Showing papers on "Multiple comparisons problem published in 1985"
1 Mar 1985
TL;DR: In this paper, a review and definitions of the types and frequency rates of statistical errors with regard to pairwise multiple comparisons are presented, and the most appropriate and most informative method of statistical analysis of the data will be that procedure which provides the best answers to those questions.
Abstract: Pairwise multiple comparisons of treatment means are appropriate in the statistical analysis of some agronomic experiments. This paper includes a review and definitions of the types and frequency rates of statistical errors with regard to pairwise multiple comparisons. Of the 10 pairwise multiple comparisons procedures described herein, the least significant difference is the procedure of choice when the appropriate contrasts among treatments each involve only two of the treatment means. This choice is based on considerations of error rates, power, and correct decision rates as well as simplicity of computation. Additional index words: Duncan’s multiple range test, Least significant difference, Statistical analysis, Waller-Duncan k-ratio t test. sons may be sensible and meaningful, and it may well be a logical part of the experimental plan to perform them. The purposes of this paper are: l) to review and define the types and frequency rates of statistical errors with regard to pairwise multiple comparisons, 2) to describe a number of the pairwise multiple comparisons procedures that are available, and 3) to suggest hat the least significant difference is always the procedure of choice when the appropriate contrasts among treatments each involve only two of the treatment means. We hope readers will find this presentation less confusing and more satisfactory than those given in statistical textbooks oridinarily used in teaching courses on the design and analysis of agronomic experiments. TYPES AND RATES OF STATISTICAL ERRORS FOR PAIRWISE COMPARISONS rI~t E OBJECTIVE of a well-designed experiment is o answer questions of concern to the experimenter. The most appropriate and most informative method of statistical analysis of the data will be that procedure which provides the best answers to those questions. Most designed experiments include treatments selected for the purpose of answering specific questions. Frequently these specific questions are best answered through the computation and testing of those meaningful, single-degree-of-freedom linear contrasts that were "built-in" to the experiment when the particular treatments were chosen by the experimenter. In many cases the set of linear contrasts will be orthogonal as well as meaningful. For examples of experiments for which the design and objectives suggest meaningful, perhaps orthogonal, single-degree-of-freedom linear contrasts to explain variation among treatments, see Bryan-Jones and Finney (1983), Carmer (1978), Chew (1976, 1977), Dawkins (1983), Johnson and Berger (1982), Little (1978, 1981), Mead and Pike (1975), Nelson Rawlings (1983), or Petersen (1977). There are, on the other hand, some experiments that the experimenter designs with the intent of examining the differences between members of each pair of treatments. Common examples of such a situation are performance trials to evaluate sets of crop cultivars. Other examples include herbicide, fungicide, insecticide, and other pesticide screening trials. Here pairwise compari’ Contribution from the Dep. of Agronomy, Univ. of Illinois, 1102 S. Goodwin Ave., Urbana, IL 61801. 2 Professor f biometry and professor f biometry and soil fertility, Dep. of Agronomy, Univ. of Illinois, Urbana. Let the true difference between two treatment means be represented by: where ri and zj represent he true effects of the ith and flh treatments, respectively. With the use of a pairwise multiple comparisons procedure one of three possible decisions is made concerning each pair of means; i.e., each &ij. The possible decisions are: 1) t~ij < 0; or 2) ~ij = 0; or 3) t~ij > 0. The correctness of a particular decision based on a pair of observed means depends on the true or parameter values of the means. The latter are, in general, unknown. Several kinds of incorrect decisions or errors are possible (Table 1). If the parameter values of two means are really equal, i.e., ~j = 0, reaching decision 1 or 3 on the basis of observed means results in a Type I error, which occurs when a true null hypothesis i rejected. On the other hand a Type II error occurs when a false null hypothesis not rejected. Thus reaching decision 2 on the basis of observed means results in a Type II error if the two true means really are not equal, i.e., 6~j ~ 0. Still another kind of error is committed if decision 1 is reached, but decision 3 is actually correct, or if decision 3 is reached, but decision 1 is actually correct. These are called reverse decisions or Type III errors. In summary then, for any given pair of treatments, the experimenter will either make the correct decision or one of the three types of errors. Table 1. Types of statistical errors possible when comparing two observed treatment means. Decision based True situation on observed means fiij < 0 6ij = 0 6ij > 0 I. t~ij < 0 Correct decision Type I error Type Ul error 2. ~ij = 0 Type 11 error Correct decision Type 11 error 3. ~ij > 0 Type Ill error Type I error Correct decision
190 citations
TL;DR: In this paper, a procedure for constructing exact simultaneous confidence intervals for a finite set of multiple comparisons among three or four groups is presented, which is based on an unbiased estimator of the mean values having a normal distribution with covariance matrix σ2V, where V is an arbitrary known positive definite matrix.
Abstract: A procedure for constructing exact simultaneous confidence intervals for a finite set of multiple comparisons among three or four groups is presented. These confidence intervals are hyperbolic and based on an unbiased estimator of the mean values having a normal distribution with covariance matrix σ2V, where V is an arbitrary known positive definite matrix. This procedure is used to study the amount of conservativeness of Tukey—Kramer intervals for pairwise comparisons in an unbalanced one-way analysis of variance design.
28 citations
1 Jan 1985
TL;DR: In this paper, the experimental design may be any one to which an analysis of variance model is applied; here, for simplicity, we assume the design is a one-way classification.
