The case-crossover design ^{1} was developed to study transient effects of exposure on acute events. Only subjects who have failed need to be observed. Inferences about risk are made in accordance with a matched case-control design. Each subject corresponds to a stratum, with the exposure at or just before failure corresponding to the “case,” and exposures sampled at other times serving as “controls.” Because comparisons are made within-subject, all time-invariant confounders are inherently controlled for.

In a standard matched case-control study, the controls that are matched to a case are sampled from a population of eligible controls. To control for confounding, it is common to restrict the set of eligible controls to those who match the case with respect to one or more confounders. In many situations matching is done on a surrogate; in this way, some control can be obtained even when levels of a potential confounder cannot be measured directly. This idea can be applied to case-crossover designs as well, to help control for confounders that vary predictably with time. For example, if control times are matched to the case time by having the same day of the week, confounders that vary on a weekly cycle are controlled for.

The potential problem with restricting the set of eligible control times in case-crossover studies is that bias can then occur from time trends in the exposure of interest itself. The existence of such bias has been noted by Greenland ^{2} and by Navidi. ^{3} For example, unidirectional sampling, in which control times are required to be earlier than the case time, can cause bias if exposures at earlier times are systematically higher or lower than those at later times. This bias is simply an example of selection bias that can occur in any case-control study, when controls are sampled in a way that causes their exposures to differ systematically from the exposure of the case. For example, time trends in exposure can cause bias in a case-control study if control exposures are assessed at a time earlier than that of the case to which they are matched.

The theory of risk set sampling ^{4} has been used to develop methods of sampling controls in matched case-control studies. Bias-free effect estimates are obtained by constructing a likelihood function that is adapted to the sampling method. The theory was originally developed for nested case-control designs , but carries over to population-based matched case-control studies as well. ^{5} The most well-known method of risk set sampling is probably the method of countermatching. ^{5} An excellent nontechnical discussion of risk set sampling is given by Langholz and Goldstein. ^{5} This theory can be applied to case-crossover designs as well, to develop sampling plans that provide good control of unmeasured confounding while avoiding time-trend bias.

Below we describe the principle of risk set sampling and apply it to develop a modification of the symmetric bidirectional (SBI) design of Bateson and Schwartz ^{6} that retains control of unmeasured confounding while eliminating time-trend bias. We also discuss the relation between control sampling methods in case-crossover designs and the use of overdispersion and autocorrelation parameters in Poisson regression. We use as a starting point for this discussion the fact that the bidirectional case-crossover design of Navidi ^{3} turns out to provide effect estimates identical to those of ordinary Poisson regression. ^{7}

Risk Set Sampling
To make the discussion concrete, assume that a given subject could potentially be observed to fail at times T _{1} ,..., T _{N} , with exposure levels X _{1} ,..., X _{N} , respectively. Let T _{k} be the actual failure time. Under a rare disease assumption, with the hazard at exposure level X given by λe ^{βX} , it is appropriate to treat the times T _{1} ,..., T _{N} as if they were individuals in a matched case-control stratum, with X _{k} the exposure of the case, and the other N − 1 values of X as control exposures. This setting is the bidirectional case-crossover design , ^{3} which in the absence of unmeasured confounding, provides unbiased estimates of risk. Following Bateson and Schwartz, ^{6} we will refer to this as the full-stratum bidirectional (FSBI) case-crossover design . The FSBI case-crossover design is a case-control design in which all eligible controls are matched to the case. This design may provide poor control for unmeasured seasonal confounding. Better control can be obtained if control times can be sampled in such a manner that values of unmeasured confounders at the sampled control times are likely to be approximately equal to the value at the case time.

