28.4 Case-Crossover Study Design

A case-crossover study is an observational epidemiological method designed primarily to assess the transient effects of acute exposures on the risk of sudden-onset outcomes. Introduced by Maclure (1991), this design utilizes individuals as their own controls, thus inherently adjusting for stable, within-subject confounders.

For example, consider assessing whether vigorous exercise increases the risk of myocardial infarction (MI) (Mittleman et al. 1993):

• Hazard Period: 1-hour preceding MI onset.
• Control Periods: Identical 1-hour windows on other days.

Applying conditional logistic regression, we estimate the odds ratio (OR) indicating how acute exercise influences MI risk.

Implementation in R: Z. Zhang (2016) and season package

Advantages

• Self-Matching: Controls individual-level confounding, including genetics, demographics, and chronic health status (Maclure 1991).
• Efficiency: Requires fewer subjects due to within-person matching (Mittleman, Maclure, and Robins 1995).
• Applicability: Highly suitable for studying acute triggers (e.g., air pollution, physical exertion, drug use) (Lumley and Levy 2000).

Limitations

• Carryover Effects: Difficulties arise if exposure has prolonged effects (Maclure and Mittleman 2000).
• Time-Varying Confounding: Cannot inherently control for confounders changing within short timeframes unless explicitly modeled (Navidi 1998).
• Selection Bias: Incorrect selection of control periods can bias results (Lumley and Levy 2000).

28.4.1 Mathematical Foundations

The case-crossover design compares exposure status during a “hazard” or “case” period immediately preceding the event to exposure status during “control” periods when the event did not occur (Maclure and Mittleman 2000).

Formally, let:

• $X_{i,t}$ denote the exposure status (binary or continuous) of individual $i$ at time $t$.
• $Y_{i,t}$ indicate the occurrence (1) or absence (0) of an acute event at time $t$.

The probability of an event occurring at time $t$ given exposure is modeled by a conditional logistic regression:

$$\text{logit}(P(Y_{i,t}=1|X_{i,t})) = \alpha_i + \beta X_{i,t}$$

Here:

• $\alpha_i$ is a subject-specific fixed effect that controls for all time-invariant confounding within subjects.

• $\beta$ is the log-odds ratio quantifying the exposure effect.

The conditional logistic regression explicitly considers matched strata (individual cases) and compares the distribution of exposure during the case and control periods within each individual:

$$L(\beta) = \prod_{i=1}^{n} \frac{\exp(\beta X_{i,case})}{\exp(\beta X_{i,case}) + \sum_{j \in C_i} \exp(\beta X_{i,j})}$$

• $X_{i,case}$ represents exposure during the hazard period for case $i$.
• $X_{i,j}$ represents exposure during control periods $C_i$ for case $i$.

28.4.2 Selection of Control Periods

28.4.2.1 Bidirectional

Bidirectional (symmetrical) selection involves choosing control periods before and after the event equally (Navidi 1998):

$$\text{Control Periods} = \{t - k, t + k\}, k \in \mathbb{N}$$

This method mitigates biases related to temporal trends in exposure.

28.4.2.2 Unidirectional

Unidirectional selection (either exclusively prior or subsequent to the event) might introduce biases if exposure trends are strong (Lumley and Levy 2000):

$$\text{Control Periods} = \{t - k\}, k \in \mathbb{N}$$

28.4.3 Assumptions

1. Transient Effect Assumption: Exposure effects are acute and short-lived (Maclure 1991).
2. Exchangeability of Hazard and Control Periods: Conditional on individual fixed effects, control periods accurately represent exposure probabilities during hazard periods (Mittleman, Maclure, and Robins 1995).
3. No Carryover Effects: Exposure in one period does not affect subsequent periods (Maclure and Mittleman 2000).

References

Lumley, Thomas, and Drew Levy. 2000. “Bias in the Case–Crossover Design: Implications for Studies of Air Pollution.” Environmetrics: The Official Journal of the International Environmetrics Society 11 (6): 689–704.

Maclure, Malcolm. 1991. “The Case-Crossover Design: A Method for Studying Transient Effects on the Risk of Acute Events.” American Journal of Epidemiology 133 (2): 144–53.

Maclure, Malcolm, and MA Mittleman. 2000. “Should We Use a Case-Crossover Design?” Annual Review of Public Health 21 (1): 193–221.

Mittleman, Murray A, Malcolm Maclure, and James M Robins. 1995. “Control Sampling Strategies for Case-Crossover Studies: An Assessment of Relative Efficiency.” American Journal of Epidemiology 142 (1): 91–98.

Mittleman, Murray A, Malcolm Maclure, Geoffrey H Tofler, Jane B Sherwood, Robert J Goldberg, and James E Muller. 1993. “Triggering of Acute Myocardial Infarction by Heavy Physical Exertion–Protection Against Triggering by Regular Exertion.” New England Journal of Medicine 329 (23): 1677–83.

Navidi, William. 1998. “Bidirectional Case-Crossover Designs for Exposures with Time Trends.” Biometrics, 596–605.

Zhang, Zhongheng. 2016. “Case-Crossover Design and Its Implementation in r.” Annals of Translational Medicine 4 (18): 341.