Confidence Intervals for Proportions and Related Measures of Effect Size

By Robert G. Newcombe

© 2013 – CRC Press

468 pages | 45 B/W Illus.

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Hardback: 9781439812785
pub: 2012-08-24
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About the Book

Confidence Intervals for Proportions and Related Measures of Effect Size illustrates the use of effect size measures and corresponding confidence intervals as more informative alternatives to the most basic and widely used significance tests. The book provides you with a deep understanding of what happens when these statistical methods are applied in situations far removed from the familiar Gaussian case.

Drawing on his extensive work as a statistician and professor at Cardiff University School of Medicine, the author brings together methods for calculating confidence intervals for proportions and several other important measures, including differences, ratios, and nonparametric effect size measures generalizing Mann-Whitney and Wilcoxon tests. He also explains three important approaches to obtaining intervals for related measures. Many examples illustrate the application of the methods in the health and social sciences. Requiring little computational skills, the book offers user-friendly Excel spreadsheets for download at, enabling you to easily apply the methods to your own empirical data.


"…this is a fantastic book, and I would recommend it highly, especially for medical researchers and statisticians in the medical field."

—Vance W. Berger, Journal of Biopharmaceutical Statistics, 2014

"This is an interesting and well-written book, with a lot to recommend it. … the examples alone comprise a valuable teaching resource. … discussions are enlivened by reference to real-life practical issues, examples, and interesting perspectives. … there will be something of value, to think about or enjoy, for almost all readers who like statistics in general or data analysis in particular. It is a pleasure to recommend it."

—Bruce Brown, Australian & New Zealand Journal of Statistics, 2014

"This book offers an excellent summary on how to construct and use confidence intervals for proportions and related measures of effect size (proportion difference, ratio of proportions, and odds ratio etc.). A unique feature of the book is that the most materials stem from the author’s own methodology research and collaborative work in medical research. The author put a lot effort to bridge the gap between statistical methodology and application through detailed real examples and insightful comments."

—Journal of Agricultural, Biological, and Environmental Statistics, Volume 19, Number 2, 2013

Table of Contents

Hypothesis Tests and Confidence Intervals

Sample and Population

Hypothesis Testing and Confidence Intervals: The Fundamentals

Why Confidence Intervals Are Generally More Informative Than p-Values

Measures of Effect Size

When Are Point and Interval Estimates Less Helpful?

Frequentist, Bayesian and Likelihood Intervals

Just What Is Meant by the Population?

The Unit of Data

Sample Size Planning

Means and Their Differences

Confidence Interval for a Mean

Confidence Interval for the Difference between Means of Independent Samples

Confidence Interval for the Difference between Two Means Based on Individually Paired Samples

Scale Transformation

Non-Parametric Methods

The Effect of Dichotomising Continuous Variables

Confidence Intervals for a Simple Binomial Proportion


The Wald Interval

Boundary Anomalies

Alternative Intervals

Algebraic Definitions for Several Confidence Intervals for the Binomial Proportion

Implementation of Wilson Score Interval in MS Excel

Sample Size for Estimating a Proportion

Criteria for Optimality

How Can We Say Which Methods Are Good Ones?


Expected Width

Interval Location

Computational Ease and Transparency

Evaluation of Performance of Confidence Interval Methods


An Example of Evaluation

Approaches Used in Evaluations for the Binomial Proportion

The Need for Illustrative Examples

Intervals for the Poisson Parameter and the Substitution Approach

The Poisson Distribution and Its Applications

Confidence Intervals for the Poisson Parameter and Related Quantities

Widening the Applicability of Confidence Interval Methods: The Substitution Approach

