Medical Statistics at a Glance - Aviva Petrie - ebook

Medical Statistics at a Glance ebook

Aviva Petrie

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Opis

Medical Statistics at a Glance is a concise and accessible introduction and revision aid for this complex subject. The self-contained chapters explain the underlying concepts of medical statistics and provide a guide to the most commonly used statistical procedures. This new edition of Medical Statistics at a Glance: * Presents key facts accompanied by clear and informative tables and diagrams * Focuses on illustrative examples which show statistics in action, with an emphasis on the interpretation of computer data analysis rather than complex hand calculations * Includes extensive cross-referencing, a comprehensive glossary of terms and flow-charts to make it easier to choose appropriate tests * Now provides the learning objectives for each chapter * Includes a new chapter on Developing Prognostic Scores * Includes new or expanded material on study management, multi-centre studies, sequential trials, bias and different methods to remove confounding in observational studies, multiple comparisons, ROC curves and checking assumptions in a logistic regression analysis * The companion website at www.medstatsaag.com contains supplementary material including an extensive reference list and multiple choice questions (MCQs) with interactive answers for self-assessment. Medical Statistics at a Glance will appeal to all medical students, junior doctors and researchers in biomedical and pharmaceutical disciplines. Reviews of the previous editions "The more familiar I have become with this book, the more I appreciate the clear presentation and unthreatening prose. It is now a valuable companion to my formal statistics course." -International Journal of Epidemiology "I heartily recommend it, especially to first years, but it's equally appropriate for an intercalated BSc or Postgraduate research. If statistics give you headaches - buy it. If statistics are all you think about - buy it." -GKT Gazette "...I unreservedly recommend this book to all medical students, especially those that dislike reading reams of text. This is one book that will not sit on your shelf collecting dust once you have graduated and will also function as a reference book." -4th Year Medical Student, Barts and the London Chronicle, Spring 2003

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Table of Contents

Title page

Copyright page

Preface

Learning objectives

Handling data

1 Types of data

Data and statistics

Categorical (qualitative) data

Numerical (quantitative) data

Distinguishing between data types

Derived data

Censored data

2 Data entry

Formats for data entry

Planning data entry

Categorical data

Numerical data

Multiple forms per patient

Problems with dates and times

Coding missing values

3 Error checking and outliers

Typing errors

Error checking

Handling missing data

Outliers

4 Displaying data diagrammatically

One variable

Two variables

Identifying outliers using graphical methods

The use of connecting lines in diagrams

5 Describing data: the ‘average’

Summarizing data

The arithmetic mean

The median

The mode

The geometric mean

The weighted mean

6 Describing data: the ‘spread’

Summarizing data

The range

Ranges derived from percentiles

The standard deviation

Variation within- and between-subjects

7 Theoretical distributions: the Normal distribution

Understanding probability

The rules of probability

Probability distributions: the theory

The Normal (Gaussian) distribution

The Standard Normal distribution

8 Theoretical distributions: other distributions

Some words of comfort

More continuous probability distributions

Discrete probability distributions

9 Transformations

Why transform?

How do we transform?

Typical transformations

Sampling and estimation

10 Sampling and sampling distributions

Why do we sample?

Obtaining a representative sample

Point estimates

Sampling variation

Sampling distribution of the mean

Interpreting standard errors

SD or SEM?

Sampling distribution of the proportion

11 Confidence intervals

Confidence interval for the mean

Confidence interval for the proportion

Interpretation of confidence intervals

Degrees of freedom

Bootstrapping and jackknifing

Study design

12 Study design I

Experimental or observational studies

Defining the unit of observation

Multicentre studies

Assessing causality

Cross-sectional or longitudinal studies

Controls

Bias

13 Study design II

Variation

Replication

Sample size

Particular study designs

Choosing an appropriate study endpoint

14 Clinical trials

Treatment comparisons

Primary and secondary endpoints

Subgroup analyses

Treatment allocation

Sequential trials

Blinding or masking

Patient issues

The protocol

15 Cohort studies

Selection of cohorts

Follow-up of individuals

Information on outcomes and exposures

Analysis of cohort studies

Advantages of cohort studies

Disadvantages of cohort studies

Study management

Clinical cohorts

16 Case–control studies

Selection of cases

Selection of controls

Identification of risk factors

Matching

Analysis of unmatched or group-matched case–control studies

Analysis of individually matched case–control studies

Advantages of case–control studies

Disadvantages of case–control studies

Hypothesis testing

17 Hypothesis testing

Defining the null and alternative hypotheses

Obtaining the test statistic

Obtaining the P-value

Using the P-value

Non-parametric tests

Which test?

Hypothesis tests versus confidence intervals

Equivalence and non-inferiority trials

18 Errors in hypothesis testing

Making a decision

Making the wrong decision

Power and related factors

Multiple hypothesis testing

Basic techniques for analysing data

Numerical data

19 Numerical data: a single group

The problem

The one-sample t-test

The sign test

20 Numerical data: two related groups

The problem

The paired t-test

The Wilcoxon signed ranks test

21 Numerical data: two unrelated groups

The problem

The unpaired (two-sample) t-test

The Wilcoxon rank sum (two-sample) test

22 Numerical data: more than two groups

The problem

One-way analysis of variance

The Kruskal–Wallis test

Categorical data

23 Categorical data: a single proportion

The problem

The test of a single proportion

The sign test applied to a proportion

24 Categorical data: two proportions

The problems

Independent groups: the Chi-squared test

Related groups: McNemar’s test

25 Categorical data: more than two categories

Chi-squared test: large contingency tables

Chi-squared test for trend

Regression and correlation

26 Correlation

Introduction

Pearson correlation coefficient

Spearman’s rank correlation coefficient

27 The theory of linear regression

What is linear regression?

The regression line

Method of least squares

Assumptions

Analysis of variance table

Regression to the mean

28 Performing a linear regression analysis

The linear regression line

Drawing the line

Checking the assumptions

Failure to satisfy the assumptions

Outliers and influential points

Assessing goodness of fit

Investigating the slope

Using the line for prediction

Improving the interpretation of the model

29 Multiple linear regression

What is it?

Why do it?

