Patricia de Winter and Peter Cahusac STARTING OUT IN STATISTICS An Introduction for Students of Human Health, Disease, and Psychology Starting Out in Statistics Starting Out in Statistics An Introduction for Students of Human Health, Disease, and Psychology Patricia de Winter University College London, UK Peter M B Cahusac Alfaisal University, Kingdom of Saudi Arabia This edition first published 2014 C ⃝ 2014 by John Wiley & Sons, Ltd Registered office: John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK Editorial offices: 9600 Garsington Road, Oxford, OX4 2DQ, UK The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK 111 River Street, Hoboken, NJ 07030-5774, USA For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com/wiley-blackwell The right of the author to be identified as the author of this work has been asserted in accordance with the UK Copyright, Designs and Patents Act 1988 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher Designations used by companies to distinguish their products are often claimed as trademarks All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners The publisher is not associated with any product or vendor mentioned in this book Limit of Liability/Disclaimer of Warranty: While the publisher and author(s) have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the publisher nor the author shall be liable for damages arising herefrom If professional advice or other expert assistance is required, the services of a competent professional should be sought Library of Congress Cataloging-in-Publication Data De Winter, Patricia, 1968– Starting out in statistics : an introduction for students of human health, disease and psychology / Patricia de Winter and Peter Cahusac pages cm Includes bibliographical references and index ISBN 978-1-118-38402-2 (hardback) – ISBN 978-1-118-38401-5 (paper) Medical statistics–Textbooks I Cahusac, Peter, 1957– II Title RA409.D43 2014 610.2′ 1–dc23 2014013803 A catalogue record for this book is available from the British Library Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic books Set in 10.5/13pt Times Ten by Aptara Inc., New Delhi, India 2014 To Glenn, who taught me Statistics Patricia de Winter Dedicated to the College of Medicine, Alfaisal University, Riyadh Peter M B Cahusac Contents Introduction – What’s the Point of Statistics? Basic Maths for Stats Revision 1.5 1.6 1.7 xv Statistical Software Packages xxiii About the Companion Website xxv Introducing Variables, Populations and Samples – ‘Variability is the Law of Life’ 1.1 1.2 1.3 1.4 xiii Aims Biological data vary Variables Types of qualitative variables 1.4.1 Nominal variables 1.4.2 Multiple response variables 1.4.3 Preference variables Types of quantitative variables 1.5.1 Discrete variables 1.5.2 Continuous variables 1.5.3 Ordinal variables – a moot point Samples and populations Summary Reference 1 4 5 6 10 10 Study Design and Sampling – ‘Design is Everything Everything!’ 11 2.1 2.2 2.3 2.4 2.5 2.6 11 11 13 13 14 15 Aims Introduction One sample Related samples Independent samples Factorial designs viii 2.7 2.8 2.9 2.10 CONTENTS Observational study designs 2.7.1 Cross-sectional design 2.7.2 Case-control design 2.7.3 Longitudinal studies 2.7.4 Surveys Sampling Reliability and validity Summary References 17 17 17 18 18 19 20 21 23 Probability – ‘Probability So True in General’ 25 3.1 3.2 3.3 3.4 3.5 3.6 25 25 26 31 35 36 37 Aims What is probability? Frequentist probability Bayesian probability The likelihood approach Summary References Summarising Data – ‘Transforming Data into Information’ 39 4.1 4.2 4.3 39 39 41 41 47 54 55 55 56 57 58 59 59 60 62 63 64 64 64 66 66 4.4 