1st Edition

Exact Analysis of Discrete Data

By Karim F. Hirji Copyright 2006
    1066 Pages 140 B/W Illustrations
    by Chapman & Hall

    Researchers in fields ranging from biology and medicine to the social sciences, law, and economics regularly encounter variables that are discrete or categorical in nature. While there is no dearth of books on the analysis and interpretation of such data, these generally focus on large sample methods. When sample sizes are not large or the data are otherwise sparse, exact methods--methods not based on asymptotic theory--are more accurate and therefore preferable.

    This book introduces the statistical theory, analysis methods, and computation techniques for exact analysis of discrete data. After reviewing the relevant discrete distributions, the author develops the exact methods from the ground up in a conceptually integrated manner. The topics covered range from univariate discrete data analysis, a single and several 2 x 2 tables, a single and several 2 x K tables, incidence density and inverse sampling designs, unmatched and matched case -control studies, paired binary and trinomial response models, and Markov chain data. While most chapters focus on statistical theory and applications, three chapters deal exclusively with computational issues. Detailed worked examples appear throughout the book, and each chapter includes an extensive problem set.

    Written at an elementary to intermediate level, Exact Analysis of Discrete Data is accessible to anyone having taken a basic course in statistics or biostatistics, bringing to them valuable material previously buried in specialized journals.

    DISCRETE DISTRIBUTIONS
    Introduction
    Discrete Random Variables
    Probability Distributions
    Polynomial Based Distributions
    Binomial Distribution
    Poisson Distribution
    Negative Binomial Distribution
    Hypergeometric Distribution
    A General Representation
    The Multinomial Distribution
    The Negative Trinomial
    Suffcient Statistics
    The Polynomial Form

    ONE-SIDED UNIVARIATE ANALYSIS
    Introduction
    One Parameter Inference
    Tail Probability and Evidence
    Exact Evidence Function
    Mid-p Evidence Function
    Asymptotic Evidence Function
    Matters of Significance
    Confidence Intervals
    Illustrative Examples
    Design and Analysis
    Exercises

    TWO-SIDED UNIVARIATE ANALYSIS
    Introduction
    Two-Sided Inference
    Twice the Smaller Tail Method
    Examples
    The Likelihood Function
    The Score Method
    Additional Illustrations
    Likelihood Ratio and Wald Methods
    Three More Methods
    Comparative Computations
    The ABC of Reporting
    Additional Comments
    At the Boundary
    Equivalent Statistics

    COMPUTING FUNDAMENTALS
    Introduction
    Computing Principles
    Combinatorial Coeocients
    Polynomial Storage and Evaluation
    Computing Distributions
    Roots of Equations
    Iterative Methods

    ELEMENTS OF CONDITIONAL ANALYSIS
    Introduction
    Design and Analysis
    Modes of Inference
    The 2 x 2 Table
    The One Margin Fixed Design
    The Overall Total Fixed Design
    The Nothing Fixed Design
    A Retrospective Design
    The Inverse Sampling Design
    Unconditional Analysis
    Conditional Analysis
    Comparing Two Rates
    Points to Ponder
    Derivation of Test Statistics

    TWO 2 x 2 TABLES
    Introduction
    Sources of Variability
    On Stratification
    Data Examples
    Statistical Models
    Conventional Analysis
    Conditional Analysis
    An Example
    A Second Example
    On Case-Control Sampling
    Anatomy of Interactions

    ASSESSING INFERENCE
    Introduction
    Exact Unconditional Analysis
    Randomized Inference
    Exact Power
    Exact Coverage
    The Fisher and Irwin Tests
    Some Features
    Desirable Features
    On Unconditional Analysis
    Why the Mid-p?

    SEVERAL 2 x 2 TABLES: I
    Introduction
    Three Models
    Exact Distributions
    The COR Model
    Conditional Independence
    Trend In Odds Ratios
    Recommendations

    SEVERAL 2 x 2 TABLES: II
    Introduction
    Models for Combining Risk
    Testing for Homogeneity
    Test Statistics
    A Worked Example
    Checking the TOR Model
    An Incidence Density Study
    Other Study Designs
    Exact Power
    Additional Issues
    Derivation

    THE 2 x K TABLE
    Introduction
    An Ordered Table
    An Unordered Table
    Test Statistics
    An Illustration
    Checking Linearity
    Other Sampling Designs
    Incidence Density Data
    An Inverse Sampling Design
    Additional Topics
    Extensions
    Derivation

    POLYNOMIAL ALGORITHMS: I
    Introduction
    Exhaustive Enumeration
    Monte-Carlo Simulation
    Recursive Multiplication
    Exponent Checks
    Applications
    The Fast Fourier Transform

    POLYNOMIAL ALGORITHMS: II
    Introduction
    Bivariate Polynomials
    A Conditional Polynomial
    Backward Induction
    Conditional Values
    Applications
    Trivariate Polynomials
    An Extension
    Network Algorithms
    Power Computation
    Practical Implementation

    MULTINOMIAL MODELS
    Introduction
    Compositions and Partitions
    A Single Multinomial
    Trinary Response Models
    Conditional Polynomials
    Several 3 x K Tables
    J x K Tables

    MATCHED AND DEPENDENT DATA
    Introduction
    Matched Designs
    Paired Binary Outcomes
    Markov Chain Models

    REFLECTIONS ON EXACTNESS
    Introduction
    Inexact Terminology
    Bayesians and Frequentists
    Design and Analysis
    Status Quo Exactness
    Practical Inexactness
    Formal Exactness
    In Praise of Exactness

    References
    Index

    Each chapter also contains Relevant Literature and Exercises sections.

    Biography

    Karim F. Hirji

    "The book’s infrastructure makes it a good candidate as a teaching resource. The chapters offer detailed worked example. Furthermore, each chapter includes exercises . . . some practitioners in discrete data analysis also will find this book useful."

    – In Technometrics, August 2008, Vol. 50, No. 3