Preface to First Edition - lib

ANALYSIS USING R 257mate ˆθ, K is the matrix defining linear functions of the elemental parametersresulting in our parameters of interest ϑ and m is a k-vector of constants. Thenull hypothesis states that ϑ j = m j for all j = 1, ...,k, where m j is somepredefined scalar being zero in most applications. The global hypothesis H 0 isclassically tested using an F-test in linear and ANOVA models (see Chapter 5and Chapter 6). Such a test procedure gives only the answer ϑ j ≠ m j for atleast one j but doesn’t tell us which subset of our null hypotheses actuallycan be rejected. Here, we are mainly interested in which of the k partial hypothesesH j 0 : ϑ j = m j for j = 1, ...,k are actually false. A simultaneousinference procedure gives us information about which of these k hypothesescan be rejected in light of the data.The estimated elemental parameters ˆθ are normally distributed in classicallinear models and consequently, the estimated parameters of interest ˆϑ = Kˆθshare this property. It can be shown that the t-statistics( ˆϑ1 − m 1se(ˆϑ 1 ) , ..., ˆϑ)k − m kse(ˆϑ k )follow a joint multivariate k-dimensional t-distribution with correlation matrixCor. This correlation matrix and the standard deviations of our estimated parametersof interest ˆϑ j can be estimated from the data. In most other models,the parameter estimates ˆθ are only asymptotically normal distributed. In thissituation, the joint limiting distribution of all t-statistics on the parametersof interest is a k-variate normal distribution with zero mean and correlationmatrix Cor which can be estimated as well.The key aspect of simultaneous inference procedures is to take these jointdistributions and thus the correlation among the estimated parameters ofinterest into account when constructing p-values and confidence intervals. Thedetailed technical aspects are computationally demanding since one has tocarefully evaluate multivariate distribution functions by means of numericalintegration procedures. However, these difficulties are rather unimportant tothe data analyst. For a detailed treatment of the statistical methodology werefer to Hothorn et al. (2008a).14.3 Analysis Using R14.3.1 Genetic Components of AlcoholismWe start with a graphical display of the data. Three parallel boxplots shownin Figure 14.1 indicate increasing expression levels of alpha synuclein mRNAfor longer NACP-REP1 alleles.In order to model this relationship, we start fitting a simple one-way ANOVAmodel of the form y ij = µ + γ i + ε ij to the data with independent normalerrors ε ij ∼ N (0, σ 2 ), j ∈ {short, intermediate, long}, and i = 1, ...,n j . Theparameters µ + γ short , µ + γ intermediate and µ + γ long can be interpreted asthe mean expression levels in the corresponding groups. As already discussed© 2010 by Taylor and Francis Group, LLC