A spatiotemporal case crossover model of asthma attacks in the City of Houston

Julia Schedler

12 maps, one for each month, showing the distribution of asthma attacks on weekdays by superneighborhood. The "hot spots" show some variability across months but the overall spatial pattern throughout the city is similar.


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


  • 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

\[ \begin{aligned} Y\vert X, Z, \beta &\sim quasiPoisson(\mu)\\ \mu = E(Y \vert X, Z, \beta) &= X\beta + Z \\ Z\sim N (0, \Sigma &= (I-\rho W)^{-1} ), \end{aligned} \]

  1. quasi-Poisson due to zero-inflation

  2. \(X\) is a vector of covariates, \(\beta\) is the regression parameter

  3. \(Z\) 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

0 if Saturday or Sunday

1 otherwise



Month x Time of day

Month x Weekday

Time of day x Weekday

12 monthly maps of the predicted number of asthma attacks by superneighborhood. Similar to the observed maps at the beginning of the presentation, but with more "color", indicating the smoothing due to the spatial structure. The overall pattern is visually similar to the observed values.

Parameter estimates

If I forget to add this please send me an email to harass me for not doing it

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!