A Test for Two-Sample Repeated Measures Designs: Effect of High-Dimensional Data Boonyarit Choopradit and Samruam Chongcharoen

Problem statement: High-dimensional repeated measures data are increa singly encountered in various areas of modern science sinc classical multivariate statistics, e.g,. Hotellin g’s T, are not well defined in the case of high-dimensio al data. Approach: In this study, the test statistics with no specific form of covariance matrix for anal yzing high-dimensional two-sample repeated measures designs with common equal covariance are p roposed. The asymptotic distributions of the proposed test statistics also were derived. Results: A simulation study exposes the approximated Type I errors in the null case very well even though the n umber of subjects of each sample as small as 10. Numerical simulations study indicates that the prop osed test have good power. Application of the new tests is demonstrated using data from the body-weig ht of male Wistar rats example. Conclusion: The proposed test statistics have an asymptotically dis tributed as standard normal distributions, under the null hypothesis. Simulation studies show that t ese test statistics still accurately control Type I error and have quite good power for any the covar iance matrix pattern considered.


INTRODUCTION
Repeated measurements across time (occasion, drug or treatment) on the same subject (e.g., patient, animal, cell culture, block) are frequently observed in several scientific fields, for example in medicine, pharmaceutical, agriculture, industrial and social sciences (Zangiabadi and Ahrari, 2005;Sharifi et al., 2008;Arbabi et al., 2009;Tu and Koh, 2010;Mosallanejad et al., 2011). This type of data commonly called univariate repeated measures data (Davis, 2002). A design contain such a data is called repeated measures design. A main advantage of this design is that test results are more powerful since subjects serve as their own controls and then variability among the subjects due to individual differences is eliminated. The simplest repeated measures design is when data is collected as a sequence of equally spaced points in time. There exist several methods of analyzing repeated measurement data. For an overview see Everitt (1995) and Keselman et al. (2001). However, these studies pertain only to the case when the dimension of repeated measurements is not exceeds the number of subjects. Let x 1k = (x 1k1 , x 1k2 , …, x 1kp ) T and x 2l = (x 2l1 , x 2l2 , …, x 2lp ) T be multivariate random vectors each of p repeated observations measured (times) on kth subject in sample 1 and lth subject in sample 2, respectively, where the samples are drawn independently from two populations (groups) and k = 1, 2, …, n 1 , l = 1, 2, …, n 2 . Suppose E(x 1k ) = µ 1 , E(x 2l ) = µ 2 , Var(x 1k ) = Σ 1 and Var(x 2l ) =Σ 2 , where µ i is the population mean vector and Σ i is the population covariance matrix of group i for i = 1, 2. Then covariance matrix in group i for j = 1, 2, …, n i , i = 1, 2.
The notation ⊕ is the direct sum and a sample from each population with sample sizes denoted by n 1 and n 2 . Traditionally, if data with fixed dimension are normally distributed, people would use the famous classical Hotelling's T 2 test for two-sample which is defined as ( ) ( ) 1 2 1 2 T n n 2 1 1 2 1 2 n nT x x is the ith sample mean vector as defined above, for i = 1, 2 and the pooled sample covariance matrix Σ be defined by Σ = . It is well known that under the null hypothesis, n p 1 2 np T − + has a central F-distribution (Davis, 2002;Anderson, 2003) with p and n-p+1 degrees of freedom where n = n 1 + n 2 -2. Nowadays, however, modern society demands that the use of subjects (e.g., animals, patients) in scientific experiments must be adequately controlled and reduced. This position is enforced by ethical committees who authorize or deny permission for these kind of experiments. As a consequence the statistician sometimes has to work with very small subjects, but the large dimension of repeated measurements on each subject. Unfortunately, if n<p, called high-dimensional data, the Hotelling's T 2 test is not well defined because the sample covariance matrix becomes singular.