The theory of risk set sampling ^{5} shows how to compute unbiased estimates of relative risk (RR) for any informative method of sampling control information. To explain this method, let us assume that m controls are to be matched to a case with failure time T _{k} . The risk set consists of the m controls that are selected, along with the case. Let R denote the risk set selected, and let π(R |T _{k} ) be the probability that R was selected, given that the failure time was T _{k} . Note that the probability that a given risk set is selected can (and should) depend on the identity of the case. The appropriate contribution to the likelihood from the risk set R is P (T _{k} |R ), the probability that T _{k} was the failure time given that R is the risk set. This value is given by

This likelihood is a weighted version of the standard conditional logistic regression likelihood, with the quantities π(R |T _{j} ) serving as the weights. Risk set sampling theory ensures that use of this likelihood will provide unbiased estimates of RR so long as the sampling plan is informative. A sampling plan is informative if it is not possible to determine the identity of the case simply from knowledge of the risk set. For noninformative sampling plans, terms involving the parameter β will cancel out, so that no estimate can be computed.

This theory can be used to develop a wide variety of case-crossover sampling plans. As an example, we develop a modification of the SBI design of Bateson and Schwartz, ^{6} which retains control of unmeasured seasonal confounding while eliminating bias.

The Symmetric Bidirectional Case-Crossover Design
The SBI case-crossover design of Bateson and Schwartz uses both day of the week and time itself as matching surrogates to help control for unmeasured confounding. This design has been used in studies of air pollution and mortality in Philadelphia, ^{8} Seoul, ^{9} and Barcelona. ^{10} In this design, each stratum consists of two control times with the same day of the week as the case time. In this way, the day of the week is a surrogate for unmeasured confounders that vary on a weekly cycle. Bateson and Schwartz also suggest that the two control times be equally spaced before and after the case time, and that the spacing be fairly close, perhaps 7–28 days. This approach provides additional control, because the close spacing results in an approximate match on time itself (an exact match is impossible), with the result that time is used as a surrogate for all slowly varying unmeasured confounders. In addition, by choosing two controls on opposite sides of the case, it is hoped that the value of any specific confounder will be slightly higher than that of the case in one control, and slightly lower in the other, so that the effects of the imperfect match will tend to cancel out. Simulation studies show that the SBI design can provide good control for many forms of time-varying confounders. ^{6} It is subject to bias, however, as we will discuss below.

As mentioned above, a sampling design is informative only if it is not possible to determine the identity of the case from knowledge of the risk set. The SBI design of Bateson and Schwartz ^{6} is in this sense noninformative, because given knowledge of the risk set, the case is always the middle time. In fact, the contribution P (T _{k} |R ) to the likelihood 1 (Eq 1 ) from such a risk set will always equal 1. To see this point, assume that a subject fails on day T _{k} . The controls to be matched would be T _{k–l} and T _{k+l} , for some short lag l , for example, 7 or 28 days. The risk set R thus consists of the times T _{k–l} , T _{k} , and T _{k+l} . The quantity π(R |T _{k} ) is equal to 1, because this is the only risk set that is ever chosen when the failure time is T _{k} . The quantities π(R |T _{k–l} ) and π(R |T _{k+l} ) are equal to 0, because this risk set would never be chosen if the failure time had been T _{k–l} or T _{k+l} . Thus P (T _{k} |R ) is equal to 1, and cannot be used to estimate the risk parameter β. Bateson and Schwartz ^{6} use the standard conditional logistic regression likelihood with risk set {T_{k−1} , T_{k} , T_{k+1} } . This likelihood is

Eq 2 is not the probability P (T _{k} |R ), which, as discussed above, is equal to 1. It does not properly represent the method by which the risk set is chosen. For this reason, the estimator it produces is subject to bias.

The Semi-Symmetric Bidirectional Case-Crossover Design
Fortunately, with a slight modification of the sampling procedure, the bias in the SBI design can be eliminated, while retaining control of unmeasured seasonal confounding. Instead of using both T _{k–l} and T _{k+l} as controls, we select one of the two at random. In this way each risk set contains one case and one control, and it is no longer possible to determine which of the two members is the case from knowledge of the pair. We might call this method the semi-symmetric bidirectional (SSBI) design. The sampling scheme is complicated somewhat by the fact that for subjects failing near the beginning or near the end of the observation time, only one of the two potential control times can be observed. A detailed description of the scheme is as follows:

Choose a short lag time l . Make l a multiple of 7 if it is desired to control for day-of-the-week effects. Let T _{k} be the failure time.

If k ≤l , choose T _{k+l} as the control (T _{k–l} will not exist).

