Index

Fn1;n2 , 284H0, 285H1, 285Φ(x), 284𝛼, 284𝛼-quantile, 284, 285𝜒2 Goodness-of-fit test, 164𝜒2-test

variance (mean unknown), 39variance (mean known), 36

𝜒𝛼;n2, 284

x, 284|x|, 284f𝛼;n1,n2 , 285n, 285t𝛼;n, 284z𝛼 , 284

Absolute value, 284Agreement, 197Alternative hypothesis, 3, 285Anderson–Darling test, 142, 157ANOVA

one-way, 253two-way, 255

Ansari–Bradley test, 125

Bartels’ test, 181Bartlett test, 258Binomial k-sample test, 62Binomial test, 51

Statistical Hypothesis Testing with SAS and R, First Edition. Dirk Taeger and Sonja Kuhnt.© 2014 John Wiley & Sons, Ltd. Published 2014 by John Wiley & Sons, Ltd.

Bowker’s test, 203Brown–Forsythe test, 260

Cohen’s kappa, 197Continuity correction, 284Correlation coefficient

partial, 91Pearson’s product moment, 81Spearman’s rank, 86Spearman’s rho, 86two correlation coefficients, 94

Cramér–von Mises test, 145, 160Critical region, 4

Datasetsblood pressure, 265coin, 268crop, 270harvest, 268health, 270hospital infections, 267iq, 266malfunction, 267pvalues, 269silicosis, 269students, 268waiting, 269workpieces, 266

datasets, 264David–Hartley–Pearson test,

223

288 INDEX

Distribution functionR, 13SAS, 11

Dixon’s tests, 225

EDF, 284EDF tests, 139, 154Empirical distribution function, 284Error

Type I, 4Type II, 4

Exponential distribution, 76

F-distribution, 284F-test

variance (two independentsamples), 41

Fisher’s exact test, 189Fisher–Irwin test, 189Freeman–Halton test, 189

Goodness-of-fit test𝜒2, 164Anderson–Darling, 142, 157Cramér–von Mises, 160, 145Jarque–Bera, 150Kolmogorov–Smirnov, 139, 154Lilliefors normality, 139Shapiro–Wilk, 148

Grubbs’ test, 219

Homogeneity, 189Homogeneity of variances, 258

Independence, 189

Jarque–Bera test, 150

Kappa, 197Kolmogorov–Smirnov test, 139, 154Kolmogorov–Smirnov two, sample

test, 132Kruskal–Wallis rank-sum test, 116Kruskal–Wallis test, 116

Large sample test on the odds ratio, 205Large sample test on the relative risk, 210

Levene test, 260Likelihood-ratio 𝜒2-test, 195Lilliefors test for normality, 139Linear regression, 239, 246

test on an individual regressioncoefficient, 247

test on the intercept, 243test on the slope, 239

Mann–Whitney U test, 110Mann–Whitney–Wilcoxon, 110McNemar’s test, 200Mid ranks, 285Monte–Carlo simulation, 13Mood test, 128Multiple linear regression, 246

test on an individual regressioncoefficient, 247

Non EDF tests, 148, 164Normality test, 139, 154

Anderson–Darling, 142Cramér–von Mises, 145Jarque–Bera, 150Kolmogorov–Smirnov, 139Lilliefors normality, 139Shapiro–Wilk, 148

Null hypothesis, 3, 285

Odds ratio, 205One-way ANOVA, 253Outliers from Gaussian distributions, 219Outliers from non Gaussian null

distributions, 229Overall F-test, 250

p-value, 4, 11, 285calculation, 11one- and two-sided, 11

Partial correlation coefficients, 91Partial F-test, 247Pearson’s 𝜒2 test, 192Pearson’s product moment correlation

coefficient, 81Poisson distribution, 67Poisson parameter

exact test, 70

INDEX 289

test on difference, 72z-test, 67

Probability value, 285Programming philosophy of SAS and R, 5Proportion tests

binomial test, 51difference of proportions (unpooled

variances), 55equal proportions (pooled

variances), 59K-sample test, 62

Randomness testBartels’ test, 181runs test for serial randomness, 174runs up and down test, 174single-sample runs test, 169von Neumann rank test, 181von Neumann test, 178Wald–Wolfowitz runs test, 169

