Basic statistics and the inconsistency of multiple comparison procedures

This paper has two main themes. First, the various statistical measures used in this journal are summarized and their interrelationships described by way of a flow chart. These are the pooled standard deviation, the pooled variance or mean square error (MSE), the standard error of each treatment mean (SEM) and of the difference between two treatment means (SED), and the least difference between two means which is significant at (e.g.) the 5% level of significance (LSD(5%)). The last three measures can be displayed as vertical bars in graphs, and the relationship between the lengths of these bars is graphically illustrated. It is suggested that the LSD is the most useful of these three measures. Second, when the experimenter has no prior hypotheses to be tested using analysis of variance contrasts, a multiple comparison procedure (MCP) that examines all pair-wise differences between treatment means, may be appropriate. In this paper a fictitious experimental data set is used to compare several well-known MCPs by focussing on a particular operating characteristic, the consistency of the results between an overall analysis of all treatments and an analysis of a subset of the experimental treatments. The procedure that behaves best according to this criterion is the unrestricted least significant difference (LSD) procedure. The unrestricted LSD is therefore recommended with the proviso that it be used as a method of generating hypotheses to be tested in subsequent experimentation, not as a method that attempts to simultaneously formulate and test hypotheses.