{"id":3282,"date":"2013-05-22T02:12:13","date_gmt":"2013-05-22T07:12:13","guid":{"rendered":"http:\/\/www.thejuliagroup.com\/blog\/?p=3282"},"modified":"2013-05-22T02:12:43","modified_gmt":"2013-05-22T07:12:43","slug":"a-quick-introduction-to-interpretation-of-mplus-exploratory-factor-analysis","status":"publish","type":"post","link":"https:\/\/www.thejuliagroup.com\/blog\/a-quick-introduction-to-interpretation-of-mplus-exploratory-factor-analysis\/","title":{"rendered":"A quick introduction to interpretation of Exploratory Factor Analysis: Mplus Example"},"content":{"rendered":"<p>Last week I wrote a bit about <a href=\"http:\/\/www.thejuliagroup.com\/blog\/?p=3266\">how to get an exploratory factor analysis using Mplus<\/a>. The question now, is what does that output MEAN ?<\/p>\n<p>First, you just get some information on the programming statements or defaults that produced your output:<\/p>\n<p>INPUT READING TERMINATED NORMALLY<\/p>\n<p>Exploratory Factor Analysis ;<\/p>\n<p>SUMMARY OF ANALYSIS<br \/>\nNumber of groups\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1<br \/>\nNumber of observations\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 730<\/p>\n<p>Number of dependent variables\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 6<br \/>\nNumber of independent variables\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0<br \/>\nNumber of continuous latent variables\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0<\/p>\n<p>Observed dependent variables<\/p>\n<p>Continuous<br \/>\nQ1F1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Q2F1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Q3F1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Q1F2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Q2F2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Q3F2<\/p>\n<p>Estimator\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 ML<br \/>\nRotation\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 GEOMIN<br \/>\nRow standardization\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 CORRELATION<br \/>\nType of rotation\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 OBLIQUE<\/p>\n<p>This tells us we our analyzing all of the data as one group, and not, for example, separate analyses for males and females. We have 730 records, six variables, all of which are continuous and listed above. The maximum likelihood method (ML) of estimation is used and the default rotation, GEOMIN, which is an oblique method, that is it allows the factors to be correlated.<\/p>\n<p>Here we have a list of our eigenvalues<\/p>\n<p>RESULTS FOR EXPLORATORY FACTOR ANALYSIS<\/p>\n<p>EIGENVALUES FOR SAMPLE CORRELATION MATRIX<br \/>\n1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 &#8230;&#8230;&#8230;\u00a0 2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 &#8230;&#8230;&#8230; \u00a0\u00a0 3\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 4\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 5<br \/>\n________\u00a0\u00a0\u00a0\u00a0\u00a0 ________\u00a0\u00a0\u00a0\u00a0\u00a0 _____\u00a0\u00a0\u00a0\u00a0 ________\u00a0\u00a0\u00a0\u00a0\u00a0 ________<br \/>\n1.866\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1.262\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.866\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.750\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.716<\/p>\n<p>EIGENVALUES FOR SAMPLE CORRELATION MATRIX<br \/>\n6<br \/>\n________<br \/>\n0.539<\/p>\n<p>In this case, you could go ahead with the eigenvalue greater than one rule, but let&#8217;s take a look at a couple of other statistics. First, we have the results from the one factor solution.\u00a0 Here we have the chi-square testing the goodness of fit of the model<\/p>\n<p>Chi-Square Test of Model Fit<\/p>\n<p>Value\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 96.228<br \/>\nDegrees of Freedom\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 9<br \/>\nP-Value\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0000<\/p>\n<p>We want this test to be non-significant because our null hypothesis is there is no difference between the observed data and our hypothesized one-factor model. This null is soundly rejected.