Ranks, 285mid ranks, 285ties, 285

Relative risk, 210Risk measures, 205Run, 285Run tests, 169Runs test for serial randomness, 174Runs up and down test, 174

Sample mean, 284Sample size, 285Shapiro–Wilk test, 148Siegel–Tukey test, 120Sign test, 101Signed-rank test, 105Significance level, 4, 284Simple kappa coefficient, 197Simple linear regression, 239

test on the intercept, 243test on the slope, 239

Single-sample runs test, 169Smirnov test, 132Spearman’s rank correlation

coefficient, 86Standard normal distribution

function, 284

Statistical testsalternative hypothesis, 3critical region, 4directional hypothesis, 9Fisher theory, 1hypothesis testing, 3Neyman–Pearson theory, 1non directional hypothesis, 9null hypothesis, 3p-value, 4, 11R, 4SAS, 4significance level, 4test problem, 3test statistic, 3theory of, 3two- and one-sided p-values, 11two- and one-sided tests, 9type I error, 4type II error, 4

t-testmean (one-sample), 22mean (paired), 33mean (two-sample), 26, 28, 33mean (unequal variances), 28mean (unpaired), 26, 28variance (two dependent

samples), 44Tables

𝜒2-distribution (critical values), 273,274

F-distribution (criticalvalues), 275–283

Gaussian distribution (criticalvalues), 271

normal distribution (criticalvalues), 271

t-distribution (critical values), 272Test

𝜒2-test on the variance (meanknown), 36

𝜒2 Goodness-of-fit, 164𝜒2-test on the variance (mean

unknown), 39t-test on variances of two dependent

populations, 44

290 INDEX

Test (continued)agreement, 197

Anderson–Darling, 142, 157

ANOVA (one-way), 253

ANOVA (two-way), 255

Ansari–Bradley test, 125

Bartels’ test, 181

Bartlett, 258

based on the EDF, 139, 154

beta distribution, 156, 159, 162

binomial test, 51

Bowker’s test for symmetry, 203

Brown–Forsythe, 260

Cohen’s kappa, 197

correlation, 81

correlation coefficient, 81, 86, 91

Cramér–von Mises, 145, 160

David–Hartley–Pearson test, 223

dispersion difference, 120

Dixon’s tests, 225

exact test on Poisson parameter, 70

exponential distribution, 76, 156,

160, 163

Fisher’s exact test, 189

Fisher–Irwin test, 189

for significance of regression, 250

Freeman–Halton test, 189

gamma distribution, 156

Gaussian distribution, 139

generalized Pareto distribution, 156,

159, 162

Grubbs’ test, 219

Gumbel distribution, 156, 159, 162

homogeneity, 189

homogeneity of variances, 258

independence, 189

inverse Gaussian distribution, 156,

159, 162

Jarque–Bera test, 150

Johnson SB distribution, 156, 159,

162

Johnson SU distribution, 156, 159,

162

k-sample binomial test, 62

Kappa, 197

Kolmogorov–Smirnov, 139, 154

Kolmogorov–Smirnov two sampletest, 132

Kruskal–Wallis rank-sum test, 116Kruskal–Wallis test, 116Levene, 260likelihood-ratio 𝜒2 test, 195Lilliefors normality, 139linear regression, 239, 246location, 101lognormal distribution, 156, 159,

160, 162, 163Mann–Whitney U test, 110Mann–Whitney–Wilcoxon test, 110marginal homogeneity, 200McNemar’s test, 200mean, 19mean (one-sample), 19, 22mean (paired), 31, 33mean (two-sample), 23, 26, 28, 31,

