分类数据分析：逻辑回归与线性模型在医学研究中的应用

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"Categorical Data Analysis" 是一本由Alan Agresti教授编著的书籍，主要探讨了分类数据的分析方法，特别关注了逻辑回归模型和线性模型的应用、检验及参数估计。这本书对于医学研究，尤其是医生研究疾病具有重要的参考价值。在数据分析领域，分类数据是一种常见的数据类型，包括离散的类别或者等级，如性别、疾病类型等。本书深入浅出地介绍了如何处理和分析这类数据。逻辑回归模型是分析二项或多元分类响应变量与连续或分类预测变量关系的重要工具，广泛应用于医学研究中的疾病预测和诊断。例如，医生可能利用患者的一些特征（如年龄、性别、症状等）构建逻辑回归模型，以预测患者是否患有某种疾病。线性模型，如方差分析（ANOVA）和线性回归，是处理连续数值型数据与分类变量关系的经典方法。在书中，作者详细阐述了如何构建和验证这类模型，以及如何进行参数估计，以了解分类变量如何影响连续变量。这对于理解疾病与风险因素之间的定量关系至关重要。此外，书中的内容可能还包括模型选择、模型假设的检查、残差分析以及模型的解释和报告。这些方法有助于确保模型的准确性和可靠性。书中还可能讨论了处理缺失数据、异常值和多重比较问题的策略，这些都是实际数据分析中经常遇到的问题。 Alan Agresti教授的这部作品不仅涵盖了理论基础，还提供了实际案例和应用示例，使得读者能够将所学知识应用到实际研究中。无论是统计初学者还是经验丰富的研究者，都能从中受益，提升在分类数据分析方面的技能。 “Categorical Data Analysis”是一本全面介绍分类数据统计分析的权威著作，对于医学研究人员、公共卫生专家以及其他需要处理分类数据的科学家来说，是一本不可或缺的参考书。通过学习本书，读者将能够更好地理解和运用逻辑回归和线性模型，从而在疾病研究和预测中做出更科学、更精确的决策。

CATEGORICAL RESPONSE DATA 3

An inter®al ®ariable is one that does have numerical distances between any

two values. For example, blood pressure level, functional life length of

television set, length of prison term, and annual income are interval vari-

ables. An internal variable is sometimes called a ratio ®ariable if ratios of

values are also valid.

The way that a variable is measured determines its classification. For

example, ‘‘education’’ is only nominal when measured as public school or

private school; it is ordinal when measured by highest degree attained, using

the categories none, high school, bachelor’s, master’s, and doctorate; it is

interval when measured by number of years of education, using the integers

0, 1, 2, . . . .

A variable’s measurement scale determines which statistical methods are

appropriate. In the measurement hierarchy, interval variables are highest,

ordinal variables are next, and nominal variables are lowest. Statistical

methods for variables of one type can also be used with variables at higher

levels but not at lower levels. For instance, statistical methods for nominal

variables can be used with ordinal variables by ignoring the ordering of

categories. Methods for ordinal variables cannot, however, be used with

nominal variables, since their categories have no meaningful ordering. It is

usually best to apply methods appropriate for the actual scale.

Since this book deals with categorical responses, we discuss the analysis of

nominal and ordinal variables. The methods also apply to interval variables

Ž.

having a small number of distinct values e.g., number of times married or

for which the values are grouped into ordered categories e.g., education

measured as - 10 years, 10᎐12 years, ) 12 years .

1.1.3 Continuous–Discrete Variable Distinction

Variables are classified as continuous or discrete, according to the number of

values they can take. Actual measurement of all variables occurs in a discrete

manner, due to precision limitations in measuring instruments. The continu-

ous᎐discrete classification, in practice, distinguishes between variables that

take lots of values and variables that take few values. For instance, statisti-

cians often treat discrete interval variables having a large number of values

Ž.

such as test scores as continuous, using them in methods for continuous

responses.

Ž.

This book deals with certain types of discretely measured responses: 1

Ž. Ž.

nominal variables, 2 ordinal variables, 3 discrete interval variables having

Ž.

relatively few values, and 4 continuous variables grouped into a small

number of categories.

1.1.4 Quantitative–Qualitative Variable Distinction

Nominal variables are qualitati®eᎏdistinct categories differ in quality, not in

quantity. Interval variables are quantitati®eᎏdistinct levels have differing

amounts of the characteristic of interest. The position of ordinal variables in

INTRODUCTION: DISTRIBUTIONS AND INFERENCE FOR CATEGORICAL DATA4

the quantitative᎐qualitative classification is fuzzy. Analysts often treat

them as qualitative, using methods for nominal variables. But in many

respects, ordinal variables more closely resemble interval variables than they

resemble nominal variables. They possess important quantitative features:

Each category has a greater or smaller magnitude of the characteristic than

another category; and although not possible to measure, an underlying

continuous variable is usually present. The political philosophy classification

Ž.

liberal, moderate, conservative crudely measures an inherently continuous

characteristic.

