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Skills and Expertise
Publications
Publications (98)
Benford's Law is a probability distribution for the likelihood of the leading digit in a set of numbers. This book seeks to improve and systematize the use of Benford's Law in the social sciences to assess the validity of self-reported data. The authors first introduce a new measure of conformity to the Benford distribution that is created using pe...
Chapter 7 describes connections, equivalencies, and relationships relating to one-way randomized-blocks analysis of variance designs. First, Fisher’s conventional one-way randomized-blocks analysis of variance is described. Second, a permutation test is presented for randomized-blocks data and the connection linking the two approaches is establishe...
Chapter 13 is the first of two chapters describing connections, equivalencies, and relationships relating to fourfold contingency tables. First, Pearson’s mean-square contingency coefficient, \(\phi ^{2}\), and tetrachoric correlation coefficient, \(r_{\text{tet}}\), are described. Second, Yule’s Q and Y measures are described, and the connections...
Chapter 4 describes connections, equivalencies, and relationships relating to two-sample tests of null hypotheses. First, Student’s conventional two-sample t-test is described. Second, a permutation two-sample test is presented and the connection linking the two tests is established. An example analysis illustrates the differences in the two approa...
Chapter 12 is the second of two chapters describing connections, equivalencies, and relationships relating to measures of nominal association. First, Cohen’s unweighted kappa measure of inter-rater agreement is described. Second, Mielke and Berry’s \(\Re \) chance-corrected measure of agreement is introduced and the connection linking Cohen’s kappa...
Chapter 8 describes connections, equivalencies, and relationships relating to bivariate linear correlation and regression. First, Pearson’s product-moment correlation coefficient is described. Second, a permutation alternative to Pearson’s correlation coefficient is presented for bivariate data and the connection linking the two measures is describ...
Chapter 5 describes connections, equivalencies, and relationships relating to matched-pair tests of null hypotheses. First, Student’s conventional matched-pair t-test is described. Second, a permutation matched-pair test is presented and the connection linking the two matched-pair tests is established. An example analysis illustrates the difference...
Chapter 2 describes two models of statistical inference: the population model first put forward by J. Neyman and E. Pearson in 1928 and the permutation model developed by R.A Fisher, R.C. Geary, T. Eden, F. Yates, H. Hotelling, M. R. Pabst, and E.J.G. Pitman in the 1920s and 1930s. First, the Neyman–Pearson population model of statistical inference...
Chapter 9 is the first of two chapters describing connections, equivalencies, and relationships relating to measures of ordinal association. First, Spearman’s rank-order correlation coefficient is presented and the connection linking Spearman’s rank-order correlation coefficient and Pearson’s product-moment correlation coefficient is described. Nex...
Chapter 14 is the second of two chapters describing connections, equivalencies, and relationships relating to fourfold (\(2 \times 2\)) contingency tables. First, the connections linking Pearson’s \(\chi ^{2}\), Pearson’s \(\phi ^{2}\), Tschuprov’s \(T^{2}\), and Cramér’s \(V^{2}\) for symmetrical \(2 \times 2\) contingency tables are described and...
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 approa...
Chapter 11 is the first of two chapters describing connections, equivalencies, and relationships relating to measures of nominal association. First, Pearson’s chi-squared test of independence is described and illustrated with an example analysis. Second, Pearson’s \(\phi ^{2}\), Pearson’s C, Tschuprov’s \(T^{2}\), and Cramér’s \(V^{2}\) measures of...
Chapter 10 is the second of two chapters describing connections, equivalencies, and relationships relating to measures of ordinal association. First, Kendall’s \(\tau _{b}\) measure of ordinal association and Stuart’s \(\tau _{c}\) measure of ordinal association are described and the connection linking Kendall’s \(\tau _{b}\), Stuart’s \(\tau _{c}\...
Chapter 6 describes connections, equivalencies, and relationships relating to multi-sample tests of null hypotheses. First, Fisher’s conventional one-way completely randomized analysis of variance is described. Second, a multi-sample permutation test is presented and the connection linking the two tests is established. An example analysis illustrat...
