Conundrum in pharmacoepidemiology: Contextualising biases in case-crossover studies

Dr Kiyoshi Kubota, physician and pharmacoepidemiologist based at the NPO Drug Safety Research Unit Japan, has pioneered innovative methods for analysing data in case-crossover studies. By tackling biases inherent to case-crossover studies, including within-subject exposure dependency and lack of pair-wise exchangeability, his methodologies offer a clearer lens for evaluating medication safety and effectiveness. The contributions of Kubota and his collaborator, Dr Thu-Lan Kelly, will encourage the effective use of case-crossover studies in pharmacoepidemiology.

The case-crossover (CXO) study design belongs to a set of self-controlled designs which include the CXO design proposed by Maclure in 1991 and the self-controlled case-series (SCCS) design by Farrington in 1995. Maclure proposed the CXO design as ‘a case-control design involving only cases where brief exposure causes a transient change in risk of a rare acute-onset disease’ and used the method to examine the link between physical exercise and acute myocardial infarction. Because the CXO and SCCS designs use data from cases only, that is, subjects who experience the event or disease that is being studied, they are also called ‘case-only’ designs.

CXO method assumptions

As in Maclure’s original description above, one of the key assumptions of the CXO method is that the effect of an exposure is ‘transient’. This is later rephrased as the assumption of no ‘carry-over’ effect, where the effect of an exposure disappears completely before the next period begins. As indicated by Maclure, the CXO design is usually analysed by the method for matched case-control studies and the exposure status is compared between the ‘case period’, the period when the event occurs, and one or more ‘control periods’ (which precede the case period) of the same subject.

Another key assumption of the CXO method is that exposure should be ‘brief’. The definition of ‘brief’ was, however, not very clear. One may agree that the drug use is ‘brief’ when local anaesthetic is used for local surgery, or when an analgesic is used ‘as needed’ by a patient with periodic headaches. However, the use of most drugs is not that ‘brief’ but rather repetitive or chronic. So, should we withhold the CXO method for most drug treatments, or is the CXO method usually useless in pharmacoepidemiology? On the other hand, it is well known that in the ‘real world’, patients frequently stop and restart antihypertensive drugs or lipid-lowering drugs, even though they are assumed to be used without interruptions. Then, is the use of the CXO method acceptable if a period is defined as one month or even one year? Do gaps between time periods matter? Most importantly, what problems emerge when the drug exposure is not ‘brief’?

Discrepancy of the odds ratios between the Mantel-Haenszel and standard conditional logistic methods

Around 2017, Dr Kiyoshi Kubota, a Japanese pharmacoepidemiologist, was astonished to find that the estimate in a CXO study was starkly different when using the Mantel-Haenszel (MH) method compared with the standard conditional logistic (SCL) regression method. Both the MH and SCL methods have been widely used to analyse data in matched case-control and CXO studies. Dr Kubota found that the result of the same CXO data showed the odds ratio (OR) 2.6 (95% confidence interval 1.1-6.2) when analysed by the MH method, but 11.5 (3.1-42.9) by the SCL method. He reported this finding in an international meeting for Pharmacoepidemiology held in Brisbane, Australia in 2017 which led to an encounter with Dr Kelly.

Dr Kubota sought to understand why such discrepancy could occur between the MH and SCL methods and discovered a paper by Vines and Farrington. In brief, they showed that when exposure status between two periods is not independent (or when ‘within-subject exposure dependency’ exists), the results of the SCL method can be biased even when the MH method is not. This non-independence is common in drug studies when medicines are taken continuously for extended periods.

How to detect and remove two types of biases inherent to the CXO study

Another key message by Vines and Farrington was that the method used to analyse CXO data should be different from the SCL used for the matched case-control studies. In the CXO study, probabilities for many ‘permutations’ of exposure in the case and control periods should be incorporated into the conditional probability as weights (see Equation (2) in their paper) while in the matched case-control studies, there is no need to consider the effect of such ‘permutation’ because there is no natural ‘order’ in case and control subjects. Vines and Farrington indicated that when the probabilities of all permutations are the same (when ‘global exchangeability’ holds), the conditional probability for the CXO study is equivalent to the SCL.

Drs Kubota and Kelly have indicated that when the SCL is used in a CXO study where ‘global exchangeability’ does not hold, at least two types of biases can occur, and those two types of biases may occur separately or simultaneously (Figure 1). The first type is ‘bias due to lack of pairwise exchangeability’ or bias when exposure is not exchangeable between periods. This type of bias occurs, for example, when trends in drug use change over the study time period and can be removed by the ‘case-time-control’ method of Suissa (1995) by adding ‘time-controls’ (a sample of non-cases) to estimate the exposure trends. This bias can also occur without exposure time trends when some exposed patients are censored (see Figure 1 A2).

The second type is ‘bias due to within-subject exposure dependency’ due to extended, continuous periods of drug use. As illuminated by Vines and Farrington, using SCL in this situation produces bias which can be detected and removed by using the cases only. Drs Kubota and Kelly showed that when the drug use is successive and the number of control periods is 2, 10, and 60, the ORs by the SCL are overestimated as 4.8, 6.6, and 7.8 respectively, when the true OR equals 4 (Figure 2A). The MH method is unbiased if the first type of bias does not exist. However, the bias from the SCL method can be exaggerated if the first type of bias exists. In some of the previous studies analysing real-world CXO data, an increase in ORs with increasing control periods was observed. Those observations may be at least in part explained by the second type of bias due to within-subject exposure dependency.

Drs Kubota and Kelly have shown that the first type of bias can be detected when the OR of time-controls differs from 1.0. When this happens, two ORs are estimated by the MH method using the cases and ’time-controls’ separately, and the former OR is divided by the latter OR to estimate an unbiased OR. They have also shown that the second type of bias can be detected by comparing ORs by the Mantel-Haenszel (MH) method with that by the SCL. If the discrepancy is more than a pre-specified level (eg, 5% or 10% of the MH OR), the MH method should be used but not the SCL method.

One of the shortcomings of the MH method is difficulty in handling a time-varying confounder. In 2021, Dr Kubota and his colleagues proposed a novel weighting method (inspired by the paper by Greenland in 1999) where weights are applied to the case and control periods to remove the second type of bias. Using this method, a time-varying confounder can be incorporated as a co-variate. When the first type of bias also exists, time-controls can be included to estimate an unbiased OR by the weighting method.

Currently, Dr Kubota is working to further develop the method for the exposure with three or more levels and for a time-varying confounder not independent between periods.