Handbook of Parametric and Nonparametric Statistical Procedures

Transcription

1 Fourth Edition Handbook of Parametric and Nonparametric Statistical Procedures David J. Sheskin Chapman & Hall/CRC Taylor & Francis Group Boca Raton london New York Chapman & Hall/CRC is an imprint of the

2 Table of Contents with Summary of Topics Introduction 1 Descriptive versus inferential statistics 1 Statistic versus parameter 2 Levels of measurement 2 Continuous versus discrete variables 4 Measures of central tendency (mode, median, mean, weighted mean, geometric mean, and the harmonic mean) 4 Measures of variability (range; quantites, percentiles, deciles, and quartiles; variance and standard deviation; the coefficient of variation) 10 Measures of skewness and kurtosis 16 Visual methods for displaying data (tables and graphs, exploratory data analysis (stem-and-leaf displays and boxplots)) 29 The normal distribution 45 Hypothesis testing 56 A history and critique of the classical hypothesis testing model 68 Estimation in inferential statistics 74 Relevant concepts, issues, and terminology in conducting research (the case study method; the experimental method; the correlational method) 75 Experimental design (pre-experimental designs; quasi-experimental designs true experimental designs; single-subject designs) 82 Sampling methodologies 98 Basic principles of probability 100 Parametric versus nonparametric inferential statistical tests 108 Univariate versus bivariate versus multivariate statistical procedures 109 Selection of the appropriate statistical procedure 110 Outline of Inferential Statistical Tests and Measures of Correlation/Association 125 Guidelines and Decision Tables for Selecting the Appropriate Statistical Procedure 133 Inferential Statistical Tests Employed with a Single Sample 141 Test 1 : The Single-Sample ζ Test 143 I. Hypothesis Evaluated with Test and Relevant Background Information 143 II. Example 143 III. Null versus Alternative Hypotheses 143 IV. Test Computations 144 V. Interpretation of the Test Results 145 VI. Additional Analytical Procedures for the Single-Sample Test and/or Related Tests 146

3 VIII Handbook of Parametric and Nonparametric Statistical Procedures VII. Additional Discussion of the Single-Sample ζ Test The interpretation of a negative z value The standard error of the population mean and graphical representation of the results of the single-sample z test Additional examples illustrating the interpretation of a computed ζ value The z test for a population proportion 152 VIII. Additional Examples Illustrating the Use of the Single-Sample ζ Test 153 Test 2: The Single-Sample / Test 157 I. Hypothesis Evaluated with Test and Relevant Background Information 157 II. Example 158 III. Null versus Alternative Hypotheses 158 IV. Test Computations 158 V. Interpretation of the Test Results 160 VI. Additional Analytical Procedures for the Single-Sample t Test and/or Related Tests Determination of the power of the single-sample / test and the singlesample ζ test, and the application of Test 2a: Cohen's d index Computation of a confidence interval for the mean of the population represented by a sample 173 VIL Additional Discussion of the Single-Sample / Test Degrees of freedom 183 VIII. Additional Examples Illustrating the Use of the Single-Sample t Test 184 Test 3: The Single-Sample Chi-Square Test for a Population Variance 191 I. Hypothesis Evaluated with Test and Relevant Background Information 191 II. Example 192 III. Null versus Alternative Hypotheses 192 IV. Test Computations 193 V. Interpretation of the Test Results 194 VI. Additional Analytical Procedures for the Single-Sample Chi-Square Test for a Population Variance and/or Related Tests Large sample normal approximation of the chi-square distribution Computation of a confidence interval for the variance of a population represented by a sample Sources for computing the power of the single-sample chi-square test for a population variance 200 VII. Additional Discussion of the Single-Sample Chi-Square Test for a Population Variance 200 VIII. Additional Examples Illustrating the Use of the Single-Sample Chi- Square Test for a Population Variance 200 Test 4: The Single-Sample Test for Evaluating Population Skewness 205 I. Hypothesis Evaluated with Test and Relevant Background Information 205 IL Example III. Null versus Alternative Hypotheses 206 IV. Test Computations V. Interpretation of the Test Results 2 0 9

4 Table of Contents ix VI. Additional Analytical Procedures for the Single-Sample Test for Evaluating Population Skewness and/or Related Tests 210 VII. Additional Discussion of the Single-Sample Test for Evaluating Population Skewness Exact tables for the single-sample test for evaluating population skewness Note on a nonparametric test for evaluating skewness 210 VIII. Additional Examples Illustrating the Use of the Single-Sample Test for Evaluating Population Skewness 211 Test 5: The Single-Sample Test for Evaluating Population Kurtosis 213 I. Hypothesis Evaluated with Test and Relevant Background Information 213 II. Example 214 III. Null versus Alternative Hypotheses 214 IV. Test Computations 215 V. Interpretation of the Test Results 217 VI. Additional Analytical Procedures for the Single-Sample Test for Evaluating Population Kurtosis and/or Related Tests Test 5a: The D'Agostino-Pearson test of normality Test 5b: The Jarque-Bera test of normality 219 VII. Additional Discussion of the Single-Sample Test for Evaluating Population Kurtosis Exact tables for the single-sample test for evaluating population kurtosis Additional comments on tests of normality 220 VIII. Additional Examples Illustrating the Use of the Single-Sample Test for Evaluating Population Kurtosis 221 Test 6: The Wilcoxon Signed-Ranks Test 225 I. Hypothesis Evaluated with Test and Relevant Background Information 225 IL Example 225 III. Null versus Alternative Hypotheses 226 IV. Test Computations 226 V. Interpretation of the Test Results 228 VI. Additional Analytical Procedures for the Wilcoxon Signed-Ranks Test and/or Related Tests The normal approximation of the WilcoxonTstatistic for large sample sizes The correction for continuity for the normal approximation of the Wilcoxon signed-ranks test Tie correction for the normal approximation of the Wilcoxon test statistic Computation of a confidence interval for a population median 234 VII. Additional Discussion of the Wilcoxon Signed-Ranks Test Power-efficiency of the Wilcoxon signed-ranks test and the concept of asymptotic relative efficiency Note on symmetric population concerning hypotheses regarding median and mean 236 VIII. Additional Examples Illustrating the Use of the Wilcoxon Signed- Ranks Test 237

