Bootstrap confidence intervals in multi-level simultaneous component analysis

Multi-level simultaneous component analysis (MLSCA) was designed for the exploratory analysis of hierarchically ordered data. MLSCA specifies a component model for each level in the data, where appropriate constraints express possible similarities between groups of objects at a certain level, yielding four MLSCA variants. The present paper discusses different bootstrap strategies for estimating confidence intervals (CIs) on the individual parameters. In selecting a proper strategy, the main issues to address are the resampling scheme and the non-uniqueness of the parameters. The resampling scheme depends on which level(s) in the hierarchy are considered random, and which fixed. The degree of non-uniqueness depends on the MLSCA variant, and, in two variants, the extent to which the user exploits the transformational freedom. A comparative simulation study examines the quality of bootstrap CIs of different MLSCA parameters. Generally, the quality of bootstrap CIs appears to be good, provided the sample sizes are sufficient at each level that is considered to be random. The latter implies that if more than a single level is considered random, the total number of observations necessary to obtain reliable inferential information increases dramatically. An empirical example illustrates the use of bootstrap CIs in MLSCA.