4.5 4.6 4.7 4.8 4.9 Aims Why summarise? Summarising data numerically – descriptive statistics 4.3.1 Measures of central location 4.3.2 Measures of dispersion Summarising data graphically Graphs for summarising group data 4.5.1 The bar graph 4.5.2 The error plot 4.5.3 The box-and-whisker plot 4.5.4 Comparison of graphs for group data 4.5.5 A little discussion on error bars Graphs for displaying relationships between variables 4.6.1 The scatter diagram or plot 4.6.2 The line graph Displaying complex (multidimensional) data Displaying proportions or percentages 4.8.1 The pie chart 4.8.2 Tabulation Summary References Statistical Power – ‘ Find out the Cause of this Effect’ 67 5.1 5.2 5.3 67 67 70 Aims Power From doormats to aortic valves 270 APPENDIX B: STATISTICAL SOFTWARE OUTPUTS SPSS for patients versus non-patients, two-way chi-squared CROSSTABS /TABLES=status BY handedness /FORMAT=AVALUE TABLES /STATISTICS=CHISQ /CELLS=COUNT /COUNT ROUND CELL status ∗ handedness Crosstabulation Count Non-patients Patients Status Total Right handed 245 348 593 Handedness Mixed handed 25 42 67 Chi-Square Tests Value df Pearson Chi-Square 456a Likelihood Ratio 459 Linear-by-Linear Association 286 N of Valid Cases 706 Left handed 18 28 46 Total 288 418 706 Asymp Sig (2-sided) 796 795 593 a cells (0.0%) have expected count less than The minimum expected count is 18.76 Minitab output for Gender and handedness (patients and non-patients combined), two-way chi-squared test Data in Table 8.4 Gender and handedness, patients and non-patients combined Using frequencies in Counts Rows: Gender Columns: Handedness All 330 344.38 -0.7747 44 38.91 0.8161 36 26.71 1.7967 410 410.00 * APPENDIX B: STATISTICAL SOFTWARE OUTPUTS 263 248.62 0.9118 23 28.09 -0.9605 10 19.29 -2.1145 296 296.00 * All 593 593.00 * 67 67.00 * 46 46.00 * 706 706.00 * Cell Contents: 271 Count Expected count Standardized residual Pearson Chi-Square = 10.719, DF = 2, P-Value = 0.005 Likelihood Ratio Chi-Square = 11.391, DF = 2, P-Value = 0.003 SPSS for Gender and handedness (patients and non-patients combined), two-way chi-squared test CROSSTABS /TABLES=gender BY handedness /FORMAT=AVALUE TABLES /STATISTICS=CHISQ /CELLS=COUNT /COUNT ROUND CELL gender ∗ handedness Crosstabulation Count gender Total Men Women Right handed 330 263 593 handedness Mixed handed 44 23 67 Chi-Square Tests Value df Pearson Chi-Square 10.719a Likelihood Ratio 11.391 Linear-by-Linear Association 10.672 N of Valid Cases 706 Left handed 36 10 46 Total 410 296 706 Asymp Sig (2-sided) 005 003 001 a cells (0.0%) have expected count less than The minimum expected count is 19.29 272 APPENDIX B: STATISTICAL SOFTWARE OUTPUTS Minitab output for Fisher’s exact test Tabulated statistics: gender, colour Using frequencies in frequency Rows: Gender All Columns: Colour All 11 4 15 Cell Contents: Count Fisher’s exact test: P-Value = 0.0769231 SPSS for Fisher’s exact test WEIGHT BY freq CROSSTABS /TABLES=gender BY colour /FORMAT=AVALUE TABLES /STATISTICS=CHISQ /CELLS=COUNT EXPECTED /COUNT ROUND CELL Gender Total gender ∗ colour Crosstabulation colour Blue Pink Boys Count Expected count 5.1 1.9 Girls Count 4 Expected count 5.9 2.1 Count 11 Expected count 11.0 4.0 Pearson Chi-Square Continuity Correctionb Likelihood Ratio Fisher’s Exact Test Linear-by-Linear Association N of Valid Cases a b Chi-Square Tests Asymp Sig (2-sided) Value Df 4.773a 029 2.558 110 6.307 012 4.455 15 Total 7.0 8.0 15 15.0 Exact Sig (2-sided) Exact Sig (1-sided) 077 051 035 cells (50.0%) have expected count less than The minimum expected count is 1.87 Computed only for a x table APPENDIX B: STATISTICAL SOFTWARE OUTPUTS 273 SPSS for odds ratio, effects of