Biostatistics the bare essentials

BIOSTATISTICS The Bare Essentials Fourth Editon Testimonials from Readers “I’m actually enjoying a stats book! When you said someone was laughing so hard they dropped your textbook in the bath, I was skeptical, but now, I truly understand! I often comment that your text is very funny and people are surprised that I did not mean the phrase as an oxymoron Thank you for taking this approach to teaching stats!” Le-anh Ngo “I hate stats, loath stats But, I have just discovered Biostatistics, and just had to say thank you for making me laugh I have never had things explained so simply and appropriately (this is my third go at stats, as an undergraduate, in my graduate dip, and now my PhD) Keep writing you are making a difference to some poor stats students Karen Munk Yes, the marginalia is very popular with faculty and students alike; a colleague and I giggled hysterically over the humorous examples and marginalia (those passing her office thought we’d lost our marbles) My favorite is Chapter 13—the yuppie patients Usually statistics books put me to sleep, but Biostatistics is the exception.” Christine Marton I thought Biostatistics looked interesting—and it definitely is! Without it, I don’t know if I would ever have made sense of the subject I’m in the midst of writing up my PhD thesis, and it’s been an invaluable reference and—perhaps more miraculous—I never read a page without a chuckle The great mix of humor and statistics are too often thought to be mutually exclusive I’m recommending this book to all my fellow graduate students.” Joe Brown “Biostatistics: The Bare Essentials would have been life-changing had it been published when I was in graduate school in at University of Chicago It now must settle for being life- enhancing Statistics doesn’t have to be that god-awful boring! And it can be simply described as well I thought Howard Wainer was clear, but you guys are even better.” Rebecca J McCauley I just wanted you to know that you and Dr Streiner are the only people on the planet who can make me laugh when trying to figure out statistics.” Monica McHenry A NOTE ON THE FRONT COVER The cover depicts the famous “Study of Human Proportion in the Manner of Vitruvius “by Leonardo da Vinci, drawn about 1490, and done to death 500 years later in 2000 Those with a classical bent may wish to know the origin of the idea According to Renaissance notions, the “Perfect Man” was based on geometric principles The arms outstretched, the top of the head, and the tip of the feet defined a square, and the tips of the arms and legs outstretched in a fanlike position inscribed a circle centered on the navel What da Vinci failed to notice is that the legs fit precisely on a normal curve, with the mean between the two heels and the apex at the crotch, one standard deviation falling exactly on the two kneecaps, and the asymptotes at the corners of the inscribed square The centers of the two feet, at the point where they intersect the arc of the circle, then determine the conventional criterion for statistical significance at ± two standard deviations from the mean Leonardo da Vinci can be forgiven, however Statistics hadn’t been invented yet in 1492 BIOSTATISTICS The Bare Essentials Fourth Edition Geoffrey R Norman, PhD Professor, Department of Clinical Epidemiology and Biostatistics McMaster University Hamilton, Ontario, Canada David L Streiner, PhD Professor, Department of Psychiatry University of Toronto Toronto, Ontario, Canada Professor Emeritus, Department of Clinical Epidemiology and Biostatistics Professor Emeritus, Department of Psychiatry and Behavioural Neurosciences McMaster University Hamilton, Ontario, Canada Master Woodworker Halton County Radial Railway Museum Milton, Ontario, Canada 2014 PEOPLE'S MEDICAL PUBLISHING HOUSE—USA SHELTON, CONNECTICUT People’s Medical Publishing House-USA 2 Enterprise Drive, Suite 509 Shelton, CT 06484 Tel: 203-402-0646 Fax: 203-402-0854 E-mail: info@pmph-usa.com © 2014 PMPH-USA, Ltd