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

Julia Schedler

## 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

• has issues when trends are present in outcome variable

• partitions the study period-- no overlap bias

Symmetric bi-directional:

• 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}

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
Weekday

0 if Saturday or Sunday

1 otherwise

Strata

Moth

Month x Time of day

Month x Weekday

Time of day x Weekday

## 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!