Calibrating fit-index cutoffs for sparse ordinal CFA: Scenario-specific thresholds and operating characteristics

Sparse ordinal response patterns can shift the sampling distributions of common confirmatory factor analysis (CFA) fit indices, making fixed heuristic cutoffs difficult to interpret across design conditions. We conducted a Monte Carlo simulation to calibrate scenario-specific cutoffs for the comparative fit index (CFI), Tucker-Lewis index (TLI), root mean square error of approximation (RMSEA), and standardized root mean square residual (SRMR) in sparse five-category ordinal CFA and to evaluate their operating characteristics. Data were generated from a three-factor population model with one cross-loading. For each dataset, we fitted a Correct model, a Local Misfit model omitting the cross-loading, and a Severe Misfit model combining two factors. We varied estimator (robust maximum likelihood [MLR], mean- and variance-adjusted maximum likelihood [MLMV], weighted least squares mean- and variance-adjusted maximum likelihood [WLSMV]), sample size (N = 250, 500, 1,000), loading strength ( $\lambda$  =  .50 vs. .80), sparsity pattern, sparse-item scope, and category handling (original vs. collapsed). Cutoffs were calibrated from Correct-model distributions at $\alpha$  =  .05, using lower-tail quantiles for CFI and TLI and upper-tail quantiles for RMSEA and SRMR. Calibrated thresholds were strongly scenario-dependent. RMSEA and SRMR cutoffs were generally far below conventional benchmarks, whereas CFI and TLI cutoffs varied with sample size and loading strength, falling below .95 in lower-information settings and approaching boundary values in larger-sample and high-loading settings. Detection was substantially lower for Local Misfit than for Severe Misfit. A calibrated CFI-RMSEA AND rule provided a transparent multi-index operating profile, with false-rejection and detection trade-offs that varied across scenarios. Collapsing produced modest changes for robust ML estimators but increased WLSMV detection in lower-information settings. We provide cutoff tables, operating-characteristic summaries, an interpolation aid, and an applied illustration to support scenario-calibrated fit evaluation in sparse ordinal CFA.