Abstract
The popularity of the classical general linear model (CGLM) is attributable mostly to its ease of fitting and validating; however, the CGLM is inappropriate for correlated observations. In this paper we explore linear models for correlated observations with an exchangeable structure (Arnold in J Am Stat Assoc 74:194–199, 1979). For the case of \(N>1\) repeated measures observations having site-dependent or site-independent covariates, the maximum likelihood estimates (MLEs) of the model’s parameters are derived, likelihood ratio tests are obtained for relevant model building hypotheses, and some Monte Carlo simulation studies are performed to illuminate important aspects of the models and their tests of hypotheses. For the case of site-independent covariates, closed-form solutions exist for the MLEs and exact tests can be constructed for the model building hypotheses. Simulations revealed that these exact tests remain robust in the presence of moderate skewness or outliers. However, these fortuitous closed-form occurrences vanish for the case of site-dependent covariates. In order to ameliorate this deficiency, some Monte Carlo simulations are performed to estimate the bias of these MLEs, the probability of a multimodal likelihood, and the suitability of the limiting chi-squared approximation to the model building hypotheses. These simulations reveal that the estimated biases of the slope parameters are negligible for sample size combinations (n, N) as small as (2, 6). Likewise, this sample size combination resulted in only an approximate 1% estimated probability of a multimodal likelihood, which drastically decreased with the increase of either n or N. Moreover, the limiting \(\chi ^2\) distributional assumption appears to hold reasonably well for a sample size of \(N=100\), regardless of the value of n. Finally, we provide examples of fitting our model and conducting tests of hypotheses using two medical datasets.
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References
Akdemir D, Gupta AK (2011) Array variate random variables with multiway Kronecker delta covariance matrix structure. J Algebr Stat 2(1):98–113
Anderson TW (2003) An introduction to multivariate statistical analysis, 3rd edn. Wiley, Hoboken
Arnold SF (1973) Application of the theory of products of problems to certain patterned covariance matrices. Ann Stat 1(4):682–699
Arnold SF (1976) Applications of products to the generalized compound symmetry problem. Ann Stat 3:227–233
Arnold SF (1979) Linear models with exchangeably distributed errors. J Am Stat Assoc 74:194–199
Azzalini A, Capitanio A (1999) Statistical applications of the multivariate skew normal distribution. J R Stat Soc Ser B 61:579–602
Casella G, Berger RL (1990) Statistical inference. Duxbury Press, Pacific Grove
Coelho CA, Roy A (2017) Testing the hypothesis of a block compound symmetric covariance matrix for elliptically contoured distributions. TEST 26(2):308–330
Consul PC (1966) On the exact distributions of the likelihood ratio criteria for testing linear hypotheses about regression coefficients. Ann Math Stat 37:1319–1330
Craig RG (1979) Autocorrelation in LANDSAT data. In: Proceedings of the 13th international symposium on remote sensing of environment, pp 1517–1524
Dutilleul P (1999) The MLE algorithm for the matrix normal distribution. J Stat Comput Simul 64:105–123
Fletcher R (1987) Practical methods of optimization, 2nd edn. Wiley, New York
Leiva R (2007) Linear discrimination with equicorrelated training vectors. J Multivar Anal 98:384–409
Liang Y, von Rosen D, von Rosen T (2015) On estimation in hierarchical models with block circular covariance structures. Ann Inst Stat Math 67(4):773–791
Lin TI, Ho HJ, Chen CL (2009) Analysis of multivariate skew normal models with incomplete data. J Multivar Anal 100:2337–2351
Lin TI, Wang WL (2013) Multivariate skew-normal linear mixed models for multi-outcome longitudinal data. Stat Model 13:199–221
Lu N, Zimmerman DL (2005) The likelihood ratio test for a separable covariance matrix. Stat Probab Lett 73:449–457
Mardia KV (1970) Measures of multivariate skewness and kurtosis with applications. Biometrika 57:519–530
Mardia KV (1974) Applications of some measures of multivariate skewness and kurtosis in testing normality and robustness studies. Sankhya B 36:115–128
Olkin I, Press S (1969) Testing and estimation for a circular stationary model. Ann Math Stat 40:1358–1373
Olkin I (1973) Testing and estimation for structures which are circularly symmetric in blocks. In: Kabe DG, Gupta RP (eds) Multivariate statistical inference. North-Holland, Amsterdam, pp 183–195
Opheim T, Roy A (2019) Revisiting the linear models with exchangeably distributed errors. In: Proceedings of the American Statistical Association, pp 2677–2686
Perlman MD (1987) Group symmetry covariance models. Stat Sci 2:421–425
Rao CR (1945) Familial correlations or the multivariate generalizations of the intraclass correlation. Curr Sci 14:66–67
Roy A, Leiva R (2011) Estimating and testing a structured covariance matrix for three-level multivariate data. Commun Stat Theory Methods 40(10):1945–1963
Roy A, Filipiak K, Klein D (2018) Testing a block exchangeable covariance matrix. Statistics 52(2):393–408
SAS Institute Inc. SAS/STAT User’s Guide Version 9.3, Cary, NC: SAS Institute Inc. 2009
Sperling MR, Gur RC, Alavi A, Gur RE, Resnick S, O’Connor MJ, Reivich M (1990) Subcortical metabolic alterations in partial epilepsy. Epilepsia 31(2):145–155
Srivastava M, von Rosen T, von Rosen D (2008) Models with a Kronecker product covariance structure: estimation and testing. Math Methods Stat 17:357–370
Szatrowski TH (1982) Testing and estimation in the block compound symmetry problem. J Educ Stat 7(1):3–18
Tsukada S (2018) Hypothesis testing for independence under blocked compound symmetric covariance structure. Commun Math Stat 6:163–184
Viroli C (2012) On matrix-variate regression analysis. J Multivar Anal 111:296–309
Wang WL, Lin TI (2016) Maximum likelihood inference for the multivariate mixture model. J Multivar Anal 149:54–64
Wang WL, Lin TI, Lachos VH (2018) Extending multivariate-t linear mixed models for multiple longitudinal data with censored responses and heavy tails. Stat Methods Med Res 27(1):48–64
Žežula I, Klein D, Roy A (2018) Testing of multivariate repeated measures data with block exchangeable covariance structure. TEST 27(2):360–378
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The authors want to thank the Editor-in-Chief and the two anonymous reviewers for their careful reading, valuable comments and suggestions that led to a quite improved version of the manuscript.
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Opheim, T., Roy, A. Linear models for multivariate repeated measures data with block exchangeable covariance structure. Comput Stat 36, 1931–1963 (2021). https://doi.org/10.1007/s00180-021-01064-9
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DOI: https://doi.org/10.1007/s00180-021-01064-9