Statistics in Medicine

Description

Statistics in Medicine, Fourth Edition, helps medical and biomedical investigators design and answer questions about analyzing and interpreting data and predicting the sample size required to achieve useful results. It makes medical statistics easy for the non-biostatistician by outlining common methods used in 90% of medical research. The text covers how to plan studies from conception to publication, what to do with data, and follows with step-by-step instructions for biostatistical methods from the simplest levels, to more sophisticated methods now used in medical articles. Examples from almost every medical specialty, and from dentistry, nursing, pharmacy and health care management are provided. This book does not require background knowledge of statistics or mathematics beyond high school algebra and provides abundant clinical examples and exercises to reinforce concepts. It is a valuable source for biomedical researchers, healthcare providers and anyone who conducts research or quality improvement projects. Cover Statistics in Medicine Copyright Dedication HTU1 Contents Foreword to the fourth edition Abstracts of Forewords to Prior Editions Acknowledgments How to use this book? 1 Goals Purpose of the book The most used statistical methods and concepts are included Using a few data sets repeatedly allows the user to focus on the method 2 Use as a Textbook Versus as a Reference 3 Possible Schedules for Courses Short course A three-credit semester course A year course (six-semester credits) Reference 1 Planning studies: from design to publication 1.1 Organizing a Study 1.2 Stages of Scientific Investigation Stages The causative process is of interest, not the data 1.3 Science Underlying Clinical Decision-Making The scientific method Jargon in science Evidence Evidence versus proof Evidence-based medicine 1.4 Why Do We Need Statistics? Primary objective Population versus sample Objective restated What statistics will not do for us What statistics will do for us 1.5 Concepts in Study Design Components of a study Control groups and placebos Variables Moving from sample to population Representativeness and bias Experimental design can reduce bias 1.6 Study Types Registry Cohort study Case–control study Case–control contrasted with cohort studies Randomized controlled trial Paired and crossover designs 1.7 Convergence With Sample Size 1.8 Sampling Schemes in Observational Studies Purpose of sampling schemes Simple random sampling Systematic sampling Caution Stratified sampling Cluster sampling Nonuniform weighted sampling 1.9 Sampling Bias A pictorial example of bias Increasing representativeness by random samples Sources of bias 1.10 Randomizing a Sample Haphazard versus random assignment Generating random numbers 1.11 How to Plan and Conduct a Study Professional guides 1.12 Mechanisms to Improve Your Study Plan Tricks of the trade Work backward through the logical process Analyze dummy data Play the role of Devil’s advocate 1.13 Reading Medical Articles Two primary goals Ways to improve efficiency in reading medical articles 1.14 Where Articles May Fall Short Confusing statistical versus clinical significance Violating assumptions underlying statistical methods Generalizing from poorly behaved data Failure to define data formats, symbols, or statistical terms Using multiple related tests that inflate the probability of false results Failure to clearly distinguish a priori analyses from exploratory data-driven analyses Choosing inappropriate statistical tests 1.15 Writing Medical Articles Format Level of composition Selecting content 1.16 Statistical Ethics in Medical Studies Ethics in the conduct of medical studies is a broad topic Patient protection requirements Patient identifiers in data sheets Statistical control parameters and sample sizes are at issue Examples of ethical considerations in specifying sample size test parameters Relationship among the statistical parameters Implications of α and β Effect on patients from the xerostomia study Effect on patients from the breast cancer study The choice of α and β should involve clinical implications Effect of test sidedness on the patient Choosing sidedness Selection of the clinical difference δ Effect of the clinical difference δ on the patient 1.17 Conclusion Appendix to Chapter 1 Glossary of statistical terms used in Chapters 1 and 2 2 Planning analysis: how to reach my scientific objective 2.1 What Is in This Chapter 2.2 Notation (or Symbols) Purpose of symbols Categories of symbols Formulas Becoming familiar with symbols A first formula Indicator symbols Symbols for ranks 2.3 Quantification and Accuracy Statistics and quantification Quantifying data Accuracy versus precision How much precision? 2.4 Data Types Types of data Distinguishing between types Rounding Ratings String (or alphabetic) data 2.5 Multivariable Concepts and Types of Adjustment Variables Univariate versus multivariate measures Multiple, multivariate, and multivariable Types of adjustment variables in multiple regression analyses 2.6 How to Manage Data Make a plan Raw data Data in format for analysis Data quality Missing data Keeping records Software for data management Software for statistical analysis Choosing software Specialty software 2.7 Defining the Scientific Goal: Description, Association Testing, Prediction 2.8 Reporting Statistical Results 2.9 A First-Step Guide to Descriptive Statistics Statistical terms Accessing computer help Univariate descriptive statistics Bivariate and trivariate descriptive statistics for confounding and effect modification 2.10 An Overview of Association Testing Setting up a test within a study A priori specification of a hypothesis Choosing the right test What do you want your data to tell you? First-step guides What to expect from these guides A first-step guide to tests of rates or averages The table format Information required to choose a test An example from prostate cancer Limitations of table Large sample tests 2.11 A Brief Discussion of Prediction Modeling A structured approach to forming a prediction tool 3 Probability and relative frequency 3.1 Probability Concepts Probability defined Probability can vary from 0 to 1 The additive rule of probability The multiplicative rule of probability Bayes’ rule Combinations 3.2 Probability and Relative Frequency A graph of probability Relative frequencies estimate probabilities Relative frequency in medicine The accuracy of estimation increases as the sample size grows larger 3.3 Graphing Relative Frequency 3.4 Continuous Random Variables 3.5 Frequency Distributions for Continuous Variables Start with a tally Frequencies expressed as a histogram Relative frequencies in continuous distributions Effect of increasing sample size Choosing intervals 3.6 Probability Estimates From Continuous Distributions The relationship Estimated probability and the term “parameter” 3.7 Probability as Area Under the Curve Concept The graphical relationship of relative frequency and probability References 4 Distributions 4.1 Characteristics of a Distribution 4.2 Greek Versus Roman Letters 4.3 What Is Typical Averages Convergence with increasing sample size 4.4 The Spread About the Typical Types