Assumptions

Categorical explanatory variables

Analysis of covariance

Choice of explanatory variables

Analysis

Outliers and influential points

30 Binary outcomes and logistic regression

Introduction

Reasoning

The logistic regression equation

The explanatory variables

Assessing the adequacy of the model

Comparing the odds ratio and the relative risk

Multinomial and ordinal logistic regression

Conditional logistic regression

31 Rates and Poisson regression

Rates

Poisson regression

32 Generalized linear models

Which type of model do we choose?

Likelihood and maximum likelihood estimation

Assessing adequacy of fit

Regression diagnostics

33 Explanatory variables in statistical models

Nominal explanatory variables

Ordinal explanatory variables

Numerical explanatory variables

Selecting explanatory variables

Interaction

Collinearity

Important considerations

34 Bias and confounding

Bias

Confounding

35 Checking assumptions

Why bother?

Are the data Normally distributed?

Are two or more variances equal?

Are variables linearly related?

What if the assumptions are not satisfied?

Sensitivity analysis

36 Sample size calculations

The importance of sample size

Requirements

Methodology

Altman’s nomogram

Quick formulae

Power statement

Adjustments

Increasing the power for a fixed sample size

37 Presenting results

Introduction

Numerical results

Tables

Diagrams

Presenting results in a paper

Additional chapters

38 Diagnostic tools

Reference intervals

Diagnostic tests

39 Assessing agreement

Measurement variability and error

Reliability

Categorical variables

Numerical variables

40 Evidence-based medicine

1 Formulate the problem

2 Locate the relevant information (e.g. on diagnosis, prognosis or therapy)

3 Critically appraise the methods in order to assess the validity (closeness to the truth) of the evidence

4 Extract the most useful results and determine whether they are important

5 Apply the results in clinical practice

6 Evaluate your performance

41 Methods for clustered data

Displaying the data

Comparing groups: inappropriate analyses

Comparing groups: appropriate analyses

42 Regression methods for clustered data

Aggregate level analysis

Robust standard errors

Random effects models

Generalized estimating equations (GEE)

43 Systematic reviews and meta-analysis

The systematic review

Meta-analysis

44 Survival analysis

Censored data

Displaying survival data

Summarizing survival

Comparing survival

Problems encountered in survival analysis

45 Bayesian methods

The frequentist approach

The Bayesian approach

Diagnostic tests in a Bayesian framework

Disadvantages of Bayesian methods

46 Developing prognostic scores

Why do we do it?

Assessing the performance of a prognostic score

Developing prognostic indices and risk scores for other types of data

Appendices

Appendix A: Statistical tables

Appendix B: Altman’s nomogram for sample size calculations (Chapter 36)

Appendix C: Typical computer output

Appendix D: Glossary of terms

Appendix E: Chapter numbers with relevant multiple-choice questions and structured questions from Medical Statistics at a Glance Workbook

Index

This edition first published 2009 © 2000, 2005, 2009 by Aviva Petrie and Caroline Sabin

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Library of Congress Cataloging-in-Publication Data

Petrie, Aviva.

 Medical statistics at a glance / Aviva Petrie, Caroline Sabin. – 3rd ed.

p.; cm. – (At a glance series)

 Includes bibliographical references and index.

 ISBN 978-1-4051-8051-1 (alk. paper)

 1. Medical statistics. I. Sabin, Caroline. II. Title. III. Series: At a glance series (Oxford, England)

 [DNLM: 1. Statistics as Topic. 2. Research Design. WA 950 P495m 2009]

 R853.S7P476 2009

 610.72′7–dc22

2008052096

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.

Preface

Medical Statistics at a Glance is directed at undergraduate medical students, medical researchers, postgraduates in the biomedical disciplines and at pharmaceutical industry personnel. All of these individuals will, at some time in their professional lives, be faced with quantitative results (their own or those of others) which will need to be critically evaluated and interpreted, and some, of course, will have to pass that dreaded statistics exam! A proper understanding of statistical concepts and methodology is invaluable for these needs. Much as we should like to fire the reader with an enthusiasm for the subject of statistics, we are pragmatic. Our aim in this new edition, as it was in the earlier editions, is to provide the student and the researcher, as well as the clinician encountering statistical concepts in the medical literature, with a book which is sound, easy to read, comprehensive, relevant, and of useful practical application.

We believe Medical Statistics at a Glance will be particularly helpful as an adjunct to statistics lectures and as a reference guide. The structure of this third edition is the same as that of the first two editions. In line with other books in the At a Glance series, we lead the reader through a number of self-contained two-, three- or occasionally four-page chapters, each covering a different aspect of medical statistics. We have learned from our own teaching experiences and have taken account of the difficulties that our students have encountered when studying medical statistics. For this reason, we have chosen to limit the theoretical content of the book to a level that is sufficient for understanding the procedures involved, yet which does not overshadow the practicalities of their execution.

Medical statistics is a wide-ranging subject covering a large number of topics. We have provided a basic introduction to the underlying concepts of medical statistics and a guide to the most commonly used statistical procedures. Epidemiology is closely allied to medical statistics. Hence some of the main issues in epidemiology, relating to study design and interpretation, are discussed. Also included are chapters which the reader may find useful only occasionally, but which are, nevertheless, fun­damental to many areas of medical research; for example, evidence-based medicine, systematic reviews and meta-analysis, survival analysis, Bayesian methods and the development of prognostic scores. We have explained the principles underlying these topics so that the reader will be able to understand and interpret the results from them when they are presented in the literature.

The chapter titles of this third edition are identical to those of the second edition, apart from Chapter 34 (now called ‘Bias and confounding’ instead of ‘Issues in statistical modelling’); in addition, we have added a new chapter (Chapter 46 – ‘Deve­loping prognostic scores’). Some of the first 45 chapters remain unaltered in this new edition and some have relatively minor changes which accommodate recent advances, cross-referencing or re-organization of the new material. We have expanded many chapters; for example, we have included a section on multiple comparisons (Chapter 12), provided more information on different study designs, including multicentre studies (Chapter 12) and sequential trials (Chapter 14), emphasized the importance of study management (Chapters 15 and 16), devoted greater space to receiver operating characteristic (ROC) curves (Chapters 30, 38 and 46), supplied more details of how to check the assumptions underlying a logistic regression analysis (Chapter 30) and explored further some of the different methods to remove confounding in observational studies (Chapter 34). We have also reorganized some of the material. The brief introduction to bias in Chapter 12 in the second edition has been omitted from that chapter in the third edition and moved to Chapter 34, which covers this topic in greater depth. A discussion of ‘interaction’ is currently in Chapter 33 and the section on prognostic indices is now much expanded and contained in the new Chapter 46.

