Exploratory Factor Analysis

Psychology 522/622 Winter 2008

Lab Lecture #10 Exploratory Factor Analysis

Last week we explored Principle Components Analysis. This week we explore Exploratory Factor Analysis. We continue with the humor scale data set. We’ll use Principal Axis Factor Analysis to conduct the EFA. We’ll compare the results of the PCA from last week with the EFA from this week.

Last week, we decided that there were two dimensions underlying the humor scale, one dimension relating to a Rickles style of humor and the other dimension related to an Allen style of humor. Using Principal Components Analysis, we forced these two dimensions (or components) to be uncorrelated. However, are they really uncorrelated? We’ll use a different approach and rotation method to explore whether these two dimensions are really uncorrelated.

Conceptually, scores for each item can be separated into two distinct sources, one source due to true differences on the underlying dimension and one source due to measurement error. Principal Components Analysis makes no attempt to separate the variance in each item due to these two sources. In other words, PCA analyzes all of the variance in the set of items. In contrast, EFA (and factor analysis in general) attempts to separate item variance into these two sources, variance due to true scores and variance due to error. We change the names slightly in factor analysis. Error variance is called “unique variance” and true score variance is called “communality.” An item’s communality is the percentage of variance that the item shares with the underlying factor. Naturally, we want this to be high as we hope that the items are at least moderately related to the underlying factor(s).

Now the difficulty is in determining each item’s communality. There are several ways to do this as outlined in Tabachnick and Fidell. The details of how this is accomplished are less important than the big picture idea that differentiates PCA from EFA: PCA ignores measurement error and EFA attempts to estimate the measurement error in each item by estimating its communality. Another way to think of this is that PCA bases its solution on the total variance in each item whereas EFA bases its solution on the variance in each item due to the underlying factors. Thus EFA is more closely aligned with issues of reliability and measurement error in conjunction with data reduction and PCA is seen more strictly as a data reduction technique.

So there are two issues we’ll focus on today: 1) conducting an EFA and focusing on communalities and output interpretation and 2) rotating the factor solution so factors can be correlated. EFA of the Humor Scale

DATAFILE: HUMOR.SAV

We conduct an EFA of the humor scale using Principal Axis Factoring. Analyze  Data Reduction  Factor Analysis Select all 10 items and move them to the Variables box. Click Descriptives, initial solution should be selected by default; in the “Correlation Matrix” section select Reproduced. Click Continue. Click Extraction, select Principal Axis Factoring from the “Method” drop down menu, select scree plot (unrotated factor solution should be selected by default); in the “Extract” section click on the button next to Number of factors and type “2” in the box. Click Continue. Click Rotation, select Direct Oblimin (in the Display box, rotated solution should be selected by default). Click Continue. Click Options, in the “Coefficient Display Format” section select Sort by size. Click Continue. Click OK.

Factor Analysis Communalities

Initial Extraction I like to make fun of .147 .148 others. I make people laugh by .424 .403 making fun of myself. People find me funny when I make jokes about .475 .700 others. I talk about my problems .462 .395 to make people laugh. I frequently make others .395 .472 the target of my jokes. People find me funny when I tell them about .404 .401 my failings. I love to get people to .248 .282 laugh by using sarcasm. I am funniest when I talk about my own .339 .343 weaknesses. I make people laugh by exposing other people's .209 .199 stupidities. I am funny when I tell others about the dumb .397 .312 things I have done. Extraction Method: Principal Axis Factoring. These variables are in the order in which you specified them in the analysis (i.e., they are not sorted by size yet). We see both the initial and after-the-extraction-and-rotation communality estimates. They range from .148 for item 1 to .700 for item 3. The communalities are the percentage of variance in each item accounted for by the factors. Thus, variability in the responses to item 3 is well described by the two factors whereas variability in the responses to item 1 is less well described by the two factors. It seems that item 1 has a lot of error variance/measurement error. Total Variance Explained

Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Factor Total % of Variance Cumulative % Total % of Variance Cumulative % SumsTotal of Squared 1 2.510 25.103 25.103 1.872 18.724 18.724 1.864a Loadings 2 2.319 23.186 48.290 1.782 17.822 36.546 1.796 3 1.130 11.297 59.587 4 .960 9.600 69.187 5 .759 7.585 76.772 6 .649 6.487 83.259 7 .583 5.826 89.085 8 .445 4.447 93.532 9 .334 3.337 96.869 10 .313 3.131 100.000 Extraction Method: Principal Axis Factoring. a. When factors are correlated, sums of squared loadings cannot be added to obtain a total variance. We see here the initial eigenvalues, the extraction sums of squared loadings, and their values after rotation. The initial eigenvalues are the same as they were for the PCA last week. Note that the extraction sums of squared loadings are smaller compared to last week (Factor 1: 1.87 v. 2.51; Factor 2: 1.78 v. 2.32). This is due to the difference between PCA and EFA in how it handles (or ignores) error variance as discussed above. So it is no surprise that these values are lower compared to last week. The rotation sums of squared loadings differ by only a little from the extraction sums of squared loadings. As the extraction sums of squared loadings are computed assuming uncorrelated (orthogonal factors) and these loadings differ little from the after-rotation values, we might surmise that the two factors are close to being uncorrelated. We’ll see if this conjecture holds later in the output. Factor Matrixa

Factor 1 2 I make people laugh by .634 .031 making fun of myself. I talk about my problems .576 .252 to make people laugh. I am funniest when I talk about my own .559 .175 weaknesses. People find me funny when I tell them about .546 .321 my failings. I am funny when I tell others about the dumb .537 .153 things I have done. I like to make fun of -.290 .253 others. People find me funny when I make jokes about -.201 .812 others. I frequently make others -.143 .672 the target of my jokes. I love to get people to -.270 .458 laugh by using sarcasm. I make people laugh by exposing other people's -.153 .420 stupidities. Extraction Method: Principal Axis Factoring. a. 2 factors extracted. 11 iterations required. This table includes the factor loadings for the 10 items onto the two factors pre-rotation (i.e., the two factors are still forced to be uncorrelated). Note that the order of the items has been changed so that the items with the largest loadings on each factor occur first. You’ll recognize the first five items as belonging to the Allen scale and the last four items belonging to the Rickles Scale. The item “I like to make fun of others” loads about equally on each scale and might be deemed “complex.” Again, though, this matrix represents pre-rotation factor loadings, so we won’t worry about it too much. Reproduced Correlations

I am funny People find People find I love to I make people when I tell I make me funny I talk about my I frequently me funny get people I am funniest laugh by others about people laugh when I make problems to make others when I tell to laugh by when I talk exposing the dumb I like to make by making jokes about make people the target of them about using about my own other people's things I have fun of others. fun of myself. others. laugh. my jokes. my failings. sarcasm. weaknesses. stupidities. done. Reproduced Correlation I like to make fun of b .148 -.176 .263 -.103 .211 -.077 .194 -.117 .150 -.117 others. I make people laugh by b -.176 .403 -.102 .373 -.070 .356 -.157 .359 -.084 .345 making fun of myself. People find me funny b when I make jokes about .263 -.102 .700 .089 .574 .151 .426 .030 .372 .017 others. I talk about my problems b -.103 .373 .089 .395 .087 .395 -.040 .366 .018 .348 to make people laugh. I frequently make others b .211 -.070 .574 .087 .472 .138 .346 .038 .304 .026 the target of my jokes. People find me funny b when I tell them about -.077 .356 .151 .395 .138 .401 -.001 .361 .051 .342 my failings. I love to get people to b .194 -.157 .426 -.040 .346 -.001 .282 -.071 .233 -.075 laugh by using sarcasm. I am funniest when I talk b about my own -.117 .359 .030 .366 .038 .361 -.071 .343 -.012 .327 weaknesses. I make people laugh by b exposing other people's .150 -.084 .372 .018 .304 .051 .233 -.012 .199 -.018 stupidities. I am funny when I tell b others about the dumb -.117 .345 .017 .348 .026 .342 -.075 .327 -.018 .312 things I have done. Residual a I like to make fun of -.042 .005 -.013 .010 -.012 .039 .093 -.079 -.019 others. I make people laugh by -.042 .002 .202 .046 -.062 -.007 -.115 -.046 -.061 making fun of myself. People find me funny when I make jokes about .005 .002 .027 .017 .023 -.006 .011 -.034 -.073 others. I talk about my problems -.013 .202 .027 .076 -.134 -.017 .068 -.027 -.166 to make people laugh. I frequently make others .010 .046 .017 .076 -.066 -.049 .000 .041 -.069 the target of my jokes. People find me funny when I tell them about -.012 -.062 .023 -.134 -.066 .057 .020 .006 .212 my failings. I love to get people to .039 -.007 -.006 -.017 -.049 .057 -.075 .041 .065 laugh by using sarcasm. I am funniest when I talk about my own .093 -.115 .011 .068 .000 .020 -.075 -.059 .042 weaknesses. I make people laugh by exposing other people's -.079 -.046 -.034 -.027 .041 .006 .041 -.059 .115 stupidities. I am funny when I tell others about the dumb -.019 -.061 -.073 -.166 -.069 .212 .065 .042 .115 things I have done. Extraction Method: Principal Axis Factoring. a. Residuals are computed between observed and reproduced correlations. There are 19 (42.0%) nonredundant residuals with absolute values greater than 0.05. b. Reproduced communalities Here we see the residual matrix. The output alerts you that 42% of the residuals are larger than .05. The largest residual is .202 between the item “I talk about my problems to make people laugh” and “I make people laugh by making fun of myself.” On the whole the size of the residuals are on the small side and there are no apparent patterns to the residuals to warrant fitting a third factor. Things look okay so we’ll continue.