of spread indicators Example: Prostate volumes from Table DB1.1 4.5 The Shape Most common shapes Example: Prostate volumes from DB1 Standardizing a distribution 4.6 Sampling Distribution of a Variable Versus a Statistic 4.7 Statistical Inference Desirable properties of a sample statistic Inference via a confidence interval Inference about a difference Inference about equivalence Steps in inference Effect of violated assumptions; robustness 4.8 Distributions Commonly Used in Statistics Normal distribution Shorthand for the normal t Distribution The standard t Degrees of freedom Chi-square (χ2) distribution Chi-square in a hypothesis test: critical value F Distribution Rank-order methods Binomial distribution Large-sample approximations to the binomial Poisson distribution Joint distributions of two variables Example: Effect of immunoglobulin on ITP Joint frequency distribution Relationship between two variables 4.9 Approximate Distribution of the Mean (Central Limit Theorem) Sample means of continuous data are distributed normal Measure of variability in the sample mean Population standard error of the mean Sample standard error of the mean 4.10 Approximate Distribution of a Sample Quantile 5 Descriptive statistics 5.1 Purpose of Descriptive Statistics 5.2 Numerical Descriptors, One Variable Section format Quantiles Rate Mean Example: Protein-specific antigen from Table DB1.1 Median Example: Protein-specific antigen from Table DB1.1 Mode Example: Protein-specific antigen from Table DB1.1 Range Variance and standard deviation Example: Protein-specific antigen from Table DB1.1 Standard error of the mean Example: Protein-specific antigen from Table DB1.1 Standard error of the mean for two independent samples Example: Protein-specific antigen from Table DB1.1 Interquartile range Example: Effect of exercise on eNO 5.3 Numerical Descriptors, Two Variables Covariance Example: Prostate volume with age Interpretation Correlation coefficient for continuous variables Interpretation Example: Prostate volume with age Caution Correlation coefficient for rank-order variables Example: Performance measure on an operated leg Correlation coefficient for categorical variables: tetrachoric correlation Example posed: Effect of drug on postoperative nausea A 2×2 contingency table Example continued: Effect of drug on postoperative nausea 5.4 Numerical Descriptors, Three Variables 5.5 Graphical Descriptors, One Variable Common types Making a histogram Histograms with differing interval widths Example: Prostate volumes Pie chart Example: Accuracy of digital rectal examination Line chart Example: Prostate abnormalities by age Relation of a line chart to a probability distribution Mean-and-standard-error chart Showing the means Showing the variability about the means Example: Distribution of prostate volumes Effect of irregular data Box-and-whisker chart Effect of no assumptions about the distribution Example: Distribution of protein-specific antigen levels Showing sample size 5.6 Graphical Descriptors, Two Variables Depicting the relationship between variables Scatterplot Example: Prostate volume by age 5.7 Graphical Descriptors, Three Variables Two-dimensional frequency distribution Challenges in three-dimensional imaging Example of low correlation: Prostate volume by age Example of high correlation: Protein-specific antigen by prostate-specific antigen density Statistical graphs in exploring data 5.8 Principles of Informative Descriptive Tables and Figures Good table formatting practices Good graphing practices References 6 Finding probabilities 6.1 Probability and Area Under the Curve 6.2 The Normal Distribution The standard normal Table of the standard normal Probability of certain ranges occurring Using the standard normal Critical value Example: Proportion of carinal resection patients under 30 years of age 6.3 The t Distribution Why we need t The nature of t The t pictured Table of t probabilities Example of a tolerance interval using t Critical value for t Example: Proportion of leg surgery patients (DB10) who can perform 6.4 The Chi-Square Distribution Why we need chi-square Chi-square pictured Tables of chi-square Example: Variability of prostate volumes Example: Variability in platelet count An effect of asymmetry in the chi-square distribution 6.5 The F-Distribution The concept of F F and standard deviations F and df Using the F table Example: Variability of prostate volumes for positive versus negative biopsies F and sample size 6.6 The Binomial Distribution Binomial events defined Binomial table Example: Has the success of laser trabeculoplasty improved? The effect of sample size The binomial for larger samples 6.7 The Poisson Distribution Poisson events described Poisson table Example: Rate of Alzheimer’s disease in Aluminum City versus Seattle The appearance of the Poisson distribution The Poisson for larger samples References 7 Hypothesis testing: concept and practice 7.1 Hypotheses in Inference A clinical hypothesis Decision theory introduced Operations research Decision-making by testing a statistical hypothesis How we get from observed data to a test The null hypothesis The alternative hypothesis Forms the hypotheses can take Why the null hypothesis is null A numerical example: Hypotheses about prostate volumes The critical value The most common types of statistics being tested and their associated probability distributions Confidence intervals are closely related Assumptions in hypothesis testing The meaning of “error” The assumption of independence of errors The assumption of normality of errors The assumption of equality of standard deviations 7.2 Error Probabilities Method presented through an example: Mean prostate-specific antigen between cancerous and healthy patients Type I (α) error Type II (β) error; power of a test p-Value Relation among truth, decision, and errors The critical value and the errors illustrated Choosing α but not β is a common reality Providing a confidence interval Accepting H0 versus failing to reject H0 Testing for a difference versus equivalence 7.3 Two Policies of Testing A bit of history Calculation by computer provides a new option Contrasting two approaches Other factors must be considered 7.4 Distinguishing Between Statistical and Clinical Significance 7.5 Controversies Regarding the Rigid Use and Abuse of p-Values 7.6 Avoiding Multiplicity Bias 7.7 Organizing Data for Inference The first step: Identify the type of data Categorical data Entering the data Rank data Converting continuous data to rank data Continuous measurement data What appears to be a data type may not act that way in analysis 7.8 Evolving a Way to Answer Your Data Question Fundamentally, a study is asking a question Description and prediction Testing Additional example: Duration of the common cold Reference 8 Tolerance, prediction, and confidence intervals 8.1 Overview The basis of tolerance, prediction, and confidence intervals Confusion between tolerance, prediction, and confidence intervals Error rate Where do these intervals come from? Other uses for probabilities 8.2 Tolerance Intervals for Patient Measurements Accounting for uncertainty in distributional parameters Example: A tolerance interval for patients’ extent