CHAPTER 1: Introduction to Design CHAPTER 2: Principles of Estimation and Inference: Means and Variance CHAPTER 3: Design and Analysis of Single-Factor Experiments: Completely Randomized Design CHAPTER 4: Single-Factor Experiments Having Repeated Measures on the Same Element CHAPTER 5: Design and Analysis of Factorial Experiments: Completely-Randomized Design CHAPTER 6: Factorial Experiments: Computational Procedures and Numerical Example CHAPTER 7: Multifactor Experiments Having Repeated Measures on the Same Element CHAPTER 8: Factorial Experiments in which Some of the Interactions are Confounded CHAPTER 9: Latin Squares and Related Designs CHAPTER 10: Analysis of Covariance

Statistical Principles in Experimental Design

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

Extension of multiple range tests to group means with unequal numbers of replications

It is sometimes useful in an analysis of variance to split the treatments into reasonably homogeneous groups. Multiple comparison procedures are often used for this purpose, but a more direct method is to use the techniques of cluster analysis. This approach is illustrated for several sets of data, and a likelihood ratio test is developed for judging the significance of differences among the resulting groups.

A Cluster Analysis Method for Grouping Means in the Analysis of Variance

The relationship between antioxidant activity and antimutagenicity of various tea extracts (green tea, pouchong tea, oolong tea, and black tea) was investigated. All tea extracts exhibited markedly antioxidant activity and reducing power, especially oolong tea, which inhibited 73.6% peroxidation of linoleic acid. Tea extracts exhibited a 65-75% scavenging effect on superoxide at a dose of 1 mg and 30 - 60% scavenging effect on hydrogen peroxide at a dose of 400 microgram. They scavenged 100% hydroxyl radical at a dosage of 4 mg except the black tea. Tea extracts also showed 50 - 70% scavenging effect on alpha, alpha-diphenyl-beta-picrylhydrazyl radical. The antioxidant activity and the scavenging effects on active oxygen decreased in the order semifermented tea > nonfermented tea > fermented tea. Tea extracts showed strong antimutagenic action against five indirect mutagens, i.e., AFB1, Trp-P-1, Glu-P-1, B[a]P, and IQ, especially oolong and pouchong teas. The antioxidant effect of tea extracts was well correlated to their antimutagenicity in some cases but varied with the mutagen and antioxidative properties.

Antioxidant activity of various tea extracts in relation to their antimutagenicity

Multiple range and multiple f tests

Miultiple range tests have been developed by several writers, for example, D. Newmiian [8], M. Keuls [5], J. W. Tukey [10] anid D. B. Duincani [3], for testinig differenices between several treatmenlt means in cases inl which all such differences are of equal a priori interest. These tests, which are also described in recent textbooks, for example, W. T. Federer [4, chapter 2], have been worked out for data in which the treatment means are homoscedastic (have equal variances) and are uncorrelated. Recently, C. Y. Kramer [6] has presented a simple miethod for extending these procedures to give useful tests for differences between means with unequal replications, the method being applicable to aniy set of heteroscedastic unicorrelated means. In a subsequent paper [7], the same authorl has given further extensions to tests of mneanis which are also correlated, such as the adjusted means from analyses of covariance or from incomplete block designs. Similar work has also been done by E. Bleicher [1] and P. G. Sanders [9] in extendinig a multiple F test to making tests in lattice and rectangular lattice designs. One purpose of this paper is to present a more complete method for these extensions which lnecessarily sacrifices a little in simplicity but is more powerful, especially in cases in which the differences between the mneans have appreciably different varianices. Another purpose is to incdicate briefly the closeniess of the properties of these complete tests of heteroscedastic and correlated meanis to those of the corresponding tests of homoscedastic and unicorrelated mieans. Incidcental to these main ptirposes, a shor t-cut skippinig priniciple, useful in applyilng multiple ranige tests to a large lntum-ber of treatmenit means (or totals), is also presented.

Multiple range tests for correlated heteroscedastic means

SUMMARY The problem of testing a comparison c stuggested by the data is first identified with the mtultiple comparisons problem for the class C of a priori equally plauisible comparisonis. A multiple comparisons dilemma is then illustrated by a simple example E emphasizing the impossibility of developing an acceptable a-level Neyman-Peaison test for c. A recent empirical Bayes additive losses and exchangeable priors (EBALEP) approach is discussed and illutstrated in getting a simple solution for E. A brief review is giveii of the use of the same approach by Waller, Dixon and Dunican in getting the k-ratio t test and interval for the all-pairwise differences problem. These rules are showii to apply to the all contrasts problem as well and several worked examples are given for their application. The k-ratio rtules are seen to have considerably increased power and low type-i error probabilities in dealiing with heterogeneous and near honmogeneous treatments, respectively.

David B. Duncan

Papers

Multiple range and multiple f tests

Multiple range tests for correlated heteroscedastic means

t TESTS AND INTERVALS FOR COTIPARISONS SUGGESTED BY THE DATA

273. Query: Multiple Comparison Methods for Comparing Regression Coefficients

Multiple Regression Combining within- and between-Plot Information