lecture8_categorical_LB_2011

lecture8_categorical_LB_2011 - Lecture 8: Logit Models for...

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Unformatted text preview: Lecture 8: Logit Models for Nominal and Ordinal Responses and Introduction to Methods for Repeated Categorical Data Laurent Briollais Samuel Lunenfeld Research Institute, Mount Sinai Hospital Dalla Lana School of Public Health, UofT November 14, 2001 Outline Fixed effects logit models for ordinal/nominal responses Overview of various approaches for repeated/mutliple categorical data analysis MLE with constraints for the comparison of marginal distributions Modeling transition probabilities between multiple binary/categorical re- sponses Examples in SAS Concluding remarks 1 CHL 5210 Fall 2011 Categorical data analysis Lecture #11 2 1 Fixed effects logit models for ordinal/nominal re- sponses 1.1 Baseline category logit models for nominal responses Let Y be categorical with J levels. Let j ( x ) = P ( Y = j | x ) with j j ( x ) = 1. Logit models pair each response Y = j with the baseline category, here Y = J . log j ( x ) J ( x ) = j + j x, for j = 1 , , J- 1 The parameters are = ( 1 , ..., J- 1 ) and ( 1 , ..., J- 1 ). if each j is p- 1 dimensional, then there are ( J- 1) + ( p- 1)( J- 1) = ( J- 1) p parameters to estimate. For a fixed x , the ratio of probabilities Y = a versus Y = b is given by a ( x ) b ( x ) = exp { ( a- b ) + ( a- b ) x } . Note: this model reduces to ordinary logistic regression when J = 2. The response probability for the j-th category can be expressed as j ( x ) = exp( j + j x ) 1 + J- 1 h =1 exp( h + h x ) with J = 0 and J = 0. CHL 5210 Fall 2011 Categorical data analysis Lecture #11 3 Multicategory logit models are special GLMs . Let y i = ( y i 1 , y i 2 , ..., y iJ ). The multivariate GLM has the following form g ( i ) = X i , where i = E ( Y i ) The baseline-category logit model is a multivariate GLM with y i = ( y i 1 , ...., y i,J- 1 since y iJ is redundant. Then i = ( 1 ( x i ) , ..., J- 1 ( x i )) and g j ( u i ) = log { ij / [1- ( i 1 + ... + i,J- 1 )] } . The model matrix for observation i is X i = 1 x i 1 x i 1 x i with 0 entries in other locations, and = ( 1 , 1 , ..., J- 1 , J- 1 ). CHL 5210 Fall 2011 Categorical data analysis Lecture #11 4 1.2 Cumulative logit models for ordinal responses Why analyze a response as ordinal? Efficiency: Armstrong & Sloan (1989, Am. J. Epid.) report efficiency losses between 89% to 99% comparing an ordinal to continuous outcome, depending on the number of categories and distribution within the ordi- nal categories. Bias: continuous model can yield correlated residuals and regressors when applied to ordinal outcomes, because the continuous model does not take into account the ceiling and floor effects of the ordinal outcome....
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This note was uploaded on 02/23/2012 for the course CHL 5210H taught by Professor Leisun during the Fall '11 term at University of Toronto- Toronto.

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lecture8_categorical_LB_2011 - Lecture 8: Logit Models for...

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