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# Confidence Intervals for Proportions and Related Measures of Effect Size

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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
Introduction
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?
Coverage
Expected Width
Interval Location
Computational Ease and Transparency

Evaluation of Performance of Confidence Interval Methods
Introduction
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
Introduction
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
Implementation

Screening and Diagnostic Tests
Background
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
Background
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
Evaluation
The Thorny Issue of Monotonicity
Some Issues Relating to MOVER and PropImp Approaches

Several Applications of the MOVER and PropImp Approaches
Introduction
Additive-Scale Interaction for Proportions
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
Evaluation
Results of the Evaluation
Implementation in MS Excel
Interpretation

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
Discussion

References

Appendices

CRC Press; August 2012
463 pages; ISBN 9781439812792