All rights reserved Without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form or by any means (electronic, mechanical, photocopying, recording, or otherwise), without the prior written permission of the publisher 14 15 16 17/King/9 8 7 6 5 4 3 2 1 Biostatistics 4th edition without SPSS software: ISBN-13: 978-1-60795-178-0 ISBN-10: 1-60795-178-9 eISBN-13: 978-1-60795-279-4 Biostatistics 4th edition with SPSS software: ISBN-13: 978-1-60795-190-2 ISBN-10: 1-60795-190-8 Printed in the United States of America by [to come] Executive Editor: Linda Mehta; Typesetter: diacriTech; Cover Designer: Mary McKeon Library of Congress Cataloging-in-Publication Data On file Sales and Distribution Canada Login Canada 300 Saulteaux Cr., Winnipeg, MB R3J 3T2 Phone: 1.800.665.1148 Fax: 1.800.665.0103 www.lb.ca Foreign Rights John Scott & Company International Publisher’s Agency P.O Box 878 Kimberton, PA 19442 USA Tel: 610-827-1640 Fax: 610-827-1671 Japan United Publishers Services Limited 1-32-5 Higashi-Shinagawa Shinagawa-ku, Tokyo 140-0002 Japan Tel: 03-5479-7251 Fax: 03-5479-7307 Email: kakimoto@ups.co.jp United Kingdom, Europe, Middle East, Africa McGraw Hill Education Shoppenhangers Road Maidenhead Berkshire, SL6 2QL England Tel: 44-0-1628-502500 Fax: 44-0-1628-635895 www.mcgraw-hill.co.uk Singapore, Thailand, Philippines,Indonesia, Vietnam, Pacific Rim, Korea McGraw-Hill Education 60 Tuas Basin Link Singapore 638775 Tel: 65-6863-1580 Fax: 65-6862-3354 www.mcgraw-hill.com.sg Australia, New Zealand Elsevier Australia Locked Bag 7500 Chatswood DC NSW 2067 Australia Tel: 161 (2) 9422-8500 Fax: 161 (2) 9422-8562 www.elsevier.com.au Brazil SuperPedido Tecmedd Beatriz Alves, Foreign Trade Department R Sansao Alves dos Santos, 102 | 7th floor, Brooklin Novo Sao Pãulo 04571-090, Brazil Tel: 55-16-3512-5539 www.superpedidotecmedd.com.br India, Bangladesh, Pakistan, Sri Lanka, Malaysia Jaypee Brothers Medical Publishers (P) Ltd 4838, 24 Ansari Road, Daryaganj New Delhi- 110002, India Desk Phone: +91 11 43574357 People’s Republic of China People’s Medical Publishing House International Trade Department No 19, Pan Jia Yuan Nan Li Chaoyang District Beijing 100021 PR China Tel: 8610-67653342 Fax: 8610-67691034 www.pmph.com/en/ To two people whose hard work, patience, diligence, and, most important, unflagging good humor, have made it possible: Geoff R Norman and David L Streiner Too many people confuse being serious with being solemn John Cleese One of the first symptoms of an approaching nervous breakdown is the belief that one’s work is terribly important Bertrand Russell Most researchers use statistics the way a drunkard uses a lamppost—more for support than illumination Winifred Castle Newman-Keuls Test, 89, 92 normally distributed, 88 orthogonal comparisons, 94t parts of, 86–87 planned comparisons, 88 orthogonal, 93–94 post-hoc comparisons, 88, 89 range tests, 89 repeated-measures, 117 sample size and power, 95–96 Scheffe method, 89, 93 signal-to-noise ratio, 87 strength of relationship, 95 Studentized range, 90–91 sums of squares, 86 tests of significance for ranked data, 278 Tukey’s HSD, 89, 91, 91t, 92t Tukey’s wholly significant difference, 92 OR see Odds Ratio (OR) Ordered medians tests of significance for ranked data, 279–280 Ordinal data graphing, 9 Ordinal variable, 4–5, 340, 341, 341t, 343t Ordinary stepwise regression multiple regression, 160 Orthogonal comparisons one-way analysis of variance, 94t Orthogonal decomposition measuring change, 197, 198 Orthogonal designs factorial ANOVA, 106 Orthogonal underlying factors, 212–213 Orthogonal vs oblique rotations, 218–219, 218f, 219f Outer fence, 26 Outliers, 26, 27f, 338 far, 26, 27f histogram of, 329f multiple regression, 164 statistical identification, 329 visual identification, 329 Overall relative risk survival analysis, 302 Overdispersion, 178 Over-identified path models, 232 P Paired and matched observations tests of significance for categorical frequency data, 257–258 Paired bar chart, 