ABSTRACT
In Chapter 2 we considered procedures for comparing two distributions, using independent samples. Here we consider a sim ilar problem in relation to k(> 2) d istributions. We denote these d istributions as F i , F2, ■■■, Fk . The general ksample problem is th a t of testing the null hypothesis of homogeneity, namely,
Ho : Ci = F-2 = • ■ ■ = Fk-(4.1)
F irst we en terta in all alternatives, which are called global alternatives. These are sta ted as
H gl '■ not Ho, i.e., there exists at least one pair (Fi, F j) such tha t Fi ^ Fj. (4.2)
This discussion will be followed by procedures for certain types of restricted alternatives, which will be defined now. Unlike the case of two distributions, here two types of one-sided alternatives, called completely ordered alternatives and partially ordered alternatives, are of interest. The completely ordered al ternatives are
H c o i ■ Fi s t < F2 st < . . . st < Fk, (4.3)
or
H Co 2 : Fi s t > F2 s t > . . . s t > Fk . (4.4)
The partially ordered alternatives is a very large class. We consider only one type of partia lly ordered alternatives, which are called simple tree alterna tives. These are appropriate ones to en terta in when one of the trea tm en ts is a control trea tm en t and we want to compare each of the o ther k — 1 tre a t ments w ith the control. In th is context, w ithout loss of generality, F\ is taken as the d istribu tion of the response variable under the control trea tm ent. The alternatives of in terest are
H t r i ■ F \ s t < Fi, for a t least one i(i = 2 , . . . , k), (4.5)
or
H t R 2 ■ Fi s t > Fi, for a t least one i(i = 2 , . . . , k). (4-6)
It may be noted th a t for k = 2 . H c o i is the same as H t r i and H ccn is the same as H t r 2 -
As in C hap ter 2, we will consider bo th complete samples and samples w ith right-censored observations. In the case of complete samples, the two situa tions of in terest are ( 1 ) binary response and (2 ) continuous response. Of course, in the discussion about samples th a t may contain some censored observations, the response variable is a continuous variable. Procedures for the binary s tud ies and categorical da ta situations will be considered first. This discussion is lim ited to tests for global alternatives; however, the case of continuous re sponse restricted alternatives also will be discussed. We will also consider complete and censored d a ta settings w ith a continuous response variable.