folic acid on neural tube defects WEIGHT BY freq CROSSTABS /TABLES=treatment BY outcome /FORMAT=AVALUE TABLES /STATISTICS=CHISQ RISK /CELLS=COUNT /COUNT ROUND CELL treatment ∗ Neural tube defects Crosstabulation Count Neural tube defects Yes No Total Folic acid 587 593 treatment None 21 581 602 Total 27 1168 1195 Pearson Chi-Square Continuity Correctionb Likelihood Ratio Fisher’s Exact Test Linear-by-Linear Association N of Valid Cases a b Chi-Square Tests Asymp Sig (2-sided) Value Df 8.297a 004 7.213 007 8.789 003 8.290 1195 Exact Sig (2-sided) Exact Sig (1-sided) 005 003 004 cells (0.0%) have expected count less than The minimum expected count is 13.40 Computed only for a x table Risk Estimate 95% Confidence Interval Value Lower Upper Odds Ratio for treatment 283 113 706 (Folic acid/None) For cohort Neural tube 290 118 714 defects = Yes For cohort Neural tube 1.026 1.008 1.043 defects = No N of Valid Cases 1195 274 APPENDIX B: STATISTICAL SOFTWARE OUTPUTS Chapter Minitab output for binomial test Sign test for median: WBC Sign test of median = WBC N 10 Below Equal 7.000 versus not = 7.000 Above P 0.0215 Median 4.500 SPSS for binomial test NPAR TESTS /BINOMIAL (0.50)=WBC (7) /MISSING ANALYSIS WBC Category Group Group Total N 7 Binomial Test Observed Prop Test Prop 90 10 10 1.00 Exact Sig (2-tailed) 50 Minitab output for Mann–Whitney test Data from Table 9.4 Crocus Placebo N 10 Median 0.00 3.00 Point estimate for ETA1-ETA2 is -2.00 95.5 Percent CI for ETA1-ETA2 is (-3.99,-0.99) W = 62.0 Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at 0.0247 The test is significant at 0.0225 (adjusted for ties) SPSS for Mann–Whitney test NPAR TESTS /M-W= spots BY treatment(1 2) /MISSING ANALYSIS .021 APPENDIX B: STATISTICAL SOFTWARE OUTPUTS treatment Crocus Placebo Total Spots Ranks N Mean Rank 6.89 10 12.80 19 275 Sum of Ranks 62.00 128.00 Test Statisticsa Mann-Whitney U Wilcoxon W Z Asymp Sig (2-tailed) Exact Sig [2*(1-tailed Sig.)] a b spots 17.000 62.000 −2.323 020 022b Grouping variable: treatment Not corrected for ties Minitab output for Wilcoxon signed rank test for paired samples (actually a one-sample Wilcoxon on the column of differences between the two treatments) Wilcoxon signed rank test: differences Test of median = 0.000000 versus median not = 0.000000 Differences N N for Test Wilcoxon Statistic 7.0 P 0.141 Estimated Median -5.000 SPSS for Wilcoxon signed rank test for paired samples NPAR TESTS /WILCOXON=HotWater WITH Coffee (PAIRED) /STATISTICS QUARTILES /MISSING ANALYSIS HotWater Coffee Descriptive Statistics Percentiles N 25th 50th (Median) 1.00 2.00 4.50 7.50 75th 3.75 9.75 276 APPENDIX B: STATISTICAL SOFTWARE OUTPUTS Ranks Coffee - HotWater a b c Negative Ranks Positive Ranks Ties Total N 1a 7b 0c Mean Rank 7.00 4.14 Sum of Ranks 7.00 29.00 Coffee < Hotwater Coffee > Hotwater Coffee = Hotwater Test Statisticsa Coffee - HotWater Z −1.556b Asymp Sig (2-tailed) 120 a b Wilcoxon signed ranks test Based on negative ranks Minitab output for Kruskal–Wallis one-way analysis of variance Kruskal–Wallis test: jump height versus flea species Kruskal-Wallis Test on Jump Height Flea species cats dogs humans Overall H = 7.01 N 9 26 Median 20.000 36.000 5.000 DF = Ave Rank 14.1 17.8 8.0 13.5 Z 0.30 2.08 -2.44 P = 0.030 SPSS for Kruskal–Wallis one-way analysis of variance DATASET ACTIVATE DataSet13 NPAR TESTS /K-W=Jump BY Flea(1 3) /MISSING ANALYSIS Height in cm Ranks Species of flea Cat Dog Human Total N 9 26 Mean Rank 14.11 17.78 8.00 APPENDIX B: STATISTICAL SOFTWARE OUTPUTS Test Statisticsa,b Height in cm Chi-Square 7.031 df Asymp Sig .030 a b Kruskal–Wallis test Grouping Variable: species of flea Minitab output for