Structural Modelling of Nonlinear Exposure- Response Relationships for Longitudinal Data Xiaoshu Lu and Esa-Pekka Takala Finnish I nstitute of Occupational Health, Finland
� Background � Mathematical Model � Model Validity and Illustration � Conclusions Paris-08,2010
Background � Many research is grounded on exposure and risk assessment � Linear model often used to assess exposure- response relationship � Standard methods provide few theories for nonlinear exposure-response studies � See an example in the following Paris-08,2010
4 Exposure Response 3 2 1 0 -1 0 5 10 15 20 25 30 Time (days) Linear mixed-effects model shows parameter estimate of -0.61 is statistically discernible at 5% level. Response is negatively associated with exposure which is incorrect. Paris-08,2010
Mathematical Model • Model equations • Model estimation Paris-08,2010
Model Equations • Methodological framework for model buildup • Model is quivalent to a mixed- effects model Paris-08,2010
Model Buildup • Let {x} t and {y} t be exposure and response measures for any subject • Use Hodrick-Prescott (HP) filter technique to extract the trend-cycle component • Obtain structural mapping of exposure to response for individual subject • Extend the mapping to group subjects by adding random subject effects Paris-08,2010
Model Equations t + ε y y t = y trend trend (1) t t+ 1 = 2 y trend t – y trend t- 1 + ε y cycle y trend (2) t similarly t + ε x x t = x trend trend (3) t t+ 1 = 2 x trend t – x trend t- 1 + ε x cycle x trend (4) t Paris-08,2010
y t = x t α + � ( t - j ) η j Where η t ∼ N(0, σ η 2 ) This is for individual subject Paris-08,2010
y it = x it α + x it u i + � ( t - j ) η j + ε it Where ε it ∼ N(0, σ ε 2 ) This is for group subjects where u i is inserted to account for subject-specific variation from the group mean Paris-08,2010
In matrix form ( mixed-effects model) Y = Xa + Z u u + Z η η + ε Σ u 0 0 0 u η = Σ N ( 0 , 0 0 ) η ε Σ 0 0 0 ε Σ u = σ u 2 I , Σ ε = σ ε 2 I V = Var( Y ) = σ u T + Z η Σ η Z η T + σ ε 2 Z u Z u 2 I Paris-08,2010
Model Estimation • If V is known the estimates are the best linear unbiased predictors (BLUPs) of the model • If V is unknown the estimates of parameters and V are jointly using iterative methods • In SAS's MIXED procedure, for example, modified Newton-Raphson method is adopted Paris-08,2010
Model Validity and Illustration • Consider the hypothetical data in Fig. 1 • Define y t as the response and x t the time- varying exposure at the t th day • The proposed model has the following exposure-response form t + ε t y trend t = a 0 + a 1 x trend where y trend t and x trend t are calculated according to HP decomposition Paris-08,2010
Results and comparison of model fit to the hypothetical data Response Pr > χ 2 Proposed model Linear mixed-effects model Exposure ( a 1 ) 1.86 *** -0.61 *** AIC (smaller *** -6.3 89.7 better) p *** <0.001; p ** <0.005; p * <0.1 Paris-08,2010
Conclusions • Exposure measures are common in many fields • We present some ways to structural modelling nonlinear longitudinal data that can not easily be modeled by traditional statistical methods • The proposed approach includes the deseasoning method as a special case which is often limited to a time series only. • The developed model is computationally attractive as various software packages and routines exist to perform the final obtained mixed-effects model Paris-08,2010
Thank You Fo Thank Y ou For You r Your Atten r Attention tion Thank Y Thank You Fo ou For You r Your Atten r Attention tion Paris-08,2010
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