This chapter introduces conventional and permutation methods for multiple matched samples, i.e., randomized-blocks designs. The chapter contains example analyses illustrating the computation of exact permutation probability values for randomized-blocks designs, calculation of measures of effect size for randomized-blocks designs, exact and Monte Ca...
This chapter describes two models of statistical inference: the population model first put forward by Jerzy Neyman and Egon Pearson in 1928 and the permutation model developed by R.A Fisher, R.C. Geary, T. Eden, F. Yates, H. Hotelling, M. R. Pabst, and E.J.G. Pitman in the 1920s and 1930s. The remainder of the chapter presents a brief history of th...
This chapter provides an overview of the concepts of central tendency and variability. For measures of central tendency, the sample mode, median, and mean are described and illustrated. For measures of variability, the sample standard deviation, sample variance, and mean absolute deviation are described and illustrated. An alternative to the mean a...
This chapter provides a brief overview of the R programming language. The R programming language is a popular statistical computing language available as open-source free software. This Chapter provides 13 sections including a general introduction to the R programming language, an introduction to R and RStudio, vectors in R, basic R data types, mat...
This chapter introduces conventional and permutation methods for two-sample tests. The chapter contains example analyses illustrating computation of exact permutation probability values for two-sample tests, calculation of measures of effect size for two-sample tests, exact and Monte Carlo permutation procedures for two-sample tests, and the applic...
This chapter introduces conventional and permutation methods for one-sample tests. Included in this chapter are example analyses illustrating computation of exact permutation probability values for one-sample tests, calculation of measures of effect size for one-sample tests, exact and Monte Carlo permutation procedures for one-sample tests, and ap...
In this chapter presents exact and Monte Carlo permutation statistical methods for multi-sample tests. Multi-sample tests are of two types: tests for experimental differences among three or more independent samples (fully- or completely-randomized designs) and tests for experimental differences among three or more dependent samples (randomized-bloc...
This chapter introduces conventional and permutation methods for matched-pairs tests. The chapter contains example analyses illustrating computation of exact permutation probability values for matched-pairs tests, calculation of measures of effect size for matched-pairs tests, exact and Monte Carlo permutation procedures for matched-pairs tests, an...
In this chapter introduces exact and Monte Carlo permutation statistical methods for Pearson’s chi-squared goodness-of-fit test, Wilks’ likelihood-ratio goodness-of-fit test, and Pearson’s chi-squared test of independence, along with selected measures of effect size based on the chi-squared test statistic and the likelihood-ratio test statistic.
In this chapter presents exact and Monte Carlo permutation statistical methods for measures of linear correlation and association. Also presented in this chapter is a permutation-based measure of effect size for a variety of measures of correlation and association. Simple linear correlation between two variables constitutes the foundation for a lar...
This book takes a unique approach to explaining permutation statistics by integrating permutation statistical methods with a wide range of classical statistical methods and associated R programs. It opens by comparing and contrasting two models of statistical inference: the classical population model espoused by J. Neyman and E.S. Pearson and the p...
This chapter introduces permutation methods for matched-pairs tests. Included in this chapter are six example analyses illustrating computation of exact permutation probability values for matched-pairs tests, calculation of measures of effect size for matched-pairs tests, the effect of extreme values on conventional and permutation matched-pairs te...
This chapter introduces permutation methods for the analysis of contingency tables. Included in this chapter are six example analyses illustrating computation of permutation methods for goodness-of-fit tests, analysis of contingency tables composed of two nominal-level (categorical) variables, analysis of contingency tables composed of two ordinal-...
This chapter introduces permutation methods for one-sample tests. Included in this chapter are six example analyses illustrating computation of exact permutation probability values for one-sample tests, calculation of measures of effect size for one-sample tests, the effect of extreme values on conventional and permutation one-sample tests, exact a...
This chapter provides an overview of the concepts of central tendency and variability. For measures of central tendency, the sample mode, median, and mean are described and illustrated. For measures of variability, the sample standard deviation, sample variance, and mean absolute deviation are described and illustrated. An alternative to the mean a...
This chapter provides a brief history and overview of the early beginnings and subsequent development of permutation statistical methods, organized by decades from the 1920s to the present.
