{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecture8_categorical_LB_2011

lecture8_categorical_LB_2011 - Lecture 8 Logit Models for...

This preview shows pages 1–6. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 π j ( x ) = P ( Y = j | x ) with j π j ( x ) = 1. Logit models pair each response Y = j with the baseline category, here Y = J . log π j ( x ) π J ( x ) = α j + β 0 j x, for j = 1 , · · · , J - 1 The parameters are α = ( α 1 , ..., α J - 1 ) and ( β 1 , ..., β J - 1 ). if each β 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 π a ( x ) π b ( x ) = exp { ( α a - α b ) + ( β a - β b ) 0 x } . Note: this model reduces to ordinary logistic regression when J = 2. The response probability for the j -th category can be expressed as π j ( x ) = exp( α j + β 0 j x ) 1 + J - 1 h =1 exp( α h + β 0 h x ) with α J = 0 and β 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 ( μ i ) = X i β, where μ 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 μ i = ( π 1 ( x i ) , ..., π J - 1 ( x i )) 0 and g j ( u i ) = log { μ ij / [1 - ( μ i 1 + ... + μ i,J - 1 )] } . The model matrix for observation i is X i = 1 x 0 i 1 x 0 i · · · 1 x 0 i with 0 entries in other locations, and β 0 = ( α 1 , β 0 1 , ..., α J - 1 , β 0 J - 1 ).

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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. This can result in biased estimates of regression coefficients and is most critical when the ordinal variables is highly skewed. Logic: continuous model can yield predicted values outside of the range of the ordinal variable.
CHL 5210 Fall 2011 Categorical data analysis Lecture #11 5 One way to use category ordering forms logits of cumulative probabilities.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 35

lecture8_categorical_LB_2011 - Lecture 8 Logit Models for...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online