5 Handbook of Parametric and Nonparametric Statistical Procedures Test 7: The Kolmogorov-Smirnov Goodness-of-Fit Test for a Single Sample 241 I. Hypothesis Evaluated with Test and Relevant Background Information 241 IL Example 242 III. Null versus Alternative Hypotheses 243 IV. Test Computations 244 V. Interpretation of the Test Results 248 VI. Additional Analytical Procedures for the Kolmogorov-Smirnov Goodness-of-Fit Test for a Single Sample and/or Related Tests Computing a confidence interval for the Kolmogorov- Smirnov goodness-of-fit test for a single sample The power of the Kolmogorov-Smirnov goodness-of-fit test for a single sample Test 7a: The Lilliefors test for normality 250 VII. Additional Discussion of the Kolmogorov-Smirnov Goodness-of-Fit Test for a Single Sample Effect of sample size on the result of a goodness-of-fit test The Kolmogorov-Smirnov goodness-of-fit test for a single sample versus the chi-square goodness-of-fit test and alternative goodness-of-fit tests 253 VIII. Additional Examples Illustrating the Use of the Kolmogorov-Smirnov Goodness-of-Fit Test for a Single Sample 253 Test 8: The Chi-Square Goodness-of-Fit Test 257 I. Hypothesis Evaluated with Test and Relevant Background Information 257 II. Examples 258 III. Null versus Alternative Hypotheses 258 IV. Test Computations 259 V. Interpretation of the Test Results 261 VI. Additional Analytical Procedures for the Chi-Square Goodness-of-Fit Test and/or Related Tests Comparisons involving individual cells when k> The analysis of standardized residuals The correction for continuity for the chi-square goodness-of-fit test Computation of a confidence interval for the chi-square goodness-offit test/confidence interval for a population proportion Brief discussion of the ζ test for a population proportion (Test 9a) and the single-sample test for the median (Test 9b) Application of the chi-square goodness-of-fit test for assessing goodness-of-fit for a theoretical population distribution Sources for computing of the power of the chi-square goodness-of-fit test Heterogeneity chi-square analysis 273 VII. Additional Discussion of the Chi-Square Goodness-of-Fit Test Directionality of the chi-square goodness-of-fit test Additional goodness-of-fit tests 279 VIII. Additional Examples Illustrating the Use of the Chi-Square Goodnessof-Fit Test 2 8 0

6 Table of Contents χι Test 9: The Binomial Sign Test for a Single Sample 289 L Hypothesis Evaluated with Test and Relevant Background Information 289 II. Examples 290 III. Null versus Alternative Hypotheses 290 IV. Test Computations 291 V. Interpretation of the Test Results 293 VI. Additional Analytical Procedures for the Binomial Sign Test for a Single Sample and/or Related Tests Test 9a: The z test for a population proportion (with discussion of correction for continuity; computation of a confidence interval; procedure for computing sample size for test of specified power; additional comments on computation of the power of the binomial sign test for a single sample) Extension of the ζ test for a population proportion to evaluate the performance of m subjects on η trials on a binomially distributed variable Test 9b: The single-sample test for the median 305 VII. Additional Discussion of the Binomial Sign Test for a Single Sample Evaluating goodness-of-fit for a binomial distribution 308 VIII. Additional Example Illustrating the Use of the Binomial Sign Test for a Single Sample 310 IX. Addendum Discussion of additional discrete probability distributions and the exponential distribution 311 The multinomial distribution 311 The negative binomial distribution 313 The hypergeometric distribution 316 The Poisson distribution 319 Computation of a confidence interval for a Poisson parameter. 322 Test 9c: Test for evaluating two Poisson counts 323 Evaluating goodness-of-fit for a Poisson distribution 324 The exponential distribution 326 The matching distribution Conditional probability, Bayes' theorem, Bayesian statistics and hypothesis testing 332 Conditional probability 332 Bayes' theorem 333 Bayesian hypothesis testing 349 Bayesian analysis of a continuous variable 374 Test 10: The Single-Sample Runs Test (and Other Tests of Randomness) 389 I. Hypothesis Evaluated with Test and Relevant Background Information 389 II. Example 390 III. Null versus Alternative Hypotheses 391 IV. Test Computations 391 V. Interpretation of the Test Results 391 VI. Additional Analytical Procedures for the Single-Sample Runs Test and/or Related Tests The normal approximation of the single-sample runs test for large sample sizes The correction for continuity for the normal approximation of the