This chapter introduces permutation methods for two-sample tests. Included in this chapter are six example analyses illustrating computation of exact permutation probability values for two-sample tests, calculation of measures of effect size for two-sample tests, the effect of extreme values on conventional and permutation two-sample tests, exact a...
This chapter presents two models of statistical inference: the conventional Neyman–Pearson population model that is taught in every introductory course and the Fisher–Pitman permutation model with which the reader is assumed to unfamiliar. The Fisher–Pitman model consists of three different permutation methods: exact permutation methods, Monte Carl...
This chapter introduces permutation methods for multiple matched samples, i.e., randomized-blocks designs. Included in this chapter are six example analyses illustrating computation of exact permutation probability values for randomized-blocks designs, calculation of measures of effect size for randomized-blocks designs, the effect of extreme value...
This chapter introduces permutation methods for measures of correlation and regression, the best-known of which is Pearson’s product-moment correlation coefficient. Included in this chapter are six example analyses illustrating computation of exact permutation probability values for correlation and regression, calculation of measures of effect size...
This chapter introduces permutation methods for multiple independent variables; that is, completely-randomized designs. Included in this chapter are six example analyses illustrating computation of exact permutation probability values for multi-sample tests, calculation of measures of effect size for multi-sample tests, the effect of extreme values...
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 fe...
This chapter examines chi-squared-based measures of association designed for two categorical variables. Four measures are introduced that are notoriously difficult to interpret because they do not norm properly between the limits of 0 and 1: Pearson’s ϕ², Tschuprov’s T², Cramér’s V², and Pearson’s C coefficient of contingency. A chi-squared-based a...
This chapter provides an introduction to two models of statistical inference—the population model and the permutation model—and the three main approaches to permutation statistical methods—exact, moment approximation, and Monte Carlo resampling-approximation. Advantages of permutation statistical methods are described and recursion techniques are d...
Permutation statistical methods possess a number of advantages compared with conventional statistical methods, making permutation statistical methods the preferred statistical approach for many research situations. Permutation statistical methods are data‐dependent, do not rely on distribution assumptions such as normality, provide either exact or...
This research monograph utilizes exact and Monte Carlo permutation statistical methods to generate probability values and measures of effect size for a variety of measures of association. Association is broadly defined to include measures of correlation for two interval-level variables, measures of association for two nominal-level variables or two...
This chapter describes permutation statistical methods for measures of association designed for two or more interval-level variables. Included in this chapter are simple and multiple ordinary least squares (OLS) regression, simple and multiple least absolute deviation (LAD) regression, point-biserial correlation, and biserial correlation. Fisher’s...
This chapter examines measures of association designed for two ordinal-level variables that are not based on pair-by-pair comparisons. Included in Chap. 6 are Spearman’s rank-order correlation coefficient, Spearman’s footrule measure of agreement, Kendall’s coefficient of concordance, Kendall’s u measure of inter-rater agreement, Cohen’s weighted k...
This chapter describes measures of association for two variables at different levels of measurement, e.g., a nominal-level independent variable and an ordinal- or interval-level dependent variable, and an ordinal-level independent variable and an interval-level dependent variable. This chapter begins with discussions of three measures of associatio...
This chapter introduces measures of association for two nominal-level variables that are not based on chi-squared. Included in Chap. 4 are Goodman and Kruskal’s λ and τ proportional-reduction-in-error measures, Cohen’s unweighted κ agreement coefficient, McNemar’s test for change, Cochran’s Q statistic, the Mantel–Haenszel test of independence for...
This chapter examines measures of association designed for two ordinal-level variables that are based on pairwise comparisons of differences between rank scores. Included in Chap. 5 are Kendall’s τa and τb measures of ordinal association, Stuart’s τc measure, Goodman and Kruskal’s γ measure, Somers’ dyx and dxy measures, Kim’s dy⋅x and dx⋅y measure...
This chapter continues the discussion measures of association for 2×2 contingency tables initiated in the previous chapter, but concentrates on symmetrical 2×2 contingency tables. Included in this chapter are permutation statistical methods applied to Pearson’s ϕ², Tschuprov’s T², and Cramér’s V² coefficients of contingency, Pearson’s product-momen...
