We consider corrections in two situations: (1) when crossover cases are included in the analysis sample, and (2) when crossover cases are excluded from the analysis sample.
Corrections When Crossover Cases Are Included. When crossover cases are included in the analysis sample, impact estimates (obtained as the difference of means between original experimental cases and original control cases) will tend to be diluted, because some original control cases will have been exposed to experimental policies. A proposed correction for this dilution is the Bloom correction (Bloom 1984; Bloom et al. 1993). In its simplest form, this procedure involves dividing the uncorrected impact estimate (the difference in mean outcomes for the experimental and control groups) by one minus the sum of the crossover rates for experimental and control cases. For example, if the crossover rate is 0.05 for experimental cases and 0.15 for control cases, the Bloom correction would involve dividing impact estimates by 0.80. The crossover rate may be measured in at least four ways for experimental and control cases:
- As the fraction of experimental cases currently subject to control group polices and as the fraction of control cases currently subject to welfare reform policies. This measure is most appropriate when prior exposure to the other set of policies has little or no effect on current outcomes.
- As zero for experimental cases and as the fraction of control cases ever subject to welfare reform policies. This measure is most appropriate when any exposure to welfare reform policies is equivalent to continual exposure to welfare reform policies.
- As the fraction of experimental cases ever subject to control group policies and as zero for control cases. This measure is most appropriate when any exposure to control group policies is equivalent to continual exposure to control group policies.
- As the fraction of time experimental cases have been subject to control group policies and as the fraction of time control cases have been subject to welfare reform policies. This measure is most appropriate when the impact of welfare reform polices depends on the percentage of time cases are exposed to these policies.
The larger the crossover rate, the larger the difference between the corrected and uncorrected impact estimates.
A major advantage of the Bloom correction is that it can be calculated in a straightforward manner if original experimental/control status is known and if actual crossover behavior is measured accurately. For the Bloom correction to be used, it is not necessary to know whether non-crossover cases are crossover-type cases (that is, whether noncrossover cases would have been crossover cases if they had been assigned to the other experimental/control group).
A major disadvantage of the Bloom correction is that it relies on a restrictive assumption about the similarity of crossover-type cases and noncrossover-type cases. The Bloom correction assumes that, if there were no opportunity for crossover to occur, the impacts of welfare reform would not differ for crossover-type cases and noncrossover-type cases, after controlling for observed characteristics. If impacts would differ for crossover-type and noncrossover-type cases, the Bloom-corrected impact estimate will be biased. It is possible that the impact of welfare reform on crossover-type cases would be larger than the impact of welfare reform on noncrossover-type cases. In these instances, the Bloom-corrected impact estimate will understate the true impact of welfare reform, although not as much as the uncorrected impact estimate. If the impact of welfare reform on crossover-type cases would be smaller than the impact of welfare reform on noncrossover-type cases, the Bloom-corrected impact estimate will overstate the true impact of welfare reform, and the true impact will lie somewhere between the Bloom-corrected estimate and the uncorrected estimate.
Another disadvantage of using the Bloom correction is that the underlying statistical procedure tends to reduce the precision of impact estimates over estimates obtained using an indicator for original experimental/control status. In certain situations, this loss of precision may be substantial.
Corrections When Crossover Cases Are Excluded. When crossover and presumed crossover-type cases (all cases that migrate, merge, or split) are excluded from the research sample, two problems arise that may benefit from the use of statistical corrections. The first is that the exclusion of cases from the analysis sample may introduce sample selection bias, either because of differential crossover between experimental and control cases or because crossover is itself correlated with unmeasured determinants of outcomes. The second problem is that, even if impacts estimated for the restricted sample were unbiased, the restricted sample may not resemble the total sample of research cases and the impacts estimated for this sample may not be the same as the impacts for the full sample.
If exclusion of crossover cases and presumed crossover-type cases introduces sample selection bias, then sample selection correction procedures, such as the Heckman correction, may be employed. Proper use of these procedures requires that variables exist that influence crossover behavior but not the outcomes of interest. Such variables may be difficult to identify, since anything influencing the decision to migrate, merge, or split may also influence program participation decisions, employment, and earnings. As with the Bloom correction, sample selection corrections generally reduce the precision of impact estimates.
Even if we could assume that exclusion of crossover and crossover-type cases from the analysis sample did not generate bias in estimating the impacts on cases that remain, the resulting analysis sample may not be representative of the original research sample. To narrow the differences between these two samples, reweighting schemes may be employed to make the two populations more similar. Unfortunately, any reweighting scheme can make the populations resemble each other across only a limited number of observed dimensions. Even after reweighting the analysis sample, differences between the analysis sample and the entire research sample are likely to remain (for example, in the degree of mobility of the cases in each sample).