Recently, there has been some interest in investigating the behavior of high dimensionality n<p that can be found in the literature. A non-exact test for two sample case when n<p was first proposed by Dempster (1958). Many works have been published on hypothesis testing for means when both p and n go to infinity with the ratio p/n must remain bounded, Bai andSaranadasa (1996), Fujikoshi et al. (2004); Srivastava and Fujikoshi (2006);Srivastava (2007;2009) ;Schott (2007) and Srivastava and Du (2008). In addition, when sample covariance matrix does not have an inverse, Chongcharoen (2011) proposed one way to modify a sample covariance matrix. Yahya et al. (2011) proposed approach for feature selection in high dimensional data. For analysis of the one-sample highdimensional repeated measures designs, see Choopradit and Chongcharoen (2011). However, the mainstream attempt in this study interests in the analysis of twosample repeated measures designs for high-dimensional data. In an influential work, Ahmad (2008) proposed the modified version of the ANOVA-type statistic for analysis of two-sample repeated measures designs when the data are multivariate normal and the dimension can be large compared to the sample size using a modified Box's approximation (Box, 1954) based on quadratic and bilinear forms.
In this study, we introduce test statistics for testing interaction and time effects which follow a multivariate approach to repeated measures for analyzing highdimensional two-sample repeated measures designs. The test statistics asymptotically follows a standard normal distribution and is not affected by even a large the dimension of repeated measurements, p. The test statistics are derived with no specific form of covariance matrix but under equal population covariance assumption.
The modified ANOVA-type statistic: Here and in the following, for any natural number p, I p denotes the p×p identity matrix of dimension p, 1 p = (1,…,1) T p×1 a pdimensional column vector of ones, T p p p J 1 1 = a square matrix of ones and P p = I p -p −1 J p is the centering matrix. Note that P p is a projection matrix. Usually, the null hypothesis to be tested is H 0 : Hµ = 0. The matrix H can be formulated in distinctive settings depending on the objectives of the experimental research. As similar null hypothesis given in Brunner et al. (1999) and Ahmad (2008), we can also write H 0 : Gµ = 0 where G = H T (HH T ) − H is the general hypothesis matrix whereas (HH T ) − denoting a generalized inverse of HH T and G is a projection matrix. We note that Gµ = 0 if and only if Hµ = 0. For the two-sample repeated measures designs (Davis, 2002), the situation when repeated measurements at p time points are obtained from two independent groups of subjects is considered. Let x ij = (x ij1 , x ij2,…, x ijp ) T denote the vector of observations from the jth subject in group i for j = 1, 2, …, n i , i = 1, 2. The matrix G can be formulated to test any appropriately hypotheses as G AB = P 2 ⊗P p (interaction effect hypothesis), It is clear from the hypotheses above that the high dimensionality influences only interaction effect hypothesis and time effect hypothesis whereas group effect hypothesis is a univariate hypothesis of the means of two independently samples from normal populations and does not depend on p. Group effect hypothesis, therefore, is not of our main interest because it can be tested using the usual t-test for two independent samples (Rencher, 2002;Johnson and Wichern, 2002). When n<p, Σ is singular. Any test statistic involved the inversion of Σ will not exist. Ahmad (2008) proposed the modified ANOVA-type statistic under covariance matrices of two groups are the same, using quadratic and symmetric bilinear forms based on Box's approximation (Box, 1954) as followed. x~N µ , Σ where the samples are drawn independently from two populations. Define the differences x 1k -x 2l , for all k = 1, 2, …, n 1 , l = 1, 2, …, n 2 , where E(x 1k -x 2l ) = µ 1 -µ 2 and Var(x 1k -x 2l ) = Σ 1 +Σ 2 . Under ( ) is the symmetric bilinear form. For assuming that the two groups have common covariance matrix Σ, that is Σ 1 = Σ 2 = Σ, the estimators:  H : P µ µ 0 + = . As similar in the interaction effect, therefore, the generating matrix without any loss of generality for the time effect is G = P p . Let x 1k and x 2l defined above with where the samples are drawn independently from two populations. Define the sums x 1k + x 2l , for all k = 1, 2, …, n 1 , l = 1, 2, …, n 2 , where E(x 1k + x 2l ) = 1 2 µ µ + and is the symmetric bilinear form. For assuming that the two groups have common covariance matrix Σ, that is 1 2 Σ Σ Σ = = , the estimators:

METERIALS AND METHODS
A proposed test: When n<p, only the first n eigenvalues of Σ will be non-zero and the smallest eigenvalues will tend to zero pretty quickly as the dimensionality grows. Consequently, many of the classical techniques, e.g., the Hotelling's T 2 test, encounter a mathematical barrier and becomes inapplicable since in the case Σ is degenerate. Hence, we look for tests which do not require the nonsingularity of Σ . We will derive the asymptotic distributions of the test statistics for two-sample repeated measures design when data are high dimension (n<p) under general conditions only that Σ is a positive definite covariance matrix (denoted as Σ>0). The test statistics considered under covariance matrices of two groups are equal and both p and n go to infinity. We should note that the limiting ratio p/n is allowed to be greater than one. Consequently, these tests can be used when n<p. From hypotheses presented as before the formulations of the test statistics for interaction effect involves differences of the vector x 1k and x 2l , but those for time effect involves sums of the vector x 1k and x 2l . Finally, the asymptotic distributions of these test statistics for the interaction effect and time effect are same since covariance of differences and sums remains the same. Hence, the details in the following section will be shown only for the interaction effect while for the time effect follows the same pattern.

The interaction effect:
For testing the null hypothesis of no interaction effect as above we first consider the following result.
These assumptions are similar to those imposed by Bai and Saranadasa (1996) and Srivastava and Du (2008) for the study of their two-sample testing procedures and for the study of their testing mean vector in one-and two-sample cases, respectively.
The following theorem establishes the asymptotic normality of our test statistic for testing the hypothesis: Using cumulant generating function of quadratic and bilinear forms (Searle, 1997) and moment-cumulant relationships (Rao, 2001) we obtained ( ) Recall Theorem 1 since ( ) Now we show that (A.1) is asymptotically distributed as N (0, 1) by using Lyapounov's theorem. We usually note that ( )
To complete the construction of our test statistic, we need only find a ratio-consistent estimator of 4tr (GΣG) 2 . A natural estimator of ( ) is similarly to the test obtained in Bai and Saranadasa (1996); Srivastava and Du (2008) and Zhang and Xu (2009) when both p and n go to infinity.

The time effect:
We consider testing the null hypothesis of no time effect as above. As mentioned above, the proof of the following theorem can be obtained similar that of Theorem 1, thus we present it without proof.
Theorem 3: For assuming Σ 1 =Σ 2 =Σ, let x hj ∼N p (0,Σ), Σ>0, where j = 1, 2, …, h n , h = 1, 2 and let G be a matrix defined the same as above. Then The proof of Theorem 4 is similar to that of Theorem 2. Due to Theorem 4 the test with an α level of significance will rejects B

Simulation study:
We assess the effectiveness of the proposed two-sample tests for high dimensional data by means of a Monte Carlo simulation study. Simulation results were obtained so as to assess the accuracy of the asymptotic standard normal distribution in approximating the actual null distributions of AB B * * T and T . If the distribution is derived correctly for the proposed test statistics, then we would expect that the estimated Type I errors should be close to the nominal significance level setting. We also estimate empirical powers of our proposed test statistics. We begin with the description of the parameter selection for our simulation.

RESULTS AND DOSCUSSION
Simulation result: Table 1 show the closeness of estimated Type I errors with nominal significance level setting at α = 0.05. As can be seen from the Table 1, the estimated Type I errors of the test statistic AB * T for interaction effect and the test statistic B * T for time effect are close to the nominal significance level setting at α = 0.05 reasonably well in all cases considered including small sample size as 10. This is shown that the proposed tests are reasonable tests. Moreover, we note that the accuracy of Type I error control is not affected by changing the covariance pattern and by the increasing the dimension p.       The corresponding empirical power curves of the test statistic AB * T for the interaction effect with the aforementioned four different covariance patterns are summarized in Table 2 and Fig. 1. To make it clear, the four covariance patterns are listed separately. We observe that the empirical power functions of AB * T have a steep ascending slope and is very high for each sample size groups and all covariance patterns considered. The corresponding empirical power of AB * T test give good power when η≥0.6 for n 1 = 10 and n 2 = 20 andη≥0.4 for n 1 = 20 and n 2 = 30. As expected, the power of AB * T test increases for increasing both p and η when fixed the sample sizes for all cases considered as well. In addition, the power of AB * T test also increases for increasing the sample sizes when both p and η are fixed. Moreover, the empirical power is also unaffected by changing the covariance pattern. In a separate simulation where the test statistic B * T were considered for the time effect as displayed in Table 3 and Fig. 2, the results obtained similar as interaction effect which reported above.
Analysis of the body-weight of male Wistar rats data: The data and the experimental description for a motivating example to deal with the high-dimensional data is reported in Brunner et al. (2002). The bodyweight of male Wistar rats was observed over a period of 22 weeks to assess the toxicity of a drug. A group of ten animals was given a placebo, while a second group of ten animals was given a high dose of the drug. The main question to be addressed is whether the body-weights of the two test groups differ in their evolution over time. For this data, we have n i = 10, i = 1, 2 and p = 22.
We get test values of AB * T 0.6273 = − with a corresponding p-value is 0.7348 and B * T 6.2078 = with a corresponding p-value < 0.0001. We conclude that a p-value highly significant for time effect but a pvalue accepting the null hypothesis of no interaction effect.

CONCLUSION
In this study, we developed test statistics for analyzing high-dimensional two-sample repeated measures designs when the data are multivariate normal. We began by highlighting the previous work in the literature describing test for the hypothesis of no interaction and no time effects. We proposed test statistics AB * T for no interaction effect and B * T for no time effect which have an asymptotically distributed as standard normal distributions, under the null hypothesis, with common covariance Σ. One of the main advantages of these statistics is that they can be used for both unstructured and factorially structured repeated measures designs when the underlying hypothesis matrix G is appropriately defined. Monte Carlo simulation studies in this study show that the general behavior of these test statistics with asymptotic standard normal distribution still accurately control Type I error and have quite good power for any the covariance matrix pattern considered with a moderate sample size and any large dimension p. The strong support is provided in the simulation results. In our study, the power simulation results suggest that it may be assumed that the quality of the proposed test statistics, AB * T and B * T , are maintained even if the dimension is in the thousands and it may also be applied for the microarray data analysis.