Exchangeability in the case-crossover design

Abstract

In cohort and case-control studies, confounding that arises as a result of differences in the distribution of determinants of the outcome between exposure groups leading to non-exchangeability are addressed by restriction, matching or with statistical models. In case-only studies, this issue is addressed by comparing each individual with his/herself. Although case-only designs use self-matching and only include individuals who develop the outcome of interest, issues of non-exchangeability are identical to those that arise in traditional case-control and cohort studies. In this review, we describe one type of case-only design, the case-crossover design, and discuss how the concept of exchangeability can be used to understand issues of confounding, carryover effects, period effects and selection bias in case-crossover studies.

Keywords: Epidemiology; case-crossover.

Figure 1

Study designs to examine the causal effect of an exposure on an outcome

1. In a full cohort study (Panel A), the entire sample is followed for event occurrence; assumption of exchangeability between exposed and unexposed person-time in the entire study base.

2. In a case-control study with traditional incidence density sampling (Panel B), all or a random sample of the cases are identified and controls are sampled at random from the person-time at risk; assumption of exchangeability between exposed and unexposed person-time in the study base represented by the cases and sample of controls. Note that an individual can be randomly selected as a control and later become a case (ID#4) and an individual can be randomly selected as a control more than once (ID#16).

3. Non-exchangeability arising from changes over time can be addressed by matching on time using survival analysis methods in a cohort study (Panel C) or risk-set sampling in a case-control study (Panel D); assumption of exchangeability between exposed and unexposed person-time at the time the case occurred.

4. In a case-crossover study, non-exchangeability arising from slowly varying characteristics is eliminated by matching each case to himself/herself. If event occurrence affects subsequent exposure, a unidirectional case-crossover design (Panel E) is appropriate. Otherwise, a bidirectional case-crossover design (Panel F) can reduce exposure time trends by stratifying on the calendar month and then sampling person-time before and after the event (ID#4), before the event (ID#10) or after the event (ID#14)Assumption of exchangeability between exposed and unexposed person-time within each individual. Note that in the unidirectional design, individuals with no variation in exposure between the hazard and control periods do not contribute information to the estimator (ID#4).

5. In a case-crossover study, non-exchangeability arising from changes over time can be addressed by matching on time or by conducting a case-time control study (Panel G) that incorporates a control group. At the time the case occurs, obtain exposure information on cases and on controls for hazard periods and for control periods; exchangeability between exposed and unexposed person-time within each individual after accounting for a time-trend that is assumed to be the same for cases and controls.

Figure 2

Compared with the frequency of exposure during the control period(s), there is a higher frequency of exposure to both sexual intercourse and marijuana during the hazard period. Assuming that sexual intercourse precedes marijuana use, the association between marijuana use and myocardial infarction may be biased.

Figure 3

Rather than the beta agonist acting alone to increase death risk, asthma exacerbation may lead to higher beta agonist use and it also increases the risk of death.

Figure 4

Rather than a causal effect of physical activity triggering death, asthma exacerbation may confound the association by resulting in both reduced physical activity and higher death risk soon after.

Figure 5

Among all people theoretically eligible to be included in the study, cases who agree to participate are less likely to have recently used an illicit drug; and, by definition, recent MI affects participation in the study.

Figure 6

Among all people theoretically eligible to be included in the study, cases who were physically active immediately prior to the MI are more likely to survive and participate in the case-crossover study than cases who were not recently physically active; and, by definition, recent MI affects participation in the study.

Figure 7

Among all of the person-times theoretically eligible to be included in the study, exposed person-time is more likely to be selected for the control period (oversample times when people are more likely to speak on the phone) or unexposed person-time is more likely to be selected for the control period (e.g. oversample times when people are less likely to speak on the phone). Therefore, control time is related to the probability of exposure and, by definition, it is directly related to the outcome.

Figure 8

The recall of exposure is affected by recent symptom onset; patients may over-estimate or underestimate exposure in the hazard period compared with their recall of exposure during control period(s).

Figure 9

Over time, there are changes in the probability of exposure, and in a unidirectional case-crossover study, the selected hazard period is always later in time. Therefore, the person-time selected for the control period has a different probability of exposure from that of the selected hazard period.

Figure 10

If there are no changes over time in the probability of exposure but there are changes over time in the probability of the outcome, the person-time selected for the control period has a different probability of outcome risk compared with the selected hazard period, but the exposure distribution is correctly represented.

Figure 11

The person-time selected for the control period has a different probability of exposure and a different probability of outcome risk compared with the selected hazard period.