Friedman’s related samples test Data from Table 9.10 Treatments – hot water, – decaff coffee, – coffee S = 6.75 DF = Treatments N 8 P = 0.034 Est Median 1.833 2.000 7.167 Sum of Ranks 13.0 13.0 22.0 Grand median = 3.667 SPSS for Friedman’s related samples test NPAR TESTS /FRIEDMAN=HotWater DecaffCoffee Coffee /STATISTICS QUARTILES /MISSING LISTWISE Descriptive Statistics Percentiles N 25th 50th (Median) HotWater 1.00 2.00 Decaffcoffee 25 2.00 Coffee 4.50 7.50 Ranks Mean rank HotWater 1.63 Decaffcoffee 1.63 Coffee 2.75 Test Statisticsa N Chi-Square 6.750 Df Asymp Sig .034 a Friedman Test 75th 3.75 3.00 9.75 277 Index 𝛼, 70, 80, 90, 104, 112, 230 absolute risk reduction, 188, 191 alternative hypothesis, 81 analysis of variance, ANOVA one-way, 112 one-way nested hierarchical design, 123 two-way, 126 randomised complete block design, 130 repeated measures, 133 Anderson–Darling test, 109 archival data, 18 arithmetic mean, 42 association, test for, 186 asymmetrical distribution, 53 average, 42 𝛽, 69, 70, 81, 90 bar graph, 55 Bayes factor, 32, 85 Bayesian approach, 31–36, 89 Bayesian probability, 31, 37 Bayesian statistics, 34 between variation, 124 bias, 18, 223 bimodal distribution, 46, 94 binary distribution, 77 binomial test, 195 bivariate normal distribution, 153 blind double, 12, 15 single, 12, 15 Bonferroni’s correction, 88, 112, 120, 210, 213, 230 bootstrap sample, 223 bootstrap, two sample test, 227 bootstrapping, 218, 222 box plot, 57 Box-and-whisker plot, 57 categorical, categorical data, 175 causation, 161 centering, 172 central limit theorem, 77, 97, 225 chi-squared, 176 chi-squared assumptions, 184 chi-squared table, 177 chi-squared test one-way, 175 two-way, 179 clinical trials, 112 clustered bar chart, 183 Cohen’s d, 69 colour map, 64 combinations, 27 confidence interval, 85, 87, 91, 105, 168, 222–224 confounding variable, confound, 17, 21 consumer’s risk, 70, 71 contingency table, 179 controls, 15, 65 Starting Out in Statistics: An Introduction for Students of Human Health, Disease, and Psychology First Edition Patricia de Winter and Peter M B Cahusac C ⃝ 2014 John Wiley & Sons, Ltd Published 2014 by John Wiley & Sons, Ltd Companion Website: www.wiley.com/go/deWinter/spatialscale 280 correlation Pearson’s product-moment correlation coefficient, 70, 153, 156 Spearman’s rank correlation test, 162, 194 correlation coefficient, 157 counterbalancing, 13, 14 covariance, 154, 157 criterion variable, 170 critical value, 80, 81, 104, 180 cross-tabulation, 179 cross-validation, 172 data transformations, 110 degrees of freedom, 51, 104, 118, 159, 177, 209 dependent t-test, 109 dependent variable, 61, 138, 164 descriptive statistics, 41, 45, 47 designs between participants, 15 case-control, 17 cohort, 18 correlative, 17 cross-over, 13, 16 cross-sectional, 17 experimental, 12 factorial, 15 independent, 15 independent samples, 15 longitudinal, 18 matched participants, 13 mixed, 17 observational study, 17 one sample, 13 parallel groups, 15 quasi-experimental, 16 randomised control trial, 15 related samples, 13 repeated measures, 13 retrospective, 18 survey, 18 deviations from the mean, 49 diagnostic plots, 149, 172 displaying proportions, 64 distribution, 42, 44, 45 distribution-free tests, 194, 219 dot plot, 41 dummy variable coding, 171–173 Dunnett’s post hoc test, 121 INDEX effect size, 68, 69, 83, 84, 86 effective sample size, 174 epidemiology, 17, 18, 190 error, 113, 117 random, 21 systematic, 21 error plot, 56 error, statistical, 21 𝜂 , 70 Excel, xiv, xvi, xx, xxi, 42, 52, 80, 110, 141, 180, 189, 191, 195, 209, 212, 214 expected frequencies, 176 experimental studies, 54 explanatory variable, 138 exploratory studies, 17 