DIFFERENT ESTIMATION STRATEGIES FOR THE COMMON FACTOR MODEL

Transcription

1 Measurement, Scaling, and Dimensional Analysis Summer 2017 Bill Jacoby DIFFERENT ESTIMATION STRATEGIES FOR THE COMMON FACTOR MODEL As discussed in class, the technical objective in a factor analysis is to obtain a pattern matrix that reproduces the adjusted correlation matrix as closely as possible, using the scalar products between the variables shadow vectors within the common factor space. There are several different strategies and estimation procedures that can be used to bring the scalar products as close as possible to the correlations. In our initial factor analysis of the 1992 economic perceptions data, we used principal axis factoring, with iterated estimation of communalities. This produces a least-squares solution that is, the scalar products are a least-squares fit to the corresponding correlations. There are two other strategies that commonly appear in the research literature: (1) a maximum likelihood solution, which produces the factor pattern matrix that has the highest probability of having generated the adjusted correlation matrix; and (2) principal components analysis, in which the empirical correlation matrix is used (i.e., ones on the main diagonal) and the first m components are used as estimates of the unobserved factors. This handout shows the log of a Stata session in which these alternative estimation strategies are applied to the 1992 economic perceptions data.. Change delimiter to semicolon and set MORE condition off.. #delimit ; delimiter now ;. set more off; > Retrieve dataset on > 1992 public opinion > about economic conditions.. use econ; > Perform factor analysis, > specifying maximum > likelihood estimation, > and two common factors. factor bet1yr inccol unem1yr > infl1yr nec1yr necmos nec4yr > jobworr, ml factors(2); (obs=1665) Iteration 0: log likelihood = Iteration 1: log likelihood = Iteration 2: log likelihood = Iteration 3: log likelihood = Iteration 4: log likelihood = Iteration 5: log likelihood =

2 Page 2 Method: maximum likelihood Retained factors = 2 Rotation: (unrotated) Number of params = 15 Schwarz's BIC = Log likelihood = (Akaike's) AIC = Factor Eigenvalue Difference Proportion Cumulative Factor Factor LR test: 2 factors vs. saturated: chi2(13) = Prob>chi2 = Factor loadings (pattern matrix) and unique variances bet1yr inccol unem1yr infl1yr nec1yr necmos nec4yr jobworr > The next command displays the > residual correlation matrix > obtained using the ML factors. estat residuals; Raw residuals of correlations (observed-fitted) Variable bet1yr inccol unem1yr infl1yr nec1yr necmos bet1yr inccol unem1yr infl1yr nec1yr necmos nec4yr jobworr Variable nec4yr jobworr nec4yr jobworr > Perform an orthogonal, > varimax rotation > (See Figure 1 for plot of > rotated common factor space)

3 Page 3. rotate, varimax; Method: maximum likelihood Retained factors = 2 Rotation: orthogonal varimax (Kaiser off) Number of params = 15 Schwarz's BIC = Log likelihood = (Akaike's) AIC = Factor Variance Difference Proportion Cumulative Factor Factor LR test: 2 factors vs. saturated: chi2(13) = Prob>chi2 = Rotated factor loadings (pattern matrix) and unique variances bet1yr inccol unem1yr infl1yr nec1yr necmos nec4yr jobworr Factor rotation matrix Factor1 Factor2 Factor Factor > The next command performs > an oblique, promax rotation. rotate, promax; Method: maximum likelihood Retained factors = 2 Rotation: oblique promax (Kaiser off) Number of params = 15 Schwarz's BIC = Log likelihood = (Akaike's) AIC = Factor Variance Proportion Rotated factors are correlated Factor Factor LR test: 2 factors vs. saturated: chi2(13) = Prob>chi2 =

4 Page 4 Rotated factor loadings (pattern matrix) and unique variances bet1yr inccol unem1yr infl1yr nec1yr necmos nec4yr jobworr Factor rotation matrix Factor1 Factor2 Factor Factor > The next two commands display the > correlation matrix for the factors > and the factor structure matrix, giving > correlations between variables and > factors. estat common; Correlation matrix of the promax(3) rotated common factors -- Factors Factor1 Factor2 -- Factor1 1 Factor estat structure; Structure matrix: correlations between variables and promax(3) rotated common factors -- Variable Factor1 Factor2 -- bet1yr inccol unem1yr infl1yr nec1yr necmos nec4yr jobworr

5 Page 5 > Perform principal components > analysis, and use first two > components as estimates > (See Figure 2 for scree plot > to determine the number of > factors). factor bet1yr inccol unem1yr > infl1yr nec1yr necmos nec4yr > jobworr, pcf factors(2); (obs=1665) Method: principal-component factors Retained factors = 2 Rotation: (unrotated) Number of params = 15 Factor Eigenvalue Difference Proportion Cumulative Factor Factor Factor Factor Factor Factor Factor Factor Factor loadings (pattern matrix) and unique variances bet1yr inccol unem1yr infl1yr nec1yr necmos nec4yr jobworr > The next command displays the > residual correlation matrix > obtained using the two > principal components. estat residuals;

6 Page 6 Raw residuals of correlations (observed-fitted) Variable bet1yr inccol unem1yr infl1yr nec1yr necmos bet1yr inccol unem1yr infl1yr nec1yr necmos nec4yr jobworr Variable nec4yr jobworr nec4yr jobworr > Performs an orthogonal, > varimax rotation (see Figure 3). rotate, varimax; Method: principal-component factors Retained factors = 2 Rotation: orthogonal varimax (Kaiser off) Number of params = 15 Factor Variance Difference Proportion Cumulative Factor Factor Rotated factor loadings (pattern matrix) and unique variances bet1yr inccol unem1yr infl1yr nec1yr necmos nec4yr jobworr Factor rotation matrix Factor1 Factor2 Factor Factor

7 Page 7 > The next command performs > an oblique, promax rotation. rotate, promax; Method: principal-component factors Retained factors = 2 Rotation: oblique promax (Kaiser off) Number of params = 15 Factor Variance Proportion Rotated factors are correlated Factor Factor Rotated factor loadings (pattern matrix) and unique variances bet1yr inccol unem1yr infl1yr nec1yr necmos nec4yr jobworr Factor rotation matrix Factor1 Factor2 Factor Factor > The next two commands display the > correlation matrix for the factors > and the factor structure matrix, giving > correlations between variables and > factors. estat common; Correlation matrix of the promax(3) rotated common factors -- Factors Factor1 Factor2 -- Factor1 1 Factor

8 Page 8. estat structure; Structure matrix: correlations between variables and promax(3) rotated common factors -- Variable Factor1 Factor2 -- bet1yr inccol unem1yr infl1yr nec1yr necmos nec4yr jobworr > Close the log and > end the STATA session. log close; Figure 1: Common factor space, with factors estimated via maximum likelihood and rotated to varimax orientation inccol Varimax rotated factor bet1yr 0.2 jobworr infl1yr unem1yr necmos nec4yr nec1yr Varimax rotated factor 1

9 Page 9 Figure 2: Scree plot for principal components analysis of 1992 economic perceptions data Eigenvalue Principal component Figure 3: Common factor space, with factors estimated via principal components and rotated to varimax orientation inccol bet1yr Varimax rotated factor jobworr infl1yr 0.2 unem1yr nec1yr necmos nec4yr Varimax rotated factor 1