G89.2247 Lecture 7 Confirmatory Factor Analysis CFA as measurement models Illustration with POMS data Issues in Confirmatory Factor Analysis G89.2247 Lecture 7

Confirmatory FA POMS Example G89.2247 Lecture 7

Psychometric Perspective on CFA Suppose X1 is a measure of a psychological construct The measure might not be perfect Subject Carelessness and error Transient variation in construct Special features or bias of specific measure Error is often addressed using multiple measures of the same construct (items, raters) Supplement X1, with X2, X3 G89.2247 Lecture 7

A Parallel Measure Model Suppose the average X is E(X)=T If X1, X2, X3 are "parallel" measures Then X1 = T + E1 X2 = T + E2 X3 = T + E3 The reliability of any X is R=[Var(T)/Var(X)] Any X will be a "fallible" measure of T True correlations of some Y with T will be underestimated by observed correlations between Y and X. G89.2247 Lecture 7

Logic of Latent Variable (Factor) Model Any set of variables might be highly correlated because they have a common cause Examples A series of items related to distress A series of behaviors that suggest impulsive tendencies A common cause would leave a pattern in the data, even if the cause itself is not measured G89.2247 Lecture 7

Path Formulation of Factor Analysis Model UNIQUE LATENT VAR MANEFEST VAR l11 l21 MANEFEST VAR UNIQUE LATENT VAR COMMON LATENT VAR l31 MANEFEST VAR UNIQUE LATENT VAR l41 MANEFEST VAR UNIQUE LATENT VAR G89.2247 Lecture 7

Matrix Formulation of Factor Analysis Model Let the vector of manifest variables be X One factor model: X = L f1 + d G89.2247 Lecture 7

Expected Variance Implied by Factor Model Let's assume E(X)=0 Then Var(X) = E(XX') E(XX')=E[(L f1 + d)(L f1 + d)'] = LFL' + Y where Var(f1) = F, Var(d) = Y We can look at the implications of this pattern in a general covariance structure fitting context. G89.2247 Lecture 7

Notes on Variance Patterns Shown in Excel Program When only one factor is considered, the fitted covariance matrix S(q) has rows and columns that are proportional to each other (except on diagonal). We can estimate the coefficients relating the factor to the manefest variables, and these are unique up to their sign. The approach generalizes to more than one factor. G89.2247 Lecture 7

Multiple Factor Analysis Suppose we believe there are three factors F=[f1 , f2 , f3] The factor model states X = L F + d Var(X) = LFL' + Y where Var(F) = F, Var(d) = Y There is no unique representation of F L F = L(M -1M)F = L*F * G89.2247 Lecture 7

More than one factor: CFA vs EFA When more than one factor is considered, it is impossible to specify a unique coordinate system for the latent variable space Factor rotations are arbitrarily chosen representations that aid interpretations Orthogonal vs oblique rotations CFA can specify a unique representation of the latent space based on theory Model identification still an issue G89.2247 Lecture 7

CFA Issues Model identification Number of factors To avoid interdermanency issues, one must fix several loadings for each variable and factor. Number of factors For CFA usually we compare the fit of models with k and k+1 factors (competing theories) Interpretation issues Scaling of latent variables Naming of latent variables Reification of latent variables Overly simple factor models (e.g. McCrae et al, 1996) G89.2247 Lecture 7

Second order factor models Just as latent variables might explain correlation among items, second order latent variables might explain correlation among factors 1 FA FB FC A1 A2 A3 B1 B2 B3 C1 C2 C3 F2 G89.2247 Lecture 7

Other Issues Influence of parts of model on overall fit The measurement model can be influenced by structural aspects of the hybred model. Recommend that the measurement model be examined in isolation Items, Item parcels Averages of items can often be useful Domain representative vs factor representative Kishton and Widaman (1994) G89.2247 Lecture 7

Covariance Structure Models in Multiple Groups Covariance structure model fitting is well adapted to exploration of multiple groups Each group generates a separate sample covariance matrix to be fit The groups can be estimated in a joint estimation phase If we minimize the sum of LR criteria across the groups, the minimum is the solution that minimizes each group Estimates can be constrained to be the same within as well as between groups G89.2247 Lecture 7

Utility of Multiple Group Analyses For structural models, tests of categorical variables and moderation of structure according to group For measurement models, tests of measurement equivalence across groups See: Reise,SP; Widaman,KF; Pugh,RH (1993) Confirmatory factor analysis and item response theory: Two approaches for exploring measurement invariance. Psychological Bulletin. 1993 114(3): 552-566 G89.2247 Lecture 7