Structural Modelling of Nonlinear Exposure- Response Relationships - - PowerPoint PPT Presentation

Structural Modelling of Nonlinear Exposure- Response Relationships for Longitudinal Data

Xiaoshu Lu and Esa-Pekka Takala

Finnish I nstitute of Occupational Health, Finland

Paris-08,2010

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

• 1

1 2 3 4 5 10 15 20 25 30 Time (days)

Exposure Response

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.

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Mathematical Model

• Model equations
• Model estimation

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Model Equations

• Methodological framework for

model buildup

• Model is quivalent to a mixed-

effects model

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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

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Model Equations

yt = ytrend

t+ εy trend t

(1) ytrend

t+ 1 = 2 ytrend t – ytrend t-1+ εy cycle t

(2) similarly xt = xtrend

t+ εx trend t

(3) xtrend

t+ 1 = 2 xtrend t – xtrend t-1+ εx cycle t

(4)

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yt = xtα + (t

• j)ηj

Where ηt ∼ N(0, ση

2)

This is for individual subject

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yit = xitα + xitui+ (t

• j)ηj + εit

Where εit ∼ N(0, σε

2)

This is for group subjects where ui is inserted to account for subject-specific variation from the group mean

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In matrix form ( mixed-effects model)

Y = Xa + Zuu + Zηη + ε

) , (           Σ Σ Σ           =          

ε η

ε η

u

N u

Σu = σ u

2I, Σε = σε 2I

V = Var(Y) = σ u

2Zu Z u T + ZηΣη Zη T+ σε 2I

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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

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Model Validity and Illustration

• Consider the hypothetical data in Fig. 1
• Define yt as the response and xt the time-

varying exposure at the tth day

• The proposed model has the following

exposure-response form ytrend

t = a0 + a1xtrend t + εt

where ytrend

t and xtrend t are calculated according

to HP decomposition

Paris-08,2010

Results and comparison of model fit to the hypothetical data

***

89.7

• 6.3

AIC (smaller better)

• 0.61***

1.86***

Exposure (a1)

Pr > χ2 Linear mixed-effects model Proposed model

Response

p***<0.001; p**<0.005; p*<0.1

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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

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