the R estimator which minimizes
the dispersion function Dj (θ) given in expression (2.3). We use the same rank scores as in
b(j) depends only on the information obtained
Section 2, but, in this case, the estimator θ
b
b = (θ
from center j. Let B
(1)′
b
,...,θ
(c)′
)′ denote the vector of these estimators. Note that the
26
b are independent. Under the null hypothesis that there is no interaction,
components of B
it follows from R estimation theory, (see for example, Chapter 3 of Hettmansperger and
b is asymptotically normal with mean B = (θ ′ , . . . , θ ′ )′ .
McKean, 2011), that B
For notational convenience, assume without loss of generality that the design has full
b is τ 2 times the block-diagonal
column rank p. Then the asymptotic covariance matrix of B
matrix
V B = diag{(W ′m1 W m1 )−1 , . . . , (W ′mc W mc )−1 },
where W mj = (I nj − n−1
j J nj )W j is the centered design matrix for the jth center. Select
the p(c − 1) × pc contrast matrix H so that
b(1)′ − θ
b(2)′ , . . . , θ
b(1)′ − θ
b(c)′ )′ .
b = (θ
HB
Under the null hypothesis, HB = 0. Our test statistic is the Wald type test
b ′ [HV B H ′ ]−1 (H B)/(qb
b
TB = (H B)
τ 2 ),
(8.1)
where q = p(c − 1) is the number of constraints and τb is the estimator of τ discussed in
Section 4.1. Note that the test statistic TB is invariant to the contrast matrix selected.
Under the null hypothesis, it follows that qTB converges to a χ2 distribution with q degrees
of freedom. Similar to the fixed effects linear model case, (Chapter 3 of Hettmansperger
and McKean, 2011), initial studies have indicated that a comparison of TB with F -critical
values leads to better small sample properties. The numerator degrees of freedom are the
number of constraints q. The denominator degrees of freedom depend on the estimator of τ
which is the average of the estimates for each center. Hence, we suggest using minj {nj − p}
denominator degrees of freedom.
27
Note that our test statistic is invariant to the intercept and the random effect bj . It
is, however, not invariant to the interaction between the random effect and the regression
parameters in the model.
8.2
Sub Hypotheses
In practice, we may be interested in testing for interaction between centers and a subset
of the fixed effects. For such a test, simply choose the hypothesis matrix H to select the
effects of interest. The number of constraints q is the number of independent rows of the
matrix H. This would be the numerator degrees of freedom of our test statistic TB . The
denominator degrees of freedom would remain the same. For example, suppose the fixed
effect includes a levels of a treatment. Then, traditionally, interest is in the interaction
between the treatments and the centers. In this case, there are a − 1 effects and, hence,
q = (a − 1)(c − 1).
8.3
Monte Carlo Study of Interaction Test
For this study we consider a one-way model with three treatment levels, a single covariate,
and two centers. The design is balanced with nj = 24 observations per center, for a total
sample size of 48. The model is
yijk = µ + αi + xijk β + bj + γij + eijk ,
(8.2)
for i = 1, 2, 3; j = 1, 2; and k = 1, . . . , 8. The parameters are the effects αi and the regression
coefficient β. As in the last section, bj denotes the random effect for center j while γij denotes
the random interaction effect between the center and the model. As usual, we assume that
the random variables bj , γij , and eijk are independent with variance, respectively, σb2 , σγ2 ,
28
and σe2 . For this model the null hypothesis for interaction is that σγ2 = 0. Our interest is in
this hypothesis, so, without loss of generality, for the study we set the effects to 0. Also, we
chose normal distributions for the random effects. We set σb2 = 4 and selected six settings
for σγ2 given by 0 (null case), 0.5, 0.75, 1.0, 1.5 and 2.0. For error distributions we chose the
normal with variance 0.5 and a family consisting of three contaminated normal distributions.
