Julia Schedler

Maclure (1991) introduced the case-crossover design as a way to assess the effect of a transient exposure on an acute outcome.

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”

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.

\[ \begin{aligned} p_{it} &= P(T_i , \sum_{m = 1}^{N_T}Y_{im} = 1 = t \vert X, R(t) )\\ &= \frac{\lambda_{0i}\exp(\beta X_{it} + \gamma_{it})}{\sum_{j \in R(t)}\lambda_{0i}\exp(\beta X_{ij} + \gamma_{ij})} \end{aligned} \]

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

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.

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 \(X_{st}\)?

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.

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

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

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 💟

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 |

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

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

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!