Analysts often utilize the quantitative nature of ordinal variables by

assigning numerical scores to categories or assuming an underlying continu-

ous distribution. This requires good judgment and guidance from researchers

who use the scale, but it provides benefits in the variety of methods available

for data analysis.

1.1.5 Organization of This Book

The models for categorical response variables discussed in this book resem-

ble regression models for continuous response variables; however, they

assume binomial, multinomial, or Poisson response distributions instead of

normality. Two types of models receive special attention, logistic regression

and loglinear models. Ordinary logistic regression models, also called logit

Ž.

models, apply with binary i.e., two-category responses and assume a bino-

mial distribution. Generalizations of logistic regression apply with multicate-

gory responses and assume a multinomial distribution. Loglinear models

apply with count data and assume a Poisson distribution. Certain equiva-

lences exist between logistic regression and loglinear models.

The book has four main units. In the first, Chapters 1 through 3, we

summarize descriptive and inferential methods for univariate and bivariate

categorical data. These chapters cover discrete distributions, methods of

inference, and analyses for measures of association. They summarize the

non-model-based methods developed prior to about 1960.

In the second and primary unit, Chapters 4 through 9, we introduce

models for categorical responses. In Chapter 4 we describe a class of

generalized linear models having models of this text as special cases. We focus

on models for binary and count response variables. Chapters 5 and 6 cover

the most important model for binary responses, logistic regression. In Chap-

ter 7 we present generalizations of that model for nominal and ordinal

multicategory response variables. In Chapter 8 we introduce the modeling of

multivariate categorical response data and show how to represent association

and interaction patterns by loglinear models for counts in the table that

cross-classifies those responses. In Chapter 9 we discuss model building with

loglinear and related logistic models and present some related models.

In the third unit, Chapters 10 through 13, we discuss models for handling

repeated measurement and other forms of clustering. In Chapter 10 we

DISTRIBUTIONS FOR CATEGORICAL DATA 5

present models for a categorical response with matched pairs; these apply,

for instance, with a categorical response measured for the same subjects at

two times. Chapter 11 covers models for more general types of repeated

categorical data, such as longitudinal data from several times with explana-

tory variables. In Chapter 12 we present a broad class of models, generalized

linear mixed models, that use random effects to account for dependence with

such data. In Chapter 13 further extensions and applications of the models

from Chapters 10 through 12 are described.

The fourth and final unit is more theoretical. In Chapter 14 we develop

asymptotic theory for categorical data models. This theory is the basis for

large-sample behavior of model parameter estimators and goodness-of-fit

statistics. Maximum likelihood estimation receives primary attention here

and throughout the book, but Chapter 15 covers alternative methods of

estimation, such as the Bayesian paradigm. Chapter 16 stands alone from the

others, being a historical overview of the development of categorical data

methods.

Most categorical data methods require extensive computations, and statis-

tical software is necessary for their effective use. In Appendix A we discuss

software that can perform the analyses in this book and show the use of SAS

for text examples. See the Web site www.stat.ufl.edur; aarcdarcda.html to

download sample programs and data sets and find information about other

software.

Chapter 1 provides background material. In Section 1.2 we review the key

distributions for categorical data: the binomial, multinomial, and Poisson. In

Section 1.3 we review the primary mechanisms for statistical inference, using

maximum likelihood. In Sections 1.4 and 1.5 we illustrate these by presenting

significance tests and confidence intervals for binomial and multinomial

parameters.

1.2 DISTRIBUTIONS FOR CATEGORICAL DATA

Inferential data analyses require assumptions about the random mechanism

that generated the data. For regression models with continuous responses,

the normal distribution plays the central role. In this section we review the

three key distributions for categorical responses: binomial, multinomial, and

Poisson.

1.2.1 Binomial Distribution

Many applications refer to a fixed number n of binary observations. Let

y , y ,..., y denote responses for n independent and identical trials such

12 n

Ž. Ž.

that PYs 1 s

␲

and PYs 0 s 1 y

␲

.Weuse the generic labels

‘‘success’’ and ‘‘failure’’ for outcomes 1 and 0. Identical trials means that the

probability of success

␲

is the same for each trial. Independent trials means

INTRODUCTION: DISTRIBUTIONS AND INFERENCE FOR CATEGORICAL DATA6

4

that the Y are independent random variables. These are often called

Bernoulli trials. The total number of successes, Y s Ý

Y , has the binomial

is1 i

Ž.

distribution with index n and parameter

␲

, denoted by bin n,

␲

The probability mass function for the possible outcomes y for Y is

nyy

pys

␲

1 y

␲

, y s 0, 1, 2,...,n,1.1

Ž. Ž . Ž .