7 Handbook of Parametric and Nonparametric Statistical Procedures single-sample runs test Extension of the runs test to data with more than two categories Test 10a: The runs test for serial randomness 395 VII. Additional Discussion of the Single-Sample Runs Test Additional discussion of the concept of randomness 398 VII. Additional Examples Illustrating the Use of the Single-Sample Runs Test 399 IX. Addendum The generation of pseudorandom numbers (The midsquare method; the midproduct method; the linear congruential method) Alternative tests of randomness Test 10b: The frequency test 407 Test 10c: The gap test 409 Test 10d: The poker test 413 Test 10e: The maximum test 413 Test 10f: The coupon collector's test 414 Test 10g: The mean square successive difference test (for serial randomness 417 Additional tests of randomness (Autocorrelation; The serial test; The d 2 square test of random numbers; Tests of trend analysis/time series analysis) 419 Inferential Statistical Tests Employed with Two Independent Samples (and Related Measures of Association/Correlation) 425 Test 11: The / Test for Two Independent Samples 427 I. Hypothesis Evaluated with Test and Relevant Background Information 427 II. Example 427 III. Null versus Alternative Hypotheses 428 IV. Test Computations 428 V. Interpretation of the Test Results 431 VI. Additional Analytical Procedures for the / Test for Two Independent Samples and/or Related Tests The equation for the / test for two independent samples when a value for a difference other than zero is stated in the null hypothesis Test 11a: Hartley's F mju test for homogeneity of variance/ F test for two population variances: Evaluation of the homogeneity of variance assumption of the / test for two independent samples Computation of the power of the t test for two independent samples and the application of Test 11b: Cohen's d index Measures of magnitude of treatment effect for the t test for two independent samples: Omega squared (Test 11c) and Eta squared (Test 11d) Computation of a confidence interval for the / test for two independent samples Test 11e: The ζ test for two independent samples 450 VII. Additional Discussion of the / Test for Two Independent Samples Unequal sample sizes Robustness of the / test for two independent samples 453

8 Table of Contents XIII 3. Outliers (Box-and-whisker plot criteria; Standard deviation score criteriai; Test 11f: Median absolute deviation test for identifying outliers and Test 11g: Extreme Studentized deviate test for identifying outliers; trimming data; Winsorization) and data transformation Missing data Clinical trials Tests of equivalence (Test 11h: The Westlake-Schuirmann test of equivalence of two independent treatments (and procedure for computing sample size in reference to the power of the test) and Test Hi: Tryon's test of equivalence of two independent treatments) Hotelling's T VIII Additional Examples Illustrating the Use of the t Test for Two Independent Samples 499 Test 12: The Mann-Whitney t/test 513 I. Hypothesis Evaluated with Test and Relevant Background Information 513 II. Example 514 III. Null versus Alternative Hypotheses 514 IV. Test Computations 515 V. Interpretation of the Test Results 518 VI. Additional Analytical Procedures for the Mann-Whitney U Test and/or Related Tests The normal approximation of the Mann-Whitney U statistic for large sample sizes The correction for continuity for the normal approximation of the Mann-Whitney t/test Tie correction for the normal approximation of the Mann-Whitney U statistic Computation of a confidence interval for a difference between the medians of two independent populations 521 VII. Additional Discussion of the Mann-Whitney (/Test Power-efficiency of the Mann-Whitney U test Equivalency of the normal approximation of the Mann-Whitney t/test and the t test for two independent samples with rank-orders Alternative nonparametric rank-order procedures for evaluating a design involving two independent samples 524 VIII. Additional Examples Illustrating the Use of the Mann-Whitney t/test 525 IX. Addendum Computer-intensive tests (Randomization and permutation tests: Test 12a: The randomization test for two independent samples; Test 12b: The bootstrap; Test 12c: The jackknife; Final comments on computer-intensive procedures) Survival analysis (Test 12d: Kaplan-Meier estimate) Procedures for evaluating censored data in a design involving two independent samples (Permutation test based on the median for censored data, Gehan's test for censored data (Test 12e), and the log-rank test (Test 12f)) 550

9 Handbook of Parametric and Nonparametric Statistical Procedures Test 13: The KoImogorov-Smirnov Test for Two Independent Samples I. Hypothesis Evaluated with Test and Relevant Background Information 577 IL Example 578 III. Null versus Alternative Hypotheses 578 IV. Test Computations 580 V. Interpretation of the Test Results 582 VI. Additional Analytical Procedures for the Kolmogorov-Smirnov test for two independent samples and/or Related Tests Graphical method for computing the Kolmogorov-Smirnov test statistic Computing sample confidence intervals for the Kolmogorov- Smirnov test for two independent samples Large sample chi-square approximation for a one-tailed analysis of the Kolmogorov-Smirnov test for two independent samples 584 VII. Additional Discussion of the Kolmogorov-Smirnov Test for Two Independent Samples Additional comments on the Kolmogorov-Smirnov test for two independent samples 585 VIII. Additional Examples Illustrating the Use of the Kolmogorov-Smirnov Test for Two Independent Samples 585 Test 14: The Siegel-Tukey Test for Equal Variability 589 I. Hypothesis Evaluated with Test and Relevant Background Information IL Example 590 III. Null versus Alternative Hypotheses 590 IV. Test Computations 591 V. Interpretation of the Test Results 594 VI. Additional Analytical Procedures for the Siegel-Tukey Test for Equal Variability and/or Related Tests The normal approximation of the Siegel-Tukey test statistic for large sample sizes The correction for continuity for the normal approximation of the Siegel-Tukey test for equal variability Tie correction for the normal approximation of the Siegel-Tukey test statistic Adjustment of scores for the Siegel-Tukey test for equal variability when 596 VII. Additional Discussion of the Siegel-Tukey Test for Equal Variability Analysis of the homogeneity of variance hypothesis for the same set of data with both a parametric and nonparametric test, and the powerefficiency of the Siegel-Tukey Test for Equal Variability Alternative nonparametric tests of dispersion 599 VIII. Additional Examples Illustrating the Use of the Siegel-Tukey Test for Equal Variability 600 Test 15: The Moses Test for Equal Variability 605 I. Hypothesis Evaluated with Test and Relevant Background Information 605 IL Example 606