New to this third edition is a set of learning objectives for each chapter, all of which are displayed together at the beginning of the book. Each set provides a framework for evaluating understanding and progress. If you are able to complete all the bulleted tasks in a chapter satisfactorily, you will have mastered the concepts in that chapter.

As in previous editions, the description of most of the statistical techniques is accompanied by an example illustrating its use. We have generally obtained the data for these examples from collaborative studies in which we or colleagues have been involved; in some instances, we have used real data from published papers. Where possible, we have used the same data set in more than one chapter to reflect the reality of data analysis, which is rarely restricted to a single technique or approach. Although we believe that formulae should be provided and the logic of the approach explained as an aid to understanding, we have avoided showing the details of complex calculations – most readers will have access to computers and are unlikely to perform any but the simplest calculations by hand.

We consider that it is particularly important for the reader to be able to interpret output from a computer package. We have therefore chosen, where applicable, to show results using extracts from computer output. In some instances, where we believe individuals may have difficulty with its interpretation, we have included (Appendix C) and annotated the complete computer output from an analysis of a data set. There are many statistical packages in common use; to give the reader an indication of how output can vary, we have not restricted the output to a particular package and have, instead, used three well-known ones – SAS, SPSS and Stata.

There is extensive cross-referencing throughout the text to help the reader link the various procedures. A basic set of statis­tical tables is contained in Appendix A. Neave, H.R. (1995) Elemementary Statistical Tables, Routledge: London, and Diem, K. (1970) Documenta Geigy Scientific Tables, 7th edition, Blackwell Publishing: Oxford, amongst others, provide fuller versions if the reader requires more precise results for hand calculations. The glossary of terms in Appendix D provides readily accessible explanations of commonly used terminology.

We know that one of the greatest difficulties facing non-statisticians is choosing the appropriate technique. We have therefore produced two flow charts which can be used both to aid the decision as to what method to use in a given situation and to locate a particular technique in the book easily. These flow charts are displayed prominently on the inside back cover for easy access.

The reader may find it helpful to assess his/her progress in self-directed learning by attempting the interactive exercises on our website (www.medstatsaag.com). This website also contains a full set of references (some of which are linked directly to Medline) to supplement the references quoted in the text and provide useful background information for the examples. For those readers who wish to gain a greater insight into particular areas of medical statistics, we can recommend the following books:

Altman, D.G. (1991)

Practical Statistics for Medical Research

. London: Chapman and Hall/CRC.

Armitage, P., Berry, G. and Matthews, J.F.N. (2001)

Statistical Methods in Medical Research.

4th edition. Oxford: Blackwell Science.

Kirkwood, B.R. and Sterne, J.A.C. (2003)

Essential Medical Statistics.

2nd Edn. Oxford: Blackwell Publishing.

Pocock, S.J. (1983)

Clinical Trials: A Practical Approach

. Chichester: Wiley.

We are extremely grateful to Mark Gilthorpe and Jonathan Sterne who made invaluable comments and suggestions on aspects of the second edition, and to Richard Morris, Fiona Lampe, Shak Hajat and Abul Basar for their counsel on the first edition. We wish to thank everyone who has helped us by providing data for the examples. Naturally, we take full responsibility for any errors that remain in the text or examples. We should also like to thank Mike, Gerald, Nina, Andrew and Karen who tolerated, with equanimity, our preoccupation with the first two editions and lived with us through the trials and tribulations of this third edition.

Aviva PetrieCaroline SabinLondon

Also available to buy now!

Medical Statistics at a Glance Workbook

A brand new comprehensive workbook containing a variety of examples and exercises, complete with model answers, designed to support your learning and revision.

Fully cross-referenced to Medical Statistics at a Glance, this new workbook includes:

Over 80 MCQs, each testing knowledge of a single statistical concept or aspect of study interpretation

29 structured questions to explore in greater depth several statistical techniques or principles

Templates for the appraisal of clinical trials and observational studies, plus full appraisals of two published papers to demonstrate the use of these templates in practice

Detailed step-by-step analyses of two substantial data sets (also available at

www.medstatsaag.com

) to demonstrate the application of statistical procedures to real-life research

Medical Statistics at a Glance Workbook is the ideal resource to improve statistical knowledge together with your analytical and interpretational skills.

Learning Objectives

By the end of the relevant chapter you should be able to:

