The influence of sample size (N) and model parsimony on a set of 22 goodness of fit indices was investigated, including those typically used in confirmatory factor analysis and some recently developed indices. For sample data simulated from 2 known population data structures, values for 6 of 22 fit indices were reasonably independent of N and were not significantly affected by the addition of freely estimated parameters known to have 0 values in the population. Indices recommended for evaluating goodness of fit were: (1) two indices based on non-centrality described by R. P. McDonald (1990); (2) a new relative (incremental) index based on non-centrality (P. M. Bentler, 1990 and R. P. McDonald and H. W. Marsh 1990); (3) unbiased estimates of goodness of fit (K. G. Joerskog and D. Sorbom, 1988) presented by J. H. Steiger that are based on non-centrality; and (4) the widely known relative index developed by L. R. Tucker and C. Lewis (1973). Penalties for model complexity designed to control sampling fluctuations and to facilitate the inevitable compromise between goodness of fit and model parsimony were evaluated. Seven tables present analysis results and there is a 36-item list of references. Appendix 1 describes the indices used in the study, and Appendix 2 gives mean values for each model. (Author/SLD)