This chapter provides an introduction measures of association for 2×2 contingency tables. Included in this chapter are discussion of Pearson’s ϕ² coefficient of contingency, Pearson’s tetrachoric correlation coefficient, Yule’s Q and Yule’s Y measures, Leik and Gove’s \(d_{N}^{\,c}\) measure, the odds ratio, and Kendall’s τb measure of ordinal asso...
Chapter 4 continues Chap. 3, utilizing the multi-response permutation procedures developed in Chap. 2 for analyzing completely randomized data at the interval level of measurement. In Chap. 4, multi-response permutation procedures are used to analyze regression residuals generated by ordinary least squares (OLS) and least absolute deviation (LAD) r...
Chapter 5 utilizes the multi-response permutation procedures (MRPP) developed in Chap. 2 for analyzing completely randomized data at the ordinal level of measurement. The structure of the MRPP test statistic, δ, depends on the choice of v in the generalized Minkowski distance function. A variety of tests are described in this chapter, including the...
Chapter 7 utilizes the Multi-Response Permutation Procedures (MRPP) developed in Chap. 2 for analyzing completely randomized data at the nominal (categorical) ordinal level of measurement. The structure of the MRPP test statistic, δ, depends on the choice of v in the generalized Minkowski distance function. A variety of tests are described in this...
Chapter 9 utilizes the Multivariate Randomized Block Permutation (MRBP) procedures developed in Chap. 8 for analyzing randomized-block data at the interval level of measurement. The structure of the MRBP test statistic, δ, depends on the choice of v in the generalized Minkowski distance function. Four tests are examined in this chapter: (1) Student...
Chapter 10 utilizes the multivariate randomized-block permutation procedures (MRBP) developed in Chap. 8 for analyzing randomized-block data at the ordinal level of measurement. The structure of the MRBP test statistic, δ, depends on the choice of v in the generalized Minkowski distance function. A variety of tests are described in this chapter, in...
Chapter 11 utilizes the multivariate randomized-block permutation procedures (MRBP) developed in Chap. 8 for analyzing randomized-block data at the nominal level of measurement. The structure of the MRBP test statistic, δ, depends on the choice of v in the generalized Minkowski distance function. A variety of tests are described in this chapter, in...
Chapter 2 introduces a generalized Minkowski distance function that is the basis for a set of multi-response permutation procedures for univariate and multivariate completely randomized data. Multi-response permutation procedures constitute a class of permutation methods for one or more response measurements that are designed to distinguish possibl...
Chapter 3 utilizes the Multi-Response Permutation Procedures (MRPP) presented in Chap. 2 to develop the relationships between the test statistics of MRPP, δ and \(\mathfrak{R}\), and selected conventional tests and measures designed for the analysis of completely randomized data at the interval level of measurement. The structure of the MRPP test s...
Chapter 6 utilizes the Multi-Response Permutation Procedures (MRPP) developed in Chap. 2 to establish relationships between the test statistics of MRPP, δ and \(\mathfrak{R}\), and multivariate generalizations of selected conventional tests and measures designed for the analysis of completely randomized data at the ordinal level of measurement. Con...
Chapter 8 utilizes a generalized Minkowski distance function as the basis for a set of multivariate block permutation procedures for univariate and multivariate randomized-block data. Multivariate block permutation procedures constitute a class of permutation methods for one or more response measurements in each block that are designed to distingui...
This research monograph provides a synthesis of a number of statistical tests and measures, which, at first consideration, appear disjoint and unrelated. Numerous comparisons of permutation and classical statistical methods are presented, and the two methods are compared via probability values and, where appropriate, measures of effect size.
Permut...
This chapter chronicles the period from 2001 to 2010. By the beginning of this period, permutation statistical methods had come of age and advances were comprised more of application and expansion into new fields and disciplines than the development of new permutation methods that characterized earlier years. In this period computing power was suff...
The focus of this book is on the birth and historical development of permutation statistical methods from the early 1920s to the near present. Beginning with the seminal contributions of R.A. Fisher, E.J.G. Pitman, and others in the 1920s and 1930s, permutation statistical methods were initially introduced to validate the assumptions of classical s...