1 Types of dataDistinguish between a sample and a populationDistinguish between categorical and numerical dataDescribe different types of categorical and numerical dataExplain the meaning of the terms: variable, percentage, ratio, quotient, rate, scoreExplain what is meant by censored data
2 Data entryDescribe different formats for entering data on to a computerOutline the principles of questionnaire designDistinguish between single-coded and multi-coded variablesDescribe how to code missing values
3 Error checking and outliersDescribe how to check for errors in dataOutline the methods of dealing with missing dataDefine an outlierExplain how to check for and handle outliers
4 Displaying data diagrammaticallyExplain what is meant by a frequency distributionDescribe the shape of a frequency distributionDescribe the following diagrams: (segmented) bar or column chart, pie chart, histogram, dot plot, stem-and-leaf plot, box-and-whisker plot, scatter diagramExplain how to identify outliers from a diagram in various situationsDescribe the situations when it is appropriate to use connecting lines in a diagram
5 Describing data: the ‘average’Explain what is meant by an averageDescribe the appropriate use of each of the following types of average: arithmetic mean, mode, median, geometric mean, weighted meanExplain how to calculate each type of averageList the advantages and disadvantages of each type of average
6 Describing data: the ‘spread’Define the following terms: percentile, decile, quartile, median, and explain their inter-relationshipExplain what is meant by a reference interval/range, also called the normal rangeDefine the following measures of spread: range, interdecile range, variance, standard deviation (SD), coefficient of variationList the advantages and disadvantages of the various measures of spreadDistinguish between intra- and inter-subject variation
7 Theoretical distributions: the Normal distributionDefine the terms: probability, conditional probabilityDistinguish between the subjective, frequentist and a priori approaches to calculating a probabilityDefine the addition and multiplication rules of probabilityDefine the terms: random variable, probability distribution, parameter, statistic, probability density functionDistinguish between a discrete and continuous probability distribution and list the properties of eachList the properties of the Normal and the Standard Normal distributionsDefine a Standardized Normal Deviate (SND)
8 Theoretical distributions: other distributionsList the important properties of the t-, Chi-squared, F- and Lognormal distributionsExplain when each of these distributions is particularly usefulList the important properties of the Binomial and Poisson distributionsExplain when the Binomial and Poisson distributions are each particularly useful
9 TransformationsDescribe situations in which transforming data may be usefulExplain how to transform a data setExplain when to apply and what is achieved by the logarithmic, square root, reciprocal, square and logit transformationsDescribe how to interpret summary measures derived from log transformed data after they have been back-transformed to the original scale
10 Sampling and sampling distributionsExplain what is meant by statistical inference and sampling errorExplain how to obtain a representative sampleDistinguish between point and interval estimates of a parameterList the properties of the sampling distribution of the meanList the properties of the sampling distribution of the proportionExplain what is meant by a standard errorState the relationship between the standard error of the mean (SEM) and the standard deviation (SD)Distinguish between the uses of the SEM and the SD
11 Confidence intervalsInterpret a confidence interval (CI)Calculate a confidence interval for a meanCalculate a confidence interval for a proportionExplain the term ‘degrees of freedom’Explain what is meant by bootstrapping and jackknifing
12 Study design IDistinguish between experimental and observational studies, and between cross-sectional and longitudinal studiesExplain what is meant by the unit of observationExplain the terms: control group, epidemiological study, cluster randomized trial, ecological study, multicentre study, survey, censusList the criteria for assessing causality in observational studiesDescribe the time course of cross-sectional, repeated cross-sectional, cohort, case–control and experimental studiesList the typical uses of these various types of studyDistinguish between prevalence and incidence
13 Study design IIDescribe how to increase the precision of an estimateExplain the principles of blocking (stratification)Distinguish between parallel and cross-over designsDescribe the features of a factorial experimentExplain what is meant by an interaction between factorsExplain the following terms: study endpoint, surrogate marker, composite endpoint
14 Clinical trialsDefine ‘clinical trial’ and distinguish between Phase I/II and Phase III clinical trialsExplain the importance of a control treatment and distinguish between positive and negative controlsExplain what is meant by a placeboDistinguish between primary and secondary endpointsExplain why it is important to randomly allocate individuals to treatment groups and describe different forms of randomizationExplain why it is important to incorporate blinding (masking)Distinguish between double- and single-blind trialsDiscuss the ethical issues arising from a randomized controlled trial (RCT)Explain the principles of a sequential trialDistinguish between on-treatment analysis and analysis by intention-to-treat (ITT)Describe the contents of a protocolApply the CONSORT Statement guidelines
15 Cohort studiesDescribe the aspects of a cohort studyDistinguish between fixed and dynamic cohortsExplain the terms: historical cohort, risk factor, healthy entrant effect, clinical cohortList the advantages and disadvantages of a cohort studyDescribe the important aspects of cohort study managementCalculate and interpret a relative risk
16 Case–control studiesDescribe the features of a case–control studyDistinguish between incident and prevalent casesDescribe how controls may be selected for a case–control studyExplain how to analyse an unmatched case–control study by calculating and interpreting an odds ratioDescribe the features of a matched case–control studyDistinguish between frequency matching and pairwise matchingExplain when an odds ratio can be used as an estimate of the relative riskList the advantages and disadvantages of a case–control study
17 Hypothesis testingDefine the terms: null hypothesis, alternative hypothesis, one- and two-tailed test, test statistic, P-value, significance levelList the five steps in hypothesis testingExplain how to use the P-value to make a decision about rejecting or not rejecting the null hypothesisExplain what is meant by a non-parametric (distribution-free) test and explain when such a test should be usedExplain how a confidence interval can be used to test a hypothesisDistinguish between superiority, equivalence and non-inferiority studiesDescribe the approach used in equivalence and non-inferiority tests
18 Errors in hypothesis testingExplain what is meant by an effect of interestDistinguish between Type I and Type II errorsState the relationship between the Type II error and powerList the factors that affect the power of a test and describe their effects on powerExplain why it is inappropriate to perform many hypothesis tests in a studyDescribe different situations which involve multiple comparisons within a data set and explain how the difficulties associated with multiple comparisons may be resolved in each situationExplain what is achieved by a post hoc testOutline the Bonferroni approach to multiple hypothesis testing
19 Numerical data: a single groupExplain the rationale of the one-sample t-testExplain how to perform the one-sample t-testState the assumption underlying the test and explain how to proceed if it is not satisfiedExplain how to use an appropriate confidence interval to test a hypothesis about the meanExplain the rationale of the sign testExplain how to perform the sign test
20 Numerical data: two related groupsDescribe different circumstances in which two groups of data are relatedExplain the rationale of the paired t-testExplain how to perform the paired t-testState the assumption underlying the test and explain how to proceed if it is not satisfiedExplain the rationale of the Wilcoxon signed ranks testExplain how to perform the Wilcoxon signed ranks test
21 Numerical data: two unrelated groupsExplain the rationale of the unpaired (two-sample) t-testExplain how to perform the unpaired t-testList the assumptions underlying this test and explain how to check them and proceed if they are not satisfiedUse an appropriate confidence interval to test a hypothesis about the difference between two meansExplain the rationale of the Wilcoxon rank sum testExplain how to perform the Wilcoxon rank sum testExplain the relationship between the Wilcoxon rank sum test and the Mann–Whitney U test