Abstract: Suppose that there are several treatments to be compared in an experiment. The experimental design may be any one to which an analysis of variance model is applied; here, for simplicity, we assume the design is a one-way classification (i.e., the only factor in the design is the treatment factor). However, usually the analysis of variance F-tests do not satisfy the needs of the experimenter, who often wishes to make specific comparisons between certain treatment means. This leads to multiple tests if there are more than two treatments and the problem is how to make valid inferences concerning the treatment comparisons of interest in the experiment when there is more than one to be considered. The early work by Duncan (1951), Scheffe (1953) and Tukey (1953) laid the foundations for the subject of multiple comparisons or, as it is sometimes called, simultaneous inference. For a review, see Miller (1966 or 1981).
25 citations
25 citations
TL;DR: In this article, the authors used expanded confidence intervals for all pairwise comparisons of treatments using an honest ordering difference rather than Tukey's "honest siginificant difference" to illuminate some hypothesis testing problems.
Abstract: Confidence intervals for location parameters are expanded (in either direction) to some “crucial” points and the resulting increase in the confidence coefficient investigated.Particaular crucial points are chosen to illuminate some hypothesis testing problems.Special results are dervied for the normal distribution with estimated variance and, in particular, for the problem of classifiying treatments as better or worse than a control.For this problem the usual two-sided Dunnett procedure is seen to be inefficient.Suggestions are made for the use of already published tables for this problem.Mention is made of the use of expanded confidence intervals for all pairwise comparisons of treatments using an “honest ordering difference” rather than Tukey's “honest siginificant difference”.
23 citations
TL;DR: In this article, the studentized differences between means are used to give ordering conclusions on all pairs of populations under consideration, and a more efficient method for setting up confidence intervals using Tukey's honest significant difference is suggested.
Abstract: In multiple comparison procedures often the ordering of populations is of more interest than the testing of hypotheses. In such a situation it is important to control the probability of a Type III error (introduced by Harter 1957 to indicate that one population is concluded to be better than another when actually it is worse). The studentized differences between means may be used to give ordering conclusions on all pairs of populations under consideration. Although this may be done by setting up confidence intervals using Tukey's honest significant difference, a more efficient method (when the confidence intervals themselves are not of interest) is suggested here, and tables are provided for its implementation.
21 citations
TL;DR: A randomization approach to multiple comparisons is developed for comparing several growth curves in randomized experiments and gives all pairwise comparisons among the mean growth curves associated with four treatments in an animal experiment using a Youden square design.
Abstract: A randomization approach to multiple comparisons is developed for comparing several growth curves in randomized experiments. The exact Type I probability error rate for these comparisons may be prespecified, and a Type I error probability for each component test can be evaluated. These procedures are free of many of the standard assumptions for analyzing growth curves and for making multiple comparisons. An application of the procedure gives all pairwise comparisons among the mean growth curves associated with four treatments in an animal experiment using a Youden square design, where growth curves are obtained on monitoring hormone levels over time.
14 citations
TL;DR: There are three types of multiple comparisons: all-pairwise multiple comparisons (MCA), multiple comparisons with the best (MCB), and multiple comparison with a control (MCC) as discussed by the authors.
Abstract: There are three types of multiple comparisons: all-pairwise multiple comparisons (MCA), multiple comparisons with the best (MCB), and multiple comparisons with a control (MCC). There are also three levels of multiple comparisons inference: confidence sets, subset comparisons, test of homogeneity. In current practice, MCA procedures dominate. In correct attempts at more efficient comparisons, in the form of employing lower level MCA procedures for higher level inference, account for the most frequent abuses in multiple comparisons. A better strategy is to choose the correct type of inference at the level of inference desired. In particular, very often the simulataneous comparisons of each treatment with the best of the other treatments (MCB) suffice. Hsu (1984b) gave simultaneous confidence intervals for θi − maxj≠iθj having the simple form [− (Yi −maxj≠i Yj − C) (Yi−maxj≠i Yj + C)+]. Those intervals were constrained, sothat even if a treatment is inferred to be the best, no positive bound on how much it i...
10 citations
14 Jun 1985
TL;DR: In this paper, nonparametric quantile estimation of quantiles and density functions under censoring, discrete failure models, and multiple comparisons was investigated. But the authors focused on the problem of selecting an asymptotically optimal design for comparing several new treatments with a control.
Abstract: : Major results have been obtained in the areas of nonparametric estimation of quantiles and of density functions under censoring, discrete failure models, and multiple comparisons. In particular, smooth nonparametric estimators of quantile functions from censored data were developed which give better estimates of percentiles of the lifetime distribution than the usual product-limit quantile function. Also, smooth density estimators from censored data were investigated using maximum penalized likelihood procedures. Several parametric models were proposed for the case of discrete failure data. These models provide a better fit to such data than some previously used discrete models. Finally, new methods of constructing simultaneous confidence intervals for pairwise differences of means of normal populations were developed, and the problem of selecting an asymptotically optimal design for comparing several new treatments with a control was solved. Work is continuing on the study of properties of kernel type quantile function estimators and development of goodness-of-fit tests for the model assumptions in accelerated life testing. Keywords: Nonparametric quantile estimation; Density estimation; Right-censored data; Discrete failure models; Multiple comparisons; Accelerated life testing.
5 citations
TL;DR: In a number of papers dating from 1955, A.M. Wassef and his co-worker,F. Z. A. Messih, have demonstrated the application of the analysis of variance technique to the study of levelling discrepancies.
Abstract: In a number of papers dating from 1955,A.M. Wassef and his co-worker,F. Z. A. Messih, have demonstrated the application of the analysis of variance technique to the study of levelling discrepancies. The usual method of testing for the significance of means by the Fisher or F-test was used by these authors.
3 citations
TL;DR: In this paper, the experiment-wise error rate when orthogonal contrasts are tested following a significant F test in a balanced one-way classification is given for a balanced classification.
Abstract: Summary
Tables are given for the experiment-wise error rate when orthogonal contrasts are tested following a significant F test in a balanced one-way classification.