<\/p>\n<p>Let&#8217;s take a look at the Chi-square for our two-factor solution<br \/>\nChi-Square Test of Model Fit<\/p>\n<p>Value\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 3.016<br \/>\nDegrees of Freedom\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 4<br \/>\nP-Value\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5552<\/p>\n<p>You can clearly see that the chi-square is much smaller and non-significant.<\/p>\n<p>Let&#8217;s take a look at two other tests. The Root Mean Square Error of Approximation (RMSEA) for the one-factor solution is .115, as shown below. We would like to see an RMSEA less than .05 which is clearly not the case here.<\/p>\n<p>RMSEA (Root Mean Square Error Of Approximation)<\/p>\n<p>Estimate\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.115<br \/>\n90 Percent C.I.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.095\u00a0 0.137<br \/>\nProbability RMSEA &lt;= .05\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.000<\/p>\n<p>For the two factor solution, our RMSEA rounds to zero, as shown below<\/p>\n<p>RMSEA (Root Mean Square Error Of Approximation)<\/p>\n<p>Estimate\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.000<br \/>\n90 Percent C.I.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.000\u00a0 0.049<br \/>\nProbability RMSEA &lt;= .05\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.954<\/p>\n<p>Clearly, we are liking the two-factor solution here, yes? The eigenvalue &gt; 1 rule (which should not be TOO emphasized) points there, as does the model fit chi-square and the RMSEA.<\/p>\n<p>In their course on factor analysis, Muthen &amp; Muthen give this very nice example of a table comparing different factor solutions using the data<\/p>\n<p><a href=\"http:\/\/www.thejuliagroup.com\/blog\/wp-content\/uploads\/2013\/05\/Mplus_EFAmodel_selection.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-3283\" alt=\"Mplus_EFAmodel_selection\" src=\"http:\/\/www.thejuliagroup.com\/blog\/wp-content\/uploads\/2013\/05\/Mplus_EFAmodel_selection-300x166.png\" width=\"300\" height=\"166\" srcset=\"https:\/\/www.thejuliagroup.com\/blog\/wp-content\/uploads\/2013\/05\/Mplus_EFAmodel_selection-300x166.png 300w, https:\/\/www.thejuliagroup.com\/blog\/wp-content\/uploads\/2013\/05\/Mplus_EFAmodel_selection.png 573w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<p>They also like the scree plot, which I do, too. I also agree with them that one should never blindly follow some rule but rather have some theory or expectation about how the factors should fall out. I also agree with them in looking at multiple indicators, for example, scree plot, chi-square, RMSEA and eigen-values.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Last week I wrote a bit about how to get an exploratory factor analysis using Mplus. The question now, is what does that output MEAN ? First, you just get some information on the programming statements or defaults that produced your output: INPUT READING TERMINATED NORMALLY Exploratory Factor Analysis ; SUMMARY OF ANALYSIS Number of&#8230;<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_kad_post_classname":"","_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[9,11],"tags":[],"class_list":["post-3282","post","type-post","status-publish","format-standard","hentry","category-software","category-statistics"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.thejuliagroup.com\/blog\/wp-json\/wp\/v2\/posts\/3282","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.thejuliagroup.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.thejuliagroup.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.thejuliagroup.com\/blog\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/www.thejuliagroup.com\/blog\/wp-json\/wp\/v2\/comments?post=3282"}],"version-history":[{"count":2,"href":"https:\/\/www.thejuliagroup.com\/blog\/wp-json\/wp\/v2\/posts\/3282\/revisions"}],"predecessor-version":[{"id":3285,"href":"https:\/\/www.thejuliagroup.com\/blog\/wp-json\/wp\/v2\/posts\/3282\/revisions\/3285"}],"wp:attachment":[{"href":"https:\/\/www.thejuliagroup.com\/blog\/wp-json\/wp\/v2\/media?parent=3282"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.thejuliagroup.com\/blog\/wp-json\/wp\/v2\/categories?post=3282"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.thejuliagroup.com\/blog\/wp-json\/wp\/v2\/tags?post=3282"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}