PSCY 612, SPRING 2008Lecture 12: Exploratory Factor Analysis (cont.)Lecture Date: 4/16/2008Contents1 Preliminary iClicker Questions 12 Part I: Review and Refinement (50 minutes; 5 minute break) 12.1 Purpose: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Objectives: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Communalities Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 Factor Loadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.5 Exploratory – Confirmatory Continuum . . . . . . . . . . . . . . . . . . . . . . . . 63 Part II: Additional details for EFA 63.1 Purpose: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Objectives: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 CFA as a restricted case of EFA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.4 CFA > EFA > PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.5 EFA is sensitive to your data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Part III: Matrix Algebra - on hold for one week 71 Preliminary iClicker Questions• Have you read all the assigned reading for today (Kim and Mueller)?• Do you understand the purpose of EFA and PCA?• Were you able to interpret the EFA and PCA results in lab?2 Part I: Review and Refinement (50 minutes; 5 minutebreak)12.1 Purpose:To provide some extra review and iron out details2.2 Objectives:1. Review the concept of communalities and factor loadings2. Explain the confirmatory – exploratory continuum2.3 Communalities RevisitedI mentioned last week that EFA can be sensitive to the structure of the covariance matrix. Recallmy analogy that PCA is to variance as EFA is to covariance. We can use that analogy to betterunderstand the nature of problems that occur. The most likely place where we see problems arein the communalities. As many of you witnessed in lab on Monday, when the datasets producedill-behaved covariance matrices, the solution was not provided. You could see only one column inthe communalities table that listed only the initial values; I shall return to those initial values in abit. What I want to focus on right now is the absence of extracted values (i.e., communality valuesthat are estimated after the factor extraction). When a communality value is estimated during theiteration - multiple operations that are required to get the parameter estimates - and that valueexceeds 1.0 then the entire process stops. Many of you saw the results of that halted process witha little message following the scree plot in your SPSS output. The message said something to theeffect that during iteration X, one of the communalities was estimated to be greater than 1.0; thedefinition of a Heywood case as I mentioned in my notes last week.Communalities are defined as:h2i=kXj=1b2jwhere i refers to the variable, j refers to the paths from latent constructs, and bjrefers to thelinear weight (path coefficient) that represents the relationship between the manifest variable andthe latent construct. We interpret communalities in the same way we interpreted R2in MRC. Nowyou can see the benefit of knowing MRC before we ventured into data reduction procedures.When communalities exceed 1.0, we can be certain that something is wrong with our modelor data - which one is to blame is not clear so we must work hard to find a cure. Recall that Imentioned several potential causes. Those included:• Bad prior communality estimatesRemember that SPSS uses the R2from an MRC model that sets the target variable as thedependent variable and the remaining variables in the model as predictors. SPSS is knownto present problems to data analysts because of the rigid method it uses to specify priorcommunality estimates because we know that these estimates tend to be lower than would beexpected from a common factor solution. Other software packages allow you to set the priorestimates to different values and SPSS may allow you to set other priors but only via syntax.• Model mis-specification (i.e., too many or too few common factors)As I stated last week, we may get Heywo od cases by failing to specify the correct model byeither including irrelevant variables or excluding relevant variables that would dramatically2shift the final solution. If you think that your model is mis-specified, you ought to searchfor manifest variables that might be irrelevant to the common factor model. For example, avariable that has a negligible relationship with other variables in the model might be a suitablecandidate for exclusion from subsequent analyses.• Insufficient data to provide stable estimatesPerhaps the greatest threat to EFA is its demand for large samples. I mentioned in lab thatEFA tends to produce reasonable estimates when sample sizes exceed 100. Once you getthe requisite minimum sample size, there are other rules of thumb for sample sizes. Someresearchers argue for a conservative 20 observations for every variable in your common factormodel while others allow for a bit less stringent 10 or even 5 observations per variable. Themost important criterion, however, is to have the minimum number of observations necessaryto produce a stable covariance matrix (i.e., nobs>= 100).• Common factor model not appropriate for dataThe final explanation for Heywood cases is simply a matter of tool misuse. There are timeswhen a common factor model does not make sense. Consider the concept of socio-economicstatus - an amalgam of variables that comprise education, income, wealth, and social class.There is no reason - either theoretically or empirically - for these variable to correlate. Commonfactor models require covariance and correlation is just one form of covariance. Therefore, SESmight not be a suitable candidate for EFA but PCA would serve us quite well.Communalities are good indicators of how well our data reduction helped us summarize ourobserved data. Often you will read something about “uniqueness.” …