341t Paired t-test, 109–114, 110t, 112t, 113, 114t, 342t effect size, 112–113 equivalent of, 343 responsiveness to change, 112 sample size calculation, 114 Pairwise deletion of data, 331 Parallel analysis, 215 Parameter Estimates, 177 Parametric tests, 5, 341–342, 342t non-normality, 334 Parsimony fit index path analysis, 239 Part correlation, 155 Partial agreement and weighted kappa measures of association for categorical data, 272–274 Partial and semipartial correlations multiple regression, 155–156 Partial association tests of significance for categorical frequency data, 262 Partial correlations, 211 Partial F-tests multiple regression, 155, 155t Partial test stepwise logistic regression, 175 Path analysis, 226– 234, 227t, 229f, 237f, 241f, 242t additive variables, 234 AGFI, 239, 243t AIC, 239 assumptions, 233–234 CAIC, 239 causality, 229 coefficients, 228, 228f comparative fit indices, 238 comparing two factor analyses, 241–242 confirmatory factor analysis, 240–241, 240f constrained parameters, 236–237, 243f correlations, 228t, 235t decomposition, 228 implied, 232 reproduced, 232–233 CR, 241, 243t diagram, 228 disattenuate, 235 disturbance terms, 230–231 DV effects, 228 endogenous variables, 229 estimation, 238 exogenous variables, 229 factor loading, 235f finding your way through, 228–229 fixed parameters, 236–237 free parameters, 236 full SEM model, 242–244 fully determined models, 232 GFI, 232–233, 239, 243t identification, 236–238 incremental fit index (IFI), 239 independence model, 241 K.I.S.S., 232 Lagrange multiplier test, 240 linear regression, 227f matrix rank, 237 maximum likelihood (ML), 238 measurement model, 236 MI, 241, 243t ML, 238 models, 229–230 model specification, 236 multicollinearity of predictor variables, 234 multiple regression, 227 NFI, 238, 243t NFI2, 239 number interpretation, 227 parsimony fit index, 239 respecification, 239–240 RMSEA, 239 sample size, 234 saturated model, 241 SE, 243t structural equation modeling, 234–235 factor analysis, 234 latent variables, 234 measured variables, 234 testing the fit, 238–239 ULS, 238 Wald statistic, 240 WLS, 238 Path coefficients path analysis, 228, 228f Path diagram path analysis, 228 Path models, 230f independent, 229–230, 342t indirect, 230 just-identified, 232 mediated, 230 nonrecursive, 231, 231f path analysis, 229–230 recursive, 231 undefined, 232 Pattern matrix, FA, 216 PCA see Principal components analysis (PCA) Pearson Correlation Coefficient of linear regression, 335 simple regression and correlation, 145 transforming data, 335 Pearson’s r, 341, 341t Perinatal mortality rate, 314 Period prevalence, 310 Permutations probability, 40 Permutation test, 265 Person-years survival analysis, 294 Phi, 341t, 343t measures of association for categorical data, 270 Pie charts, 14, 14f Pillai’s trace MANOVA, 129, 130t, 132t Planned comparisons one-way analysis of variance, 88 orthogonal, 93–94 Platykurtosis, 25 Point-biserial correlation measures of association for ranked data, 285–286, 286t, 289, 291 Point prevalence, 309 Poisson distribution tests of significance for categorical frequency data, 252 Poisson regression, 153, 175–176, 176f deviant statistics, 177 parameter estimates for, 177t sample problem, 176–177 Polygons see Frequency, polygons Pooled vs separate variance estimates, 81 Population means, 47 and samples differences, 47–48 variances, 47 Positive skew, 25 Posterior probability, 68 Post-hoc comparisons one-way analysis of variance, 88, 89 Power, 53 ANCOVA, 187 MANOVA, 133 sample size, 133 series regression, 187 of study, 53 Precision, inaccurate, 16 Predicting means from differences factorial ANOVA, 100t Predictor variables multicollinearity of, 234 Prevalence, 309–310 Principal axis factoring, 213 Principal components analysis (PCA), 208, 213 Probability, 37–45, 39t, 41f, 41t and additive rule, 38–39 binomial distribution, 41–45, 43f CI, 62–63, 63f combinations and permutations, 40 conditional, 39 survival analysis, 296 conditionally dependent events, 38–39 