This section considers two analytic issues related to crossover cases:
 Should crossover cases be included or excluded from the analysis sample?
 Depending on how crossover cases are treated for analysis, should statistical corrections be employed to adjust for a possible bias to impact estimates?

a. Implications of Including or Excluding Crossover Cases from the Analysis Sample

Unless crossover cases leave the state, administrative records on these cases usually will be available even after they migrate to a nonresearch site, split from another case, or merge with another case. Specifically, as long as welfare participation and earnings information are stored in statewide data systems, it will be possible to determine if a research case is participating in welfare or has (UIcovered) earnings. Consequently, researchers will be able to include most crossover cases in the sample used to generate impact estimates.
If crossover cases are included in the analysis sample, the difference in mean outcomes between original experimental cases and original control cases will tend to understate the impact of welfare reform. This dilution results because some cases will have received a mixture of experimental and control group policies. The extent of bias in the impact estimates will depend on the extent of crossover, as well as on the manner in which the impact of welfare reform on crossovertype cases differs from the impact of welfare reform on noncrossovertype cases.
If all crossovertype cases could be identified, then these cases could be excluded from the analysis sample, and impacts could be estimated for noncrossovertype cases only. While these impact estimates would not be representative of impacts of welfare reform on crossovertype cases, they would be unbiased estimates of the impacts of welfare reform on noncrossovertype cases.
In practice, however, crossovertype cases cannot be identified perfectly, since it is uncertain which of the cases in each experimental/control group would cross over if they were subject to the other set of policies. A common practice is to exclude from the analysis sample cases that migrate to nonresearch sites, regardless of their experimental/control status. As long as migration does not depend on experimental/control status, this exclusion will eliminate from the sample both migrant crossover cases and the corresponding group of crossovertype cases. Similarly, to correct for merge/split crossover, all cases that merge or split could be deleted from the sample (although, in practice, it is often difficult to identify all merging or splitting cases within state administrative files).
If crossover behavior does depend on a case's experimental/control status, then excluding crossover cases from the analysis sample will lead to biased impact estimates. The size and direction of the resulting bias will depend on the incidence of crossover and the relationship between experimental/control status and crossover behavior. Biased impact estimates also will result from excluding crossover cases if crossover behavior is correlated with unobserved factors (such as motivation) that affect outcomes. It is not clear whether the bias in impact estimates from excluding crossover cases from the sample exceeds the bias from including these cases in the sample.


b. Statistical Corrections for Crossover

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 noncrossover cases are crossovertype 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 crossovertype cases and noncrossovertype cases. The Bloom correction assumes that, if there were no opportunity for crossover to occur, the impacts of welfare reform would not differ for crossovertype cases and noncrossovertype cases, after controlling for observed characteristics. If impacts would differ for crossovertype and noncrossovertype cases, the Bloomcorrected impact estimate will be biased. It is possible that the impact of welfare reform on crossovertype cases would be larger than the impact of welfare reform on noncrossovertype cases. In these instances, the Bloomcorrected 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 crossovertype cases would be smaller than the impact of welfare reform on noncrossovertype cases, the Bloomcorrected impact estimate will overstate the true impact of welfare reform, and the true impact will lie somewhere between the Bloomcorrected 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 crossovertype 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 crossovertype 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 crossovertype 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).
