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A Multivariate Variance Components Model for Analysis of Covariance In Statistical Science 2009, Vol. 24, No. 2, 223–237 DOI: 10.1214/09-STS294 c Institute of Mathematical Statistics, 2009 A Multivariate Variance Components Model for Analysis of Covariance in Designed Experiments James G. Booth, Walter T. Federer, Martin T. Wells and Russell D. Wolfinger Abstract. Traditional methods for covariate adjustment of treatment means in designed experiments are inherently conditional on the ob- served covariate values. In order to develop a coherent general method- ology for analysis of covariance, we propose a multivariate variance components model for the joint distribution of the response and co- variates. It is shown that, if the design is orthogonal with respect to (random) blocking factors, then appropriate adjustments to treatment means can be made using the univariate variance components model obtained by conditioning on the observed covariate values. However, it is revealed that some widely used models are incorrectly specified, leading to biased estimates and incorrect standard errors. The approach clarifies some issues that have been the source of ongoing confusion in the statistics literature. Key words and phrases: Adjusted mean, blocking factor, conditional model, orthogonal design, randomized blocks design. 1. INTRODUCTION Cochran and Cox (1957), Snedecor and Cochran (1967), as well as more recent texts such as Milliken This article concerns the adjustment of treatment and Johnson (2002). We are specifically interested means in designed experiments to account for one or in settings in which the covariates are random vari- more covariates. Analysis of covariance has a long ables, not fixed by the design. Also, we generally history, dating back to Fisher (1934). Much of its suppose that the covariate values are not affected development in the context of designed experiments by the treatments, for example, because they were arXiv:1001.3011v1 [stat.ME] 18 Jan 2010 followed soon after. Examples can be found in many measured prior to application of the treatments. The classical textbooks, including Federer (1955), multivariate mixed model we propose can be modi- fied to handle problems in which this assumption is James Booth e-mail: [email protected] and Martin not valid. However, there are additional inferential Wells e-mail: [email protected] are Professors and issues in this case. Walter Federer e-mail: [email protected] was Emeritus Professor, Department of Biological Statistics and As a canonical example we discuss in detail the Computational Biology, Comstock Hall, Cornell randomized complete blocks (RCB) design with a University, Ithaca, New York 14853, USA. Russell single covariate. In the classical analysis of this de- Wolfinger is Director of Scientific Discovery and sign the blocks are treated as fixed, and the covariate Genomics, SAS Institute Inc., Cary, North Carolina is included as a predictor. The least squares treat- 27513, USA e-mail: Russ.Wolfi[email protected]. ment means obtained from the fixed blocks model This is an electronic reprint of the original article fit adjust the arithmetic treatment means to ac- published by the Institute of Mathematical Statistics in count for differences in the average covariate mea- Statistical Science, 2009, Vol. 24, No. 2, 223–237. This surements among the treatments. Including the co- reprint differs from the original in pagination and variate block means as an additional predictor has typographic detail. no effect on the least squares fit because differences 1 2 BOOTH, FEDERER, WELLS AND WOLFINGER at the block level are already accounted for. Treating show that conditional maximum likelihood can elim- blocks as fixed effectively confines the scope of infer- inate the bias. ence to only those blocks and covariate values in the An outline of the remainder of the paper is as fol- study. In particular, the standard errors obtained lows. In the next section we look back historically from the fixed effects model for the least squares and attempt to explain why some fundamental is- treatment means do not account for repeated sam- sues in analysis of covariance are still unresolved. pling of blocks. The randomized complete blocks design is discussed However, in most applications the blocks in the in detail in Section 3. In Section 4 we show how the study can be viewed as a random sample from a pop- bivariate model for the randomized complete blocks ulation of interest, and it is of interest to extend the design can be generalized to orthogonal designs, and scope of inference to the population of blocks. An to allow adjustment for multiple covariates. In the obvious modification of the classical model in this orthogonal case the conditional model for the re- case is to simply treat the block effects as random; sponse given the covariates implied by the multivari- that is, to fit a (univariate) mixed model with two ate mixed model for the joint distribution turns out variance components. Under the standard indepen- to be