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One-Sample Tests
Abstract
Chapter 3 describes connections, equivalencies, and relationships relating to one-sample tests of null hypotheses. First, Student’s conventional one-sample t-test is described. Second, a permutation one-sample test is presented and the connection linking the two tests is established. An example analysis illustrates the differences in the two approaches and the connection linking the two tests. Third, measures of effect size for one-sample tests are presented for both Student’s one-sample t-test and the permutation one-sample test and the connections linking the various measures are set out. Fourth, Wilcoxon’s nonparametric one-sample signed-rank test is introduced for rank-score data and illustrated with an example analysis. A permutation alternative to Wilcoxon’s test is described and the connection linking the two tests is established. Finally, the connection linking a conventional one-sample z-test for proportions and Pearson’s chi-squared goodness-of-fit test is delineated.
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Confusion over the reporting and interpretation of results of commonly employed classical statistical tests is recorded in a sample of 1,645 papers from 12 psychology journals for the period 1990 through 2002. The confusion arises because researchers mistakenly believe that their interpretation is guided by a single unified theory of statistical inference. But this is not so: classical statistical testing is a nameless amalgamation of the rival and often contradictory approaches developed by Ronald Fisher, on the one hand, and Jerzy Neyman and Egon Pearson, on the other. In particular, there is extensive failure to acknowledge the incompatibility of Fisher’s evidential pvalue with the Type I error rate, •, of Neyman–Pearson statistical orthodoxy. The distinction between evidence (p’s) and errors (•’s) is not trivial. Rather, it reveals the basic differences underlying Fisher’s ideas on significance testing and inductive inference, and Neyman–Pearson views on hypothesis testing and inductive behavior. So complete is this misunderstanding over measures of evidence versus error that it is not viewed as even being a problem among the vast majority of researchers and other relevant parties. These include the APA Task Force on Statistical Inference, and those writing the guidelines concerning statistical testing mandated in APA Publication Manuals. The result is that, despite supplanting Fisher’s significance-testing paradigm some fifty years or so ago, recognizable applications of Neyman–Pearson theory are few and far between in psychology’s empirical literature. On the other hand, Fisher’s influence is ubiquitous.
Contends that both the interpretation of an effect size and the actual estimation of a coefficient of determination are partially theory-dependent. Two theoretical models for the variables cases are considered. In a variety of circumstances where the square of the correlation is used, the required assumptions are not tenable. In the alternate model, the absolute value of the correlation provides a coefficient of determination. The correlation coefficient is recommended for use as an effect-size indicator, because evaluating effect size in terms of variance accounted for may lead to interpretations that grossly underestimate the magnitude of a relation. (25 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Introduces the binomial effect size display, which displays the change in success rate (e.g., survival and improvement rates) attributable to a new treatment procedure. An example of the use of this method is presented. (11 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
The increased use of effect sizes in single studies and meta-analyses raises new questions about statistical inference. Choice of an effect-size index can have a substantial impact on the interpretation of findings. The authors demonstrate the issue by focusing on two popular effect-size measures, the correlation coefficient and the standardized mean difference (e.g., Cohen's d or Hedges's g), both of which can be used when one variable is dichotomous and the other is quantitative. Although the indices are often practically interchangeable, differences in sensitivity to the base rate or variance of the dichotomous variable can alter conclusions about the magnitude of an effect depending on which statistic is used. Because neither statistic is universally superior, researchers should explicitly consider the importance of base rates to formulate correct inferences and justify the selection of a primary effect-size statistic.
The primary purpose of this textbook is to introduce the reader to a wide variety of elementary permutation statistical methods. Permutation methods are optimal for small data sets and non-random samples, and are free of distributional assumptions. The book follows the conventional structure of most introductory books on statistical methods, and features chapters on central tendency and variability, one-sample tests, two-sample tests, matched-pairs tests, one-way fully-randomized analysis of variance, one-way randomized-blocks analysis of variance, simple regression and correlation, and the analysis of contingency tables. In addition, it introduces and describes a comparatively new permutation-based, chance-corrected measure of effect size.
Because permutation tests and measures are distribution-free, do not assume normality, and do not rely on squared deviations among sample values, they are currently being applied in a wide variety of disciplines. This book presents permutation alternatives to existing classical statistics, and is intended as a textbook for undergraduate statistics courses or graduate courses in the natural, social, and physical sciences, while assuming only an elementary grasp of statistics.
The comparison of two treatments generally falls into one of the following two categories: (a) we may have a number of replications for each of the two treatments, which are unpaired, or (b) we may have a number of paired comparisons leading to a series of differences, some of which may be positive and some negative. The appropriate methods for testing the significance of the differences of the means in these two cases are described in most of the textbooks on statistical methods.
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The efficacy of two improvement-over-chance effect sizes for two-group univariate comparisons
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