conditionally probable events, 39 cumulative, 296 of the difference, 49, 50 effect sizes, 64 empirical derivation of, 37–38 factorials, 40 forest plots, 66, 67f independent events, 40 multiplicative law, 39 mutually exclusive events, 38–39 reporting, 66 ritual and myth of p less than 05, 54–55 sample size estimation, 64–66, 66f statistical significance vs clinical importance, 63–64, 63t survival analysis, 296 theoretical derivation of, 38 two tails vs one tail, 61–62 Probability of superiority, 277 Problems with Ratios, 312–313 Product-limit method survival analysis, 296–297 Prognostic variables time-dependent, 303 time-independent, 303 Promax rotation, 219 Proportion, 5 difference between, 344 testing, 305 of variance in multiple regression, 154f Pseudo-R2 statistics, 174 Q Quadratic trends one-way analysis of variance, 95 Quadratic weights measures of association for categorical data, 273 Quartiles, 26 Quick and dirty sample sizes, 344–345 Quintiles, 23 R R2 multiple regression, 161 adjusted, 161, 161f Random and fixed factors factorial ANOVA, 104–105 Random effects regression, 201 HLM, 203 Randomized controlled trial (RCT), 345 tests of significance for categorical frequency data, 251 Random sampling, 47 Range, 9, 22 variations on, 23 Range tests one-way analysis of variance, 89 Ranked data, 276–281, 277t, 281t see also Measures of association for ranked data Friedman two-way ANOVA, 280–281 Jonckheere test, 279 Kruskal-Wallis one-way ANOVA, 278 Mann-Whitney U test, 277 median test, 319 more than two groups, 278, 278t multiple comparisons, 279 ordered medians, 279–280 repeated measures, 280–281 sample size and power, 281 two grouping factors, 280 two independent groups, 277–278 Wilcoxon rank sum test, 277 Wilcoxon signed rank test, 280–281 Rank order, 9t Rank variables, 341, 343t Rate, 5 Ratio data, 20 Ratios problems with, 312–313 standard error of, 313 Ratio variables, 4–5, 235, 316, 340, 341, 341t, 342t categorization, 136 RCT see Randomized controlled trial (RCT) Rectangular distribution, 32 Recursive path model, 231 Reduction in deviance, 177 Regression, 235 see also Logistic regression; Multiple regression; Simple regression and correlation advanced topics in, 181–190 coefficient, standard, 143t equations, extrapolation, 237 line, 141, 144 linear path analysis, 227f transforming data, 335 mean, 193, 194f and ANCOVA, 194–195 differential, 195 mixed effects, 201 HLM, 203 nonlinear, ANCOVA, 186–187 ordinary stepwise multiple regression, 159t, 160 Poisson see Poisson regression power series, ANCOVA, 187 random effects, 201 HLM, 203 residual, 143t stepwise logistic partial test, 175 stepwise multiple regression, 158–159, 159f Reject the null hypothesis, 51 Related samples t-test, 341 Related variables, 343t Relative improvement over chance (RIOC) measures of association for categorical data, 272 Relative odds, 311 logistic regression, 172 Relative risk (RR), 310–311 confidence intervals for, 313–314 vs OR, 312 survival analysis, 301 Relative risk reduction (RRR), 311 Repeated measures, 110, 136 tests of significance for ranked data, 281 Repeated-measures analysis of variance, 115–124, 120t, 123t, 138, 342t assumptions and limitations of designs, 121–122 between-subjects and within-subjects factors, 119–120, 119t central limit theorem, 121 crossed and nested factors, 120 Intraclass Correlation Coefficient (ICC), 121 Latin Square, 120 measurement reliability, 121 sample size estimation, 121 Reporting guidelines, 83–84, 96 p-levels, 345 probability, 66 writing up, 345 Reproduced correlation path analysis, 232–233 Resampling methods, 265–266 Residual multiple regression, 163 Respecification path analysis, 239–240 Response formats, 223 Responsiveness to change paired t-test, 112 Reverse J transforming data, 336 Rho, 341t, 343, 343t measures of association for ranked data, 284–285, 290 RIOC see Relative improvement over chance (RIOC) Ritual and myth of p less than 05 probability, 54 RMSEA see Root mean square error of approximation (RMSEA) Robust