Robustness is Not Dimensionality: On the Sensitivity of Component Comparability Coefficients to Sample Size

Everett (1983) has proposed that, under certain conditions, replicability provides an answer to the question of the number of dimensions to retain in component analysis. But replicability must logically be a function of sample size as well as dimensionality. In the present study, the effects of sample size and sample composition are systematically examined on the replicability of principal components. Using observer ratings of personality from the California Adult Q-Set, comparability coefficients are examined in 192 series of principal components analyses. Results indicate that (a) once one has 20 or more subjects per item, the most comparable solution typically has as many components as items; (b) if these full solutions are ignored, there is still a substantial relationship between prescribed number of components and sample size when one uses either the .90 or .85 threshold decision rules and (c) other criteria for determining the number of components to retain, such as the Minimum Average Partial (MAP) rule, do not show the same relationship with sample size. These results indicate that dimensionality cannot be inferred from component robustness, as these are empirically as well as logically separate matters.