Item factor analysis: current approaches and future directions

Abstract

The rationale underlying factor analysis applies to continuous and categorical variables alike; however, the models and estimation methods for continuous (i.e., interval or ratio scale) data are not appropriate for item-level data that are categorical in nature. The authors provide a targeted review and synthesis of the item factor analysis (IFA) estimation literature for ordered-categorical data (e.g., Likert-type response scales) with specific attention paid to the problems of estimating models with many items and many factors. Popular IFA models and estimation methods found in the structural equation modeling and item response theory literatures are presented. Following this presentation, recent developments in the estimation of IFA parameters (e.g., Markov chain Monte Carlo) are discussed. The authors conclude with considerations for future research on IFA, simulated examples, and advice for applied researchers.

Figure 1

Latent response distribution for a single dichotomous item representing the latent distribution of interest. τ₁₁ marks the latent cut-point between observed responses.

Figure 2

Bivariate and marginal latent response distributions for two dichotomous items. The bivariate latent response distribution, with a correlation of .70, represents the distribution of interest. The ellipses represent the .01, .05, and .10 regions. The threshold parameters τ₁₁ and τ₂₁ denote the cut-points for Items 1 and 2, respectively.

Figure 3

Three-dimensional bivariate latent response distribution for two items with a correlation of .70.

Figure 4

A: Two-parameter logistic (2PL) model trace line for a single dichotomous item with a difficulty (i.e., b) equal to −1.67. The slope of the trace line (a = 1.28) describes the strength of the relationship between the underlying latent construct (i.e., θ) and the probability of an individual endorsing the item. B: Graded response model trace lines for an item with C = 5 response categories. Each line represents the corresponding probability of endorsing the c^th category given θ. Category 0 is denoted with a long-dashed line (– – –), Category 1 is denoted with dash–dot–dash line (– · –), Category 2 is denoted with a dotted line (…), Category 3 is denoted with a short-dashed line (- - -), and Category 4 is denoted with a solid line (―).

Figure 5

A: Weighted least squares weight matrix where p = 3 and u = 6. B: Modified weighted least squares for categorical data weight matrix where p = 3 and u = 6.