Beyond the Basics

11.6 Beyond the Basics

The confirmatory factor analysis function presented here is the lessR function cfa . The iterated

estimation procedure that cfa implements is iterated centroid estimation or ICE , revised from centroid estimation: the original Fortran code in John Hunter’s program PACKAGE (Hunter & Cohen, 1969). This Factor analysis procedure is the computational basis of the first factor analyses with multiple factors ever done based on centroid factors modified in the 1930s because it is computationally straightforward and so made factor analysis accessible with iteration for

in the pre-computer era of statistics. A communalities. centroid factor is based on a total score, in this case the sum of the corresponding items that define the factor. This relation to the total score of the

item analysis provides a conceptual similarity of factor analysis and item analysis.

centroid factor:

The ICE procedure as implemented here for confirmatory factor analysis is limited to Derived from the total score of a set multiple indicator measurement models, defined as measurement models in which the measured of items. variables are partitioned into groups with each group of items postulated as a unidimensional set. The construction of multi-item scales, where each item on the scale measures the same shared or common factor, applies to a measurement model that is applicable to much social science research. Other models, however, are also applicable in other situations, such as the evaluation of multi-trait, multi-method measurement.

Confirmatory factor analysis is a specific implementation of the more general concept of structural equation modeling. More general confirmatory factor analysis models, and more general models beyond confirmatory factor analysis, require more general structural equation modeling or SEM software. An ambitious, comprehensive system with many developers is openmx website, OpenMx (Boker et al., 2012). The installation of this software does not follow the usual openmx.psyc.virginia.edu install.packages function, so check their documentation for installation directions. A more recent R SEM package is lavaan (Rossell, 2012a, 2012b).

lavaan website,

These packages use more advanced estimation procedures than cfa ’s ICE , such as full http://lavaan.ugent.be

information maximum likelihood. And, statistical tests of fit are generally available for the estimation procedures implemented in these packages. ICE , however, does surprisingly well at estimation precision, almost as well as full information maximum likelihood (Gerbing & Hamilton, 1994). Also, ICE does not spread misspecification errors throughout the system, instead localizing such errors within the misspecified factor. Formal statistical tests are useful,

276 Factor/Item Analysis

but are literally only appropriate when a respecified model is tested on new data. Re-building

a model on the same data means that the statistical tests become heuristic aids to assessing fit, such as the residual analysis provided by cfa , and not formal statistical tests. As such, the cfa of the ICE procedure is a useful, easy to use procedure applicable to the construction of multiple indicator measurement models for analysis with a confirmatory factor analysis.

Worked Problems

The following Read statement reads the attitude data analyzed in Hunter et al. (1982). The ID takes up the first four columns, then Gender coded as 0 for Male and 1 for Female in one column. Then 74 columns of responses, one to each of 74 attitude items. All the responses are already reversed scored where appropriate.

> mydata <- Read("http://lessRstats.com/data/Mach4Plus.fwd", widths=c(4,1,rep(1,74)), col.names=c("ID", "Gender", to("m",20), to("d",20), to("e",10),

to("i",8), to("p",8), to("c",8)))

The 74 columns of responses to the attitude scales consist of, in this order, the Christie and Geis (1970) 20-item Mach IV scale, the Rokeach (1960) 20-item dogmatism scale, the Rosenberg (1965) 10-item self-esteem scale, an 8-item internal locus of control scale, an 8-item powerful others external locus of control scale, and an 8-item chance external locus of control scale from Levenson (1976).

1 Rotter (1966) proposed the concept of Locus of Control. Those with Internal Locus of Control perceive themselves to be in control of their destiny. Those with an External Locus of Control perceive themselves to have their destiny determined by forces outside of their own control. Levenson (1976) proposed to expand the concept of External Locus of Control to two different concepts, control by Powerful Others and Chance. Here we investigate the structure of Locus of Control.

(a) Read the Mach4Plus data file into R ( Section 2.3.4 ). (b) Create a subset of the data file that retains just the 24 Locus of Control items ( Section 3.6 ).

(c) Compute the correlation matrix of the 24 Locus of Control items ( Section 8.3 ). (d) Examine the heat map of the correlation matrix. How many groups of items appear to

be on the basis of the strength of their correlation? Why? (e) Run the confirmatory factor analysis on the three-factor structure that corresponds to the three Locus of Control concepts. How well does the model fit? Answer in terms of the residuals and pattern coefficients.

(f) Revise the three-factor model to improve fit. Interpret this solution, including the correlation between the two External Locus of Control factors. (g) For the 24 × 24 item correlation matrix, obtain the scree plot of the eigenvalues and the plot of the differences of successive eigenvalues. What is the smallest number of factors that appear to reasonably well account for the correlations among the observed variables? Why?

(h) Extract the specified number of factors from the scree plot, and, secondarily, the heat map. Interpret the meaning of the factors.

Factor/Item Analysis 277

(i) Run the corresponding confirmatory factor analysis for this number of factors and interpret. (j) Compare the two competing measurement models. What is your conclusion regarding the measurement model that underlies these Locus of Control items?

2 Consider the Rosenberg (1965) Self-Esteem scale. Does this scale measure one dimension of self-esteem, or are multiple aspects of self-esteem assessed with the scale?

(a) Read the Mach4Plus data file into R ( Section 2.3.4 ). (b) Create a subset of the data file that retains just the 10 self-esteem items ( Section 3.6 ).

(c) Compute the correlation matrix of the 10 self-esteem items ( Section 8.3 ). (d) Examine the heat map of the correlation matrix. How many groups of items appear to

be on the basis of the strength of their correlation? Why? (e) Run the confirmatory factor analysis on the one factor structure. How well does the model fit? Answer in terms of the residuals and pattern coefficients. (f) What revisions, if any, would you make to this model? (g) Interpret the analysis and provide your conclusion regarding the number of dimensions measured by this self-esteem scale.

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APPENDIX