Difference between Independent Proportions and the Square-and-Add Approach

The Ordinary 2 x 2 Table for Unpaired Data

The Wald Interval

The Square-and-Add or MOVER Approach

Other Well-Behaved Intervals for the Difference between Independent Proportions

Evaluation of Performance

Number Needed to Treat

Bayesian Intervals

Interpreting Overlapping Intervals

Sample Size Planning

Difference between Proportions Based on Individually Paired Data

The 2 x 2 Table for Paired Binary Data

Wald and Conditional Intervals

Intervals Based on Profile Likelihoods

Score-Based Intervals

Evaluation of Performance

Methods for Triads of Proportions


Trinomial Variables on Equally Spaced Scales

Unordered Trinomial Data: Generalising the Tail-Based p-Value to Characterise Conformity to Prescribed Norms

A Ternary Plot for Unordered Trinomial Data

Relative Risk and Rate Ratio

A Ratio of Independent Proportions

Three Effect Size Measures Comparing Proportions

Ratio Measures Behave Best on a Log Scale

Intervals Corresponding to the Empirical Estimate

Infinite Bias in Ratio Estimates

Intervals Based on Mesially Shrunk Estimated Risks

A Ratio of Proportions Based on Paired Data

A Ratio of Sizes of Overlapping Groups

A Ratio of Two Rates

Implementation in MS Excel

The Odds Ratio and Logistic Regression

The Rationale for the Odds Ratio

Disadvantages of the Odds Ratio

Intervals Corresponding to the Empirical Estimate

Deterministic Bootstrap Intervals Based on Median Unbiased Estimates

Logistic Regression

An Odds Ratio Based on Paired Data


Screening and Diagnostic Tests


Sensitivity and Specificity

Positive and Negative Predictive Values

Trade-Off between Sensitivity and Specificity: The ROC Curve

Simultaneous Comparison of Sensitivity and Specificity between Two Tests

Widening the Applicability of Confidence Interval Methods: The Propagating Imprecision Approach


The Origin of the PropImp Approach

The PropImp Method Defined

PropImp and MOVER Wilson Intervals for Measures Comparing Two Proportions

Implementation of the PropImp Method


The Thorny Issue of Monotonicity

Some Issues Relating to MOVER and PropImp Approaches

Several Applications of the MOVER and PropImp Approaches


Additive-Scale Interaction for Proportions

Radiation Dose Ratio

Levin’s Attributable Risk

Population Risk Difference and Population Impact Number

Quantification of Copy Number Variations

Standardised Mortality Ratio Adjusted for Incomplete Data on Cause of Death

RD and NNT from Baseline Risk and Relative Risk Reduction

Projected Positive and Negative Predictive Values

Estimating Centiles of a Gaussian Distribution

Ratio Measures Comparing Means

Winding the Clock Back: The Healthy Hearts Study

Grass Fires

Incremental Risk-Benefit Ratio

Adjustment of Prevalence Estimate Using Partial Validation Data

Comparison of Two Proportions Based on Overlapping Samples

Standardised Difference of Proportions

Generalised Mann–Whitney Measure

Absolute and Relative Effect Size Measures for Continuous and Ordinal Scales

The Generalised Mann–Whitney Measure

Definitions of Eight Methods

Illustrative Examples


Results of the Evaluation

Implementation in MS Excel


Generalised Wilcoxon Measure

The Rationale for the Generalised Wilcoxon Measure ψ

Paired and Unpaired Effect Size Measures Compared

Estimating the Index ψ

Development of a Confidence Interval for ψ

Evaluation of Coverage Properties: Continuous Case

Results of Evaluation for the Continuous Case

Coverage Properties for Discrete Distributions




About the Author

Robert G. Newcombe is a professor in the Institute of Primary Care and Public Health at Cardiff University School of Medicine, where he teaches medical statistics and epidemiology and is involved in medical and dental research. Dr. Newcombe is a member of the editorial board of Statistical Methods in Medical Research and serves on the Cardiff & Vale Research Review Service and Wales Ambulance Service Trust Research & Development panels.

About the Series

Chapman & Hall/CRC Biostatistics Series

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Probability & Statistics / General