The first of these has percentage of contamination 10% with the ratio of the variance of the
contaminated part to good part at 30, CN(0.10, 30). For the second, the percentage of
contamination was increased to 15%, CN(0.15, 30); while for the third the variance ratio
was increased to 50, CN(0.10, 50). Two thousand simulations were run for each situation.
In order to compare our test with the traditional test for interaction between the centers
and the treatment effects, we used the sub-hypothesis matrix H which selects only the
differences in the treatment effects, as discussed in Section 8.2. Hence, we considered the
test based on the statistic TB , using the F -critical values with q = (3 − 1)(2 − 1) = 2 and
nj − p = 21 degrees of freedom. For the traditional test we chose the likelihood ratio test
based on fitting the full model (8.2) and the reduced model (full model without γij ). We
used the lme function in R to fit the models, followed by the anova function to obtain the
likelihood ratio test.
Table 9 displays the empirical levels and power of the three tests for the nominal 0.05
level. Over all four distributions, including the normal, the LRT is conservative. The test
based qTB based on χ2 -critical values is moderately liberal. On the other hand, the test
based on TB based on F -critical values has empirical levels quite close to the nominal values.
In terms of empirical power, for the normal errors situation, the LRT and the robust
tests have similar power. For the contaminated normal distributions, though, the gap in
empirical powers between the LRT and the robust procedures is quite large, between 20
and 30 percentage points in many of the cases. Of the two robust procedures, we would
29
Table 9: Simulation Results for Tests of Interaction for Normal and Contaminated Normal
Errors
Normal Errors
σγ2
Level 0.00 0.50 0.75 1.00 1.50 2.00
TB
.05
.051 .670 .757 .794 .854 .896
qTB
.05
.078 .708 .803 .821 .872 .914
LRT
.05
.027 .630 .753 .778 .841 .887
CN(0.10, 30) Errors
σγ2
Level 0.00 0.50 0.75 1.00 1.50 2.00
TB
.05
.043 .465 .569 .643 .723 .784
qTB
.05
.069 .528 .619 .693 .755 .818
LRT
.05
.019 .254 .347 .404 .505 .579
CN(0.15, 30) Errors
Level 0.00 0.50 0.75 1.00 1.50 2.00
TB
.05
.038 .409 .521 .590 .673 .752
qTB
.05
.057 .467 .577 .638 .709 .787
LRT
.05
.017 .149 .203 .277 .353 .441
CN(0.10, 50) Errors
Level 0.00 0.50 0.75 1.00 1.50 2.00
TB
.05
.038 .422 .540 .614 .690 .736
qTB
.05
.067 .477 .594 .673 .735 .770
LRT
.05
.013 .183 .251 .306 .398 .464
recommend the test based on TB . It possessed validity of level while displaying robustness
of power.
9
Conclusion
In this article, we have presented a highly efficient analysis of a mixed model that is commonly
used in multi-center clinical trials. For such models, center is a random effect and the fixed
effects include treatment and possibly one or more covariates. Our estimates include rankbased estimation of the fixed effects and robust estimates of the variance components. We
30
developed studentized residuals which correct for both variance of the errors and location
in factor space. These serve as a useful diagnostic tool in checking quality of fit and in the
discovery of potential outliers. Our estimates of the fixed effects are for general rank scores.
The inference presented is complete, including, besides of fixed effects estimation, estimation
of standard errors, confidence intervals for contrasts, tests of general linear hypotheses,
prediction of random effects, and estimates of the variance components. The estimates of
the fixed effects minimize a convex objective function and are easily computed. We have
also presented a nonparametric test based on center R estimators for the interaction between
the random effect and the fixed effects and in the accompanying simulation study showed
that it possesses robustness in validity and power. We have illustrated the robustness of
our procedures on an example in a multicenter-center clinical trail setting where the data
contained outliers.
As our discussion on asymptotic relative efficiency shows, these MR estimates are highly
efficient compared to the traditional maximum likelihood estimates. In our small sample
simulation study assuming heavy tailed error structure, the MR analyses maintained level
and were much more powerful than the maximum likelihood analysis. The insensitivity to
outliers makes our complete MR analysis an important tool in detecting outliers and as a
robust alternative to the traditional maximum likelihood analysis.