10 Table of Contents xv III. Null versus Alternative Hypotheses 606 IV. Test Computations 608 V. Interpretation of the Test Results 610 VI. Additional Analytical Procedures for the Moses Test for Equal Variability and/or Related Tests The normal approximation of the Moses test statistic for large sample sizes 611 VII. Additional Discussion of the Moses Test for Equal Variability Power-efficiency of the Moses Test for equal variability Issue of repetitive resampling Alternative nonparametric tests of dispersion 613 VIII. Additional Examples Illustrating the Use of the Moses Test for Equal Variability 613 Test 16: The Chi-Square Test for r x c Tables (Test 16a: The Chi-Square Test for Homogeneity; Test 16b: The Chi-Square Test of Independence (employed with a single sample)) 619 I. Hypothesis Evaluated with Test and Relevant Background Information 619 II. Examples 621 III. Null versus Alternative Hypotheses 622 IV. Test Computations.625 V. Interpretation of the Test Results 626 VI. Additional Analytical Procedures for the Chi-Square Test for r χ c Tables and/or Related Tests Yates' correction for continuity Quick computational equation for a 2 χ 2 table Evaluation of a directional alternative hypothesis in the case of a 2 χ 2 contingency table Test 16c: The Fisher exact test Test 16d: The ζ test for two independent proportions (and computation of sample size in reference to power) Computation of a confidence interval for a difference between two proportions Test 16e: The median test for independent samples Extension of the chi-square test for r χ c tables to contingency tables involving more than two rows and/or columns, and associated comparison procedures The analysis of standardized residuals Sources for computing the power of the chi-square test for r χ c tables Measures of association for r χ c contingency tables 655 Test 16f: The contingency coefficient 657 Test 16g: The phi coefficient 659 Test 16h: Cramer's phi coefficient 660 Test 16i: Yule's Q 661 Test 16j: The odds ratio (and the concept of relative risk) (and Test 16j-a: Test of significance for an odds ratio and computation of a confidence interval for an odds ratio) 662 Test 16k: Cohen's kappa (and computation of a confidence interval for kappa, Test 16k-a: Test of significance for

11 xvi Handbook of Parametric and Nonparametric Statistical Procedures Cohen's kappa, and Test 16k-b: Test of significance for two independent values of Cohen's kappa) Combining the results of multiple 2x2 contingency tables: 673 Heterogeneity chi-square analysis for a 2 χ 2 contingency table 673 Test 161: The Mantel-Haenszel analysis (Test 161-a: Test of homogeneity of odds ratios for Mantel-Haenszel analysis, Test 161-b: Summary odds ratio for Mantel- Haenszel analysis, and Test 161-c: Mantel-Haenszel test of association) 676 VII. Additional Discussion of the Chi-Square Test for r χ c Tables Equivalency of the chi-square test for r χ c tables when c = 2 with the t test for two independent samples (when r = 2) and the single-factor between-subjects analysis of variance (when r 2) Test of equivalence for two independent proportions: Test 16m: The Westlake-Schuirmann test of equivalence of two independent proportions (and procedure for computing sample size in reference to the power of the test) Test 16n: The log-likelihood ratio Simpson's Paradox Analysis of multidimensional contingency tables through use of a chi-square analysis Test 16o:Analysis of multidimensional contingency tables with log-linear analysis 713 VIII. Additional Examples Illustrating the Use of the Chi-Square Test for r x c Tables 727 Inferential Statistical Tests Employed with Two Dependent Samples (and Related Measures of Association/Correlation) 741 Test 17: The t Test for Two Dependent Samples 743 I. Hypothesis Evaluated with Test and Relevant Background Information 743 II. Example 744 III. Null versus Alternative Hypotheses 744 IV. Test Computations 745 V. Interpretation of the Test Results 747 VI. Additional Analytical Procedures for the t Test for Two Dependent Samples and/or Related Tests Alternative equation for the t test for two dependent samples The equation for the t test for two dependent samples when a value for a difference other than zero is stated in the null hypothesis Test 17a: The / test for homogeneity of variance for two dependent samples: Evaluation of the homogeneity of variance assumption of the t test for two dependent samples Computation of the power of the t test for two dependent samples and the application of Test 17b: Cohen's d index Measure of magnitude of treatment effect for the t test for two dependent samples: Omega squared (Test 17c) 761

12 Table of Contents 6. Computation of a confidence interval for the / test for two dependent samples Test 17d: Sandler's/I test Test 17e: The ζ test for two dependent samples 765 VII. Additional Discussion of the t Test for Two Dependent Samples The use of matched subjects in a dependent samples design Relative power of the t test for two dependent samples and the / test for two independent samples Counterbalancing and order effects Analysis of a one-group pretest-posttest design with the t test for two dependent samples Tests of equivalence: Test 17f: The Westlake-Schuirmann test of equivalence of two dependent treatments (and procedure for computing sample size in reference to the power of the test) and Test 17g: Tryon's test of equivalence of two dependent treatments 776 VIII. Additional Example Illustrating the Use of the / Test for Two Dependent Samples 785 Test 18: The Wilcoxon Matched-Pairs Signed-Ranks Test 791 I. Hypothesis Evaluated with Test and Relevant Background Information 791 II. Example 792 III. Null versus Alternative Hypotheses 792 IV. Test Computations 793 V. Interpretation of the Test Results 794 VI. Additional Analytical Procedures for the Wilcoxon Matched-Pairs Signed-Ranks Test and/or Related Tests The normal approximation of the Wilcoxon T statistic for large sample sizes The correction for continuity for the normal approximation of the Wilcoxon matched-pairs signed-ranks test Tie correction for the normal approximation of the Wilcoxon test statistic Computation of a confidence interval for a median difference between two dependent populations 799 VII. Additional Discussion of the Wilcoxon Matched-Pairs Signed-Ranks Test Power-efficiency of the Wilcoxon matched-pairs signed-ranks test Probability of superiority as a measure of effect size Alternative nonparametric procedures for evaluating a design involving two dependent samples 801 VIII. Additional Examples Illustrating the Use of the Wilcoxon Matched- Pairs Signed-Ranks Test 802 Test 19: The Binomial Sign Test for Two Dependent Samples 805 I. Hypothesis Evaluated with Test and Relevant Background Information 805 II. Example 806 III. Null versus Alternative Hypotheses 806 IV. Test Computations 807 V. Interpretation of the Test Results 809