of tracheal carina resection 8.3 Concept of a Confidence Interval for a Parameter Confidence interval for a parameter defined With what statistics can we use confidence intervals? Caution: Confidence intervals and asymmetry 8.4 Confidence Interval for a Population Mean, Known Standard Deviation Example posed: Upper confidence limit for mean tracheal resection Example completed: Upper confidence limit on mean tracheal resection Additional Example 1: Confidence interval for mean age of tracheal resection Additional Example 2: Confidence interval for mean treatment time for arm fractures 8.5 Confidence Interval for a Population Mean, Estimated Standard Deviation Example posed: Confidence interval for mean prostate volume Example completed: Confidence interval for mean prostate volume Additional Example 1: Confidence interval for mean ratings of postpartum stretch marks Additional Example 2: Confidence interval for mean theophylline levels 8.6 Confidence Interval for a Population Proportion Proportions fall into two types Example posed, common event: Confidence interval for proportion of positive biopsies Example posed, rare event: Confidence interval for proportion of children with high lead levels Example completed, common event: Confidence interval for proportion of positive biopsies Example completed, rare event: Confidence interval for proportion of children with high lead levels Additional example: Patient satisfaction with anesthesia Rule of three: If number of rare events is 0 Performance of Wald-based confidence intervals Accounting for the dependence of the variance on the mean Example revisited: Patient satisfaction with anesthesia 8.7 Confidence Interval for a Population Median Example posed: Confidence interval on median viral load in HIV patients Example completed: Confidence interval for median viral load in HIV patients Additional example: Confidence interval for median LOS for a group A strep outbreak 8.8 Confidence Interval for a Population Variance or Standard Deviation Example posed: Confidence interval for standard deviation of prostate volumes Example completed: Confidence interval for standard deviation of prostate volumes Additional Example 1: Confidence interval for precision of thermometer Additional Example 2: Confidence interval for variability of plasma silicon level 8.9 Confidence Interval for a Population Correlation Coefficient Example posed: Confidence interval for correlation between age and plantar flexion Example completed: Confidence interval for correlation between age and plantar flexion Additional example: Confidence interval for correlation between Hct and erythropoietin References 9 Tests on categorical data 9.1 Categorical Data Basics Terms and symbols Example: Rate of thrombosis recurrence Organizing data for categorical methods Contingency table symbols Choosing categorical methods Categories versus ranks 9.2 Tests on Categorical Data: 2×2 Tables What does a test tell us? Example: Is prediction (diagnosis) of prostate cancer by digital rectal examination better than chance? A test of independence 9.3 The Chi-Square Test of Contingency A more exact statement of the question being asked about digital rectal examinations The basis of the chi-square test of contingency Example posed: Is digital rectal examination independent of biopsy outcome? Calculation for the chi-square test of contingency Yates’ correction Example continued: Digital rectal examination and biopsy outcome The effect of sample size on contingency tests What to do if we have percentages but not counts 9.4 Fisher’s Exact Test of Contingency Another test of independence between two categorical variables Example continued: Digital rectal examination and biopsy outcomes Additional example: Improvement with surgical experience 9.5 Tests on r×c Contingency Tables Example posed: Is the use of smokeless tobacco related to ethnic origin? Method: Fisher’s exact test for r×c contingency tables Large tables with violated chi-square assumptions Example continued: Smokeless tobacco and ethnic origin Additional example: Choice of therapist for psychiatry residents 9.6 Tests of Proportion What a test of proportion will do and why it is needed When a proportion is very small Basis of the test A different method is used if n is large Test of two proportions Test of three or more proportions An adjustment factor to the approximation Examples of tests of probabilities giving rise to a sample against theoretical proportions, small n Example: Rate of positive biopsy results Example: Does radiation therapy improve the proportion survival? Examples of tests of probabilities giving rise to a sample against theoretical proportions, large n Example: The issue of physician cleanliness 150 years ago Example: Absorbable mesh in abdominal wall closures Example of test of two population proportions International normalized ratio range in a Coumadin Clinic 9.7 Tests of Rare Events (Proportions Close to Zero) What a test of a small proportion will do and why it is needed What to do if π is very close to 1 rather than 0 What to do if no events are observed A different method is used for large samples Example posed, small λ: Are large prostates too frequent? Example posed, large λ: Does a new drug cause birth defects? Example completed, small λ: are large prostates too frequent? Additional example, small λ: site of injection and incidence of sarcoma Example completed, large λ: does a new drug cause birth defects? Additional example, large λ: anorexia in the military versus the public 9.8 McNemar’s Test: Matched Pair Test of a 2×2 Table Example posed: Out-of-range readings for clinic versus laboratory Additional Example 1 posed: Is smoking associated with lung cancer? Example completed: Out-of-range readings for clinic versus laboratory Additional Example 1 completed: Is smoking associated with lung cancer? Additional Example 2: Is there a genetic predisposition to stomach cancer? 9.9 Cochran’s Q: Matched Pair Test of a 2×r Table Example posed: Consistency of cancer rating severity Example completed: Consistency of cancer rating severity 9.10 Three or More Ranked Samples With Two Outcome Categories: Royston’s Ptrend Test Example posed: Mortality and extent of carinal resection Example completed: Mortality and extent of carinal resection References 10 Risks, odds, and receiver operating characteristic curves 10.1 Association Measures for Categorical Data: Risks and Odds Method and example combined: Accuracy and errors in the diagnosis of appendicitis Truth table format Example data: Exposure to dust and occurrence of coccidioidomycosis Probability and odds compared Truth table statistics False-positive rate and false-negative rate Sensitivity Specificity Accuracy Positive and negative predictive values Relative risk Odds ratio The relationship between relative risk and odds ratio Likelihood ratio Negative likelihood ratio Be wary of the term likelihood ratio Attributable risk The relationship between relative risk and attributable risk Additional example: Truth tables as related to cancer survival rates Pancreatic cancer data False-positive and false-negative rates Sensitivity, specificity, and accuracy Positive and negative predictive values Relative