a univariate mixed model. This implies that dence and normality assumptions, adjusted treat- appropriate adjustment of treatment means can be ment means obtained from the mixed model have accomplished using standard software. The method- the same form as the classical least squares means, ology is applied to some standard examples of or- except that the covariate regression coefficient is thogonal designs. The nonorthogonal case is discussed a weighted average of the estimates obtained from in Section 5. Here we show that appropriate adjust- intra- and inter-block regressions. ment cannot be accomplished by fitting a univari- A key point made in this paper is that both uni- ate mixed model. However, an EM algorithm for variate fixed and mixed models for analysis of co- fitting a general multivariate linear variance com- variance are inherently conditional on the measured ponents model is developed using arguments that covariate values. An obvious question is, therefore, parallel those for the univariate case, as discussed what joint distribution for response and covariate in Searle, Casella and McCulloch (1992), Chapter 8. leads to the conditional models. We show that by The methodology is applied to balanced incomplete simply treating the block effects as random in the block designs and unbalanced designs. We conclude randomized complete blocks design setting, one is with some discussion in Section 6. implicitly assuming that the marginal block vari- ance for the covariate is zero; that is, an implied 2. HISTORICAL PERSPECTIVE model for the joint distribution that is not realistic. As a result, the adjusted treatment means obtained Since analysis of covariance in designed experi- from the naive, univariate mixed model are biased. ments has such a long history, readers may won- However, by starting with a bivariate variance com- der why confusion over such basic modeling issues ponents model for the joint distribution of response persists even today. We attempt to explain this by and covariate, one is led to a sensible univariate con- discussing the topic in its historical context. Ar- ditional model, which properly accounts for the de- guably, the heyday of analysis of covariance was pre- sign with respect to the covariate. The idea of a 1960, predating the widespread use of matrix alge- bivariate model is suggested in Cox and McCullagh bra in statistics and clearly before the availability [(1982), Section 7], but not fully developed. Multi- of high-speed computing. Matrices are two hundred variate variance components models are discussed and some years old but their use in statistics only be- in Khuri, Mathew and Sinha [(1998), Chapter 10], came commonplace in the late 1950s. The very first but the application to analysis of covariance is not paper in the first issue of the Annals of Mathematical considered. The fully conditional approach was also Statistics by Wichsell (1930) is entitled “Remarks advocated by Neuhaus and McCulloch (2006) who on Regression,” yet it has no matrices. It is likely consider the situation where random effects in a that the lateness of adoption of matrices arose from generalized linear mixed model may be correlated their being treated as a topic in pure mathematics with one of the predictors. Classical likelihood ap- and, hence, their practical use in statistical mod- proaches lead to inconsistent estimators in this set- eling remained hidden. In the classic design books ting. Results in Neuhaus and McCulloch (2006) Cochran and Cox (1957) and Federer (1955), there COVARIANCE IN DESIGNED EXPERIMENTS 3 is no mention of matrices, whereas in Kempthorne Some statisticians, however, have advo- (1952) the design problem is formulated as a general cated a more general model which allows linear model but is not applied to analyze any ad- the intra-block regression coefficients to vanced designs. Kempthorne [(1952), page 66] even be different from the inter-block regres- notes that, at the time, there did not appear to sion. In this paper, all models are such be a complete matrix formulation of the general that the intra-block regression is the same linear model anywhere in the literature. In unbal- as that for the inter-block regression. It is anced data it was a longtime puzzle why statistical difficult for this writer to visualize situa- methods gave two different least squares estimates tions allowing separate regressions. of fixed effects in a one-way classification depending It turns out that the “[s]ome statisticians” Zelen upon whether one assumed that one effect was zero, or that all the effects summed to zero. The literature referred to were right, but why? Clearly their argu- had certainly not kept up with R. C. Bose’s (1949) ments were not entirely convincing at the time. The concept of estimability. Nor were 1950s design and bivariate variance components model described in linear model researchers aware that Penrose’s (1955) Section 3.3 reveals that Zelen’s two statements are generalized inverse matrix could be used to solve incompatible.
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