Exact Analysis of Discrete Data

ISBN-13: 9781584880707 / Angielski / Twarda / 2005 / 1066 str.

Karim F. Hirji

Opis

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.

Spis treści

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.

O autorze

Karim F. Hirji