Cox model survival analysis, 305 Robust estimators, 28 Robustness, 27 Root mean square error of approximation (RMSEA) path analysis, 239 Rotated factor loading matrix, 238t Rotating, FA, 216 Roy’s largest root MANOVA, 129, 130t, 132t RR see Relative risk (RR) RRR see Relative risk reduction (RRR) S Sample mean, 47, 49 alternative hypothesis, 65f null hypothesis, 65f Samples and populations difference between them, 47–48 inferential statistics, 48–49 Sample size, 63t ANCOVA, 189 calculation factorial ANOVA, 107 multiple regression, 165–166 paired t-test, 114 equal t-test, 79 estimation hypothesis testing, 148–149 probability, 65, 65f repeated-measures ANOVA, 121 simple regression and correlation, 148–149, 149f tests of significance for categorical frequency data, 264–265, 264t, 248t FA, 222 HLM, 206 path analysis, 233 and power one-way analysis of variance, 95–96 survival analysis, 305–306 tests of significance for ranked data, 281 t-test, 83 quick and dirty, 344–345 Sample standard deviations, 47 Sample t-test related, 341 Sample variances, 47 Saturated model path analysis, 241 Scale reliability, FA, 223 Scatter plots, 341t multiple regression, 163f simple regression and correlation, 146f Scheffé method one-way analysis of variance, 89, 93 Score coefficient matrix, FA, 222 Score observed measuring change, 192 Scree test, FA, 214 SD see Standard deviation (SD) SDS see Self-Rating Depression Scale (SDS) SE see Standard error (SE) Secular trend survival analysis, 300 Self-Rating Depression Scale (SDS), 33 SEM see Standard error (SE), of mean; Structural equation modeling (SEM) Sensitivity, 68 Severe skew transforming data left, 336 right, 336 Shrinkage multiple regression, 161 Sigma, 19 Signal, 60, 79 Signal-to-noise ratio one-way analysis of variance, 87 statistical inference, 60 Significance factor loading, 219t simple regression and correlation, 147–148 writing up, 345 Sign test tests of significance for categorical frequency data, 263–264 Simple regression and correlation, 140–166, 141f, 142t, 143t, 146t, 147f, 342t coefficient of determination, 144–145 confidence intervals, 147–148 correlation coefficient, 144–145 interpretation of, 145–146 covariance, 145 effect size, 147–148 Fisher’s z, 147 F-test, 143 least-squares analysis, 142 making r normal, 147 Pearson Correlation Coefficient, 145 regression line, 144 errors and confidence intervals, 141 regression residual, 143t sample size estimation, 148–149, 149f hypothesis testing, 148–149 scatter plots, 146f significance tests, 147–148 standard regression coefficient, 143t sum of squares, 142 Skewed distribution, 74, 341t histograms, 28f mean, 28f median, 28f mode, 28f Skewness, 19, 25, 26 left, 25 normal curve, 34 right, 25 Small numbers tests of significance for categorical frequency data, 255 SMC see Squared multiple correlation (SMC) Spearman rank correlation measures of association for ranked data, 285 Spearman’s rho, 288, 343, 343t measures of association for ranked data, 284–285, 290–291 vs Tau, interpretation of, 287 Specificity, 68 Sphericity doubly repeated MANOVA, 131 SPSS data entry, 347–350 defining variables, 348–349 getting started with, 347–357 reading existing files, 350–353 transforming data, 353–357 Squared multiple correlation (SMC), 236 FA, 213 Square root transforming data, 337 SSCP see Sum of squares and cross-products (SSCP) Stacked bar charts, 15f Stacked graphs, 15, 15f Stacked line graphs, 15f Standard deviation (SD), 23–24, 47, 176, 341t correlated, 336f independent, 336f sample, 47 t-test, 80 Standard error (SE), 26, 79 difference in proportion, 263 inferential statistics, 48–49 of mean, 50, 237 path analysis, 243t of proportion, 263 of ratios, 313 survival analysis, 298–299 tests of significance for categorical frequency data, 263 Standardized mean difference, 82 Standardized regression coefficient multiple regression, 156 simple regression and correlation, 143t Standard scores, 32–33, 33t Statistical distributions, 223 Statistical etiquette, 57–59 