IBM® SPSS® Amos™ 28

Amos uses the parameter estimates from the original sample as initial estimates in the iterative estimation procedure for each bootstrap sample. An alternative procedure, not implemented in Amos, would be to repeat for each bootstrap sample the same procedure for choosing initial values that was used in the analysis of the original sample. In principle, this approach would provide the most faithful replication of the analysis of the original sample.

The correctness of Amos's strategy for choosing initial values depends on whether the initial values affect the final values, and there are two issues here. One issue concerns the possible existence of multiple local minima of the discrepancy function. If there are multiple local minima, the choice of initial values will determine which local minimum appears as the final solution. For this reason, it may be that using the same initial values for every bootstrap replication would tend to produce unusually small estimated standard errors. Amos's choice of initial values in bootstrap replications is thus problematical in the presence of multiple local minima. On the other hand, it is not clear that computing fresh initial estimates for each bootstrap replication would be worth the trouble. If multiple local minima are suspected, the dependability of the entire estimation procedure is open to question, so that it would be cold comfort in any case to have estimates of standard errors even if they could be had.

A second issue in the choice of initial values for bootstrap replications concerns the numerical accuracy of Amos estimates. Neglecting the possibility of multiple local minima, it remains true that the choice of initial values will have at least a marginal effect on the final parameter estimates in each bootstrap replication. This is partly due to round-off error and partly due to the fact that Amos uses an iterative procedure that terminates at a more or less arbitrary point (see the documentation of the Crit1 and Crit2 methods).

There is thus the possibility that using the same initial estimates for each bootstrap replication will systematically influence the parameter estimates in each replication in such a way as to affect the bootstrapped standard errors. Numerical experiments have shown, however, that variability in parameter estimates resulting from the manipulation of initial values is negligible compared to variability from one bootstrap sample to another. Of course, for a statistic with a very small standard error, numerical inaccuracies may be the primary source of variability from one bootstrap sample to another. The behavior of Amos in such extreme cases has not been investigated.