If k >N −l , choose T _{k–l} as the control (T _{k+l} will not exist).

For other values of k , choose half of the cases at random, and assign T _{k–l} as the control. For the other half of the cases, assign T _{k+l} as the control.

We now use Eq 1 to derive the appropriate likelihood for this sampling design. The risk set R for a subject who fails at time T _{k} is either {T _{k} , T _{k–1} }or {T _{k} , T _{k+1} }. The contribution to the likelihood of the subject will be either

MATH

depending on whether T _{k–l} or T _{k+l} was chosen as the control. We will now specify the sampling probabilities π(R |T _{k} ), π(R |T _{k–l} ), and π(R |T _{k+l} ). These quantities depend on the value of k :

If k ≤l , the risk set R will contain the times T _{k} and T _{k+l} . The quantity π(R |T _{k+l} ) is equal to 1, because we always choose T _{k+l} as the control if the case is T _{k} . The quantity π(R |T _{k+l} ) is equal to 1/2, because if T _{k+l} were the case, we would choose T _{k} as the control with probability 1/2 (and T _{k+2l} with probability 1/2).

If l <k ≤ 2 l , the risk set R will contain the time T _{k} , and either T _{k–l} or T _{k+l} , each with probability 1/2. The quantity π(R |T _{k} ) is therefore equal to 1/2. The quantity π(R |T _{k+l} ) is equal to 1/2, because if T _{k+l} were the case, we would choose T _{k} as the control with probability 1/2 (and T _{k+2l} with probability 1/2). The quantity π(R |T _{k–l} ) is equal to 1, because if T _{k–l} were the case, we would always choose T _{k} as the control, because T _{k–2l} does not exist when k ≤ 2 l .

If 2 l <k ≤N − 2 l , the risk set R will contain the time T _{k} , and either T _{k–l} or T _{k+l} , each with probability 1/2. The quantity π(R |T _{k} ) is therefore equal to 1/2. The quantities π(R |T _{k+l} ) and π(R |T _{k–l} ) are both equal to 1/2, because if either T _{k+l} or T _{k–l} were the case, we would choose T _{k} as the control with probability 1/2.

If N − 2 l <k ≤N −l , the risk set R will contain the time T _{k} , and either T _{k–l} or T _{k+l} , each with probability 1/2. The quantity π(R |T _{k} ) is therefore equal to 1/2. The quantity π(R |T _{k–l} ) is equal to 1/2, because if T _{k–l} were the case, we would choose T _{k} as the control with probability 1/2 (and T _{k–2l} with probability 1/2). The quantity π(R |T _{k+l} ) is equal to 1, because if T _{k+l} were the case, we would always choose T _{k} as the control, because T _{k+2l} does not exist when k >N − 2 l .

If k >N −l , the risk set R will contain the times T _{k} and T _{k–l} . The quantity π(R |T _{k} ) is equal to 1, because we always choose T _{k–l} as the control if the case is T _{k} . The quantity π(R |T _{k–l} ) is equal to 1/2, because if T _{k–l} were the case, we would choose T _{k} as the control with probability 1/2 (and T _{k–2l} with probability 1/2).

Case-Crossover Methods and Poisson Regression
Case-crossover methods are closely related to Poisson regression methods. The two approaches arise from two differing views of mortality count data. One point of view is to consider each failure as an independent event, producing a stratum consisting of a case exposure and a population of control exposures. This view leads to the case-crossover approach. The second point of view is to consider each day as producing an independent count. This view leads to ordinary Poisson regression, in which daily counts are considered to be Poisson random variables of which the means are functions of daily exposure.

Both points of view are equally valid. In fact, it is known that ordinary Poisson regression and the FSBI case-crossover design are equivalent, in that they have identical likelihood functions, and thus produce identical estimates. ^{7} It is useful to have two points of view from which to derive the same estimate, because each point of view provides its own opportunities for refinement, to help adjust for unmeasured confounding.