extrapolation, 145, 148 F statistic, 118 F table, 119 factor, 16 false discovery rate, 229, 230 false negative, 30, 32 false positive, 30, 32 Fisher’s Exact test, 185 Fisher, R.A., xxiii, 7, 29, 30 fitted values, 149 fixed factor, 112, 119, 171 frequentist probability, 26, 33, 34, 36 Friedman test for correlated samples, 211 G∗ Power, 84 genomics, xvi, 39, 63, 111, 218 geometric mean, 152 goodness of fit test, 179 Gosset, William Sealy, 101 graphical display, 54 H0 , 29, 81, 91 H1 , 29, 81, 91 heat map, 63 heterogeneity of subsamples, 165 histogram, 42, 43, 45, 46, 52, 55, 94, 225 hypothesis testing, 36 hypothesis testing approach, 88, 89 IgNobel prize, 219 importance, practical or clinical, 29 independence, test for, 186 INDEX independent t-test, 98 independent variable, 61, 138, 164 inference, 8, 47 inter-quartile range, 53, 57 interaction, 16, 127, 172 intercept, 136, 148 Kolmogorov-Smirnov test, 109 Kruskal–Wallis one-way analysis of variance, 207 latent variable, 21 levels, of factor or variable, 16 Levene’s test, 109 leverage, 173 likelihood, 35, 36 likelihood approach, 35, 37, 88–91 likelihood ratio, 32, 35, 36, 85, 90, 91 Likert scale, 18 Lilliefors test, 109 line graph, 62 linear regression, 136 linearity, 172 log scale, 64 Log transformation, 110 log-linear analysis, 186 logistic regression, 171 magnitude of the effect, 69 main effects, 16 Mann–Whitney U test, 202, 228 Mann–Whitney test, 199 margin of error, 19 marginal totals, 180 matching cases, 17 mean, 42–45, 47, 53, 54 mean squares, 118, 144 measures of central location, 41 measures of dispersion, 47, 48 median, 42–45, 53, 54, 57 microarrays, 64 minimum size of effect, 69 Minitab, xx, xxiii, xxiv, 97, 105, 120, 122, 126, 145, 148, 158, 160, 166 mode, 42–45 multicollinearity, collinearity, 167, 174 multidimensional data, 63 multiple regression, 164 multiple testing, 35, 36, 88, 111, 230 multivariate analysis, 17 281 negative control, 65, 112 negatively skewed distribution, 43 nested hierarchical design, 123, 124 new statistics, the, 218 Neyman–Pearson hypothesis testing, 30, 37 nominal data, 175 non-parametric tests, 193 normal distribution, 45, 46, 72, 94, 101 normally distributed, 148, 172, 193 Not Proven verdict, 30 null hypothesis, 29, 81, 88, 99, 109, 112, 176 null hypothesis testing approach, 91 number needed to treat, 188, 191 numerical variable, observed frequencies, 176 odds, 26 odds ratio, 186 one-tailed test, 88, 105, 186 one-way analysis of variance, ANOVA, 112, 210 one-way chi-squared test, 175 order effects, 13 carry-over, 13 ordinary least squares regression, 138, 152 outcome variable, 170 outliers, 57, 148, 162, 173, 194 overfitting, 172 p value, 28, 37, 77, 84, 86, 88 p values, problem with, 84 parametric tests, 45, 193 partial regression plot, 170 Pearson residuals, 182 Pearson’s product-moment correlation coefficient, 156 percentile method, 223 percentiles, 53 pie chart, 64 placebo, 15, 112 Poisson distribution, 53, 54, 110 pooled standard deviation, 102 population parameters, 96, 223 populations, 6, 47, 96 positively skewed, 194 positively skewed distribution, 44, 110 posterior probability, 32 power, 31, 67, 79, 86, 88, 217 practical importance, 69, 84 precision, 85 282 INDEX prediction, 136, 145 prevalence, 31 principal components analysis, 61 prior probability, 32 probability, 25, 36, 68, 104, 111, 180, 196, 221 Bayesian, 31 frequentist, 26 producer’s risk, 70, 71 profile, 21 proportions, 175 proving hypotheses, 111 Pythagoras’s theorem, 113 Q–Q plot, quantile–quantile plot, 107, 225, 227 qualitative variable, 3, quantile, 107 quantititative variable, quartile, 53, 57, 77 questionnaire, 18 R, xx, xxiii, xxiv, 85, 166, 177, 180, 