Statistical identification outliers, 329 Statistical inference, 46–69 elements of, 48–49 signal-to-noise ratio, 60–61 Statistical significance, 63 vs clinical importance, 63–64, 63t Statistics see also Inferential statistics descriptive, 2, 30, 340, 341t multivariate, 344 need for, 2 nonparametric, 343–344, 343t tests of significance for categorical frequency data, 252 univariate, 341 Stem-and-leaf plots, 10–11, 11t, 341t SPSS, 18 Stepwise regression logistic partial test, 175 multiple regression, 158–159, 159f Stopping rules, 57–59 Strength of relationship factorial ANOVA, 107 one-way analysis of variance, 95 Structure matrix, FA, 216 Structural equation modeling (SEM), 226 path analysis, 234 factor analysis, 234–236 full, 242–244 latent variables, 234 measured variables, 234 Studentized range, 90–91 Student’s t-test, 79 Subject-to-variable ratio, 238 Subscript notation, 19 Substitution game, 320 Sufficiency, 26 Summarizing data survival analysis, 293–294 Sum of squares between, 86 within, 86 factorial ANOVA, 101–102 factor loadings, 216–217 and mean squares, 101–102, 101f one-way analysis of variance, 86 residual, 142 simple regression and correlation, 142 total, 87 Sum of squares and cross-products (SSCP), 129, 129t Survival analysis, 292–307, 293f, 293t, 295f, 295t–296t, 300f, 300t, 302t–304t actuarial approach, 294, 307 adjusting for covariates, 302–303 assumptions of, 299–300 average hazard rate, 298 Breslow test, 302 comparing two or more groups, 300 conditional failure, 298 conditional probability, 296 confidence intervals, 287 contingency tables, 320 Cox proportional hazards model, 303, 307 cumulative probability, 296 end point, 299 exponential curve, 304 goodness-of-fit chi-squared, 305 hazard function, 297–298 HR, 301 increasing Weibull function, 298 independent censoring, 299 index case, 303 Kaplan-Meier approach, 294, 296–297, 297t, 307 left censoring, 299 life-table, 294 log-log plot, 305 log-minus-log plot, 305 loss to follow-up, 299 Mantel-Cox chi-squared, 302 Mantel-Cox log-rank test, 301–302 mean survival, 293–294 median survival time, 298 negative Weibull function, 298 overall relative risk, 302 person-years, 294 product-limit method, 296–297 proportionality testing, 305 robust Cox model, 305 RR, 301 sample size and power, 305–306 secular trends, 300 standard error, 298–299 starting point, 299 summarizing data, 293–294 summary measures, 298–299 survival curve, 296, 297f survival rate, 294 survival-table, 294 Tarone-Ware test, 302 techniques, 294–295 time-dependent prognostic variables, 303 time-independent prognostic variables, 303 when to use, 292–293 z-test, 300–301 Survival curve analysis, 296, 297f Survival rate analysis, 294 Survival tables, 294 Symmetric distribution mean, 27f median, 27f mode, 27f T Tables confusing, 17t contingency, 252 improved, 17t making better, 15 survival, 294 2x2, 252, 269 Tarone-Ware test survival analysis, 302 Tetrachoric correlation measures of association for ranked data, 289 Theoretical derivation of probability, 38 Theoretical distribution, 32f Three-dimensional bar charts, 14, 14f Threshold values logistic regression, 170 Time dependent prognostic variables, 303 HLM, 201 independent prognostic variables, 303 metric for measuring, 201 survival analysis, 303 TLI see Tucker-Lewis Index (TLI) Transformation, 51 Transforming data, 334–335, 333t, 338f confidence intervals, 338 guidelines for, 336 interpretation of, 338 J-shaped distribution, 334f, 337 linear regression, 335 logarithm, 337 moderate skew right, 336 Pearson correlation, 335 reverse J, 336 severe skew, 336 SPSS, 353–357 square root, 337 Trimmed mean, 28, 30 Trimodal distribution, 21 True score measuring change, 192 Truncated mean, 28 Truncated zeros, 178 T-test, 46, 78–84, 82f, 342t, 344 Cohen’s d, 82 degrees of freedom, 80 effect size, 82 equal sample sizes, 79 extended, 81 independent samples, 84, 341 mean greater than zero, 80f multiple, 137 ordinal equivalent, 343 pooled vs separate variance estimates, 81 related samples, 