ž/

w Ž.x Ž. Ž .

where the binomial coefficient s n!r y! n y y !.Since EY s EY

ž/

Ž.

s 1 =

␲

q 0 = 1 y

␲

EY s

␲

and var Y s

␲

1 y

␲

Ž. Ž. Ž .

The binomial distribution for Y s Ý Y has mean and variance

␮

s EY s n

␲

and

␴

s var Y s n

␲

1 y

␲

Ž. Ž. Ž .

Ž. Ž .

The skewness is described by EYy

␮

␴

s 1 y 2

␲

r n

␲

1 y

␲

Ž.

The distribution converges to normality as n increases, for fixed

␲

There is no guarantee that successive binary observations are independent

or identical. Thus, occasionally, we will utilize other distributions. One such

case is sampling binary outcomes without replacement from a finite popula-

tion, such as observations on gender for 10 students sampled from a class of

size 20. The hypergeometric distribution, studied in Section 3.5.1, is then

relevant. In Section 1.2.4 we mention another case that violates these

binomial assumptions.

1.2.2 Multinomial Distribution

Some trials have more than two possible outcomes. Suppose that each of n

independent, identical trials can have outcome in any of c categories. Let

y s 1iftrial i has outcome in category j and y s 0 otherwise. Then

ij ij

Ž.

y s y , y ,..., y represents a multinomial trial, with Ý y s 1; for

ii1 i2 ic j i j

Ž.

instance, 0, 0, 1, 0 denotes outcome in category 3 of four possible categories.

Note that y is redundant, being linearly dependent on the others. Let

n s Ý y denote the number of trials having outcome in category j. The

jiij

Ž.

counts n , n ,...,n have the multinomial distribution.

12 c

Ž.

Let

␲

s PY s 1 denote the probability of outcome in category j for

jij

each trial. The multinomial probability mass function is

nn n

12 c

pn, n ,...,n s

␲␲

⭈⭈⭈

␲

.1.2

Ž. Ž.

12 cy112c

ž/

n ! n ! ⭈⭈⭈ n !

12 c

DISTRIBUTIONS FOR CATEGORICAL DATA 7

Ž. Ž

Since Ý n s n, this is cy1 -dimensional, with n s n y n q ⭈⭈⭈

jj c 1

qn . The binomial distribution is the special case with c s 2.

cy1

For the multinomial distribution,

En s n

␲

, var n s n

␲

1 y

␲

, cov n , n syn

␲␲

Ž. Ž. Ž .

Ž.

jj jjj jk jk

1.3

Ž.

We derive the covariance in Section 14.1.4. The marginal distribution of each

n is binomial.

1.2.3 Poisson Distribution

Sometimes, count data do not result from a fixed number of trials. For

instance, if y s number of deaths due to automobile accidents on motorways

in Italy during this coming week, there is no fixed upper limit n for y as you

are aware if you have driven in Italy . Since y must be a nonnegative integer,

its distribution should place its mass on that range. The simplest such

distribution is the Poisson. Its probabilities depend on a single parameter,

Ž.

the mean

␮

. The Poisson probability mass function Poisson 1837, p. 206 is

␮

pys , y s 0, 1, 2, . . . . 1.4

Ž. Ž .

Ž. Ž.

It satisfies EY s var Y s

␮

.Itisunimodal with mode equal to the

Ž.

integer part of

␮

. Its skewness is described by EYy

␮

␴

s 1r

␮

. The

distribution approaches normality as

␮

increases.

The Poisson distribution is used for counts of events that occur randomly

over time or space, when outcomes in disjoint periods or regions are inde-

pendent. It also applies as an approximation for the binomial when n is large

and

␲

is small, with

␮

s n

␲

.Soifeach of the 50 million people driving in

Italy next week is an independent trial with probability 0.000002 of dying in a

Ž.

fatal accident that week, the number of deaths Y is a bin 50000000, 0.000002

Ž.

variate, or approximately Poisson with

␮

s n

␲

s 50,000,000 0.000002 s 100.

A key feature of the Poisson distribution is that its variance equals its

mean. Sample counts vary more when their mean is higher. When the mean

number of weekly fatal accidents equals 100, greater variability occurs in the

weekly counts than when the mean equals 10.

1.2.4 Overdispersion

In practice, count observations often exhibit variability exceeding that pre-

dicted by the binomial or Poisson. This phenomenon is called o®erdispersion.

We assumed above that each person has the same probability of dying in a

fatal accident in the next week. More realistically, these probabilities vary,

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