last observation analysis in anova and ancova - Institute of Statistical ...

860 BIN CHENG, JUN SHAO AND BOB ZHONG2. One-Way ANCOVAConsider a clinical trial consisting of I treatments, n i patients randomized totreatment group i, and T scheduled post-baseline visits for each patient. Supposethat under treatment i, n it patients drop out after visit t. Hence, n i1 +· · ·+n iT =n i and (n i1 , . . . , n iT ) has the multinomial(n i , p i1 , . . . , p iT ) distribution, where p itis the population proportion of patients dropping out after visit t under treatmenti. We assume a pattern-mixture one-way ANCOVA model, i.e., for patients whodrop out after visit t, their last observed responses y itk ’s are independent withmeans µ it + b ′ z itk , where k = 1, . . . , n it , t = 1, . . . , T , i = 1, . . . , I, the µ it ’sare unknown fixed treatment effects, z itk is a q-vector of covariates observed foreach patient, and b is a q-vector of unknown parameters. Unlike the y-responsevariable, the z-covariate for a patient does not vary with t, although we usethe notation z itk to specify it. No other condition is imposed on the dropoutmechanism (i.e., dropout may be nonignorable).When there is no dropout, testing for treatment effect may be carried outby using responses from the end of the study and the method of ANCOVA.When there are dropouts, as we discussed in Section 1, the LOAN considers thehypothesisH 0 : µ 1 = · · · = µ I , (1)where µ i = p i1 µ i1 + · · · + p iT µ iT .The LOCF test is the ANCOVA test that treats y itk as the observation atthe end of the trial. Since µ it usually changes with t, one wonders what theLOCF tests for. If (1) is the hypothesis of interest, there is the question ofvalidity of the LOCF test. The following result shows when the LOCF test isasymptotically valid for (1). Since Shao and Zhong (2003) showed that whenI ≥ 3, the LOCF test is asymptotically wrong for testing (1) in a one-wayANOVA without covariates, we only consider the case of I = 2 treatments.In this paper, χ 2 ddenotes the chi-square distribution with d degree of freedom,while χ 2 d,α and F l,m,α denote, respectively, the 1 − α quantiles of χ 2 dand the F-distribution with degrees of freedom l and m, where α is a given nominal level.Let[MSTR = (ȳ 1.. − ˆb ′¯z 1.. ) − (ȳ 2.. − ˆb ′¯z ] / 2 2∑ T∑ ∑n it2.. )a 2 itk ,i=1 t=1 k=112∑ T∑ ∑n it [MSE =(y itk − ȳ i.. ) −n 1 + n 2 − q − 2ˆb ′ 2(z itk − ¯z i.. )],i=1 t=1 k=1where z itk is the covariate value associated with y itk , ȳ i.. and ¯z i.. are, respectively,the averages of y itk and z itk over the indexes t and k,2∑ T∑ˆb = (˜Z ′ ∑n it−1 ˜Z) (y itk − ȳ i.. )(z itk − ¯z i.. ),i=1 t=1 k=1