The Misspecification of Components and Factors

Abstract

Factor and component analysis are two similar statistical procedures widely-used to reduce a set of p variables to a smaller set of m variables. This reduced set of m components or factors can be interpreted as an overall pattern structure or used in the derivation of factor and component scores. A commonly occurring and potentially serious problem concerns the misspecification of the number of factors and components (m). Misspecifications can take the form of extracting too many or too few factors or components. A series of simulation studies was undertaken to determine the practical effects of such misspecifications within and between the methods of maximum likelihood factor analysis (MLFA) and principal component analysis (PCA). Computer-simulated data sets, representing baseline factor and component patterns, were generated to represent a wide range of conditions. Item saturation, (aij - .4, .6 & .8), sample size (N - 75, 150, 225 & 450), and the variable to component and factor ratios (p:m - 4:1, 6:1 & 12:1) were systematically varied to create the baseline patterns prior to deliberate misspecifications. The problem was examined from several perspectives by investigating relationships within MLFA and PCA during both overextraction and underextraction, and by investigating relationships between MLFA and PCA during overextraction, underextraction and for the correct structural patterns. Results indicated an overall degradation in the MLFA and PCA solutions during both overextraction and underextraction. Although degradation within methods occurred during overextraction, little information was lost even at maximal overextraction for the strongest (aij = .8 & .6) pattern structures during either MLFA or PCA. By contrast, underextraction was a very serious problem with much loss of information occurring at the first underextraction and continuing with each successive underextraction. Greater degradation occurred with MLFA than PCA during underextraction. High similarity between MLFA and PCA solutions occurred for the correct pattern specifications and for the overextracted solutions. Low similarity between MLFA and PCA solutions occurred during underextraction. Item saturation was the major determinant, while sample size and variable to component (factor) ratio were lesser though important determinants, of stable pattern structures during overextraction and for correct solutions, both within and between methods. No condition of interest was found to be a consistent determiner of stable pattern structures during underextraction.