risk Odds ratio Likelihood ratio Attributable risk 10.2 Inference for the Risk Ratio: The Log Risk Ratio Test Introduction The log in the log relative risk test Example posed: Significance of relative risk in predicting a biopsy result Example completed: Significance of relative risk in predicting a biopsy result 10.3 Inference for the Odds Ratio: The Log Odds Ratio Test Introduction Example posed: Significance of odds ratio in predicting a biopsy result Example completed: Significance of odds ratio in predicting a biopsy result Additional example: Radial keratotomy experience and visual acuity Considerations when choosing between the relative risk and the odds ratio Clinical relevance of the dependence between categories in a 2×2 table Confidence interval on the odds ratio Example: Confidence intervals on odds ratio to predict biopsy result from a digital rectal examination 10.4 Receiver Operating Characteristic Curves Choosing the best critical value Choosing the best weighted critical value 10.5 Comparing Two Receiver Operating Characteristic Curves Using the receiver operating characteristic curve to choose the better of two indicators (risk factors) References 11 Tests of location with continuous outcomes 11.1 Basics of Location Testing The organization of this chapter The basic question being asked The means question being asked The assumption of normality and the concept of robustness Other assumptions: Independently sampled data and equal variability The effect of violated assumptions We must specify the null and alternative hypotheses The alternative hypothesis is used to choose a one- or two-tailed test 11.2 Single or Paired Means: One-Sample Normal (z) and t Tests Single samples and paired samples are treated the same Examples posed, normal test: Early versus later members of our prostate-specific antigen sample Examples posed, t test: Does asthma training reduce acute care visits? Example completed, normal test: Average early versus average later prostate-specific antigen Example completed, t Test: Does asthma training reduce acute care visits? Additional example: Is a new dyspepsia treatment effective in the emergency department? Normal (z) test t test 11.3 Two Means: Two-Sample Normal (z) and t Tests The normal test and the t test are two forms of the two-sample means test Assumptions required The effect of differing variances and/or sample sizes Examples posed: Are age or prostate volume risk factors for prostate cancer? Example, normal (z) test, equal variances: Age and prostate cancer Could the t test have been used instead of the z test? Following the steps for the t test in the example of age and prostate cancer Example, normal (z) test, unequal sample sizes and variances: Is prostate volume a risk factor for prostate cancer (CaP)? Example posed for t test, unequal sample sizes and variances: Is prostate volume a risk factor for prostate cancer (CaP)? Suppose we had used the rank-sum test to assess prostate volume Additional examples, z and t tests, equal variances: Comparing the effectiveness of two treatments Additional example, t test, unequal variances: Pain relief from two drugs 11.4 Three or More Means: One-Factor Analysis of Variance Example posed: Prostate cancer’s risks related to age Concept of one-way analysis of variance One-way analysis of variance is like the t test generalized to three or more means Assumptions required for analysis of variance New terms in analysis of variance: Mean square and sum of squares Identifying the mean difference(s) that caused the significance: Multiple comparisons tests Example completed: Age as related to CaP risks Descriptive statistics and assumptions Analysis of variance calculations Which among the possible mean differences account(s) for the significance? Suppose we do not have a statistical software package Additional example: Does steroid decrease edema following rhinoplasty? If so, what level of steroid should be used? Identifying the steroid dosage 11.5 Three or More Means in Rank Order: Analysis of Variance Trend Test Is there a pattern of change through the means? The concept of testing for a linear pattern Patterns other than linear Implementing the method Example: Does human immunodeficiency virus patients’ CD4 count trend downward with years of disease? 11.6 The Basics of Nonparametric Tests What are ranks? Ranking categorized data that fall into a natural order How do we use ranks? When do we use ranks? Why do we use ranks? Ties in ranked data 11.7 Single or Paired Sample Distribution(s): The Signed-Rank Test Example 1 posed, single sample: Prostate-specific antigens versus a population median Example 2 posed, paired sample: Does a drug change heart rate? One tail or two? Example 1 completed: Prostate-specific antigens versus a population median Example 2 completed: Does a drug change heart rate? Additional example: Hardware to repair ankle functionality 11.8 Two Independent Sample Distributions: The Rank-Sum Test What is being tested Example: Is prostate-specific antigen different for positive versus negative biopsy? Why U is calculated from T A one-tailed test Other names for this test Additional example: Hematocrit for laparoscopic versus open pyloromyotomies 11.9 Large Sample-Ranked Outcomes When do we need a large-sample approximation? Single or paired data Example posed: Bias in early sampling of prostate biopsy patients How large is a “large sample”? Example completed: First 20 prostate-specific antigens versus remainder Two large sample-independent ranked outcomes Example posed: Prostate-specific antigen for patients with versus without BPH Example completed: Prostate-specific antigen for patients with versus without BPH 11.10 Three or More Independent Sample Distributions: The Kruskal–Wallis Test Example posed: Is prostate-specific antigen the same among BPH, CaP, and NED patients? Example completed: Compare prostate-specific antigen for three disease groups Additional example: Performance of five different surgical instruments 11.11 Three or More Matched Sample Distributions: The Friedman Test Example posed: Two skin tests for allergen sensitivity Example completed: Two skin tests for allergen sensitivity Additional example: Level of gentamicin treatment over time 11.12 Three or More Ranked Independent Samples With Ranked Outcomes: Cusick’s Nptrend Test Example posed: Location of pediatric snake bite by age Example completed: Location of pediatric snake bite by age Comparison with a contingency test 11.13 Three or More Ranked Matched Samples With Ranked Outcomes: Page’s L Test Example posed: Treating migraine-caused nausea Example completed: Migraine-caused nausea Page’s L compared to some tests with similar appearing formats 11.14 Potential Drawbacks to Using Nonparametric Tests Nontransitivity of the rank-sum test Example: Nontransitivity of the rank-sum test Choosing the test to use Reference 12 Equivalence testing 12.1 Concepts and Terms Concept Equivalence tests are similarity tests Equivalence versus noninferiority (or nonsuperiority) Equivalence of means, proportions, and other parameters 12.2 Basics Underlying Equivalence Testing The null hypothesis is the key Difference symbols What the null hypothesis statement really