Introductory Medical Statistics 3rd edition Richard F Mould Institute of Physics Publishing Bristol and Philadelphia Copyright © 1998 IOP Publishing Ltd @ IOP Publishing Ltd 1998 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, without the prior permission of the publisher. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency under the terms of its agreement with the Committee of Vice-Chancellors and Pnncipals. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN 0 7503 5 13 4 Library of Congress Cataloging-in-Publication Data are available First edition 1976 Second edition 1989 Reprinted 1994, 1995 Series Editors: R F Mould, Croydon, UK C G Orton, Karamanos Cancer Institute, Detroit, USA J A E Spaan, University of Amsterdam, The Netherlands J G Webster, University of Wisconsin-Madison. USA Published by Institute of Physics Publishing, wholly owned by The Institute of Physics, London Institute of Physics Publishing, Dirac House, Temple Back, Bristol BSI 6BE, UK US Office: Institute of Physics Publishing, The Public Ledger Building, Suite 1035, South Independence Mall West, Philadelphia, PA 19106, USA Typeset using the 1OP Bookmaker macros, from the author’s Microsoft Word 6 files Printed in the UK by J W Arrowsmith Ltd, Bristol Copyright © 1998 IOP Publishing Ltd The Medical Science Series is the official book series of the International Federation for Medical and Biological Engineering (IFMBE) and the International Organization for Medical Physics (IOMP). IFMBE The IFMBE was established in 1959 to provide medical and biological engineering with an international presence. The Federation has a long history of encouraging and promoting international cooperation and collaboration in the use of technology for improving the health and life quality of man. 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Copyright © 1998 IOP Publishing Ltd With love to Imogen, who kept her Grandad entertained while this book was being written Copyright © 1998 IOP Publishing Ltd Contents Preface xvii 1 Data Presentation 1 1.8 Graphs: Linear and Logarithmic Axes 7 1.10 Venn and Euler Diagrams 16 1.11 Box and Whiskers Plot 17 2 Describing Curves and Distributions 18 2.2 Mean, Mode and Median 18 2.4 Standard Deviation and Variance 24 2.5 Coefficient of Variation 27 2.6 Probability Density Function 28 3 The Normal Distribution Curve 30 3.4 Normal Probability Graph Paper 37 3.5 Testing for Normality using the Chi-squared Test 40 3.6 The Lognormal Curve 42 4 Introduction to Sampling, Errors, Accuracy and Precision 51 4.2 Sample Distribution and Sampling Distribution of the Sample Mean 53 ix Copyright © 1998 IOP Publishing Ltd X Contents 4.3 Central Limit Theorem 53 4.4 Formulae for Standard Errors 55 4.5 Gross Accidental Errors 56 4.7 Errors of Interpretation 56 4.9 Combination of Errors 58 4.10 Root Mean Square Error 58 4.1 1 Confidence Limits 59 5 Introduction to Probability 60 5.1 Descriptions of Probability 60 5.2 The Two Laws of Probability 62 5.3 Bayesian Probability 62 6 Binomial Probabilities 65 6.1 Permutations and Combinations 65 6.2 The Binomial Distribution 67 6.3 Examples of Binomial Problems 68 6.4 The Normal Approximation to the Binomial 71 6.5 Gambling with Dice 74 7 Poisson Probabilities 78 7.1 The Poisson Distribution 78 7.2 Examples of Poisson Problems 80 7.3 The Poisson Approximation to the Binomial 82 7.4 The Normal Approximation to the Poisson 82 7.5 Radioactive Decay 82 8 Introduction to Statistical Significance 85 8.1 The Null Hypothesis 85 8.2 Null, Positive and Negative Results of Studies 86 8.3 Probability Levels and Significance 87 8.2 Clinical and statistical significance 87 8.4 Type-I and Type-I1 Errors and Alpha and Beta Risks 88 8.5 A Generalised Schedule for Significance Testing 90 8.6 Degrees of Freedom 91 9 The Chi-squared Test 93 9.2 Goodness of Fit: Precision of Nuclear Medicine Counting Instruments 96 9.3 Goodness of Fit: A Racing Problem 97 9.4 Goodness of Fit: A Poisson Problem 100 9.5 Goodness of Fit: A Lognormal Curve Fitting Problem 101 9.6 The 2 x 2 Contingency Table 102 Copyright © 1998 IOP Publishing Ltd Contents XI 9.7 A 2 x 2 Contingency Table: A Cholera Epidemic Problem 104 9.8 The Generalised r x c Contingency Table 105 9.9 Yates Correction for Small Samples 107 9.10 Did Mendel Cheat? 