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Summary

Describing categorical random variable – chapter 1 Poisson for count data Binomial for binary data Multinomial for I>2 outcome categories Others Limitation: one parameter only, can be adjusted by

scale parameter inference

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Summary

Two-way contingency table – chapters 2, 3 Parameters: risk, odds Comparison: relative risk, odds ratio Estimation: delta method Tests: chi-square, fisher’s exact test Ordered two-way tables:

assign scores - Trend test M2=(n-1)r2

uses an ordinal measure of monotone trend:

SAS: proc freq with option relarisk, chisq, exact, etc.

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Summary

Three-way (multi-way) tables – chapter 2, 3 Partial tables Conditional and marginal odds ratio Conditional and marginal independence Inference – chapter 4-9:

Third or others variables are considered as covariates modeling

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Summary – generalized linear models

Random component is exponential family (not necessary normal)

Systematic component – linear model Link function – connect mean to Systematic component

xbeta Log Logit Identity

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Logistic regression

Chapters 5-7 SAS proc logistic, genmod Binary outcome – logistic regression Multinomial response

Nominal-baseline-category logit models Ordinal – cumulative logit models

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Log-linear model

Chapters 8-9 Two-way table Three-way tables Multi-way tables

Model selection Ordinal responses

Log-linear model for rates

SAS: genmod

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By far – cross sectional data

If the data are collected over time, the data for the same subject in different time points will be correlated. Longitudinal data

Multivariate responses *

Non-linear models *

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Longitudinal data

Chapter 10 – two time points: matched pairs

Chapter 11 – repeated measures using marginal models (no random effects)

Chapter 12 – random effect model or generalized linear mixed models

Recent developments – publications for categorical responses since 2002 (final project) Read one or two recent papers 20 minutes presentation

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models

Linear model (LMs) (t-tests, ANOVA, ANCOVA) SAS: proc TTEST, ANOVA, REG, GLM

Generalized linear models (GLMs) SAS: proc GENMOD, LOGISTIC, CATMOD

Linear mixed model (LMMs) – permitting heterogeneity of variance, variance structure is based on random effects and their variance components SAS: proc MIXED

Generalized linear mixed models (GLMMs) SAS: proc NLMIXED

Non-linear mixed model SAS: proc NLMIXED

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Models for matched pairs

In this chapter, we introduce methods for comparing categorical responses for two samples when each observation in one sample pairs with an observation in the other.

For easy understanding, we assume n independent subjects and let Yi = (Yi1,Yi2, ...,Yiti)is the observation of subject i at different time.

In statistics, {Y1,Y2, ...,Yn} are called longitudinal data

For fixed i , Yi is a time series; for fixed time j , {Y1j ,Y2j , ...,Ynj} is a sequence of independent random variables.

If ti = 2 for all i , {Y1,Y2, ...,Yn} is called matched-pairs data. Note that the two samples {Y11,Y21, ...,Yn1} and {Y12,Y22, ...,Yn2} are not independent.

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Outline

10.1 Comparing Dependent Proportions;

10.2 Conditional Logistic Regression for Binary Matched Pairs;

10.3 Marginal Models for Squared Contingency Tables;

10.4 Symmetry, Quasi-symmetry and Quasi-independence;

10.5 Measure Agreement Between Observers;

10.6 Bradley-Terry Models for Paired Preferences.

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10.1 COMPARING DEPENDENT PROPORTIONS

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10.1.2 Prime minister approval rating example

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SAS code/*section 10.1.2 page 411*/

data tmp;

p11=794/1600; p12=150/1600; p21=86/1600; p22=570/1600;

p1plus=p11+p12; pplus1=p11+p21;

se=sqrt( ((p12+p21)-(p12-p21)**2)/1600);

lci=p1plus-pplus1-1.96*se;

uci=p1plus-pplus1+1.96*se;

z0=(86-150)/(86+150)**0.5;

McNemarsTest=z0**2;

pvalue=1-cdf('chisquare',McNemarsTest,1);

se_ind=sqrt(p1plus*(1-p1plus)+pplus1*(1-pplus1))/sqrt(1600); /*assume independent*/

lci_ind=p1plus-pplus1-1.96*se_ind;

uci_ind=p1plus-pplus1+1.96*se_ind;

proc print; run;

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SAS code McNemar’s Testdata matched;

input first second count @@;

datalines;

1 1 794 1 2 150 2 1 86 2 2 570

;

proc freq; weight count;

tables first*second/ agree; exact mcnem; /*McNemars Test*/

proc catmod; weight count;

response marginals;

model first*second= (1 0 ,

1 1) ;

run;

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PROC FREQ For square tables, the AGREE option in PROC FREQ

provides the McNemar chi-squared statistic for binary matched pairs, the X2 test of fit of the symmetry model (also called Bowker’s test), and Cohen’s kappa and weighted kappa with SE values.

The MCNEM keyword in the EXACT statement provides a small-sample binomial version of McNemar’s test.

PROC CATMOD provide the confidence interval for the difference of proportions. The code forms a model for the marginal proportions

in the first row and the first column, specifying a model matrix in the model statement that has an intercept parameter (the first column) that applies to both proportions and a slope parameter that applies only to the second; hence the second parameter is the difference between the second and first marginal proportions.

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10.1.3 Increased precision with dependent samples

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Fit marginal modeldata matched1;

input case occasion response count @@;

datalines;

1 0 1 794

1 1 1 794

2 0 1 150

2 1 0 150

3 0 0 86

3 1 1 86

4 0 0 570

4 1 0 570

;

proc logistic data=matched; weight count;

model response=occasion; run;

XtXt

proc genmod data=matched1 DESCENDING;

weight count;

model response=occasion/dist=bin link=identity;

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Google calculatorln((880 * 656) / (944*720) )= -0.163294682

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Matlab code for deriving previous MLE and SE

%% page 417

syms b n21 n12

LL=log(exp(b)^n21/(1+exp(b))^(n12+n21));

simplify(diff(LL,'b'))

%result (n21-exp(b)*n12)/(1+exp(b))

%thus beta=log(n21/n12)

simplify(diff(diff(LL,'b'),'b'))

%result -exp(b)*(n12+n21)/(1+exp(b))^2

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10.2.4 Random effects in binary matched-pairs model

An alternative remedy to handling the huge number of nuisance parameters in logit model

(10.8)

treats as random effects. Assume ~

This model is an example of a generalized linear mixed model, containing both random effects and the fixed effect beta.

Fit by proc NLMIXED Chapter 12

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10.2.5 Logistic Regression for Matched Case–Control Studies

The two observations in a matched pair need not refer to the same subject.

For instance, case-control studies that match a single control with each case yield matched-pairs data.

Example: A case-control study of acute myocardial infarction (MI) among Navajo Indians matched 144 victims of MI according to age and gender with 144 people free of heart disease.

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Now, for subject t in matched pair i, consider the model

the conditional ML estimate of OR is

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10.2.6 Conditional ML for matched pairs with multiple predictors

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10.2.7 Marginal models vs. conditional models

Section 10.1 Marginal model (McNemar’s test H0: =0)

Section 10.2 conditional model

Conditional ML Random effects, NLMIXED