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VARIANCE ESTIMATION OF MICROSIMULATION MODELS THROUGH SAMPLE REUSE 252 This need for replication is a reason to restructure microsimulation models so that replications are less expensive. Use of Re sampling Techniques in Constructing Confidence Intervals A much more difficult problem than simple variance estimation is the creation of confidence intervals with high coverage probability, say 95 or 99 percent. For issues for which policy makers would like guarantees that expenditures for a revised program are not going to exceed a particular amount, it is very likely that, to provide confidence intervals with 95 or 99 percent coverage, more replicates would be needed, probably in the hundreds. However, this need depends on the approach used to generate the confidence intervals and the assumptions that one is comfortable in making. (See Efron [1981] for an introduction to this discussion.) There are currently two approaches to the solution to this problem, although this is an active area of research and new developments can be expected. First, if one is willing to assume that the output of interest follows an approximate normal distribution, adding and subtracting the usual critical values from the normal distribution times the estimated standard deviation should provide reasonable confidence intervals with coverage probability corresponding to the critical values used. This approach would obviously be feasible with a relatively small number of replications. The second approach is called the percentile method. The original idea was to use the percentiles provided by the bootstrap replications as a confidence interval, with coverage probability corresponding to the percentiles used. Clearly, to estimate these percentiles would take a large number of replications. This idea has exhibited poor performance in some situations, and various bias-correcting procedures have been presented to remedy the problem. These bias-correcting procedures are all very costly in terms of replications. If one is interested in providing confidence ellipsoids for estimates of more than one result from a microsimulation model, estimation of covariances could proceed in the same way that variances are estimated. However, the construction and use of confidence ellipsoids of two and higher dimensions have received very little investigation, so little is currently known about their performance. Finally, Johns (1987) has presented methods that have been successful in reducing the necessary number of bootstrap replicates through use of importance sampling, by identifying the pseudosamples that are likely to contribute a good deal to the variability in the results and then oversampling those pseudosamples. In the case of microsimulation models, one could oversample groups that are most affected by the proposed changes in regulations in the replication process. It is unclear to what extent this would reduce the number of replications necessary to construct useful 95 or 99 percent confidence intervals.