22 Numerical data: more than two groupsExplain the rationale of the one-way analysis of variance (ANOVA)Explain how to perform a one-way ANOVAExplain why a post hoc comparison method should be used if a one-way ANOVA produces a significant result and name some different post hoc methodsList the assumptions underlying the one-way ANOVA and explain how to check them and proceed if they are not satisfiedExplain the rationale of the Kruskal–Wallis testExplain how to perform the Kruskal–Wallis test
23 Categorical data: a single proportionExplain the rationale of a test, based on the Normal distribution, which can be used to investigate whether a proportion takes a particular value.Explain how to perform this testExplain why a continuity correction should be used in this testExplain how the sign test can be used to test a hypothesis about a proportionExplain how to perform the sign test to test a hypothesis about a proportion
24 Categorical data: two proportionsExplain the terms: contingency table, cell frequency, marginal total, overall total, observed frequency, expected frequencyExplain the rationale of the Chi-squared test to compare proportions in two unrelated groupsExplain how to perform the Chi-squared test to compare two independent proportionsCalculate the confidence interval for the difference in the proportions in two unrelated groups and use it to compare themState the assumption underlying the Chi-squared test to compare proportions and explain how to proceed if this assumption is not satisfiedDescribe the circumstances under which Simpson’s paradox may occur and explain what can be done to avoid itExplain the rationale of McNemar’s test to compare the proportions in two related groupsExplain how to perform McNemar’s testCalculate the confidence interval for the difference in two proportions in paired groups and use the confidence interval to compare them
25 Categorical data: more than two categoriesDescribe an r × c contingency tableExplain the rationale of the Chi-squared test to assess the association between one variable with r categories and another variable with c categoriesExplain how to perform the Chi-squared test to assess the association between two variables using data displayed in an r × c contingency tableState the assumption underlying this Chi-squared test and explain how to proceed if this assumption is not satisfiedExplain the rationale of the Chi-squared test for trend in a 2 × k contingency tableExplain how to perform the Chi-squared test for trend in a 2 × k contingency table
26 CorrelationDescribe a scatter diagramDefine and calculate the Pearson correlation coefficient and list its propertiesExplain when it is inappropriate to calculate the Pearson correlation coefficient if investigating the relationship between two variablesExplain how to test the null hypothesis that the true Pearson correlation coefficient is zeroCalculate the 95% confidence interval for the Pearson correlation coefficientDescribe the use of the square of the Pearson correlation coefficientExplain when and how to calculate the Spearman rank correlation coefficientList the properties of the Spearman rank correlation coefficient
27 The theory of linear regressionExplain the terms commonly used in regression analysis: dependent variable, explanatory variable, regression coefficient, intercept, gradient, residualDefine the simple (univariable) regression line and interpret its coefficientsExplain the principles of the method of least squaresList the assumptions underlying a simple linear regression analysisDescribe the features of an analysis of variance (ANOVA) table produced by a linear regression analysisExplain how to use the ANOVA table to assess how well the regression line fits the data (goodness of fit) and test the null hypothesis that the true slope of the regression line is zero.Explain what is meant by regression to the mean
28 Performing a linear regression analysisExplain how to use residuals to check the assumptions underlying a linear regression analysisExplain how to proceed in a regression analysis if one or more of the assumptions are not satisfiedDefine the terms ‘outlier’ and ‘influential point’ and explain how to deal with each of themExplain how to assess the goodness of fit of a regression modelCalculate the 95% confidence interval for the slope of a regression lineDescribe two methods for testing the null hypothesis that the true slope is zeroExplain how to use the regression line for predictionExplain how to (1) centre and (2) scale an explanatory variable in a regression analysisExplain what is achieved by centring and scaling.
29 Multiple linear regressionExplain the terms: covariate, partial regression coefficient, collinearityDefine the multiple (multivariable) linear regression equation and interpret its coefficientsGive three reasons for performing a multiple regression analysisExplain how to create dummy variables to allow nominal and ordinal categorical explanatory variables with more than two categories of response to be incorporated in the modelExplain what is meant by the reference category when fitting models that include categorical explanatory variablesDescribe how multiple regression analysis can be used as a form of analysis of covarianceGive a rule of thumb for deciding on the maximum number of explanatory variables in a multiple regression equationUse computer output from a regression analysis to assess the goodness of fit of the model, and test the null hypotheses that all the partial regression coefficients are zero and that each partial regression coefficient is zeroExplain the relevance of residuals, leverage and Cook’s distance in identifying outliers and influential points
30 Binary outcomes and logistic regressionExplain why multiple linear regression analysis cannot be used for a binary outcome variableDefine the logit of a proportionDefine the multiple logistic regression equationInterpret the exponential of a logistic regression coefficientCalculate, from a logistic regression equation, the probability that a particular individual will have the outcome of interestDescribe two ways of assessing whether a logistic regression coefficient is statistically significantDescribe various ways of testing the overall model fit, assessing predictive efficiency and investigating the underlying assumptions of a logistic regression analysisExplain when the odds ratio is greater than and when it is less than the relative riskExplain the use of the following types of logistic regression: multinomial, ordinal, conditional
31 Rates and Poisson regressionDefine a rate and describe its featuresDistinguish between a rate and a risk, and between an incidence rate and a mortality rateDefine a relative rate and explain when it is preferred to a relative riskExplain when it is appropriate to use Poisson regressionDefine the Poisson regression equation and interpret the exponential of a Poisson regression coefficientCalculate, from the Poisson regression equation, the event rate for a particular individualExplain the use of an offset in a Poisson regression analysisExplain how to perform a Poisson regression analysis with (1) grouped data and (2) variables that change over timeExplain the meaning and the consequences of extra-Poisson dispersionExplain how to identify extra-Poisson dispersion in a Poisson regression analysis
32 Generalized linear modelsDefine the equation of the generalized linear model (GLM)Explain the terms ‘link function’ and ‘identity link’Specify the link functions for the logistic and Poisson regression modelsExplain the term ‘likelihood’ and the process of maximum likelihood estimation (MLE)Explain the terms: saturated model, likelihood ratioExplain how the likelihood ratio statistic (LRS), i.e. the deviance or −2log likelihood, can be used to:assess the adequacy of fit of a modelcompare two models when one is nested within the otherassess whether all the parameters associated with the covariates of a model are zero (i.e. the model Chi-square)
33 Explanatory variables in statistical modelsExplain how to test the significance of a nominal explanatory variable in a statistical model when the variable has more than two categoriesDescribe two ways of incorporating an ordinal explanatory variable into a model when the variable has more than two categories, and:state the advantages and disadvantages of each approachexplain how each approach can be used to test for a linear trendExplain how to check the linearity assumption in multiple, Poisson and logistic regression analysesDescribe three ways of dealing with non-linearity in a regression modelExplain why a model should not be over-fitted and how to avoid itExplain when it is appropriate to use automatic selection procedures to select the optimal explanatory variablesDescribe the principles underlying various automatic selection proceduresExplain why automatic selection procedures should be used with cautionExplain the meaning of interaction and collinearityExplain how to test for an interaction in a regression analysisExplain how to detect collinearity