As discussed above, the case-crossover approach allows for confounder adjustment through sampling schemes that make it more likely that control times will closely match the case time with respect to an unmeasured seasonally varying confounder. The Poisson regression approach has been extended to allow for confounder adjustment as well, for example, by including parameters to model overdispersion and autocorrelation, using appropriate methods of analysis.

A Simulation Study
We performed two simulation studies to compare the performance of several case-crossover sampling designs, and a quasi-likelihood (QL) extension of Poisson regression. For the exposure of interest, we used 1,324 daily mean PM_{10} levels measured at a monitoring station near downtown Denver during the period 1989–1992 (Figure 1 ). We also constructed a trend variable with a cosine-linear form suggested by Bateson and Schwartz ^{6} (Figure 2 ). As was done in Bateson and Schwartz, ^{6} we assumed that the number of events on a day with a given mean PM_{10} level was distributed Poisson with mean 22 times the RR for exposure times the RR for trend. We assumed a log RR for exposure of 0.1 per 100 μg/m^{3} . In the first simulation, we assumed a log RR of 0 for trend, so that there was no seasonal confounding. In the second simulation, we assumed a log RR of 0.1 for trend as well, but did not include the trend variable in the model, so that this variable represented unmeasured seasonal confounding.

FIGURE 1: Daily mean PM10 levels in Denver for 1,324 days during the years 1989–1992. Units are μg/m3.

FIGURE 2: The trend variable is associated with both the daily exposure measure and outcome count to introduce confounding. The formula for the trend variable is (1 + day/1,324) · [1 + 0.6 · cos(2 ·π· (day/365)].

We studied four case-crossover sampling designs. The first was the FSBI design, in which all 1,323 controls were matched to each case. The second was a random matched pair (RMP) design, in which a single control time was chosen at random from the 1,323 available times. The third was the SBI design of Bateson and Schwartz, ^{6} and the fourth was the SSBI design described above. We also studied a QL extension of Poisson regression in which a parameter for overdispersion was included, the mortality counts were assumed to follow a Markov process, and counts lagged 1 and 2 days were included as covariates.

Table 1 , in the columns labeled “Without Seasonal Confounding,” presents the results of the first simulation. Because there was no unmeasured confounding, any bias is due to time trends in the exposure of interest. No bias is noticeable in the FSBI, RMP, or SSBI designs, nor in the QL method. The SBI design has a bias of approximately 60%, as a result of time trends in PM_{10} exposure. It is important to note that the magnitude of the bias in the SBI design is specific to the exposure dataset used in the simulation. The bias for other exposure datasets might well be less.

Table 1: Results of Case-Crossover and Quasi-Likelihood Analyses of Simulated Data

Table 1 , in the columns labeled “With Seasonal Confounding,” presents the results of the second simulation. Here bias may be caused by unmeasured confounding as well as by time trends. The FSBI and RMP designs, which do not control for the unmeasured seasonal trend, show considerable bias. The bias in the QL method is only slightly less. The SBI design has about the same bias as in the first simulation (Table 1 ), which shows that it controls well for seasonal confounding while still being susceptible to bias from time trends in the exposure of interest. The SSBI design is nearly unbiased. This design has retained the control of the unmeasured seasonal confounding that is characteristic of the SBI design, but is not affected by time trends in exposure.

Discussion
The theory of risk set sampling provides a means to develop a wide variety of sampling schemes for case-crossover studies. We have presented a sampling scheme that is designed to control for slowly varying unmeasured confounders, without introducing bias. Other sampling methods can be designed to achieve different goals. For example, Navidi ^{11} suggested using the strategy of countermatching ^{12,13} to increase efficiency. This approach produces a 1:1 matching scheme that is more efficient than the RMP design discussed above, but does not provide control for unmeasured confounding.

The particular extension of Poisson regression studied here provided only a small improvement over the FSBI case-crossover design (which is equivalent to ordinary Poisson regression) in the bias caused by unmeasured seasonal confounding. More elaborate extensions of Poisson regression have been proposed, among them iteratively reweighted and filtered least squares, ^{14,15} frequency domain methods, ^{16} transitional regression models, ^{17} and hierarchical modeling. ^{18} The potential for these and other methods for achieving control of unmeasured confounding will continue to be an important area of investigation.

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