09Trans - PubH8452 Longitudinal Data Analysis - Fall 2011...

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PubH8452 Longitudinal Data Analysis - Fall 2011 Transition Models Transition Models Outline Model Specification Fitting Transition Models Transition Models for Binary Responses Data Transition Models example: ICHS data 1
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PubH8452 Longitudinal Data Analysis - Fall 2011 Transition Models Transition Models The distribution of the observed response at time j , Y ij , is modeled conditionally as an explicit function of the past responses H ij = ( Y i 1 ,...,Y ij - 1 ) and covariates X ij . Typically, a Markov model is assumed, that is, Y ij only depends on q (the order of the Markov process) previous responses Pr( Y ij |H ij ) = Pr( Y ij | Y ij - 1 ,...,Y ij - q ) . For notational convenience, we assume that the observational times are equally spaced. If they aren’t, we need stronger assumptions about the functional form of the time dependence. 2
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PubH8452 Longitudinal Data Analysis - Fall 2011 Transition Models Model Specification Y ij |H ij is assumed to be from exponential family distribution: f ( y ij |H ij ) = exp { [ y ij θ ij - b ( θ ij )] + c ( y ij ) } . Conditional mean μ C ij = E( Y ij |H ij ) = b 0 ( θ ij ) satisfies g ( μ C ij ) = X T ij β + q X r =1 f r ( H ij ; α ) for some functions f r ( · ). Conditional variance v C ij = Var( Y ij |H ij ) = b 00 ( θ ij ) φ satisfies v C ij = V ( μ C ij ) φ. 3
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PubH8452 Longitudinal Data Analysis - Fall 2011 Transition Models Examples Continuous response : linear regression with autoregressive errors. Y ij = X T ij β + q X r =1 α r ( y ij - r - X T ij - r β ) + ² ij , where ² ij are iid zero-mean Gaussian r.v.’s. E [ Y ij ] = X T ij β no matter what q is. Binary responses : g ( μ C ij ) = logit( μ C ij ) = X T ij β + q X r =1 α r y ij - r . The interpretation of the regression coefficients depends on the order q (i.e. β = β q ). Count responses : q = 1 log( μ C ij ) = X T ij β + α (log y * ij - 1 - X T ij - 1 β ) 4
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PubH8452 Longitudinal Data Analysis - Fall 2011 Transition Models where y * ij - 1 = max( y ij - 1 ,c ); 0 < c < 1 which leads to μ C ij = e X T ij β ˆ y * ij - 1 exp( X T ij - 1 β ) ! α . The constant c prevents y i,j - 1 = 0 from being an absorbing state (otherwise Y ij - 1 = 0 Y ik = 0 for all k j ). For α < 0, a response at time t - 1 greater than e X T t - 1 β (not its expected value) decreases the expectation for the current response. When α > 0 the opposite occurs (positive correlation). 5
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PubH8452 Longitudinal Data Analysis - Fall 2011 Transition Models Fitting Transitional Models The likelihood of Y i is not always fully specified. In a first order Markov model, the likelihood contribution for the i th subject L i ( Y i 1 ,...,Y in i ) = f ( Y i 1 ) f ( Y i 2 | Y i 1 ) ··· ,f ( Y in i | Y in i - 1 ) = f ( Y i 1 ) n i Y j =2 f ( Y ij | Y ij - 1 ) . In a Markov model of order
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09Trans - PubH8452 Longitudinal Data Analysis - Fall 2011...

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