108 10 The Fisher Exact Probability Test 110 10.2 One-tailed Example 110 10.3 Two-tailed Example 113 11 The t-test 115 11.2 Estimate of the Population Mean p from the Sample Mean x, and Calculation of Confidence Limits 117 11.3 The One-sample t-test: Inference Based on a Single Sample Mean x, when the Reference Population from which the Sample is Drawn is Known to have a Mean of p 120 1 1.4 Paired t-test: Difference Between Means 120 1 1.5 Unpaired Two-sample t-test 121 11.6 Normal Test for the Difference Between Means of Large Samples 124 12 Difference Between Proportions for Independent and for Non- independent (McNemar’s Test for Paired Proportions) Samples 127 12.1 Independent Samples from Normal Populations 127 12.2 McNemar’s Test for Difference Between Paired Proportions 128 13 Wilcoxon, Mann-Whitney and Sign Tests 132 13.1 Non-parametric Rank Tests 132 13.2 Wilcoxon Signed Ranks Test for Matched Pairs 132 13.3 Wilcoxon Rank Sum Test for Unpaired Data 135 13.4 Mann-Whitney U Test 141 13.5 Kruskal-Wallis Test 142 13.6 Sign Test 143 14 Survival Rate Calculations 147 14.2 Basic Data for the Worked Examples 151 14.3 Kaplan-Meier Life Table Method for Non-grouped Data 152 14.1 Ranking procedure for survival times 152 14.2 Probability of dying at time 153 14.3 Calculation of overall survival rates 154 14.4 Calculation of cancer-specific survival rates 154 14.4 Life Table Method for Grouped Data 160 14.1 When to group data and when not to group data 160 Copyright © 1998 IOP Publishing Ltd xii Contents 14.2 Lost to follow-up or withdrawn alive 161 14.3 Grouping procedure for the survival times 161 14.4 Probability of dying at time T, 162 14.5 Calculation of overall survival rates 164 14.6 Overall survival rate comparisons for grouped data and Kaplan-Meier methods 164 14.7 Standard errors 164 15 The Logrank and Mantel-Haenszel Tests 168 15.2 The Logrank Test 169 15.3 The Mantel-Haenszel Test 171 15.2 Worked example 171 16 Regression and Correlation 178 16.2 Method of Least Squares for Estimation of the Slope and Intercept of a Straight Line 183 16.4 Pearson’s Correlation Coefficient 188 16.5 Testing for a Significant Correlation: an Application of the t-test 188 16.6 Spearman’s Rank Correlation Coefficient 191 16.7 Kendall’s t Rank Correlation Coefficient 192 17 Analysis of Variance 196 17.3 One-way ANOVA: Worked Example 203 17.4 Two-way ANOVA 208 17.5 Relationship Between Correlation Coefficient and a One-way ANOVA Table 21 1 18 Multivariate Analysis: The Cox Proportional Hazards Model 213 18.2 Hazard Function, Cumulative Distribution Function and Survivor Function 214 18.3 Assumptions of the Cox Proportional Hazards Model 216 18.4 Hazard Equation for Three Prognostic Factors 216 18.5 Relaxation of the Assumption of Proportionality of Hazards 217 18.6 Example: Locoregional Recurrence of Breast Cancer 218 Copyright © 1998 IOP Publishing Ltd ,.1 Relationship between survival probability and hazard 221 18.2 Likelihood ratio test 22 1 18.3 Table of results 223 18.4 Relative risk: ratio between hazards 224 18.5 Stepwise selection of variables 225 18.8 Examples: Breast Cancer 227 19 Sensitivity and Specificity 232 19.2 Problems of Defining Normality and Abnormality 233 19.3 Receiver Operating Characteristic (ROC) Curve 233 19.4 Example: Prostate Cancer Early Detection 235 19.5 Example: Nuclear Medicine Imaging Interlaboratory Comparison Studies 237 19.6 Kappa Test 239 20 Clinical Trials 242 20.2 Trial Aims and Objectives 242 20.3 Trial Description by Phases 1-111 243 20.1 Methods of randomisation 246 20.1 Number of patients required for a phase I11 trial 259 20.4 Multiple comparisons between subgroups: Bonferroni method 266 20.5 Inflation factor due to patient refusals 266 20.12 Reporting Clinical Trial Results 273 21 Cancer Treatment Success, Cure and Quality of Life 278 2 1.2 Qualitative Assessment 283 Copyright © 1998 IOP Publishing Ltd xiv Contents 21.3 Semi-quantitative Assessment: Complete, Partial and No Regression 284 21.4 Quality of Life: 10-point or four-point Scale? 284 21.5 Further Examples of Quality of Life Status Scoring 286 21.3 Head and neck 292 21 S.4 American Society of Anesthiologists scale 293 21.
