1 STA 617 – Chp10 Models for matched pairs 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

2 STA 617 – Chp10 Models for matched pairs 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 M 2 =(n-1)r 2  uses an ordinal measure of monotone trend:  SAS: proc freq with option relarisk, chisq, exact, etc.

3 STA 617 – Chp10 Models for matched pairs 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

4 STA 617 – Chp10 Models for matched pairs 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

5 STA 617 – Chp10 Models for matched pairs Logistic regression  Chapters 5-7  SAS proc logistic, genmod  Binary outcome – logistic regression  Multinomial response  Nominal-baseline-category logit models  Ordinal – cumulative logit models

6 STA 617 – Chp10 Models for matched pairs 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

7 STA 617 – Chp10 Models for matched pairs 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 *

8 STA 617 – Chp10 Models for matched pairs 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

9 STA 617 – Chp10 Models for matched pairs 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

10 STA 617 – Chp10 Models for matched pairs 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 Y i = (Y i1,Y i2,...,Y iti ) is the observation of subject i at different time.  In statistics, {Y 1,Y 2,...,Y n } are called longitudinal data  For fixed i, Y i is a time series; for fixed time j, {Y 1j,Y 2j,...,Y nj } is a sequence of independent random variables.  If t i = 2 for all i, {Y 1,Y 2,...,Y n } is called matched-pairs data. Note that the two samples {Y 11,Y 21,...,Y n1 } and {Y 12,Y 22,...,Y n2 } are not independent.

11 STA 617 – Chp10 Models for matched pairs 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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14 STA 617 – Chp10 Models for matched pairs 10.1 COMPARING DEPENDENT PROPORTIONS

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18 STA 617 – Chp10 Models for matched pairs Prime minister approval rating example

19 STA 617 – Chp10 Models for matched pairs SAS code /*section 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;

20 STA 617 – Chp10 Models for matched pairs SAS code McNemar’s Test data matched; input first second count datalines; ; 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;

21 STA 617 – Chp10 Models for matched pairs  PROC FREQ  For square tables, the AGREE option in PROC FREQ provides the McNemar chi-squared statistic for binary matched pairs, the X 2 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.

22 STA 617 – Chp10 Models for matched pairs Increased precision with dependent samples

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24 STA 617 – Chp10 Models for matched pairs Fit marginal model data matched1; input case occasion response count datalines; ; proc logistic data=matched; weight count; model response=occasion; run; Xt proc genmod data=matched1 DESCENDING; weight count; model response=occasion/dist=bin link=identity;

25 STA 617 – Chp10 Models for matched pairs Google calculator ln((880 * 656) / (944*720) )=

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31 STA 617 – Chp10 Models for matched pairs 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

32 STA 617 – Chp10 Models for matched pairs 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

33 STA 617 – Chp10 Models for matched pairs 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.

34 STA 617 – Chp10 Models for matched pairs  Now, for subject t in matched pair i, consider the model  the conditional ML estimate of OR is

35 STA 617 – Chp10 Models for matched pairs Conditional ML for matched pairs with multiple predictors

36 STA 617 – Chp10 Models for matched pairs 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