34 Bias and confoundingExplain what is meant by biasExplain what is meant by selection bias, information bias, funding bias and publication biasDescribe different forms of bias which comprise either selection bias or information biasExplain what is meant by the ecological fallacyExplain what is meant by confounding and what steps may be taken to deal with confounding at the design stage of a studyDescribe various methods of dealing with confounding at the analysis stage of a studyExplain the meaning of a propensity scoreDiscuss the advantages and disadvantages of the various methods of dealing with confounding at the analysis stageExplain why confounding is a particular issue in a non-randomized studyExplain the following terms: causal pathway, intermediate variable, time-varying confounding
35 Checking assumptionsName two tests and describe two diagrams that can be used to assess whether data are Normally distributedExplain the terms homogeneity and heterogeneity of varianceName two tests that can be used to assess the equality of two or more variancesExplain how to perform the variance ratio F-test to compare two variancesExplain how to proceed if the assumptions under a proposed analysis are not satisfiedExplain what is meant by a robust analysisExplain what is meant by a sensitivity analysisProvide examples of different sensitivity analyses
36 Sample size calculationsExplain why it is necessary to choose an optimal sample size for a proposed studySpecify the quantities that affect sample size and describe their effects on itName five approaches to calculating the optimal sample size of a studyExplain how information from an internal pilot study may be used to revise calculations of the optimal sample sizeExplain how to use Altman’s nomogram to determine the optimal sample size for a proposed t-test (unpaired and paired) and Chi-squared testExplain how to use Lehr’s formula for sample size calculations for the comparison of two means and of two proportions in independent groupsWrite an appropriate power statementExplain how to adjust the sample size for losses to follow-up and/or if groups of different sizes are requiredExplain how to increase the power of a study for a fixed sample size
37 Presenting resultsExplain how to report numerical resultsDescribe the important features of good tables and diagramsExplain how to report the results of a hypothesis testExplain how to report the results of a regression analysisIndicate how complex statistical analyses should be reportedLocate and follow the guidelines for reporting different types of study
38 Diagnostic toolsDistinguish between a diagnostic test and a screening test and explain when each is appropriateDefine ‘reference range’ and explain how it is usedDescribe two ways in which a reference range can be calculatedDefine the terms: true positive, false positive, true negative, false negativeEstimate (with a 95% confidence interval) and interpret each of the following: prevalence, sensitivity, specificity, positive predictive value, negative predictive valueConstruct a receiver operating characteristic (ROC) curveExplain how the ROC curve can be used to choose an optimal cut-off for a diagnostic testExplain how the area under the ROC curve can be used to assess the ability of a diagnostic test to discriminate between individuals with and without a disease and to compare two diagnostic testsCalculate and interpret the likelihood ratio for a positive and for a negative test result if the sensitivity and specificity of the test are known.
39 Assessing agreementDistinguish between measurement variability and measurement errorDistinguish between systematic and random errorDistinguish between reproducibility and repeatabilityCalculate and interpret Cohen’s kappa for assessing the agreement between paired categorical responsesExplain what a weighted kappa is and when it can be determinedExplain how to test for a systematic effect when comparing pairs of numerical responsesExplain how to perform a Bland and Altman analysis to assess the agreement between paired numerical responses and interpret the limits of agreementExplain how to calculate and interpret the British Standards Institution reproducibility/repeatability coefficientExplain how to calculate and interpret the intraclass correlation coefficient and Lin’s concordance correlation coefficient in a method comparison studyExplain why it is inappropriate to calculate the Pearson correlation coefficient to assess the agreement between paired numerical responses
40 Evidence-based medicineDefine evidence-based medicine (EBM)Describe the hierarchy of evidence associated with various study designsList the six steps involved in performing EBM to assess the efficacy of a new treatment, and describe the important features of each stepExplain the term number needed to treat (NNT)Explain how to calculate the NNTExplain how to assess the effect of interest if the main outcome variable is binaryExplain how to assess the effect of interest if the main outcome variable is numericalExplain how to decide whether the results of an investigation are important
41 Methods for clustered dataDescribe, with examples, clustered data in a two-level structureDescribe how such data may be displayed graphicallyDescribe the effect of ignoring repeated measures in a statistical analysisExplain how summary measures may be used to compare groups of repeated measures dataName two other methods which are appropriate for comparing groups of repeated measures dataExplain why a series of two-sample t-tests is inappropriate for analysing such data
42 Regression methods for clustered dataOutline the following approaches to analysing clustered data in a two-level structure: aggregate level analysis, analysis using robust standard errors, random effects (hierarchical, multilevel, mixed, cluster-specific, cross-sectional) model, generalized estimating equations (GEE)List the advantages and disadvantages of each approachDistinguish between a random intercepts and a random slopes random effects modelExplain how to calculate and interpret the intraclass correlation coefficient (ICC) to assess the effect of clustering in a random effects modelExplain how to use the likelihood ratio test to assess the effect of clustering
43 Systematic reviews and meta-analysisDefine a systematic review and explain what it achievesDescribe the Cochrane CollaborationDefine a meta-analysis and list its advantages and disadvantagesList the four steps involved in performing a meta-analysisDistinguish between statistical and clinical heterogeneityExplain how to test for statistical homogeneityExplain how to estimate the average effect of interest in a meta-analysis if there is evidence of statistical heterogeneityExplain the terms: fixed effects meta-analysis, random effects meta-analysis, meta-regressionDistinguish between a forest plot and a funnel plotDescribe ways of performing a sensitivity analysis after performing a meta-analysis
44 Survival analysisExplain why it is necessary to use special methods for analysing survival dataDistinguish between the terms ‘right-censored data’ and ‘left-censored data’Describe a survival curveDistinguish between the Kaplan–Meier method and lifetable approaches to calculating survival probabilitiesExplain what the log-rank test is used for in survival analysisExplain the principles of the Cox proportional hazards regression modelExplain how to obtain a hazard ratio (relative hazard) from a Cox proportional hazards regression model and interpret itList other regression models that may also be used to describe survival dataExplain the problems associated with informative censoring and competing risks
45 Bayesian methodsExplain what is meant by the frequentist approach to probabilityExplain the shortcomings of the frequentist approach to probabilityExplain the principles of Bayesian analysisList the disadvantages of the Bayesian approachExplain the terms: conditional probability, prior probability, posterior probability, likelihood ratioExpress Bayes theorem in terms of oddsExplain how to use Fagan’s nomogram to interpret a diagnostic test result in a Bayesian framework
46 Developing prognostic scoresDefine the term ‘prognostic score’Distinguish between a prognostic index and a risk scoreOutline different ways of deriving a prognostic scoreList the desirable features of a good prognostic scoreExplain what is meant by assessing overall score accuracyDescribe how a classification table and the mean Briar score can be used to assess overall score accuracyExplain what is meant by assessing the ability of a prognostic score to discriminate between those that do and do not experience the eventDescribe how classifying individuals by their score, drawing an ROC curve and calculating Harrell’s c statistic can each be used to assess the ability of a prognostic score to discriminate between those that do and do not experience the eventExplain what is meant by correct calibration of a prognostic scoreDescribe how the Hosmer–Lemeshow goodness of fit test can be used to assess whether a prognostic score is correctly calibratedExplain what is meant by transportability of a prognostic scoreDescribe various methods of internal and external validation of a prognostic score