means Assumptions One-sided versus two-sided tests Using a confidence interval instead of a test statistic 12.3 Choosing a Noninferiority or Equivalence Margin 12.4 Methods for Noninferiority Testing Example posed: Effect of silicone implants on plasma silicon levels Example completed: Silicone implants and plasma silicon levels Additional example: Exhaled nitric oxide from exercise-induced bronchospasm 12.5 Methods for Equivalence Testing Example posed: Cardiac index by bioimpedance versus thermodilution Example completed: Cardiac index by bioimpedance versus thermodilution Additional example: Effect of prior surgery on rate of intubation 12.6 Joint Difference and Equivalence Testing Example posed Terminology History Example completed References 13 Tests on variability and distributions 13.1 Basics of Tests on Variability Why should we be interested in testing variability? A test of variability serves two main purposes How are two variances compared? The true variance of a population may be compared with a postulated value or another population's variance A nonsignificant test result must be interpreted carefully 13.2 Testing Variability on a Single Sample Example posed: Prostate-specific antigen variability Example completed: Prostate-specific antigen variability Additional example: Is a treatment for dyspepsia in the emergency department too variable? 13.3 Testing Variability Between Two Samples Example posed: Does the initial unrepresentative variability in prostate-specific antigen extend to later patients? Example completed: Does the initial unrepresentative variability in prostate-specific antigen extend to later patients? Calculations Interpretation Additional example: Compare the variability of two pain-relief drugs 13.4 Testing Variability Among Three or More Samples Example posed: Can we assume equal variances in the test of classifying patients by risk of prostate cancer? Example completed: Can we assume equal variances in the test of classifying patients by risk of prostate cancer? Calculation Additional example: Medication to reduce edema following rhinoplasty 13.5 Basics on Tests of Distributions What do we usually ask about distributions? Testing normality Testing other distributions Testing equality 13.6 Test of Normality of a Distribution Types of normality tests and their comparison The role of computer software Using the Shapiro–Wilk test Using the Kolmogorov–Smirnov test (one-sample form) Example posed: Are the ages of the patients in Table DB1.1 normal? Example completed: Are the ages of the 10 patients distributed normal? Additional example: A test of normality on a potential human immunovirus vaccine Stepping through the test procedure Using the chi-square goodness-of-fit test (large sample test of normality) Characteristics of the chi-square goodness-of-fit test Example posed: Is the distribution of ages of the 301 prostate patients normal? Example completed: Is the distribution of ages normal? Additional example: Normality of human immunovirus data 13.7 Test of Equality of Two Distributions The two-sample Kolmogorov–Smirnov test Example posed: Are two prostate-specific antigen samples from DB1 distributed the same? The two-sample Kolmogorov–Smirnov on large samples Example completed: Are two prostate-specific antigen samples from DB1 distributed the same? Additional example: A potential vaccine for human immunovirus References 14 Measuring association and agreement 14.1 What Are Association and Agreement? Association Agreement Using association measures to assess agreement Reliability as agreement This chapter addresses descriptors of association and agreement 14.2 Contingency as Association Contingency tables 2×2 Tables Example: Association between suspicious dust and coccidioidomycosis 2×c Tables Example: Use of smokeless tobacco by ethnic group Larger (r×c) tables with nominal categories Example: Choosing a therapist for psychiatry residents Larger (r×c) tables with rankable categories Example: Clinic versus laboratory values of International Normalized Ratio in diabetic patients 14.3 Correlation as Association Correlation: Continuous or rank-order data Example: Hop distances between injured and healthy legs 14.4 Contingency as Agreement Agreement among contingency categories Example: Cancer severity rated by three radiologists 14.5 Correlation as Agreement Example: Estimation of patient weight in the emergency department 14.6 Agreement Among Ratings: Kappa Example: Agreement among radiologists rating cancer severity 14.7 Agreement Among Multiple Rankers Example posed: Evaluation of a proposed research protocol by an IRB Example concluded for W Riffenburgh’s A Example concluded for A 14.8 Reliability Internal consistency as reliability Example posed: Reliability of four types of thermometer Example completed: Reliability of four types of thermometer Correlation as reliability 14.9 Intraclass Correlation References 15 Linear regression and correlation 15.1 Introduction 15.2 Regression Concepts and Assumptions What is regression? The term regression Confirmatory versus exploratory uses of regression Five classic assumptions underlying linear regression 15.3 Simple Regression Concept of a line fitted to a set of points on a graph Example posed: Day 10 theophylline level predicted from baseline Example completed: Day 10 theophylline level predicted from baseline Additional Example 1: Predicting temperature of lung-congested infants Predicting temperature by age Additional Example 2: Does mental patient hospital stay relate to intelligence quotient? 15.4 Assessing Regression: Tests and Confidence Intervals Tools of assessment R2, the coefficient of determination Standard errors Example posed: Testing the theophylline prediction Methods of assessment Example completed: Testing the theophylline prediction Additional example: Infant lung congestion continued 15.5 Deming Regression History Additional Examples 1 and 2 of Section 21.3 continued 15.6 Types of Regression Classifying types of regression models Examples of the several types Regression models in statistical software Where regression types are found in this book 15.7 Correlation Concepts and Assumptions What is correlation? Assumptions underlying correlation Rank correlation Names for correlation 15.8 Correlation Coefficients Example posed: Theophylline level example continued Methods for correlation coefficients When to use correlation as opposed to regression Example completed: Theophylline level example Additional Example 1: Infant lung congestion continued Additional Example 2: Hospital stay and intelligence quotient for mental patients continued 15.9 Correlation as Related to Regression 15.10 Assessing Correlation: Tests and Confidence Intervals Example posed: Are theophylline level correlations different by day? Methods for testing correlation and finding a confidence interval Example completed: Are theophylline level correlations different by day? Significance tests of population correlation coefficients Test of two correlation coefficients A 95% confidence interval on the population correlation coefficient Additional Example 1: Infant lung congestion continued A significance test of the correlation coefficient Test of two correlation