33mean (unequal variances), 28mean (unpaired), 23, 26, 28Mood test, 128multiple linear regression, 246normality, 139not based on the EDF, 148, 164odds ratio, 205on an individual regression

coefficient, 247on non normality distributions, 154on the intercept, 243on the slope, 239other nonparametric tests, 132outliers, 219outliers for non Gaussian null

distributions, 229outliers from Gaussian

distributions, 219overall F-test, 250paired t-test, 33paired z-test, 31partial correlation, 91partial F-test, 247Pearson’s 𝜒2-test, 192Pearson’s product moment correlation

coefficient, 81Poisson distribution, 67

INDEX 291

Poisson parameter exact test, 70power function distribution, 156,

159, 162randomness, 169Rayleigh distribution, 156, 159, 162regression analysis, 239relative risk, 210risk measures, 205run tests, 169runs test for serial randomness, 174runs up and down test, 174scale difference, 120Shapiro–Wilk, 148Siegel–Tukey test, 120sign test, 101signed-rank test, 105simple kappa coefficient, 197simple linear regression, 239single-sample runs test, 169Smirnov test, 132Spearman’s rank correlation

coefficient, 86spread difference, 120symmetry, 197t-test, 22test on outliers for exponential null

distributions, 229test on outliers for uniform null

distribution, 232two correlation coefficients, 94two-sample F-test on two populations

variances, 41two-sample pooled t-test, 26variance, 36variance (mean known), 36variance (mean unknown), 39variance (two dependent

samples), 44variance (two independent

samples), 41variance homogeneity, 258von Neumann rank test, 181von Neumann test, 178Wald–Wolfowitz runs test, 169Weibull distribution, 156, 159, 160,

162, 163Welch test, 28

Wilcoxon matched-pairs signed-ranktest, 114

Wilcoxon rank-sum test, 110Wilcoxon signed-rank test, 105z-test, 19z-test for the difference of two

proportions (unpooledvariances), 55

z-test for the equality between twoproportions (pooledvariances), 59

z-test for two correlation coefficients(independent populations),94

z-test on Poisson parameter, 67z-test on the difference of two

Poisson parameters, 72Test for significance of regression, 250Test for symmetry

Bowker’s test for symmetry, 203Test on agreement

Cohen’s kappa, 197kappa, 197McNemar’s test, 200simple kappa coefficient, 197

Test on an individual regressioncoefficient, 247

Test on marginal homogeneityMcNemar’s test, 200

Test on outliers for exponential nulldistribution, 229

Test on outliers for uniform nulldistributions, 232

Test on risk measuresodds ratio, 205relative risk, 210

Test on the intercept, 243Test on the slope, 239Tests for homogeneity of variances, 258Tests in regression analysis, 239Tests on contingency tables

Bowker’s test for symmetry, 203Cohen’s kappa, 197Fisher’s exact test, 189Fisher–Irwin test, 189Freeman–Halton test, 189kappa, 197

292 INDEX

Tests on contingency tables (continued)likelihood-ratio 𝜒2-test, 195McNemar’s test, 200odds ratio, 205Pearson’s 𝜒2-test, 192relative risk, 210simple kappa coefficient, 197

Tests on heterogeneitylikelihood-ratio 𝜒2-test, 195

Tests on homogeneityFisher’s exact test, 189Fisher–Irwin test, 189Freeman–Halton test, 189Pearson’s 𝜒2-test, 192

Tests on independenceFisher’s exact test, 189Fisher–Irwin test, 189Freeman–Halton test, 189likelihood-ratio 𝜒2-test, 195Pearson’s 𝜒2-test, 192

Tests on outliers, 219David–Hartley–Pearson test, 223Dixon’s tests, 225exponential distribution, 229Gaussian distribution, 219, 223, 225Grubbs’ test, 219uniform distribution, 232

Tests on randomness, 169Ties, 285Two-way ANOVA, 255

Type I error, 4Type II error, 4

Variance homogeneity, 258von Neumann rank test, 181von Neumann test, 178

Wald–Wolfowitz runs test, 169Welch–Satterthwaite test, 29Welch test, 28Wilcoxon matched-pairs signed-rank

test, 114Wilcoxon rank-sum test, 110Wilcoxon signed-rank test, 105

z-testdifference of proportions (unpooled

variances), 55difference between two Poisson

parameters, 72equal proportions (pooled

variances), 59exponential distribution

parameter, 76mean (one-sample), 19mean (paired), 31mean (two-sample), 23, 31mean (unpaired), 23Poisson parameter, 67two correlation coefficients, 94