13 xviii Handbook of Parametric and Nonparametric Statistical Procedures VI. Additional Analytical Procedures for the Binomial Sign Test for Two Dependent Samples and/or Related Tests The normal approximation of the binomial sign test for two dependent samples with and without a correction for continuity Computation of a confidence interval for the binomial sign test for two dependent samples Sources for computing the power of the binomial sign test for two dependent samples, and comments on asymptotic relative efficiency of the test 814 VII. Additional Discussion of the Binomial Sign Test for Two Dependent Samples The problem of an excessive number of zero difference scores Equivalency of the binomial sign test for two dependent samples and the Friedman two-way analysis variance by ranks when = VIII. Additional Examples Illustrating the Use of the Binomial Sign Test for Two Dependent Samples 815 Test 20: The McNemar Test 817 I. Hypothesis Evaluated with Test and Relevant Background Information 817 II. Examples 818 III. Null versus Alternative Hypotheses 820 IV. Test Computations 822 V. Interpretation of the Test Results 822 VI. Additional Analytical Procedures for the McNemar Test and/or Related Tests Alternative equation for the McNemar test statistic based on the normal distribution The correction for continuity for the McNemar test Computation of the exact binomial probability for the McNemar test model with a small sample size Computation of the power of the McNemar test Computation of a confidence interval for the McNemar test Computation of an odds ratio for the McNemar test Additional analytical procedures for the McNemar test Test 20a: The Gart test for order effects 830 VII. Additional Discussion of the McNemar Test Alternative format for the McNemar test summary table and modified test equation The effect of disregarding matching Alternative nonparametric procedures for evaluating a design with two dependent samples involving categorical data Test of equivalence for two independent proportions: Test 20b: The Westlake-Schuirmann test of equivalence of two dependent proportions 840 VIII. Additional Examples Illustrating the Use of the McNemar Test 850 IX. Addendum 852 Extension of the McNemar test model beyond 2x2 contingency tables Test 20c:The Bowker test of internal symmetry Test 20d:The Stuart-Maxwell test of marginal homogeneity 856

14 Table of Contents Inferential Statistical Tests Employed with Two or More Independent Samples (and Related Measures of Association/Correlation) 865 Test 21: The Single-Factor Between-Subjects Analysis of Variance 867 I. Hypothesis Evaluated with Test and Relevant Background Information 867 II. Example 868 III. Null versus Alternative Hypotheses 868 IV. Test Computations 869 V. Interpretation of the Test Results 873 VI. Additional Analytical Procedures for the Single-Factor Between- Subjects Analysis of Variance and/or Related Tests Comparisons following computation of the omnibus F value for the single-factor between-subjects analysis of variance (Planned versus unplanned comparisons (including simple versus complex comparisons); Linear contrasts; Orthogonal comparisons; Test 21a: Multiple t tests/fisher's LSD test; Test 21b: The Bonferroni- Dunn test; Test 21c: Tukey'sHSD test; Test 21d: The Newman- Keuls test; Test 21e: The Scheffé test; Test 21f: The Dunnett test; Additional discussion of comparison procedures and final recommendations; The computation of a confidence interval for a comparison) Comparing the means of three or more groups when k Evaluation of the homogeneity of variance assumption of the singlefactor between-subjects analysis of variance (Test 11a: Hartley's F max test for homogeneity of variance, Test 21g: The Levene Test for homogeneity of variance«test 21 h: The Brown-Forsythe test for homogeneity of variance) Computation of the power of the single-factor between-subjects analysis of variance Measures of magnitude of treatment effect for the single-factor between-subjects analysis of variance: Omega squared (Test 21 i), Eta squared (Test 21j), and Cohen's/index (Test 21k) Computation of a confidence interval for the mean of a treatment population Trend analysis 922 VII. Additional Discussion of the Single-Factor Between-Subjects Analysis of Variance Theoretical rationale underlying the single-factor between-subjects analysis of variance Definitional equations for the single-factor between-subjects analysis of variance Equivalency of the single-factor between-subjects analysis of variance and the t test for two independent samples when k = Robustness of the single-factor between-subjects analysis of variance Equivalency of the single-factor between-subjects analysis of variance and the / test for two independent samples with the chi-square test for r χ c tables when c = 2 939