1

Types of Data

Data and Statistics

The purpose of most studies is to collect data to obtain information about a particular area of research. Our data comprise observations on one or more variables; any quantity that varies is termed a variable. For example, we may collect basic clinical and demographic information on patients with a particular illness. The variables of interest may include the sex, age and height of the patients.

Our data are usually obtained from a sample of individuals which represents the population of interest. Our aim is to condense these data in a meaningful way and extract useful information from them. Statistics encompasses the methods of collecting, summarizing, analysing and drawing conclusions from the data: we use statistical techniques to achieve our aim.

Data may take many different forms. We need to know what form every variable takes before we can make a decision regarding the most appropriate statistical methods to use. Each variable and the resulting data will be one of two types: categorical or numerical (Fig. 1.1).

Figure 1.1 Diagram showing the different types of variable.

Categorical (Qualitative) Data

These occur when each individual can only belong to one of a number of distinct categories of the variable.

Nominal data

– the categories are not ordered but simply have names. Examples include blood group (A, B, AB and O) and marital status (married/widowed/single, etc.). In this case, there is no reason to suspect that being married is any better (or worse) than being single!

Ordinal data

– the categories are ordered in some way. Examples include disease staging systems (advanced, moderate, mild, none) and degree of pain (severe, moderate, mild, none).

A categorical variable is binary or dichotomous when there are only two possible categories. Examples include ‘Yes/No’, ‘Dead/Alive’ or ‘Patient has disease/Patient does not have disease’.

Numerical (Quantitative) Data

These occur when the variable takes some numerical value. We can subdivide numerical data into two types.

Discrete data

– occur when the variable can only take certain whole numerical values. These are often counts of numbers of events, such as the number of visits to a GP in a particular year or the number of episodes of illness in an individual over the last five years.

Continuous data

– occur when there is no limitation on the values that the variable can take, e.g. weight or height, other than that which restricts us when we make the measurement.

Distinguishing between Data Types

We often use very different statistical methods depending on whether the data are categorical or numerical. Although the distinction between categorical and numerical data is usually clear, in some situations it may become blurred. For example, when we have a variable with a large number of ordered cate­gories (e.g. a pain scale with seven categories), it may be difficult to distinguish it from a discrete numerical variable. The dis­tinction between discrete and continuous numerical data may be even less clear, although in general this will have little impact on the results of most analyses. Age is an example of a variable that is often treated as discrete even though it is truly continuous. We usually refer to ‘age at last birthday’ rather than ‘age’, and therefore, a woman who reports being 30 may have just had her 30th birthday, or may be just about to have her 31st birthday.

Do not be tempted to record numerical data as categorical at the outset (e.g. by recording only the range within which each patient’s age falls rather than his/her actual age) as important information is often lost. It is simple to convert numerical data to categorical data once they have been collected.

Derived Data

We may encounter a number of other types of data in the medical field. These include:

Percentages

– These may arise when considering improvements in patients following treatment, e.g. a patient’s lung function (forced expiratory volume in 1 second, FEV1) may increase by 24% following treatment with a new drug. In this case, it is the level of improvement, rather than the absolute value, which is of interest.

Ratios

or

quotients

– Occasionally you may encounter the ratio or quotient of two variables. For example, body mass index (BMI), calculated as an individual’s weight (kg) divided by her/his height squared (m

2

), is often used to assess whether s/he is over- or underweight.

Rates

– Disease rates, in which the number of disease events occurring among individuals in a study is divided by the total number of years of follow-up of all individuals in that study (Chapter 31), are common in epidemiological studies (Chapter 12).

Scores

– We sometimes use an arbitrary value, such as a score, when we cannot measure a quantity. For example, a series of responses to questions on quality of life may be summed to give some overall quality of life score on each individual.

All these variables can be treated as numerical variables for most analyses. Where the variable is derived using more than one value (e.g. the numerator and denominator of a percentage), it is important to record all of the values used. For example, a 10% improvement in a marker following treatment may have different clinical relevance depending on the level of the marker before treatment.

