There is a rich literature concerning multiple comparison of several normal means; see for example Miller (1981), Hochberg and Tamhane (1987), Hsu (1996), and Benjamini and Braun (2002). One of the most famous procedures is Scheffé (1953) which allows any contrasts of the normal means to be assessed by using simultaneous confidence intervals. This paper focuses on the problem of assessing any contrasts of several simple linear regression models, which generalises the problem of assessing any contrasts of several normal means. Spurrier (1999) was the first to study this problem by constructing simultaneous confidence bands for all contrasts of several simple linear regression models but under some restrictive assumptions. His work was followed by Spurrier (2002), Bhargava and Spurrier (2004), and Liu et al., 2004, Liu et al., 2007 and Liu et al., 2009 among others who constructed simultaneous confidence bands for finite, such as pairwise and treatment-control, comparisons of several simple or multiple linear regression models.
Construction and application of confidence bands for one single linear regression model have been extensively studied by Working and Hotelling (1929), Gafarian (1964), Wynn and Bloomfield (1971), Bohrer and Francis (1972), Casella and Strawderman (1980), Uusipaikka (1983), Naiman (1986), Sun and Loader (1994), Sun et al. (1999), Efron (1997), Al-Saidy et al. (2002), Piegorsch et al. (2005), and Liu et al., 2005 and Liu et al., 2008 and Liu and Hayter (2007), to name just a few.
Many large clinical studies compare two or more dose levels with a placebo control using several hundred or thousand patients. The primary and secondary study objectives (and thus the comparisons of interest) are often required to be specified in advance before study begins. Multiple test procedures tailored to these objectives are applied to guarantee a strict type I error rate control. In addition, post hoc analysis (also called data snooping) are often conducted to investigate the new treatment in a variety of different subgroups, which are often defined only after the primary data analysis. Examples of subgroups include age groups, disease severity at the beginning of study, races, gender, etc. or combinations of these. Given the confirmatory environment of later phase clinical trials, simultaneous confidence bands for all contrast of several linear regression models are therefore needed. These confidence bands are also useful when comparing the treatment effect of a new compound as a function of a covariate other than dose for several subgroups of patients; see the example in Section 4.
To be specific, suppose observations (xij,yij)(xij,yij) are available from kk (k≥3k≥3) simple linear regression models
View the MathML sourceyij=αi+βixij+ϵij,j=1,…,ni,i=1,…,k,
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where View the MathML sourcebi=(αi,βi)T are the unknown regression coefficients of the iith regression line, and ϵijϵij are assumed to be independently and identically distributed N(0,σ2)N(0,σ2) random errors with σ2>0σ2>0 unknown. The design matrix for the iith regression line is given by View the MathML sourceXi=(1,xi) where View the MathML source1=(1,…,1)T and View the MathML sourcexi=(xi1,…,xini)T. Let View the MathML sourcebˆi=(αˆi,βˆi)T denote the least squares estimator of View the MathML sourcebi, and View the MathML sourceσˆ2 denotes the usual pooled error mean square with distribution View the MathML sourceσˆ2∼σ2χν2/ν, where View the MathML sourceν=∑i=1k(ni−2).
Let CC be the set of vectors View the MathML sourcec=(c1,…,ck)T such that View the MathML source∑i=1kci=0, and let View the MathML sourcex=(1,x)T. The focus of this paper is the construction of 1−α1−α level simultaneous confidence bands for all the contrasts among the kk regression lines over a given finite or infinite interval (l,u)(l,u) of the covariate xx. Specifically, we consider confidence bands of the form
equation(1)
View the MathML source∑i=1kcixTbi∈∑i=1kcixTbˆi±rσˆ∑i=1kci2xT(XiTXi)−1xfor allx∈(l,u)and allc∈C,
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where −∞≤l<u≤+∞−∞≤l<u≤+∞ are given numbers, and rr is a suitably chosen critical constant so that the simultaneous coverage probability of all the confidence bands in (1) is equal to the pre-specified level 1−α1−α. The value of rr depends on α,k,ν,(l,u)α,k,ν,(l,u), and X1,…,XkX1,…,Xk.
Spurrier (1999) provides elegant distributional results which allow rr to be computed exactly by using numerical integration but only for the special case of (l,u)=(−∞,+∞)(l,u)=(−∞,+∞) and X1=⋯=XkX1=⋯=Xk. In many applications, the requirement of equal design matrices across groups is too restrictive, however. Furthermore, confidence bands on a finite interval (l,u)(l,u) are more useful since a regression model is often a reasonable approximation only over a limited range of xx, and restricting xx to (l,u)(l,u) results in narrower confidence bands which allow sharper statistical inference. In this paper a simulation-based method is given to approximate rr so long as the design matrices are non-singular. The proposed method can achieve any desired accuracy in the approximation of rr with a sufficiently large number of replications in the simulation process. In this general setting, it is unlikely that useful distributional results can be established for exact computation of rr.
This paper is organized as follows. In Section 2 the simulation method and the required computational implementation are described. In Section 3 numerical results are provided to validate the accuracy of the simulation method. Application of the method to a real problem considered in Hewett and Lababidi (1982) is given Section 4. Finally, the Appendix contains some proofs.
