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Summary

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. Summary. Two-way contingency table – chapters 2, 3

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Summary

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  1. 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. 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.

  3. 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. 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. 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. 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. 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. 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. 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. 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 , Yiis a time series; for fixed time j , {Y1j ,Y2j , ...,Ynj} isa 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.

  11. 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.

  12. 10.1 COMPARING DEPENDENT PROPORTIONS

  13. 10.1.2 Prime minister approval rating example

  14. 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)+p1plus*(1-p1plus))/sqrt(1600); /*assume independent*/ lci_ind=p1plus-pplus1-1.96*se_ind; uci_ind=p1plus-pplus1+1.96*se_ind; procprint; run;

  15. SAS code McNemar’s Test data 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;

  16. PROC FREQ • For square tables, the AGREE option in PROC FREQ provides the McNemar chi-squared statistic for binary matched pairs, the X2test 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.

  17. 10.1.3 Increased precision with dependent samples

  18. Fit marginal model data 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 ; proclogisticdata=matched; weight count; model response=occasion; run; Xt proc genmod data=matched1 DESCENDING; weight count; model response=occasion/dist=bin link=identity;

  19. Matlab code for deriving previous MLE and SE %% page 417 syms bn21n12 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

  20. 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

  21. 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.

  22. Now, for subject t in matched pair i, consider the model • the conditional ML estimate of OR is

  23. 10.2.6 Conditional ML for matched pairs with multiple predictors

  24. 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

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