Standard error estimation in invariance alignment

The invariance alignment (IA) method enables group comparisons in factor models involving either continuous or discrete items. This article evaluates the performance of the commonly used delta method for standard error estimation against alternative bootstrap confidence interval (CI) approaches for IA using the 𝐿0.5 and 𝐿0 loss functions. For IA applied to continuous items, both the delta method and all bootstrap methods yielded acceptable coverage rates. In contrast, for dichotomous items, only bias-corrected bootstrap CIs provided reliable statistical inference in moderate to large sample sizes. In small sample sizes with dichotomous items, none of the individual methods performed consistently well. However, a newly proposed average bootstrap CI approach—based on averaging the lower and upper CI limits from two bootstrap methods—achieved acceptable coverage rates.