Extension of Multiple Range Tests to Group Means with Unequal Numbers of Replications

In many fields of research, one is faced with the task of comparing the effects of treatments which have been replicated unequally. This happens for a number of reasons. In an experiment on animals, some may get sick and have to be removed from the experiment. In some experiments, the amount of material available for certain treatments may not be as much as for other treatments. If the experimenter has specified orthogonal contrasts that he is interested in before he runs the experiment, one can test the various treatment effects by an F-test after the treatment sum of squares has been partitioned into individual degrees of freedom for each orthogonal contrast. If the experimenter has not specified orthogonal contrasts, one is faced with the problem of deciding which treatments are significantly different. Several writers, including Duncan, Keuls, Newman, and Tukey, have developed multiple range tests to show differences among treatments that have been replicated the same number of times and when nothing was specified concerning the treatments. Duncan 1 compares the above methods and gives citations. This extension to unequal numbers of replications will be exemplified with reference to Duncan's New Multiple Range Test, but is applicable to any of the above writers' tests; all one has to do is use their tabled ranges. In Duncan's test for an equal number of replications, the difference between any two ranked means is significant if the difference exceeds a shortest significant range. This shortest significant range is designated by R, and is obtained by multiplying the standard error of a mean, s,, by a given value, zn2, obtained from a table of significant studentized ranges which Duncan has tabled for both the 5% and 1% test. In Duncan's terminology, n2 is the degrees of freedom of the error mean square and p = 1, 2, * *, t is the number of means concerned. Consider an experiment with five treatments, A, B. C, D, and E, each replicated n times. Suppose on ranking the means from low to high one obtains