182, 191, 197, 205, 209, 212, 214, 223, 225, 227 R2 , 70, 145, 159, 165, 168, 172 random assignment, 13, 14 random factors, 94, 112, 113, 117, 124 random sample, 19 random sampling, 18, 107 random variation, 47 randomisation test, 218, 219, 227 randomised complete block design, 130 randomised control trials, RCT, 15, 190 range, 48, 49 reduced major axis regression, 152 regression linear, 136 multiple, 164 ordinary least squares, 136, 152 reduced major axis, 152 regression equation, 148, 167 regression line, 138 regression to the mean, 14 relative risk, 190 reliability, 20 repeated measures ANOVA, 133 representative sample, 19 resampling statistics, 217 residual variation, 113, 140, 148 residuals, 145, 172 risk, 186 risk-benefit analysis, 191 𝜌, rho, 156 robust procedures, 193 RStudio, xxiv sample, 6, 59, 96 sample size, 82, 84 sample statistics, 96, 222 sampling, 19, 96 cluster, 18 convenience, 20 non-random, 20 quota, 20 self-selecting, 20 snow-ball, 20 stratified, 19, 20 sampling distribution, 77, 89, 227 sampling frame, 18, 19 sampling with replacement, 223 scattergram, scatter plot, 60, 61, 154, 155, 163 self-experimentation, 11 sensitivity, 31, 82 sequential data collection, 87 Shapiro–Wilk test, 109 sign test, 195 significance level, 29, 80 significance testing approach, 88 significance testing, 29, 37 null hypothesis significance testing, 29 simple effects analysis, 16 simple random sampling, 19 skewed distribution, 42–44, 77 slope, 137, 148 Spearman’s rank correlation test, 162, 194 SPSS, xx, xxii–xxiv, 166, 168 spurious correlation, 17 square root transform, 110 standard deviation, 51–54 standard error of the difference, 102 standard error of the mean, 59, 79, 82 standard normal distribution, 73 standardised coefficient, 168 standardised residuals, 149 Pearson residuals, 182 statistical significance, 29, 68 stopping data collection, 35 stopping rule, 87 INDEX strata, 19 strength of a linear relationship, 156 strength of evidence, 30, 36, 89 Student’s t-test, 98 Student’s t-distribution, 101 subjectivity, 35, 37 sum of squares, 51, 116, 141, 142 t-test dependent, 109 independent, 98 t-test assumptions, 107 t-table, 105 tables, 64 three dimensional graph, 61, 170, 172 tolerance, 168 treatment, 12, 40, 93, 112, 136, 194 Tukey’s post hoc test, 120, 127 two-tailed test, 80, 88, 105, 186 two-way ANOVA, 126 two-way chi-squared test, 179 Type I error, 29, 32, 36, 71, 87, 91, 101, 111, 210, 230 Type II error, 29, 32, 68, 81, 88, 230 uniform distribution, 46, 77 unstandardised slope coefficient, 167 validity, 21 external, 19 internal, 20 variability, 1, 47, 49 variables, continuous, discrete, multiple response, nominal, ordinal, preference, variance, 51 variance inflation factor, 167 Welch’s t-test, 109 white coat effect, 113 Wilcoxon signed-rank test, 205 Wilcoxon’s matched-pairs signed-ranks test, 205 Wilcoxon’s rank sum test, 202 Wilcoxon–Mann–Whitney test, 202 within variation, 124 z, 73, 80, 86 z table, 75 z-score, 157 z-test, 96 283 WILEY END USER LICENSE AGREEMENT Go to www.wiley.com/go/eula to access Wiley’s ebook EULA ... Starting Out in Statistics Starting Out in Statistics An Introduction for Students of Human Health, Disease, and Psychology Patricia de Winter University College London,... life form appears from behind Starting Out in Statistics: An Introduction for Students of Human Health, Disease, and Psychology First Edition Patricia de Winter and Peter M B Cahusac C ⃝ 2014. .. Cataloging -in- Publication Data De Winter, Patricia, 1968– Starting out in statistics : an introduction for students of human health, disease and psychology / Patricia de Winter and Peter Cahusac pages cm Includes bibliographical