IAUPR 5/23/00 3:25 PM Page ii In memory of Professor James McGilvray (1938–1995) IAUPR 5/23/00 3:25 PM Page iii Interpretation and Uses of Medical Statistics LESLIE E. DALY MSc, PhD, HonMFPHM Associate Professor Department of Public Health Medicine and Epidemiology University College Dublin National University of Ireland, Dublin GEOFFREY J. BOURKE MA, MD, FRCPI, FFPHM, FFPHMI Emeritus Professor Department of Public Health Medicine and Epidemiology University College Dublin National University of Ireland, Dublin Formerly Consultant in Epidemiology and Preventive Medicine St Vincent’s University Hospital Dublin FIFTH EDITION IAUPR 5/23/00 3:25 PM Page iv © 2000 by distributors Blackwell Science Ltd Marston Book Services Ltd Editorial Offices: PO Box 269 Osney Mead, Oxford OX2 0EL Abingdon, Oxon OX14 4YN 25 John Street, London WC1N 2BL (Orders: Tel: 01235 465500 23 Ainslie Place, Edinburgh EH3 6AJ Fax: 01235 465555) 350 Main Street, Malden MA 02148-5018, USA USA 54 University Street, Carlton Blackwell Science, Inc. 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No part of is available from the British Library this publication may be reproduced, stored in a retrieval system, or ISBN 978-0-632-04763-5 transmitted, in any form or by any means, electronic, mechanical, Library of Congress photocopying, recording or otherwise, Cataloging-in-publication Data except as permitted by the UK Copyright, Designs and Patents Act Daly, Leslie E. 1988, without the prior permission Interpretation and uses of medical statistics / of the copyright owner. Leslie E. Daly, Geoffrey J. Bourke.—5th ed. p. cm. First published 2000 Previous editions entered under: Bourke, Geoffrey J. 4 2007 Includes bibliographical references and index. Set by Excel Typesetters Co., Hong Kong ISBN 978-0-632-04763-5 Printed and bound in Singapore 1. Medical statistics. 2. Medicine— by Markono Print Media Pte Ltd Research—Methodology. I. Bourke, Geoffrey J. (Geoffrey Joseph) II. Bourke, The Blackwell Science logo is a Geoffrey J. (Geoffrey Joseph). Interpretation and trade mark of Blackwell Science Ltd, uses of medical statistics. III. Title. registered at the United Kingdom RA409 .B65 2000 Trade Marks Registry 519.5¢02461—dc21 99–045805 For further information on Blackwell Science, visit our website: www.blackwell-science.com IAUPR 5/23/00 3:25 PM Page v Contents Note:An exclamation mark in square brackets after a chapter or section heading signifies material that is somewhat more difficult than that surrounding it. Preface, ix Structure of the Book, xii 1 Describing Data—A Single Variable, 1 1.1 Introduction, 1 1.2 Types of data, 1 1.3 Qualitative data—simple tables and bar charts, 5 1.4 Picturing quantitative data, 7 1.5 Shapes of distributions, 19 1.6 Measures of central value, 20 1.7 Other measures of location—quantiles, 31 1.8 Measures of dispersion, 37 1.9 Summary, 41 2 Probability, Populations and Samples, 43 2.1 Introduction, 43 2.2 Probability, 43 2.3 Populations and samples, 48 2.4 Sample surveys, 52 2.5 Summary, 56 3 Associations: Chance, Confounded or Causal? 57 3.1 Introduction, 57 3.2 Examining associations, 57 3.3 Interpreting associations, 65 3.4 Associations due to chance, 65 3.5 Associations due to bias or confounding, 77 3.6 Causal associations, 81 3.7 Summary, 85 4 Confidence Intervals: General Principles; Proportions, Means, Medians, Counts and Rates, 86 4.1 Introduction, 86 4.2 The normal distribution, 87 4.3 Sampling variation—proportions, 91 4.4 Confidence intervals for a proportion, 94 4.5 Sampling variation—means, 99 4.6 Confidence intervals for a mean, 101 4.7 Confidence intervals for a geometric mean [!], 107 4.8 Confidence intervals for a median [!], 109 4.9 Confidence intervals for a count or rate [!], 111 4.10 Summary, 114 v IAUPR 5/23/00 3:25 PM Page vi vi 5 Hypothesis Testing: General Principles and One-sample Tests Contents for Means, Proportions, Counts and Rates, 116 5.1 Introduction, 116 5.2 The null and alternative hypotheses, 117 5.3 The significance test, 118 5.4 Relationship with confidence intervals, 122 5.5 One-sided and two-sided tests, 123 5.6 General structure of a significance test: the one-sample ztest for a mean, 125 5.7 Non-significant results and power: type I errors, type II errors and sample size, 129 5.8 The one-sample ttest for a mean, 135 5.9 The one-sample ztest for a proportion, 136 5.10 The one-sample c2test for many proportions [!], 136 5.11 The one-sample ztest for counts or rates, 139 5.12 Small sample sizes and the validity of assumptions [!], 140 5.13 Summary, 142 6 Epidemiological and Clinical Research Methods, 143 6.1 Introduction, 143 6.2 Observational and experimental studies, 144 6.3 The cross-sectional study or survey, 145 6.4 The cohort study, 146 6.5 Measures of association in cohort studies, 149 6.6 Risk with variable-time follow-up [!], 158 6.7 The case–control study, 161 6.8 Measures of association in case–control studies—the odds ratio, 165 6.9 The analysis of cohort and case–control studies [!], 168 6.10 Comparisons of cohort and case–control