Permutation tests are a paradox of old and new. Permutation tests pre-date most traditional parametric statistics, but only recently have become part of the mainstream discussion regarding statistical testing. Permutation tests follow a permutation or ‘conditional on errors’ model whereby a test statistic is computed on the observed data, then (1)...
When the categories of the independent variable in an analysis of variance are quantitative, it is more informative to evaluate the trends in the treatment means than to simply compare differences among the treatment means. A permutation alternative to the conventional F test is shown to possess significant advantages when analyzing trend among qua...
An alternative to conventional rank tests based on a Euclidean distance analysis space is described. Comparisons based on exact probability values among classical two-sample t-tests and the Wilcoxon-Mann-Whitney test illustrate the advantages of the Euclidean distance analysis space alternative.
Two measures of effect size are described for the Mantel-Haenszel test. Both measures belong to the r-family of effect size measures. One measure is based on a maximum-corrected model, and the second measure is based on a chance-corrected model.
The authors describe exact and resampling permutation methods, with detailed advantages of permutation methods for quantitative historical analysis, and compare conventional and permutation two-sample t tests using the Poor Law Relief data set.
The reporting of measures of effect size has become increasingly important in psychology. A Monte Carlo resampling permutation procedure is introduced to find near-optimum maximum values for Stuart’s τc
measure for two-way ordinal contingency tables, also termed Kendall’s τc
since Kendall introduced τa
and τb
. Comparisons between resampling and ex...
Abstract This study extends ideas of environmental equity to large-scale hog operations. We investigate counties in 17 hog producing states to determine whether large-scale hog operations are more likely to be sited and expanded in areas that have a disproportionate number of Black, Hispanic, and/or economically disadvantaged residents. The data fo...
The measurement of the magnitude of association between a nominal independent variable and an ordinal dependent variable is an important, but neglected, component in psychological research. Two measures of nominal-ordinal association are described and compared. Resampling permutation methods are utilized to compute probability values for both measu...
A g-treatment extension of the classical two-treatment ridit analysis is introduced. Concerns regarding distributional assumptions for ridit analyses are avoided by employing data dependent resampling methods. Upper-tail P-values based on resampling permutations are obtained for the g-treatment ridit analyses.
Five procedures to calculate the probability of weighted kappa with multiple raters under the null hypothesis of independence are described and compared in terms of accuracy, ease of use, generality, and limitations. The five procedures are (1) exact variance, (2) resampling contingency, (3) intraclass correlation, (4) randomized block, and (5) res...
A permutation algorithm and associated FORTRAN program are provided for resampling weighted kappa. Program RWK provides the weighted kappa test statistic and the resampling one-sided upper-tail probability value.
A new procedure to compute weighted kappa with multiple raters is described. A resampling procedure to compute approximate probability values for weighted kappa with multiple raters is presented. Applications of weighted kappa are illustrated with an example analysis of classifications by three independent raters.
A permutation algorithm and associated FORTRAN program are provided for weighted kappa. Program EWK provides the weighted kappa test statistic and the exact one-sided upper-tail probability values.
The number of resamplings necessary to accurately estimate a probability value is an open question. One million resamplings is shown to be sufficient to ensure precision to three places under most conditions.
Weighted kappa described by Cohen in 1968 is widely used in psychological research to measure agreement between two independent raters. Everitt then provided the exact variance for weighted kappa for two raters. In this paper, Everitt's exact variance is extended to three or more raters.
A resampling algorithm is presented for analyzing multiway contingency tables with fixed marginal frequency totals. Applications are illustrated with extensions of Fisher's exact, Pearson's chi-squared, and likelihood-ratio tests to three-way contingency tables.
Measures of effect size are increasingly important in psychological research. In this paper, a chance-corrected measure of effect size is introduced for Cochran's Q test.
An algorithm and associated FORTRAN program are provided for six common measures of ordinal association: Kendall's taua and taub, Stuart's tauc, Goodman and Kruskal's gamma, and Somers' dyx and dxy. Program ROMA reports the observed data table, the values for the six test statistics, and the resampling upper- and lower-tail probability values assoc...