15 xx Handbook of Parametric and Nonparametric Statistical Procedures 6. The general linear model Fixed-effects versus random-effects models for the single-factor between-subjects analysis of variance Multivariate analysis of variance (MANOVA) 944 VIII. Additional Examples Illustrating the Use of the Single-Factor Between- Subjects Analysis of Variance 945 IX. Addendum Test 211: The Single-Factor Between-Subjects Analysis of Covariance 946 Test 22: The Kruskal-Wallis One-Way Analysis of Variance by Ranks.981 I. Hypothesis Evaluated with Test and Relevant Background Information 981 II. Example 982 III. Null versus Alternative Hypotheses 983 IV. Test Computations 983 V. Interpretation of the Test Results 985 VI. Additional Analytical Procedures for the Kruskal-Wallis One-Way Analysis of Variance by Ranks and/or Related Tests Tie correction for the Kruskal-Wallis one-way analysis of variance by ranks Pairwise comparisons following computation of the test statistic for the Kruskal-Wallis one-way analysis of variance by ranks 986 VII. Additional Discussion of the Kruskal-Wallis One-Way Analysis of Variance by Ranks Exact tables of the Kruskal-Wallis distribution Equivalency of the Kruskal-Wallis one-way analysis of variance by ranks and the Mann-Whitney {/test when k = Power-efficiency of the Kruskal-Wallis one-way analysis of variance by ranks Alternative nonparametric rank-order procedures for evaluating a design involving k independent samples 991 VIII. Additional Examples Illustrating the Use of the Kruskal-Wallis One- Way Analysis of Variance by Ranks 992 IX. Addendum Test 22a: The Jonckheere-Terpstra Test for Ordered Alternatives 993 Test 23: The Van der Waerden Normal-Scores Test for * Independent Samples 1007 I. Hypothesis Evaluated with Test and Relevant Background Information 1007 II. Example 1008 III. Null versus Alternative Hypotheses 1008 IV. Test Computations 1009 V. Interpretation of the Test Results 1011 VI. Additional Analytical Procedures for the van der Waerden Normal- Scores Test for k Independent Samples and/or Related Tests Pairwise comparisons following computation of the test statistic for the van der Waerden normal-scores test for k independent samples 1012 VII. Additional Discussion of the van der Waerden Normal-Scores Test for k Independent Samples 1015

16 Table of Contents 1. Alternative normal-scores tests 1015 VIII. Additional Examples Illustrating the Use of the van der Waerden Normal-Scores Test for k Independent Samples 1015 Inferential Statistical Tests Employed with Two or More Dependent Samples (and Related Measures of Association/Correlation) 1021 Test 24: The Single-Factor Within-Subjects Analysis of Variance 1023 I. Hypothesis Evaluated with Test and Relevant Background Information 1023 II. Example 1025 III. Null versus Alternative Hypotheses 1025 IV. Test Computations 1025 V. Interpretation of the Test Results 1030 VI. Additional Analytical Procedures for the Single-Factor Within-Subjects Analysis of Variance and/or Related Tests Comparisons following computation of the omnibus F value for the single-factor within-subjects analysis of variance (Test 24a: Multiple / tests/fisher's LSD test; Test 24b: The Bonferroni- Dunn test; Test 24c: Tukey's HSD test; Test 24d: The Newman-Keuls test; Test 24e: The Scheffé test; Test 24f: The Dunnett test; The computation of a confidence interval for a comparison; Alternative methodology for computing MS na for a comparison) Comparing the means of three or more conditions when k Evaluation of the sphericity assumption underlying the single-factor within-subjects analysis of variance Computation of the power of the single-factor within-subjects analysis of variance Measures of magnitude of treatment effect for the single-factor within-subjects analysis of variance: Omega squared (Test 24g) and Cohen's/ index (Test 24h) Computation of a confidence interval for the mean of a treatment population Test 24i: The intraclass correlation coefficient 1053 VII. Additional Discussion of the Single-Factor Within-Subjects Analysis of Variance Theoretical rationale underlying the single-factor within-subjects analysis of variance Definitional equations for the single-factor within-subjects analysis of variance Relative power of the single-factor within-subjects analysis of variance and the single-factor between-subjects analysis of variance Equivalency of the single-factor within-subjects analysis of variance and the t test for two dependent samples when k = The Latin square design 1063 VIII. Additional Examples Illustrating the Use of the Single-Factor Within- Subjects Analysis of Variance 1065

17 xxii Handbook of Parametric and Nonparametric Statistical Procedures Test 25: The Friedman Two-Way Analysis of Variance by Ranks 1075 I. Hypothesis Evaluated with Test and Relevant Background Information 1075 II. Example 1076 III. Null versus Alternative Hypotheses 1076 IV. Test Computations 1077 V. Interpretation of the Test Results 1078 VI. Additional Analytical Procedures for the Friedman Two-Way Analysis Variance by Ranks and/or Related Tests Tie correction for the Friedman two-way analysis variance by ranks Pairwise comparisons following computation of the test statistic for the Friedman two-way analysis of variance by ranks 1080 VII. Additional Discussion of the Friedman Two-Way Analysis of Variance by Ranks Exact tables of the Friedman distribution Equivalency of the Friedman two-way analysis of variance by ranks and the binomial sign test for two dependent samples when = Power-efficiency of the Friedman two-way analysis of variance by ranks Alternative nonparametric rank-order procedures for evaluating a design involving k dependent samples Relationship between the Friedman two-way analysis of variance by ranks and Kendall's coefficient of concordance 1087 VIII. Additional Examples Illustrating the Use of the Friedman Two-Way Analysis of Variance by Ranks 1087 EX. Addendum 1088 l.test 25a: The Page Test for Ordered Alternatives 1088 Test 26: The Cochran Q Test 1099 I. Hypothesis Evaluated with Test and Relevant Background Information 1099 II. Example 1100 III. Null versus Alternative Hypotheses 1100 IV. Test Computations 1100 V. Interpretation of the Test Results 1102 VI. Additional Analytical Procedures for the Cochran Q Test and/or Related Tests Pairwise comparisons following computation of the test statistic for the Cochran Q test 1102 VII. Additional Discussion of the Cochran Q Test Issues relating to subjects who obtain the same score under all of the experimental conditions Equivalency of the Cochran Q test and the McNemar test when k = Alternative nonparametric procedures with categorical data for evaluating a design involving k dependent samples 1109 VIII. Additional Examples Illustrating the Use of the Cochran Q Test 1109 Inferential Statistical Test Employed with a Factorial Design (and Related Measures of Association/Correlation) 1117