coefficients A 95% confidence interval on the population correlation coefficient Additional Example 2: Intelligence quotient versus hospital stay for mental patients continued A significance test that the correlation coefficient=0 A significance test against a correlation coefficient other than 0 A 95% confidence interval on the population correlation coefficient 15.11 Interpretation of Small-But-Significant Correlations References 16 Multiple linear and curvilinear regression and multifactor analysis of variance 16.1 Introduction Adjusting for multiple variables simultaneously Interpretation of regression coefficients Curved line models Visualizing models in two dimensions Visualizing models in three dimensions A curved surface in three dimensions More than three dimensions The case of multiple dependent variables Analysis of variance and regression 16.2 Multiple Linear Regression Example posed: Predicting length of hospital stay due to psychological problems An admonition Stepwise multiple regression Managing nominal variables Example completed: Predicting length of hospital stay due to psychological problems Choosing the model Data input Model assessment and data-driven model building Additional Example 1: Infant lung congestion Choosing the model Data input Model assessment and data-driven model building Additional Example 2: Can we predict length of hospital stay due to strep infection? Univariate regressions The effect of missing observations on df A full model regression Eliminating variables The prediction model Performing the backward stepwise regression by software A final note on model validation Additional Example 3: Can fasting cholesterol be predicted without fasting? 16.3 Model Diagnosis and Goodness of Fit Underlying assumptions of the linear regression model Normality Constant variance Influential observations Leverage Cook’s distance What to do about influential points? 16.4 Accounting for Heteroscedasticity Transformations Weighted least squares Empirical variance estimation 16.5 Curvilinear Regression Example posed: A better fit to the survival of malarial rats Choosing the model for rat survival Example completed: A better fit to the survival of malarial rats Data input Results output Interpretation An admonition Additional example: Infant lung congestion continued Choosing the model Data input Results output Interpretation 16.6 Two-Factor Analysis of Variance Orientation by example: Heart rate examined by two factors Effects of eating on mean heart rate using a t test Effect of eating on mean heart rate using one-way analysis of variance Effect of time of day on mean heart rate using one-way analysis of variance Examining the effect of eating, of time of day, and of their interaction on mean heart rate at once: two-way analysis of va... Interpretation of the analysis of variance table Two-factor heart rate example completed Interpretation Note the improvement over the t test and the one-way analysis of variance Additional example: Cooling kidneys prior to surgery 16.7 Analysis of Covariance Purpose Orientation by example: Do sex and/or age affect theophylline levels of emphysema patients? Data Means What to do about the continuous variable Additional example: Comfort of bandage lenses following eye surgery 16.8 Three-Way and Higher Way Analysis of Variance Orientation by example: Cooling kidneys extended to all factors Interpretation 16.9 Concepts of Experimental Design Purpose of design Factors influencing outcomes Assumptions and balance in multifactor designs Definition Example: Two types of influence on blood flow after orthopedic casting Balance is assumed in this chapter Missing observations More specialized designs and techniques Factorial designs Sparse-data designs References 17 Logistic regression for binary outcomes 17.1 Introduction 17.2 Extensions of Contingency Table Analyses Simple Logistic Regression Example: Estimating the probability of coronary heart disease in the Framingham study 17.3 Multiple Logistic Regression: Model Specification and Interpretation Example: Predicting the probability of coronary heart disease in the Framingham study 17.4 Inference for Association Parameters Test of association Example: Testing the association between sex and coronary heart disease in the Framingham study Confidence intervals for the odds ratio Example: Confidence interval for the odds ratio corresponding to sex in the Framingham study 17.5 Model Diagnostics and Goodness-of-Fit Assessing functional form of predictors Example: Prediction of nodal involvement in prostate cancer patients Assessing influential observations Example: Prediction of nodal involvement in prostate cancer patients Cook’s distance Assessing goodness-of-fit: The Hosmer–Lemeshow test Example: Estimating the probability of coronary heart disease in the Framingham study References 18 Poisson regression for count outcomes 18.1 Introduction 18.2 The Poisson Distribution 18.3 Means Versus Rates Example: Estimating the rate of seizures for epileptic patients 18.4 Inference for the Rate of a Poisson Random Variable Example: Hypothesis test for the rate of seizures among epileptic patients 18.5 Comparing Poisson Rates From Two Independent Samples Example: Testing the rate of seizures among treated and control epileptic patients 18.6 The Simple Poisson Regression Model Example: Comparison of the rate of word repetitions among cognitively normal and demented patients 18.7 Multiple Poisson Regression: Model Specification and Interpretation 18.8 Obtaining Predicted Rates Example: Predicting the rate of repetitions in neuropsychological testing 18.9 Inference for Association Parameters Test of association Example: Testing the association between cognitive status and the rate of repeats in neuropsychological testing Confidence intervals for the rate ratio Example: Confidence interval for the rate ratio comparing mild cognitive impairment to cognitively normal patients in neuro... Reference 19 Analysis of censored time-to-event data 19.1 Survival Concepts A broad use of the term “survival” Survival versus failure to survive Time-dependent survival Time-series and survival 19.2 Censoring 19.3 Survival Estimation: Life Table Estimates and Kaplan–Meier Curves Life table estimates of survival Data and calculations required for a life table Example posed: What is the survival pattern of diabetics Calculations for the life table for men with diabetes mellitus Kaplan–Meier estimator of the survival curve Confidence intervals on survival estimates Example: Men with diabetes mellitus 19.4 Survival Testing: The Log-Rank Test Testing the difference among survival curves Example posed: Survival of men versus women with diabetes The log-rank test compared to other tests of survival curves Example completed: Survival of men versus women with diabetes Additional example: Testing three cancer survival curves 19.5 Adjusted Comparison of Survival Times: Cox Regression Example posed: Time for acquired immunodeficiency syndrome patients to contract syphilis Method for Cox regression Terminology Example completed: Time until acquired immunodeficiency syndrome patients contract syphilis References 20 Analysis of repeated continuous measurements over time 