studies, 169 6.11 The randomized controlled trial, 173 6.12 Applicability versus validity of trial results, 186 6.13 Alternative trial designs, 192 6.14 Ethical considerations for trials, 195 6.15 Summary, 200 7 Confidence Intervals and Hypothesis Tests: Two-group Comparisons, 202 7.1 Introduction, 203 7.2 Independent and paired comparisons, 203 7.3 Parametric and non-parametric significance tests, 205 7.4 Comparison of two independent means, 207 7.5 Inferring significance from confidence intervals, 212 7.6 Comparison of two independent geometric means [!], 215 7.7 Comparison of two independent medians, 218 7.8 Comparison of paired means, 221 7.9 Comparison of paired medians, 223 7.10 Comparison of two independent proportions or risks (2 ¥2 tables), 225 7.11 Comparison of two independent sets of proportions, 239 7.12 Comparison of paired proportions, 242 7.13 Comparison of two counts, 247 7.14 Comparison of two rates, 248 7.15 Comparison of two life tables [!], 252 7.16 Significance testing versus confidence intervals, 262 7.17 Summary, 264 IAUPR 5/23/00 3:25 PM Page vii 8 Sample Size Determination, 269 vii 8.1 Introduction, 269 Contents 8.2 Factors affecting the approach to sample size estimation, 270 8.3 Sample sizes for single-group studies, 272 8.4 Sample size specifications for two-group comparisons, 276 8.5 Derivation of a formula for two-group comparisons [!], 279 8.6 Sample sizes for the comparison of two independent groups, 281 8.7 Sample sizes for the comparison of paired means, 290 8.8 Confidence intervals and sample sizes for comparative studies, 292 8.9 Summary, 295 9 Comparison of More than Two Independent Groups [!], 296 9.1 Introduction, 296 9.2 Comparison of independent means in more than two groups: one-way analysis of variance, 297 9.3 Comparison of independent proportions in more than two groups: the c2test, 309 9.4 Comparison of other statistics in more than two groups, 313 9.5 Summary, 313 10 Associations between Two Quantitative Variables: Regression and Correlation, 315 10.1 Introduction, 315 10.2 Describing relationships between two quantitative variables, 316 10.3 Correlation, 322 10.4 Calculation of regression and correlation coefficients [!], 325 10.5 Statistical inference in regression and correlation, 327 10.6 When the assumptions do not hold [!], 332 10.7 Biases and misinterpretations in regression, 335 10.8 Summary, 338 11 Multivariate Analysis and the Control of Confounding [!], 339 11.1 Introduction, 339 11.2 Quantitative dependent variable, 340 11.3 Qualitative dependent variable: (i) case–control and cohort studies, 348 11.4 Confounding versus effect modification or interaction [!], 361 11.5 Qualitative dependent variable: (ii) vital statistics approaches, 364 11.6 Multivariate analysis, 379 11.7 Summary, 380 12 Bias and Measurement Error, 381 12.1 Introduction, 381 12.2 Biases in study design, 382 12.3 Biases in selecting the sample, 382 12.4 Accuracy: measurement error for a single test, 384 12.5 Agreement: comparing two different tests or observers, 402 12.6 Validity: comparing a test with a gold standard, 411 12.7 Bias in statistical analysis and interpretation, 415 12.8 Critical reading of the literature, 417 12.9 A note on research procedures, 419 12.10 Summary, 421 IAUPR 5/23/00 3:25 PM Page viii viii Appendix A: Computational Shortcuts, 422 Contents A.1 Introduction, 422 A.2 The standard deviation, 422 A.3 The c2test for independent 2 ¥2 tables, 424 A.4 Regression and correlation, 426 A.5 Sums of squares in ANOVA, 426 Appendix B: Statistical Tables, 430 B.1 Introduction, 430 B.2 Tables, 431 Appendix C: A ‘Cookbook’ for Hypothesis Tests and Confidence Intervals, 520 C.1 Introduction, 520 C.2 The ztest for a single proportion. Confidence intervals for a proportion, 523 C.3 The zand ttests for a single mean. Confidence intervals for a mean or geometric mean, 525 C.4 Confidence intervals for a single median, 526 C.5 The ztest for a single count or rate. Confidence intervals for a count or rate, 527 C.6 The c2test for many proportions (in one group), 528 C.7 The ttest for two independent means. Confidence intervals for the difference between two means and for a ratio of geometric means, 529 C.8 The Wilcoxon rank sum test for two independent medians, 531 C.9 The ttest for paired means. Confidence intervals for a mean difference, 532 C.10 The sign test for paired medians, 533 C.11 The Wilcoxon signed rank test for paired medians, 534 C.12 The ztest for two independent proportions. Confidence intervals for a difference between proportions, 535 C.13 The c2test for a 2 ¥2 table. Confidence intervals for a relative risk (ratio of proportions) and for an odds ratio, 536 C.14 Fisher’s test for a 2 ¥2 table, 538 C.15 The c2test for a general I¥Jtable, 539 C.16 The McNemar and exact tests for paired proportions. Confidence intervals for a difference between proportions and for an odds ratio, 540 C.17 The ztest for two independent counts or rates. Confidence intervals for a difference between counts or rates, and for a ratio of counts or rates, 542 