06Binary - PubH8452 Longitudinal Data Analysis Fall 2011...

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Unformatted text preview: PubH8452 Longitudinal Data Analysis - Fall 2011 Likelihood-Based Methods for Repeated Binary Data Likelihood-Based Methods for Repeated Binary Data Outline • Multivariate normal distribution vs. joint multinomial distribution • Log-linear model • A hybrid model • Bahadur model • Modeling marginal odds ratios 1 PubH8452 Longitudinal Data Analysis - Fall 2011 Likelihood-Based Methods for Repeated Binary Data Joint Multinomial Distribution • An n-vector of binary variables Y has an exact joint multinomial distribution with 2 n points in its sample space. • In the most general case, the multinomial distribution has 2 n- 1 number of parameters. • A subset of the n-vector Y , say Y s , also has a multinomial distribution. The parameters of Pr( Y s ) are sums of the parameters of Pr( Y ). • The variances are functions of the means. • To relate covariates to the means μ , a nonlinear link function is typically used (logit, probit). 2 PubH8452 Longitudinal Data Analysis - Fall 2011 Likelihood-Based Methods for Repeated Binary Data Issues with Modeling Repeated Binary Data • Parsimony: constrains higher-order associations to be zero. • Flexibility: allows dependence on covariates. • Interpretability: e.g., odds ratio is more natural than correlation. 3 PubH8452 Longitudinal Data Analysis - Fall 2011 Likelihood-Based Methods for Repeated Binary Data The Log-Linear Model • Log-linear models (Bishop et al, 1975) have been popular in studying multiple correlated categorical (binary) variables. • The saturated log-linear model is: log Pr( Y = y ) = c ( θ ) + n X j =1 θ j y j + X j 1 <j 2 θ j 1 ,j 2 y j 1 y j 2 + ··· + θ 1 ,...,n y 1 ··· y n , (1) where c ( θ ) is a normalizing constant. • θ is a (2 n- 1)-vector of canonical parameters: θ = ( θ 1 , . . . , θ n , θ 12 , . . . , θ n- 1 ,n , . . . , θ 1 ,...,n ) T . • θ can be viewed as a log-linear transformation of the multinomial cell probabilities π (an 2 n vector), θ = C T 1 log π , 4 PubH8452 Longitudinal Data Analysis - Fall 2011 Likelihood-Based Methods for Repeated Binary Data where C 1 is an 2 n × (2 n- 1) matrix. • The elements of θ can be partitioned as: main effects θ 1 , . . . , θ n n 2-way effects θ 12 , θ 13 , . . . , θ n- 1 ,n ( n 2 ) 3-way effects θ 123 , θ 124 , . . . , θ n- 2 ,n- 1 ,n ( n 3 ) ....
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06Binary - PubH8452 Longitudinal Data Analysis Fall 2011...

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