18 Table of Contents Test 27: The Between-Subjects Factorial Analysis of Variance.1119 I. Hypothesis Evaluated with Test and Relevant Background Information 1119 II. Example 1120 III Null versus Alternative Hypotheses 1120 IV. Test Computations 1122 V. Interpretation of the Test Results 1128 VI. Additional Analytical Procedures for the Between-Subjects Factorial Analysis of Variance and/or Related Tests Comparisons following computation of the F values for the betweensubjects factorial analysis of variance (Test 27a: Multiple t tests/ Fisher's LSD test; Test 27b: The Bonferroni-Dunn test; Test 27c: Tukey's HSD test; Test 27d: The Newman-Keuls test; Test 27e: The Scheffé test; Test 27f: The Dunnett test; Comparisons between the marginal means; Evaluation of an omnibus hypothesis involving more than two marginal means; Comparisons between specific groups that are a combination of both factors; The computation of a confidence interval for a comparison; Analysis of simple effects) Evaluation of the homogeneity of variance assumption ofthe betweensubjects factorial analysis of variance Computation of the power of the between-subjects factorial analysis of variance Measures of magnitude of treatment effect for the between-subjects factorial analysis of variance: Omega squared (Test 27g) and Cohen's /index (Test 27h) Computation of a confidence interval for the mean of a population represented by a group 1150 VII. Additional Discussion of the Between-Subjects Factorial Analysis of Variance Theoretical rationale underlying the between-subjects factorial analysis of variance Definitional equations for the between-subjects factorial analysis of variance Unequal sample sizes The randomized-blocks design Additional comments on the between-subjects factorial analysis of variance (Fixed-effects versus random-effects versus mixed-effects models; Nested factors/hierarchical designs and designs involving more than two factors; Screening designs) 1158 VIII. Additional Examples Illustrating the Use of the Between-Subjects Factorial Analysis of Variance 1161 IX. Addendum Π62 Discussion of and computational procedures for additional analysis of variance procedures for factorial designs Test 27i: The factorial analysis of variance for a mixed design 1162 Analysis of a crossover design with a factorial analysis of variance for a mixed design Test 27j: Analysis of variance for a Latin square design 1182

19 xxiv Handbook of Parametric and Nonparametric Statistical Procedures 3. Test 27k: The within-subjects factorial analysis of variance Analysis of higher order factorial designs 1208 Measures of Association/Correlation 1219 Test 28: The Pearson Product-Moment Correlation Coefficient 1221 I. Hypothesis Evaluated with Test and Relevant Background Information 1221 II. Example 1225 HL Null versus Alternative Hypotheses 1225 IV. Test Computations 1226 V. Interpretation ofthe Test Results (Test 28a: Test of significance for a Pearson product-moment correlation coefficient; The coefficient of determination) 1228 VI. Additional Analytical Procedures for the Pearson Product-Moment Correlation Coefficient and/or Related Tests Derivation of a regression line The standard error of estimate Computation of a confidence interval for the value of the criterion variable Computation of a confidence interval for a Pearson product-moment correlation coefficient Test 28b: Test for evaluating the hypothesis that the true population correlation is a specific value other than zero Computation of power for the Pearson product-moment correlation coefficient Test 28c: Test for evaluating a hypothesis on whether there is a significant difference between two independent correlations Test28d: Test for evaluating a hypothesis on whether k independent correlations are homogeneous Test 28e: Test for evaluating the null hypothesis H o : p = p Tests for evaluating a hypothesis regarding one or more regression coefficients (Test 28f: Test for evaluating the null hypothesis H o : β = 0; Test 28g: Test for evaluating the null hypothesis J/ o : ß, = ß 2 ) Additional correlational procedures VII. Additional Discussion of the Pearson Product-Moment Correlation Coefficient The definitional equation for the Pearson product-moment correlation coefficient Covariance The homoscedasticity assumption of the Pearson product-moment correlation coefficient Residuals, analysis of variance for regression analysis and regression diagnostics Autocorrelation (and Test 28h: Durbin-Watson test) The phi coefficient as a special case of the Pearson product-moment correlation coefficient Ecological correlation Cross-lagged panel and regression-discontinuity designs 1288 VIII. Additional Examples Illustrating the Use of the Pearson Product- Moment Correlation Coefficient 1295