20.1 Introduction 20.2 Distinguishing Longitudinal Data From Time-Series Data 20.3 Analysis of Longitudinal Data Correlation due to repeated measures: Repeated measures analysis of variance Orientation by example: Heart rate by disease and by time of day Data and means Interpretation““““ An admonition Example completed: Heart rate by disease and by time of day Additional example: Fasciotomy and antivenin to relieve snakebite edema Experiment and data Means table Calculations Analysis of variance table Interpretation Covariance structures Common choices of covariance structures The general linear model Orientation by example: Retest effects in Alzheimer’s disease Summary of available data Writing down the general linear model Example: Interpretation of coefficient related to time in the cognitive testing study Example: Comparing slopes over time across diagnostic populations Fitting the model Tests of association Example continued: Comparing slopes over time across diagnostic populations Example completed: Retest effects in Alzheimer’s disease Assessment of the covariance/correlation structure Model estimates Linear mixed effects models Method for the random intercept model Example: Interpretation of coefficient related to time in the cognitive testing study Example: Comparing slopes over time across diagnostic populations Tests of association Example continued: Comparing slopes over time across diagnostic populations Example completed: Retest effects in Alzheimer’s disease 20.4 Time-Series Detecting patterns The time-series methods introduced here are basic The need for smoothing processes Examples posed: Prostate-specific antigen, prostate volume, and age Relation of prostate volume to prostate-specific antigen Relation of prostate-specific antigen to age Method for moving averages Types of moving average Calculation of moving average Serial correlation Cross-correlation Autocorrelation Examples completed: Prostate-specific antigen, prostate volume, and age Relation of prostate volume to prostate-specific antigen level Relation of prostate-specific antigen to age Cross-correlation of prostate-specific antigen and prostate volume as depending on age Additional example: Purity of dental wash water Autocorrelation: is the contamination periodic? Testing patterns Methods of change-point identification Periodic samples Accumulating samples Smoothed samples Moving samples Example posed: Level of medical technical support Moving F method Moving sample designations The reason for denoting a moving sample by its leading member A two-tailed test Multiple causes of variability are possible Example completed: Level of medical technical support Additional example: Heart rate by type of anesthetic in trauma References 21 Sample size estimation 21.1 Issues in Sample Size Considerations Why are we concerned with sample size? Concept of estimating the minimum required sample size The term power analysis Value of very small samples Effect of increasing the sample size Convergence Clinical relevance and patient care Sequential analysis in relation to sample size estimation Interim testing in relation to sample size estimation 21.2 Is the Sample Size Estimate Adequate? The power analysis’s sample size is just a (rather poor) estimate A “safety factor” is advisable 21.3 The Concept of Power Analysis The logic behind the method A list of inputs needed for a power analysis Choosing test sidedness Choosing test parameters 21.4 Sample Size Methods Each test has its own formula Sample size methods addressed 21.5 Test on One Mean (Normal Distribution) Example posed: Hormonal therapy for benign prostatic hyperplasia Example completed: Hormonal therapy for benign prostatic hyperplasia Additional example: Treatment of emergency dyspepsia 21.6 Test on Two Means (Normal Distribution) Example posed: Range of motion in artificial knee Example completed: Range of motion in artificial knee Additional example: Testing two treatments of dyspepsia 21.7 Tests When Distributions Are Nonnormal or Unknown Example posed: Testing drug effect on intraocular pressure Example completed: Testing drug effect on intraocular pressure 21.8 Test With No Objective Prior Data Example posed: Effectiveness of an herbal remedy in treating colds Example completed: Herbal remedy 21.9 Confidence Intervals on Means Example posed: Extent of carinal resection β is absent from confidence interval sample size estimation Example completed: Extent of carinal resection Additional example: Extent of carinal resection continued 21.10 Test of One Proportion (One Rate) Contingency tables Test of one proportion Example posed: Rate of positive prostate cancer biopsies Example completed: Rate of positive prostate cancer biopsies Additional example: Rate of schistosomiasis 21.11 Test of Two Proportions (Two Rates) Example posed: Rate of personality disorder in criminals Estimates on the borderline between binomial and Poisson Example completed: Rate of personality disorder in criminals Additional example: Sex difference in fever reporting 21.12 Confidence Intervals on Proportions (On Rates) Example posed: Rate of patients satisfied with oral surgery anesthesia Example completed: Satisfaction with oral surgery anesthesia Additional example: Efficacy of a dermatological treatment 21.13 Test on a Correlation Coefficient Example posed: Repaired ankle plantar flexion correlated with age Statistical significance and clinical meaning Example concluded: Plantar flexion and age Additional example: Correlation between surgical difficulty and duration 21.14 Tests on Ranked Data 21.15 Variance Tests, Analysis of Variance, and Regression 21.16 Equivalence Tests 21.17 Number Needed to Treat or Benefit Number needed to treat: Screening for disease Example posed: Screening for lung cancer in Baltimore Example completed: Screening for lung cancer in Baltimore Additional example: Treating oral leukoplakia Cost of number needed to treat Detection compared to other methods of efficacy, such as mortality Number needed to benefit: Assessing the benefit of a new treatment Example posed: Number needed to benefit from switching tattoo ink Example completed: Number needed to benefit from switching tattoo ink References 22 Clinical trials and group sequential testing 22.1 Introduction Why the need for clinical trials? 