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A.1
Theory
Assume Model 2.1 holds. By reparameterizing this model, we can work with a design matrix
that has full column rank. Let X j denote the reparameterized design matrix and let β
denote the new parameters. Let X ′ = [X ′1 · · · X ′c ]. Then Xβ = W θ. Hence, Model 2.1
can be written as Y j = X j β + 1nj bj + ej , for j = 1, . . . , c. For this appendix, this is the
notation that we use. Assume that X is centered.
We need the usual assumptions:
(A1): f (t) is absolutely continuous and has finite Fisher information;
′
(A2): n−1
j X j X j → Σj > 0 as n → ∞;
(A3): and lim sup1≤i≤nj hjni = 0, as n → ∞, where hjni denotes the ith leverage value of
the projection matrix onto the range of X j .
∗
−1
′
Let V x = lim n−1 X ∗′
m X m and X m = [X m1 · · · X mc ] where X mj = (I nj − nj J nj )X j .
Since Xβ = W θ, the definition of the dispersion function remains the same. We refer to
b = Argmin D(β). The negative of
the dispersion function as D(β) and the R estimator as β
Pc
P nj
the gradient of D(β) is S(β) = j=1 S j (β), where S j (β) = i=1
xm,ij aj [Rj (yij − x′m,ij β)],
and xm,ij is the (i, j)th row of X m and Rj denotes the intra-center rankings. Equivalently,
b R solves S(β
b R ) = 0.
β
A.1.0.1
Proof of (4.4)
Because of the invariances of the R estimator, we can assume that true parameter β is 0.
Under the assumptions, the asymptotic theory of rank-based linear model procedures hold
34
for each center; see Jureckova (1971), Jaeckel (1972), McKean and Hettmansperger (1976),
Puri and Sen (1985), and Hettmansperger and McKean (2011). For the gradient for the jth
√
center S j (β), (1/ nj )S j (0) is asymptotically normal with mean 0 and variance-covariance
′
matrix n−1
j X mj X mj ; hence,
1
D
√ S(0) → Np (0, V x ).
(A.1.1)
n
Further, for each center, the linearity result for the gradient implies that √1n S(β n ) =
√
√
√1 S(0) − τ −1 V x nβ + op (1), uniformly for
nβ n = O(1).
n
n
This leads to the asymptotic quadraticity result: D(β n ) = Q(β n ) + op (1), uniformly for
√
nβ n = O(1), where
Q(β n ) =
√
1√ ′
nβ n V x nβ n − β ′n S(0) + D(0).
2τ
(A.1.2)
Note that the unique minimizer of Q(β n ) is
f = τ V −1 n−1 S(0).
β
n
x
(A.1.3)
√ f
2 −1
Based on the result (A.1.1), nβ
n → N(0, τ V x ), in distribution. Using the asymptotic
quadraticity result (A.1.2) and the same convexity argument as Jaeckel (1972), we can show
√ f c
that n(β
n − β n ) → 0, in probability. Hence, (4.4) follows.
A.1.0.2
Proof of (4.10)
By reparameterizing, hypotheses (4.7) can be written as H0 : β 2 = 0 versus HA : β 2 6= 0,
b denote
where β = [β ′1 β ′2 ]′ . Let X = [X 1 X 2 ] be the corresponding partition of X. Let β
b denote the reduced model estimate, i.e., the estimate based
the full model estimate and let β
1
b ) − D(β).
b
on the fit of the model Y = X 1 β 1 + e. The reduction in dispersion is RD = D(β
1
Using the quadraticity results, we have
RD =


−1
x1
τ
 V

−
S(0)′ V −1

x
2n
O
35

O 
 S(0) + op (1).
O
By a matrix argument, it follows under H0 , that RD/(τ /2) converges in distribution to a χ2
random variable with q degrees of freedom. Hence (4.10) is true.
36