20 Table of Contents xxv IX. Addendum Bivariate measures of correlation that are related to the Pearson productmoment correlation coefficient (Test 28i: The point-biserial correlation coefficient (and Test 28i-a: Test of significance for a point-biserial correlation coefficient); Test 28j: The biserial correlation coefficient (and Test 28j-a: Test of significance for a biserial correlation coefficient); Test 28k: The tetrachoric correlation coefficient (and Test 28k-a: Test of significance for a tetrachoric correlation coefficient)) Meta-analysis and related topics (Measures of effect size; Meta-analytic procedures (Test 281: Procedure for comparing k studies with respect to significance level; Test 28m: The Stouffer procedure for obtaining a combined significance level (p value) for k studies; The file drawer problem; Test 28n: Procedure for comparing k studies with respect to effect size; Test 28o: Procedure for obtaining a combined effect size for k studies); Alternative meta-analytic procedures; Practical implications of magnitude of effect size value; Test 28p: Binomial effect size display; The significance test controversy; The minimum-effect hypothesis testing model) 1306 Test 29: Spearman's Rank-Order Correlation Coefficient 1353 I. Hypothesis Evaluated with Test and Relevant Background Information 1353 II. Example 1355 III. Null versus Alternative Hypotheses 1355 IV. Test Computations 1356 V. Interpretation ofthe Test Results (Test 29a: Test of significance for Spearman's rank-order correlation coefficient) 1357 VI. Additional Analytical Procedures for Spearman's Rank-Order Correlation Coefficient and/or Related Tests Tie correction for Spearman's rank-order correlation coefficient Spearman's rank-order correlation coefficient as a special case of the Pearson product-moment correlation coefficient Regression analysis and Spearman's rank-order correlation coefficient Partial rank correlation Use of Fisher's z r transformation with Spearman's rankorder correlation coefficient 1364 VII. Additional Discussion of Spearman's Rank-Order Correlation Coefficient The relationship between Spearman's rank-order correlation coefficient, Kendall's coefficient of concordance, and the Friedman two-way analysis of variance by ranks Power efficiency of Spearman's rank-order correlation coefficient Brief discussion of Kendall's tau: An alternative measure of association for two sets of ranks Weighted rank/top-down correlation 1367 VIII. Additional Examples Illustrating the Use of the Spearman's Rank-Order Correlation Coefficient 1368

21 Handbook of Parametric and Nonparametric Statistical Procedures Test 30: Kendall's Tau I. Hypothesis Evaluated with Test and Relevant Background Information 1371 II. Example 1373 III. Null versus Alternative Hypotheses 1373 IV. Test Computations 1374 V. Interpretation ofthe Test Results (Test 30a: Test of significance for Kendall's tau) 1376 VI. Additional Analytical Procedures for Kendall's Tau and/or Related Tests Tie correction for Kendall's tau Regression analysis and Kendall's tau Partial rank correlation Sources for computing a confidence interval for Kendall's tau 1382 VII. Additional Discussion of Kendall's Tau Power efficiency of Kendall's tau Kendall's coefficient of agreement 1382 VIII. Additional Examples Illustrating the Use of Kendall's Tau 1382 Test 31: Kendall's Coefficient of Concordance 1387 I. Hypothesis Evaluated with Test and Relevant Background Information 1387 II. Example 1388 III. Null versus Alternative Hypotheses 1388 IV. Test Computations 1389 V. Interpretation ofthe Test Results (Test 31a: Test of significance for Kendall's coefficient of concordance) 1390 VI. Additional Analytical Procedures for Kendall's Coefficient of Concordance and/or Related Tests Tie correction for Kendall's coefficient of concordance 1391 VII. Additional Discussion of Kendall's Coefficient of Concordance Relationship between Kendall's coefficient of concordance and Spearman's rank-order correlation coefficient Relationship between Kendall's coefficient of concordance and the Friedman two-way analysis of variance by ranks Weighted rank/top-down concordance Kendall's coefficient of concordance versus the intraclass correlation coefficient 1396 VIII. Additional Examples Illustrating the Use of Kendall's Coefficient of Concordance 1398 Test 32: Goodman and Kruskal's Gamma 1403 I. Hypothesis Evaluated with Test and Relevant Background Information II. Example 1404 III. Null versus Alternative Hypotheses 1405 IV. Test Computations 1406 V. Interpretation ofthe Test Results (Test 32a: Test of significance for Goodman and Kruskal's gamma) 1409 VI Additional Analytical Procedures for Goodman and Kruskal's Gamma and/or Related Tests

22 Table of Contents XXVII 1. The computation of a confidence interval for the value of Goodman and Kruskal's gamma Test 32b: Test for evaluating the null hypothesis H^ γ, = γ Sources for computing a partial correlation coefficient for Goodman and Kruskal's gamma 1412 VII. Additional Discussion of Goodman and Kruskal's Gamma Relationship between Goodman and Kruskal's gamma and Yule's Q Somers' delta as an alternative measure of association for an ordered contingency table 1412 VIII. Additional Examples Illustrating the Use of Goodman and Kruskal's Gamma 1413 Multivariate Statistical Analysis 1417 Matrix Algebra and Multivariate Analysis 1419 I. Introductory Comments on Multivariate Statistical Analysis 1419 II. Introduction to Matrix Algebra 1420 Test 33: Multivariate Regression 1433 I. Hypothesis Evaluated with Test and Relevant Background Information 1433 II. Examples 1439 ΠΙ. Null versus Alternative Hypotheses 1441 IV/V. Test Computations and Interpretation of the Test Results Test computations and interpretation of results for Example 33.1 (Computation of the multiple correlation coefficient; The coefficient of multiple determination; Test of significance for a multiple correlation coefficient; The multiple regression equation; The standard error of multiple estimate; Computation of a confidence interval for Γ; Evaluation of the relative importance of the predictor variables; Evaluating the significance of a regression coefficient; Computation of a confidence interval for a regression coefficient; Analysis of variance for multiple regression; Semipartial and partial correlation (Test 33a: Computation of a semipartial correlation coefficient; Test of significance for a semipartial correlation coefficient; Test 33b: Computation of a partial correlation coefficient; Test of significance for a partial correlation coefficient) 1442 Test computations and interpretation of results for Example 33.2 with SPSS 1459 VI. Additional Analytical Procedures for Multiple Regression and/or Related Tests Cross-validation of sample data 1470 VII. Additional Discussion of Multivariate Regression Final comments on multiple regression analysis Causal modeling: Path analysis and structural equation modeling Brief note on logistic regression 1473 VIII. Additional Examples Illustrating the Use of Multivariate Regression 1474