The Bonferroni and the Scheffe Multiple Comparison Procedures

where b = (X'X) Y'X'y is the least squares estimator of fl. It is easily shown that z is N (6, o₂ V) where V = R (X'X) 'R'. The usual unbiased estimator of o. 2 is s2= (y -Xb) ' (y -Xb) / (Tk). In situations in which we wish only to decide whether H is true or not we can use a direct test of H such as an F test. It is perhaps more common that when H is rejected we want to know which components of 6 are different from zero and of the non-zero components which are positive and which negative. In this situation we have a multiple decision problem and a natural solution is to use an induced test. As an example suppose in the case q = 2 that we wish to test the hypothesis H: 01 =02 = 0. Since H is true if and only if the separate hypotheses H1: 01 = 0 and H2: 02= 0 are both true, this suggests a sequence of separate tests which will induce a test of H. Testing the two hypotheses H1 and H2 where we are interested in whether 01 or 62 or both are different from zero induces a multiple decision problem in which the four possible decisions are