In response to substantial economic and social dislocations in the United States, many rangeland owners are changing land use and management practices. Changes in land use can significantly affect the services rangeland ecosystems provide. Decisions associated with such changes are likely mediated by landowner views regarding individual rights, soc...
A fundamental shift in editorial policy for psychological journals was initiated when the fourth edition of the Publication Manual of the American Psychological Association (1994) placed emphasis on reporting measures of effect size. This paper presents measures of effect size for the chi-squared and the likelihood-ratio goodness-of-fit statistic t...
Permutation procedures to compute exact and resampling probability values associated with measures of association for ordered r x c contingency tables are described. Comparisons with asymptotic probability values demonstrated that exact and resampling permutation procedures were preferred for sparse contingency tables.
Goodman and Kruskal's tau measure of categorical association is advanced as a replacement for conventional measures of effect size for r x c contingency tables. Goodman and Kruskal's tau is an asymmetric measure of categorical association which is based entirely on the observed data and possesses a clear interpretation in terms of proportional redu...
A permutation method is presented to calculate resampling probability values for differences between two independent indices of ordinal variation and consensus.
In response to substantial economic and social dislocations in the United States, many rangeland owners are changing land use and management practices. Changes in land use can significantly affect the services rangeland ecosystems provide. Decisions associated with such changes are likely mediated by landowner views regarding individual rights, soc...
An algorithm and associated FORTRAN program are provided for the exact variance of weighted kappa. Program VARKAP provides the weighted kappa test statistic, the exact variance of weighted kappa, a Z score, one-sided lower- and upper-tail N(0,1) probability values, and the two-tail N(0,1) probability value.
When analyzing categorical data, it is often important to assess the magnitude of variation, or consensus, among observations in unordered categories. Utilizing the theory of partitions, exact solutions for five commonly used measures of categorical variation are presented. When the number of partitions is very large, resampling methods provide clo...
Fisher's well-known continuous method for combining independent probability values from continuous distributions is compared with an exact discrete analog of Fisher's continuous method for combining independent probability values from discrete distributions using matched-pairs t-test data. Fisher's continuous method is shown to be inadequate for co...
Resampling permutation procedures provide good approximations to exact permutation procedures and are therefore the preferred alternative when large samples render exact tests intractable. Resampling permutation procedures are described for three measures of variation and the three corresponding measures of consensus among ordered categories.
Permutation procedures to compute exact and resampling probability values for weighted kappa are described. Comparisons with asymptotic probability values demonstrate that exact permutation procedures are advantageous for sparse data sets, whereas resampling permutation procedures are appropriate for both sparse and nonsparse data sets.
Permutation tests are based on all possible arrangements of observed data sets. Consequently, such tests yield exact probability values obtained from discrete probability distributions. An exact nondirectional method to combine independent probability values that obey discrete probability distributions is introduced. The exact method is the discret...
An algorithm and computer program to calculate exact goodness-of-fit tests for unordered categories with equal probabilities under the null hypothesis are presented. FORTRAN program EBGF utilizes partitions and multinomial weights to reduce computation times for Fisher's exact, exact chi-square, exact likelihood-ratio, exact Freeman-Tukey, and exac...
Traditional asymptotic probability values resulting from log-linear analyses of sparse frequency tables are often much too large. Asymptotic probability values for chi-squared and likelihood-ratio statistics are compared to nonasymptotic and exact probability values for selected log-linear models. The asymptotic probability values are all too often...
A recent trend in the psychological literature has been to include measures of effect size when reporting probability values. The several measures of effect size associated with the Student t test for two independent samples are appropriate only when the variances are homogeneous. In this paper, commonly used measures of effect size are considered...
The Fisher-Pitman and powers of ranks permutation tests are shown to provide substantial resistance to extreme values when compared with the conventional t test for analyzing matched-pairs experimental designs.
The Fisher-Pitman and powers of ranks permutation tests are shown to provide substantial resistance to extreme values when compared with the conventional t test for analyzing matched-pairs experimental designs.
Exact, resampling, and Pearson type III permutation methods are provided to compute probability values for Piccarreta's nominal-ordinal index of association. The resampling permutation method provides good approximate probability values based on the proportion of resampled test statistic values equal to or greater than the observed test statistic v...