C.18 One-way analysis of variance for means in more than two groups. Confidence intervals for means and mean differences, 544 C.19 The c2test for a trend in proportions in a 2 ¥ktable, 546 C.20 Tests for regression and correlation, 547 C.21 Spearman’s rank order correlation coefficient, 548 Appendix D: World Medical Association Declaration of Helsinki, 550 Bibliography and References, 554 Index, 559 IAUPR 5/23/00 3:25 PM Page ix Preface For a difference to be a difference, it has to make a difference. Gertrude Stein (1874–1946) The first edition of this book appeared in 1969, and was designed to introduce the basic concepts of statistics and their medical application to readers who had no formal training in statistical theory or methods. The book emphasized interpretation rather than techniques of calculation, and sought to make readers familiar with the expressions and methods commonly employed in the analysis and presentation of data in medical research. Its success demon- strated the need for such a book, and a second edition, incorporating a number of revisions and extensions, was published in 1975. The third edition, published 10 years later, while retaining the basic aim of interpretation, entailed a comprehensive revision of the book’s scope and content. Taking account of the developments in the range and sophistication of statistical tech- niques in medical research, and the need for a greater understanding of statis- tics by medical undergraduates and graduates, the book included details for calculation of the common statistical tests, and chapters on research design and methodology. The fourth edition (1991) greatly expanded on the material covered previously and more emphasis was placed on the use of confidence intervals. In the past 9 years there has been further expansion in the use of statistical analyses in the medical literature, and a number of techniques and approaches rarely seen in the 1980s have come into common use. This fifth edition of our book takes account of some of these changes, though the new topics chosen for inclusion may of course not please everybody. The book has also under- gone a major restructuring—with some sections totally rewritten – and it is now more logically organized and, we hope, substantially more user-friendly. Included in the new material are descriptions of the box plot, the stem and leaf diagram and the dot plot, and we have provided a method for calculation of a confidence interval for a single median. The geometric mean is considered in some detail in the context of log-transformed data, with emphasis on the interpretation of significance tests and confidence intervals for the compari- son of two geometric means. There is a new chapter, entitled ‘Associations: Chance, Confounded or Causal?’ which explains without any formulae the concepts underlying confounding, confidence intervals and pvalues, and the interpretation of associations observed in research investigations. We believe that understanding this descriptive chapter is central to the understanding of the place of statistics in medical research. One change we have made, informed by our teaching, is in our explanation of sampling variation. Previously we illustrated this topic with the sampling distribution of the mean. This resulted in the usual confusion between the ix IAUPR 5/23/00 3:25 PM Page x x standard error of the mean and the standard deviation of the actual observa- Preface tions. In this edition, sampling variation is introduced in the context of the sampling distribution of a proportion or percentage, and we hope that this has made the explanations much more understandable. Because of its paramount importance we have added an entirely new chapter on the calculation of sample size. This chapter considers sample size calculations in some detail and provides, in addition to the relevant formulae, useful tables that should give researchers an indication of the order of magni- tude of the number of subjects they might require in different situations. A chapter on measurement in medicine includes some material from the previ- ous edition, but much is new. In particular, we describe the quantification of agreement using the kappa statistic or the Bland and Altman techniques. Finally, for the benefit of the researcher, we have included a copy of the World Medical Association Declaration of Helsinki (1996) in an Appendix. Given the increasing number of new statistical computer programs and the rate at which upgraded versions of existing software appear, we decided to dis- pense with the chapter on statistical computing altogether. Our experience is that discussion of computer software becomes out of date in a very short time. The reader would be best advised to consult with current users regarding suit- able programs for biostatistical or epidemiological data analysis. Readers familiar with previous editions will note too the disappearance of the chapter on vital statistics. We felt that the techniques previously discussed in this chapter were better sited elsewhere