341 sample related, 341 size and power, 83 SD, 80 Student’s, 79 Tucker-Lewis Index (TLI), 239 Tukey’s honestly significant difference one-way analysis of variance, 89, 91, 91t, 92t Tukey’s wholly significant difference one-way analysis of variance, 92 Two factor repeated measures analysis of variance, 116–117, 117t Two grouping factors tests of significance for ranked data, 280 Two independent groups tests of significance for ranked data, 277–278 Two-tailed test, 61 Two tails vs one tail probability, 61–62 Two-way analysis of variance, 342t Two-way factorial analysis of variance, 106, 106t Two-way multivariate analysis of variance, 344 2x2 tables, 269 contingency, 252 Type I errors, 50, 53, 53t equivalence and non-inferiority testing, 326 multiplicity, 88–89 rates, 59–60 Type II errors, 53, 53t equivalence and non-inferiority testing, 326 U ULS see Unweighted least squares (ULS) Unbiasedness, 27 Unconditional vs conditional maximum likelihood estimation, 171–172 Undefined path models, 232 Unexplained variance, 205 Unipolar factors, 217 Univariate statistics, 341 Unstructured correlation matrix, 203, 203f Unweighted least squares (ULS) path analysis, 238 V Values, 3 imputing, 338 logistic regression, 170 missing, 338 threshold, 170 Variables, 3 see also Ratio variables dependent, 3 vs independent variable, 343 MANOVA, 127, 128f multiple regression, 156–157 dummy, 162t frequency, 341 independent, 343t vs dependent variables, 343 MANOVA, 127 multiple regression, 151–152, 153t intercorrelated, 154 interval, 4–5, 10t, 340, 341, 341t, 342t latent FA, 208 multiple regression, 153–157, 163f nominal, 4–5, 340, 341, 341t path analysis endogenous, 229 exogenous, 229 prognostic, 303 rank, 341, 343t related, 343t SPSS, 348–349 types of, 153 Variance-covariance matrix (VCV) Box’s M, 128 MANOVA, 128 Variance inflation factor (VIF) multiple regression, 165 Variances, 23–24, 236 distribution, FA, 217 MANOVA, 128 outputs, HLM, 205t percent of, 238 sample, 47 Varimax rotation, 217 VCV see Variance-covariance matrix (VCV) Vectors, MANOVA, 127 VIF see Variance inflation wfactor (VIF) Visual identification outliers, 329 W Wald statistic path analysis, 240 Wald test logistic regression, 172 Washout period, 115 Weibull function increasing survival analysis, 298 Weighted disagreement measures of association for categorical data, 274 Weighted kappa, 341t Weighted least squares (WLS) path analysis, 238 Weighted sum by sample size, 81 Whiskers, 26, 27f Wholly significant difference one-way analysis of variance, 92 Wilcoxon rank sum test, 343, 343t tests of significance for ranked data, 277 Wilcoxon signed rank test, 343t tests of significance for ranked data, 264–265 Wilks’ lambda MANOVA, 129, 130t, 132t Wilks-Shapiro Normality Test, 26 Winsorized mean, 28 Winsorized SD, 28 Within-subjects factors repeated-measures ANOVA, 119–120 WLS see Weighted least squares (WLS) Writing up, 345 CI, 345 ES, 345 reporting p-levels, 345 significance, 345 testing for baseline differences, 345–346 X X bar, 19 Y Yates’ corrected chi-squared, 255 Yates’ correction for continuity, 255 Yule’s Q, 270 Z Zero missing, 13–14, 71 should be missing, 73 Zero inflated models, 178 Zero-inflated negative binomial model, 178 Z-score, 33–34 calculating, 36 Z-test, 79 survival analysis, 300–301 ... exactly on the two kneecaps, and the asymptotes at the corners of the inscribed square The centers of the two feet, at the point where they intersect the arc of the circle, then determine the conventional... stilt in the sample, and the range can double It follows that the more people there are in the sample, the better are the chances of finding one of these folks So, the second problem is that the range... Let’s take a look and see how this is done, at the same time explaining these somewhat odd-sounding terms The “leaf” consists of the least significant digit of the number, and the “stem” is the most significant So, for the number 94, the leaf is “4” and the stem is “9.” If our data included numbers such