Censored Data

We may come across censored data in situations illustrated by the following examples.

If we measure laboratory values using a tool that can only detect levels above a certain cut-off value, then any values below this cut-off will not be detected, i.e. they are censored. For example, when measuring virus levels, those below the limit of detectability will often be reported as ‘undetectable’ or ‘unquantifiable’ even though there may be some virus in the sample. In this situation, if the lower cut-off of a tool is

x

, say, the results may be reported as ‘<

x

’. Similarly, some tools may only be able to reliably quantify levels below a certain cut-off value, say

y

; any measurements above that value will also be censored and the test result may be reported as ‘>

y

’.

We may encounter censored data when following patients in a trial in which, for example, some patients withdraw from the trial before the trial has ended. This type of data is discussed in more detail in Chapter 44.

2

Data Entry

When you carry out any study you will almost always need to enter the data into a computer package. Computers are invaluable for improving the accuracy and speed of data collection and analysis, making it easy to check for errors, produce graphical summaries of the data and generate new variables. It is worth spending some time planning data entry – this may save con­siderable effort at later stages.

Formats for Data Entry

There are a number of ways in which data can be entered and stored on a computer. Most statistical packages allow you to enter data directly. However, the limitation of this approach is that often you cannot move the data to another package. A simple alternative is to store the data in either a spreadsheet or database package. Unfortunately, their statistical procedures are often limited, and it will usually be necessary to output the data into a specialist statistical package to carry out analyses.

A more flexible approach is to have your data available as an ASCII or text file. Once in an ASCII format, the data can be read by most packages. ASCII format simply consists of rows of text that you can view on a computer screen. Usually, each variable in the file is separated from the next by some delimiter, often a space or a comma. This is known as free format.

The simplest way of entering data in ASCII format is to type the data directly in this format using either a word processing or editing package. Alternatively, data stored in spreadsheet packages can be saved in ASCII format. Using either approach, it is customary for each row of data to correspond to a different individual in the study, and each column to correspond to a different variable, although it may be necessary to go on to subsequent rows if data from a large number of variables are collected on each individual.

Planning Data Entry

When collecting data in a study you will often need to use a form or questionnaire for recording the data. If these forms are designed carefully, they can reduce the amount of work that has to be done when entering the data. Generally, these forms/questionnaires include a series of boxes in which the data are recorded – it is usual to have a separate box for each possible digit of the response.

Categorical Data

Some statistical packages have problems dealing with non-numerical data. Therefore, you may need to assign numerical codes to categorical data before entering the data into the computer. For example, you may choose to assign the codes of 1, 2, 3 and 4 to categories of ‘no pain’, ‘mild pain’, ‘moderate pain’ and ‘severe pain’, respectively. These codes can be added to the forms when collecting the data. For binary data, e.g. yes/no answers, it is often convenient to assign the codes 1 (e.g. for ‘yes’) and 0 (for ‘no’).

Single-coded

variables – there is only one possible answer to a question, e.g. ‘is the patient dead?’. It is not possible to answer both ‘yes’ and ‘no’ to this question.

Multi-coded

variables – more than one answer is possible for each respondent. For example, ‘what symptoms has this patient experienced?’. In this case, an individual may have experienced any of a number of symptoms. There are two ways to deal with this type of data depending upon which of the two following situations applies.

There are only a few possible symptoms, and individuals may have experienced many of them.

A number of different binary variables can be created which correspond to whether the patient has answered yes or no to the presence of each possible symptom. For example, ‘did the patient have a cough?’, ‘did the patient have a sore throat?’

There are a very large number of possible symptoms but each patient is expected to suffer from only a few of them.

A number of different nominal variables can be created; each successive variable allows you to name a symptom suffered by the patient. For example, ‘what was the first symptom the patient suffered?’, ‘what was the second symptom?’. You will need to decide in advance the maximum number of symptoms you think a patient is likely to have suffered.

Numerical Data

Numerical data should be entered with the same precision as they are measured, and the unit of measurement should be consistent for all observations on a variable. For example, weight should be recorded in kilograms or in pounds, but not both interchangeably.

Multiple Forms Per Patient

Sometimes, information is collected on the same patient on more than one occasion. It is important that there is some unique identifier (e.g. a serial number) relating to the individual that will enable you to link all of the data from an individual in the study.

Problems with Dates and Times

Dates and times should be entered in a consistent manner, e.g. either as day/month/year or month/day/year, but not interchangeably. It is important to find out what format the statistical package can read.

Coding Missing Values

You should consider what you will do with missing values before you enter the data. In most cases you will need to use some symbol to represent a missing value. Statistical packages deal with missing values in different ways. Some use special char­acters (e.g. a full stop or asterisk) to indicate missing values, whereas others require you to define your own code for a missing value (commonly used values are 9, 999 or −99). The value that is chosen should be one that is not possible for that variable. For example, when entering a categorical variable with four categories (coded 1, 2, 3 and 4), you may choose the value 9 to represent missing values. However, if the variable is ‘age of child’ then a different code should be chosen. Missing data are discussed in more detail in Chapter 3.

Example
As part of a study on the effect of inherited bleeding disorders on pregnancy and childbirth, data were collected on a sample of 64 women registered at a single haemophilia centre in London. The women were asked questions relating to their bleeding disorder and their first pregnancy (or their current pregnancy if they were pregnant for the first time on the date of interview). Fig. 2.1 shows the data from a small selection of the women after the data have been entered onto a spreadsheet, but before they have been checked for errors. The coding schemes for the categorical variables are shown at the bottom of Fig. 2.1. Each row of the spreadsheet represents a separate individual in the study; each column represents a different variable. Where the woman is still pregnant, the age of the woman at the time of birth has been calculated from the estimated date of the baby’s delivery. Data relating to the live births are shown in Chapter 37.

Figure 2.1 Portion of a spreadsheet showing data collected on a sample of 64 women with inherited bleeding disorders.

Data kindly provided by Dr R.A. Kadir, University Department of Obstetrics and Gynaecology, and Professor C.A. Lee, Haemophilia Centre and Haemostasis Unit, Royal Free Hospital, London.