throughout the book and that the topic did not warrant a chapter all to itself. There are a number of features to this text book that make it unique, some of which are new to this edition. A major change is the addition of ‘summary boxes’ throughout the text. These boxes give the essence of the topic under consideration and should be a great aid in revision. All relevant formulae are summarized at the end of each chapter in such boxes also. Additionally, as a guide to first-time readers, some chapters or chapter headings have an asterisk in parentheses after them ‘[!]’, signifying material that is generally a little more difficult than the material that surrounds it. The reader should note too the new guide to the choice of a statistical test on the inside front and back covers. Appendix C, which was a feature of previous editions, also provides an overview of the statistical methods covered in the text together with a detailed step-by-step description of the computational approach for each test. This appendix should be useful to those who, having previously read the detailed description in the body of the book, just wish to select and carry out a particular procedure. Looking up statistical tables can also be confusing, and the tables in Appendix B were especially designed (for the third edition) with the end-user in mind. Each table is laid out in a stan- dard format that greatly simplifies its use. As in the previous editions, derivations of formulae are avoided and instead emphasis is placed on explaining their logic. Although alternative methods are available for some of the more advanced techniques discussed, we are of the view that confusion is avoided if one only is presented. In any event, with the IAUPR 5/23/00 3:25 PM Page xi wide availability of many excellent and easy-to-use statistical computing xi packages, the necessity for hand computation has decreased greatly over a Preface number of years. Nevertheless many studies can still profit from a ‘hand analy- sis’, and computational details and formulae for tests or procedures are only given if they are feasible to undertake with a calculator. This allows researchers to perform some calculations without a computer and to gain a deeper understanding of what the computer is doing in particular situations. It also facilitates further calculation as an adjunct to computer output, where a software package might not perform all the requisite calculations (confidence intervals for some parameters of 2 ¥2 tables for instance). On the other hand, for those using more advanced techniques, the book’s concentration on the interpretation of results should help the readers understand the output from a computer package without burdening them with arcane formulae. Despite the changes in the book over the past 30 years we hope that it will continue to appeal to those who, though less interested in the actual statistical calculations, seek a basic understanding of statistical concepts and their role in medicine. This book contains much more than a ‘cookbook’ of statistical tech- niques, and much less than the detailed and often difficult texts that cover advanced methods. We have tried to steer a middle course by introducing con- cepts slowly and carefully, by avoiding any proofs or unnecessary formulae, and yet going far enough to cover most of the important topics and techniques in what we hope is an understandable manner. Parts of the book may be skipped over by those who want only a basic introduction to statistics or who wish to understand what they read in the journals, while enough detail is given to enable it to be used as a practical guide to data analysis. It should be suitable as a textbook for a range of courses taken by medical and paramedical stu- dents at both undergraduate and postgraduate levels. To assist the reader, an overview of the structure of the book is included in the following pages. Authors always appreciate feedback from their readers. While positive comments are of course always welcome, it is the suggestions for improve- ment, the identification of areas that are unclear, and the pointing out of mis- takes, misprints and misspellings that can be most useful. We would therefore like to express our thanks to those readers who have written to us in the past, to our colleagues who made valuable suggestions, and particularly to those of our students who have made their views known. To help readers contact us about this edition of Interpretation and Uses of Medical Statistics we have set up a dedicated email address ([email protected]) and a web page (http://www.ucd.ie/iums). We will use the web site to post any corrections to the text and we will include links to other sites that may be of interest. Addi- tionally we plan to include exercises based on material in the book – mainly from past examination papers that we have set. Just as we were contemplating this new edition in late 1995, Professor James McGilvray, who had been a co-author, died suddenly. This book is dedi- cated to his memory. L.D and G.J.B