22.2 Fundamentals of Clinical Trial Design Defining the target population Defining the intervention Defining the outcome Choosing a comparison group Statistical criteria for evidence 22.3 Reducing Bias in Clinical Trials: Blinding and Randomization Blinding Randomization 22.4 Interim Analyses in Clinical Trials: Group Sequential Testing Disadvantages of fixed-sample methods Examples of group sequential stopping rules References 23 Epidemiology 23.1 The Nature of Epidemiology Definitions Epidemiology compared with other branches of medicine Stages of scientific knowledge in epidemiology Epidemiology is an eclectic science 23.2 Some Key Stages in the History of Epidemiology 23.3 Concept of Disease Transmission Spectrum of disease Modes of transmission Herd immunity 23.4 Descriptive Measures Incidence and prevalence Mortality rate Example: Cervical cancer in the Acornhoek Region of the Transvaal4 The odds ratio 23.5 Types of Epidemiologic Studies Basic variables Experimental studies and intervention trials Nonexperimental or observational studies Cohort studies Case–control studies Prevalence or cross-sectional studies Inferring causation 23.6 Retrospective Study Designs: The Case–Control Study Design Measure of association in the case–control study Number and selection of controls 23.7 The Nested Case–Control Study Design 23.8 The Case–Cohort Study Design 23.9 Methods to Analyze Survival and Causal Factors Life tables list survival through time Example: Life table for infant malaria Graphing survival information Serial correlation through time in epidemiology Cross-correlation Autocorrelation 23.10 A historical note References 24 Meta-analyses 24.1 Introduction The concept of meta-analysis Steps to conduct a meta-analysis Criteria for an acceptable meta-analysis 24.2 Publication Bias in Meta-analyses 24.3 Fixed- and Random-Effects Estimates of the Pooled Effect Fixed-effects estimate Random-effects estimate 24.4 Tests for Heterogeneity of Estimated Effects Across Studies 24.5 Reporting the Results of a Meta-analysis 24.6 Further References References 25 Bayesian statistics 25.1 What Is Bayesian Statistics Frequentist statistics Bayesian statistics Revisiting the data and analytic approaches Some comments on frequentist versus Bayesian statistics 25.2 Bayesian Concepts Bayesian terminology Bayesian inference 25.3 Describing and Testing Means Example posed: Baseline glomerular filtration rate on a humanitarian mission Mean estimation methods Example continued: Glomerular filtration rate estimates on a humanitarian mission Shrinkage Mean testing methods Example concluded: Glomerular filtration rate testing on a humanitarian mission Additional example: an emergency department treatment for dyspepsia 25.4 On Parameters Other Than Means 25.5 Describing and Testing a Rate (Proportion) Example posed: Preventing nausea after gall bladder surgery Estimating the rate Example continued: Reducing post–gall bladder surgery nausea 25.6 Conclusion References 26 Questionnaires and surveys 26.1 Introduction 26.2 Surveys Design of survey Methods of collection Data and analysis Bias in surveys Survey response rate A comment on terminology 26.3 Questionnaires Comparison of interviews versus written questionnaires Characteristics of written questionnaires and their questions Open-ended versus closed-ended questions Format of questions Types of answer Instructions to respondent Wording Coding Rating scales Questionnaire bias Some mechanisms to improve questionnaires 27 Techniques to Aid Analysis 27.1 Interpreting Results Example: Question versus Data Mismatch Example: Data versus Method Mismatch Imperfect Data and the Art of Statistics 27.2 Significance in Interpretation Definition of Significance The Purpose of Estimating the Significance Level Indicators of Significance What If Two Tests Give Different Significance Levels? 27.3 Post Hoc Confidence and Power Post-Test Confidence in the One-Sample Mean Power of the Test Interpreting Post Hoc Power Calculating the Post Hoc Power 27.4 Multiple Tests and Significance The Issue Some Approaches to Solving the Problem The False Discovery Rate: An Approach to Large Scale Testing Example: Does Anesthetic Type Affect Recovery Time? 27.5 Bootstrapping, Resampling, and Simulation What Bootstrapping Is How Bootstrapping Works The Fundamental Assumption of the Bootstrap The Jackknife Example: Confidence Interval on the Median Size of a Tumor I Additional Example: Confidence Interval on the Median Size of a Tumor II Simulation 27.6 Bland-Altman Plot: A Diagnostic Tool Example: Comparing Balance With versus Without Hearing Protection Additional Example: Clinic versus Laboratory Differences in INR values 27.7 Cost Effectiveness Example: Cost Effectiveness of Pneumococcal Vaccine in the U.S. Navy References 28 Methods you might meet, but not every day 28.1 Overview 28.2 Analysis of Variance Issues Additional pairwise comparison methods following analysis of variance Equality of variances assumption Nonadditivity Nonlinear models 28.3 Regression Issues Bias in regression variables Analysis of residuals 28.4 Rates and Proportions Issues 28.5 Multivariate Methods Canonical correlation Hotelling’s T2 Discriminant analysis Mahalanobis’ distance Multivariate analysis of variance and covariance Principal components analysis Factor analysis Cluster analysis Multivariate data displays 28.6 Markov Chains: Following Multiple States through Time Markov chain modeling A shipboard flu epidemic Where This method can be found Further examples 28.7 Markov Chain Monte Carlo: Evolving Models Some history Markov chain Monte Carlo as Bayesian statistics Processing Markov chain Monte Carlo Examples of Markov chain Monte Carlo’s evolving models 28.8 Markov Chain Monte Carlo: Stationary Models Examples from ophthalmology 28.9 Further Nonparametric Tests 28.10 Imputation of Missing Data 28.11 Frailty Models in Survival Analysis 28.12 Bonferroni “Correction” 28.13 Logit and Probit 28.14 Adjusting for Outliers Sample (or mean) trimming Winsorized sample 28.15 Curve Fitting to Data Theoretical fits Spline fit 28.16 Sequential Analysis An example Advantages of sequential analysis Disadvantages of sequential analysis 28.17 Another Test of Normality 28.18 Data Mining 28.19 Data Science and The Relationship Among Statistics, Machine Learning, and Artificial Intelligence References Appendix 1 Answers to exercises: Final Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Appendix 2 Databases Purpose for Repeated Use of These Data Sets Additional Databases DB1 Indicators Of Prostate Biopsy Results18 Background Data DB2 Effectiveness Of A Drug In Reducing Nausea Following Gallbladder Removal19 Background Data DB3 Effect Of Azithromycin On Serum Theophylline Levels Of Patients With Emphysema20 Background Data DB4 Effect Of Protease Inhibitors On Pulmonary Admissions21 Background Data DB5 Effect Of Silicone Implants On Plasma Silicon22 Background Data DB6 Laser Removal Of Tattoos As Related To Type Of Ink Used23 Background Data DB7 Relation Of Bone Density To Incidence Of Femoral Neck Stress Fractures24 Background Data DB8 Comparing Two Types Of Assay On The Effect Of Gag On The Bladder Surface25 Background Data DB9 Prediction Of Growth Factors By Platelet Counts26 Background Data DB10 Tests Of Recovery After Surgery On Hamstrings Or Quadriceps27 Background Data DB11 Survival Of Malarial Rats Treated With Hemoglobin, Rbcs, Or A Placebo28 Background Data DB12 Identification Of Risk Factors For Death Following Carinal Resection29 Background Data DB13 Quality Test On Warfarin International Normalized Ratio Values30 Background Data DB14 Exhaled Nitric Oxide As An Indicator Of Exercise-Induced Bronchoconstriction31 Background Data DB15 Comparison Of Kidney Cooling Methods Used To Prolong Surgical Time Window32 Background Data DB16 Word Repetitions Among Cognitively Normal And Demented Patients33 DB17 Retest Effects In Neuropsychological Testing34 References Appendix 3 Tables of probability distributions Appendix 4 Symbol index Classes of Symbols Mathematical Symbols Statistical Symbols and Terminology Statistical Subject Index Medical Subject Index