A spatiotemporal case crossover model of asthma attacks in the City of Houston
Case-crossover
Maclure (1991) introduced the case-crossover design as a way to assess the effect of a transient exposure on an acute outcome.
A comparison of the case-crossover design and the case-control design.
Case-crossover model
The case-crossover design uses conditional logistic regression to fit the following relative risk model:
\[ \lambda_i(t, X_{it}) = \lambda_{0it} \exp(\beta X_{it}) = \lambda_{0i}\exp(\beta X_{it} + \gamma_{it}). \]
- individual, time-varying nuisance factors (\(\gamma_{it}\)) drop out of the model when conditioning on “reference windows”
Relative risk model, continued
The case-crossover assumption is important in the estimation of the probability that subject \(i\) fails at time \(t\), given that \(t\) is in a pre-specified reference window R.
\[ \begin{aligned} p_{it} &= P(T_i , \sum_{m = 1}^{N_T}Y_{im} = 1 = t \vert X, R(t) )\\ &= \frac{\lambda_{0i}\exp(\beta X_{it} + \gamma_{it})}{\sum_{j \in R(t)}\lambda_{0i}\exp(\beta X_{ij} + \gamma_{ij})} \end{aligned} \]
The case-crossover assumption
Demonstration of how appropriately chosen reference windows can capture temporary periods of constant risk which can be conditioned to “use the cases as their own controls”.
Choice of reference window
Two popular choices:
Time-stratified: divides study period into pre-specified reference windows
leads to unbiased estimates
has issues when trends are present in outcome variable
partitions the study period-- no overlap bias
Symmetric bi-directional:
leads to biased estimates
does not partition the study period, leading to overlap bias
adjustments exist (semi-symmetric bi-directional), but are complicated to implement
Equivalence with Poisson Regression
If constant exposure is assumed $(X_{it} = X_t)$, a Poisson regression model can be constructed to yield equivalent estimates, provided the structure of the nuisance term is chosen correctly for a given reference window design.
Time-stratified: include indicator variables for strata
Symmetric bi-directional: locally weighted running mean smoother
Lu and Zeger (2007) note that this equivalence can be leveraged to evaluate case crossover analysis using GLM diagnostics.
Constant Exposure Assumption
Pro: allows equivalence with Poisson regression
Con: a simplifying assumption
Given that individual exposure is tied to location, could the constant exposure assumption be relaxed by replacing $X_{it}$ with \(X_{st}\)?
Spatial case-crossover
Why it might be useful:
Account for spatial patterns in data analyzed using case-crossover
Relax constant exposure assumption
Why it can be tricky:
Assumes that subject-specific, spatially varying nuisance factors are constant within reference windows
Reference windows are sets of spatial locations
An individual is not guaranteed to be in the same super neighborhood during all reference windows
Questions
Is there an approach that will respect spatial structure while still being viewed as a case-crossover design?
- Perhaps: A spatiotemporal case-crossover where the case-crossover is in time and the autoregression is in space.
Hierarchical GLM
Fit a hierarchical generalized linear model (HGLM) to the asthma data:
Stratification by month and weekday/weekend indicators to construct case-crossover reference groups
Time-independent spatial error term to account for spatial correlation (spatial random effect) constructed using median Hausdorff distance
Ambient ozone by super neighborhood as exposure
Used R package hglm
Model structure
\[ \begin{aligned} Y\vert X, Z, \beta &\sim quasiPoisson(\mu)\\ \mu = E(Y \vert X, Z, \beta) &= X\beta + Z \\ Z\sim N (0, \Sigma &= (I-\rho W)^{-1} ), \end{aligned} \]
quasi-Poisson due to zero-inflation
\(X\) is a vector of covariates, \(\beta\) is the regression parameter
\(Z\) is the spatial random effect, where \(W\) is based on the extended 💟 Hausdorff distance 💟
Different stratifications used in model
| Time variables | Values |
|---|---|
| Time of day | Morning: 6am-10am Midday: 10am-4pm Afternoon/Evening: 4pm-8pm Night: 8pm-6am |
| Month | 1, 2, 3, …, 12 |
| Weekday | 0 if Saturday or Sunday 1 otherwise |
| Strata |
|---|
Moth Month x Time of day Month x Weekday Time of day x Weekday |
Parameter estimates
Results: Asthma-related 9-1-1 calls
Spatial modeling did not change conclusions in this case
Spatial variability in asthma attacks may not be captured in the super neighborhood level of aggregation
Results: Asthma-related 9-1-1 calls
Did not find evidence that ambient ozone impacts the prevalence of asthma attacks when stratifying by month
Only one year of data
Ozone and apparent temperature were statistically significant when month was removed from the model
Using additional years of data may allow detection of these effects while still maintaining desired reference groups
The methodology presented here…
capitalizes on the known equivalence of case-crossover design with CLR and count time series models such as Poisson time series
is simple in use and interpretation, while advancing proper statistical methodology
can extend to a wide range of count spatio-temporal processes and spatial structures
Thank you!