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Previous approaches. Assume that the dependent variable is distributed normally dynamic regression models include lagged values of x on the right hand side of the regression equation autoregressive integrated moving-average models (ARIMA) to control for residual auto-correlation Poisson regre
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1. By Carracedo-Martnez et al
Case-Crossover Analysis of Air Pollution Health Effects: A Systematic Review of Methodology and Application.
Environ Health Perspect 118:11731182 (2010). Case Cross-Over Design 1
2. Previous approaches Assume that the dependent variable is distributed normally
dynamic regression models
include lagged values of x on the right hand side of the regression equation
autoregressive integrated moving-average models (ARIMA)
to control for residual auto-correlation
Poisson regression 2
3. Poisson regression aka a log-linear model
the dependent variable is a count
does not require knowledge of the denominator (the entire population) as long as population flux is in steady state
when cases can be enumerated
the dependent variable is a rate, where the rate is a count of events occurring to a particular unit of observation
an offset 3
4. Limitation of Poisson Regression the parametric functions of time or of its sinusoidal transforms cannot be easily adapted the cyclical component of varying frequency
nonparametric Poisson regression
d.f. of the smoothed nonparametric function must be specified by the researcher
Poisson regression with the application of generalized additive models (GAMs)
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5. Case-crossover (CCO) design proposed by Maclure (1991)
to identify risk factors of acute events
each subject serves as his or her own control by assessing referent exposure at a point in time prior to the event
initially used to assess the effect of exposures measured at an individual level
not applicable to exposures with a time trend, such as air pollution
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6. bidirectional CCO developed by Navidi (1998)
having control time periods before and after the event
appropriate for ecologic-type exposures
such as air pollution, because the existence of registries means that the values of such exposure can be ascertained even after the event
pollution values are not affected by the presence of prior morbidity and mortality events 6
7. Other Varients for control of biases full-stratum bidirectional
proposed by Navidi
symmetric bidirectional case-crossover
proposed by Bateson & Schwartz
semisymmetric bidirectional case-crossover
proposed by Navidi & Weinhandl
time-stratified case-crossover
proposed by Lumley & Levy 7
8. 8
9. Bias and Power (Figueiras et al. 2005) full semisymmetric design
the least bias together with the best coverage and statistical power
but unstable when the beta value varied with respect to the usual values.
semisymmetric CCO
fewer biases than did symmetric or time-stratified CCO (both of which yielded similar results)
but a lower statistical power 9
10. Advantages of CCO Studies no confounding by time-fixed characteristics (self-matching)
control of time-varying confounders (short-interval reference-window)
directly estimate the effect of personal exposures, and assess effect modification of exposure by individual attributes
no concurvity as in Poisson GAM models (the nonparametric analogue of multicollinearity) 10
11. Problems fails to appropriately account for fluctuations in time (assumption that confound the exposure, the effect estimate is biased)
perform model-checking
Discuss this later on
Arbitrariness in the selection of reference periods or sampling method 11
12. Comparisons of varied CCO designs Most popular
Symmetric bidirectional CCO
Time-stratified CCO
Conclusions about semi-symmetric CCO are contradictory
But statistical power is no good
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13. Steps of Applying CCO Designs to Study the Relationship between Air Pollution and Health 13
14. Steps of Applying CCO Designs to Study the Relationship between Air Pollution and Health 14
15. Steps of Applying CCO Designs to Study the Relationship between Air Pollution and Health 15
16. Steps of Applying CCO Designs to Study the Relationship between Air Pollution and Health 16
17. Log-Linear Time Series Models log-linear regression model (Lu et al. 2008)
Yt is the number of events with ,
where ? is the over-dispersion parameter, Xt is the exposure such as air pollution, and St is the value of a smooth function of time at time t
Most case-crossover analyses rely on conditional logistic regression and assume that all subjects are independent
the CCO approach is equivalent to a log-linear time-series model without over-dispersion 17
18. Model Checking Standardized Residuals
use residual plot to reveal outliers, autocorrelation, or cyclic effects
Q-Q plots of can be used to check the assumption
Q-Q plot: a graphical method for comparing two probability distributions by plotting their quantiles against each other 18
19. Example: Model Checking before (top) and after (bottom) removing influential points
Method A is a time-stratified case-crossover design
Method D is a time-series method using a natural spline with 8 degrees of freedom 19
20. Model Checking Dffits (large|Dffits|? influential)
check for highly influential events in case-crossover studies
where and are the predicted outcomes at time t with or without the observation in the regression, is the SE estimated without ;
the leverage statistic for time t, which is the tth diagonal elements of the projection matrix H 20
21. Example: Model Checking before (top) and after (bottom) removing influential points
Method A is a time-stratified case-crossover design
Method D is a time-series method using a natural spline with 8 degrees of freedom
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22. Simulations of Statistical Power Analyze statistical power (Symons et al. 2006)
Estimated results from 500 simulations each for three fixed values of b (beta coefficient from conditional logistic regression model).
sufficient power to detect statistically significant effects for odds ratios greater than 1.20 for an IQR difference (i.e. OR=exp(9.2*0.02)=1.2) in 24-hour averaged PM2.5
uncertainty increased for
detecting a smaller ?=0.01
(OR=1.10) 22