Showing papers on "Pairwise comparison published in 1985"

Pairwise multiple comparisons of treatment means in agronomic research

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

Multiple criteria approach to the facilities layout problem

Abstract: Thin paper presents a multiple criteria model for combining quantitative and qualitative approaches to plant layout. The quantitative factor (work flow) is weighted by the qualitative factor (closeness rating) to form the model. The objective is to minimize the total weighted work flow volume between departments. An heuristic approach is used on an initial layout to improve it in a multiple puss step-by-step pairwise exchange. The results indicate that the model does an excellent job, especially in separating those departments which have an undesirable closeness rating.

Assessing (Software) Reliability Growth Using a Random Coefficient Autoregressive Process and Its Ramifications

Abstract: In this paper we motivate a random coefficient autoregressive process of order 1 for describing reliability growth or decay. We introduce several ramifications of this process, some of which reduce it to a Kalman Filter model. We illustrate the usefulness of our approach by applying these processes to some real life data on software failures. Finally, we make a pairwise comparison of the models in terms of the ratio of likelihoods of their predictive distributions, and identify the "best" model.

Comparing Draws for Single Elimination Tournaments

Abstract: Tournaments are used to select a single winner from a group of participants in a sporting event or a paired-comparison experiment. This study compares different draws, or pairings, of teams in single-elimination tournaments under a set of relatively nonrestricting assumptions about the participating teams' pairwise probabilities of winning. We analyze and compare draws for four-team tournaments using various criteria, then attempt to generalize the results to eight-team tournaments. For example, only one-four team draw maximizes the probability that the best team wins for all pairwise probabilities, whereas eight draws are possibly optimal for eight-team tournaments.

Multi-Objective Analysis with Subjective Information

Abstract: A method is presented for incorporating subjective information into multi‐objective evaluations. The method is based upon an eigenvalue and eigenvector analysis and structures multi‐objective evaluations into a series of hierarchies in which pairwise comparisons are made. The method is demonstrated in the design of an aquatic monitoring network. Theoretical aspects of the approach are reviewed, including measures of subjective inconsistency, the sensitivity of inconsistency to pairwise comparisons, subjective scaling factors, and sensitivity of final, multi‐objective weights. An interactive computer program for the application of the technique is described.

Pairwise versus joint ranking: Another look at the Kruskal-Wallis statistic

Abstract: SUMMARY A test statistic for the k-sample location problem is constructed by appropriately combining all pairwise two-sample Wilcoxon tests. The result is an analogue of the Kruskal-Wallis statistic in the sense that the Pitman asymptotic relative efficiency between the two is one. However, the approximate Bahadur efficiency of the analogue relative to the Kruskal-Wallis statistic is shown to be greater than or equal to one at every alternative. The standard k-sample linear rank statistics for testing equality of the location parameters of k populations are based on the joint ranking of the observations in the combined samples. It is also possible to construct k-sample test statistics in this setting by combining all {k(k -1) pairwise two-sample linear rank statistics. This can be done to yield analogues to the standard k-sample linear rank statistics in the sense that the Pitman asymptotic relative efficiency between such a pair is one. This is referred to as the pairwise rank approach, and we note that the question of pairwise versus joint ranking has recently received attention in the multiple comparisons literature as well, as by, for example, Koziol & Reid (1977), Voshaar (1980), Fligner (1984) or Fairley & Pearl (1984). In ? 2, the pairwise rank analogue of the Kruskal & Wallis (1952) statistic is developed. Although the Pitman efficiency between the procedures is one, the approximate Bahadur (1960) efficiency of the pairwise rank analogue relative to the Kruskal-Wallis test is shown to be greater than or equal to one at every alternative. A simple algebraic relationship between the Kruskal-Wallis statistic and its analogue is established which helps to explain these efficiency results. The technique for constructing pairwise rank analogues in the general scores case is also considered. Section 3 describes other problems in which it may be fruitful to create k-sample statistics through appropriate combinations of two-sample statistics.

Non-binary Social Choice: Revealed Preferential Interpretation

Abstract: In this paper, the framework of social choice theory is transformed in terms of the revealed preference relation. The reader may realize that this transformation relaxes the traditional framework without essentially changing Arrow's (1963) original requirements and that his General Possibility Theorem (GPT) and the related results so far achieved still hold. The introduction of the revealed preference relation is motivated by a traditional narrow interpretation of Arrow into a choice functional framework in which the "base relation" is used. The "base relation" pays attention to choices over pairs only, and all the choices from the set of larger-than-pairs are actually given no role in the analysis. However, because of its "binary relational" structure (see Sen, 1977), Arrow's original framework is appropriate for any size of subsets of alternatives, not only for pairs, and it does not necessarily require the choice based on all non-empty subsets of the set of all alternatives. If we are seriously to consider the generality of his General Possibility Theorem, Arrow's framework has to be transformed so as to satisfy the above points. This can be done by introducing the revealed preference relation. The motivation of the revealed preference originally comes from the realization that it is almost impossible to get complete information in an actual case and that only some preferences are revealed (Samuelson, 1938). Thus, it is obvious that the revealed preference is based on the imperfection of information for preferences and need not be induced from pairwise comparisons only.

A Series of Experiments into the Use of Pairwise Comparison Techniques to Evaluate Criteria Weights

Abstract: This paper describes a series of experiments to investigate the nature of responses to questions to elicit the relative importance of criteria in a multi-attribute evaluation. The experiments were designed to investigate the behaviour of decision makers under various questioning procedures. The subjects were students in knitwear design and in engeering at Trent Polytechnic. We feel that the pairwise comparison technique used in Saaty’s AHP is a powerful means of eliciting such judgments and the experiments were carried out within that framework.

On pairwise S-closed bitopological spaces

Abstract: The concept of pairwise S-closedness in bitopological spaces has been introduced and some properties of such spaces have been studied in this paper.

Pairwise Comparisons in a Multi-Objective Energy Model

Abstract: We propose to use pairwise comparisons within the framework of ideal-point or reference-point methods for multi-objective programming. The decision makers are requested to estimate the ratios which are acceptable for deviations from the ideal vector. Thereafter, we seek the nearest feasible solution using the weighted Tchebycheff norm. In the paper we sketch some aspects of magnitude scaling and the ideal-point methods. We show the results of our numerical experiments in long-term energy planning with nine objective functions. Finally, we present a preliminary evaluation of the combined method on the basis of its possible contribution to interactive decision analysis.

Tests for Patterned Alternatives.

Abstract: : This paper treats the problem of testing for a patterned alternative in a one- or two-way layout. Ordered and umbrella alternatives are special cases. Statistics based on combined rankings and on pairwise rankings are developed. Modifications necessary to incorporate covariates are also included. (Author)

On Enforcing Socially Best Alternatives of Binary Group Decision Rules

Abstract: We consider a large class of group decision rules based on pairwise comparisons. We show that if a socially best alternative is defined by the group decision rule, then one can devise a game form such that at least one such alternative can be realized under the appropriate notion of equilibrium.

A Pascal Computer Program for Calculating the Tukey-Kramer Pairwise Multiple Comparison Procedure

Abstract: This paper describes a Pascal microcomputer program that computes all pairwise comparisons of means using the Tukey-Kramer test. Input to the program are means, sample sizes, mean-square error from a one-way analysis of variance, and the 95th percentile point on the studentized range distribution.