Pattern Matrixa

Factor 1 2 I talk about my problems .630 .061 to make people laugh. People find me funny when I tell them about .630 .136 my failings. I make people laugh by .598 -.168 making fun of myself. I am funniest when I talk about my own .585 -.007 weaknesses. I am funny when I tell others about the dumb .556 -.021 things I have done. People find me funny when I make jokes about .129 .837 others. I frequently make others .129 .685 the target of my jokes. I love to get people to -.073 .521 laugh by using sarcasm. I make people laugh by exposing other people's .022 .448 stupidities. I like to make fun of -.170 .331 others. Extraction Method: Principal Axis Factoring. Rotation Method: Oblimin with Kaiser Normalization. a. Rotation converged in 4 iterations. After an oblique rotation, Tabachnick and Fidell suggest interpreting the pattern matrix. (For discussion of the issues and dangers, see page 602.) The pattern matrix gives coefficients that describe the unique relationship between each item and each factor (controlling for the other factors). After rotation, the five Allen items all hang together on the first factor and the last five Rickles items all hang together on the second factor. All of these coefficients are above the |.30| level to suggest a “salient” loading. We would call this a “clean” solution as there are no complex items and the factor loadings for each item onto its primary factor is above the salient threshold. Structure Matrix

Factor 1 2 I talk about my problems .626 .009 to make people laugh. People find me funny when I tell them about .619 .085 my failings. I make people laugh by .612 -.217 making fun of myself. I am funniest when I talk about my own .586 -.054 weaknesses. I am funny when I tell others about the dumb .558 -.066 things I have done. People find me funny when I make jokes about .061 .827 others. I frequently make others .073 .675 the target of my jokes. I love to get people to -.115 .527 laugh by using sarcasm. I make people laugh by exposing other people's -.015 .446 stupidities. I like to make fun of -.197 .345 others. Extraction Method: Principal Axis Factoring. Rotation Method: Oblimin with Kaiser Normalization. The structure matrix yields the correlation of each item with each factor. Though people tend not to report the structure matrix, it is a good idea to inspect it to see whether the interpretation from the pattern matrix also holds for the structure matrix. We see that these two matrices yield similar conclusions.

Factor Correlation Matrix

Factor 1 2 1 1.000 -.082 2 -.082 1.000 Extraction Method: Principal Axis Factoring. Rotation Method: Oblimin with Kaiser Normalization. Here we see the correlation between the Allen and Rickles factors. Note that the correlation is not zero (though it is close to zero). The correlation is negative which suggests that people who score high on the Rickles scale tend to score lower on the Allen scale. Summary We’ve conducted an EFA of the humor scale. We found evidence of a good fit of a two- factor solution to the data. The two factors are interpretable as Allen-type humor and Rickles-type humor. These factors correlated negatively and weakly. The two-factor solution is clean with no complex items and all items are salient on their respective factors. These results suggest that we could create two scales based on the 10 items, one an Allen scale and one a Rickles scale, and include these scale scores in later analyses, if desired.

The table below summarizes the different PCA/EFA procedures that we’ve covered. FA Procedure Extraction Components are Rotation Matrix of Interest PCA unrotated Principal Orthogonal None Component Matrix solution Components (i.e., uncorrelated) PCA rotated Principal Orthogonal Varimax Rotated Component solution Components (i.e., uncorrelated) Matrix EFA rotated Principal Axis Oblique Direct Pattern Matrix solution Factoring (i.e., correlated) Oblimin

APPENDIX

Review: Creating Composite Scores A common method of computing composite scores is to calculate the mean of the item responses. We learned this waaaaaay back in the first week of PSY 522/622, but let’s practice by creating a mean composite score for the Rickles scale.

Using the “mean” command: COMPUTE rickles = mean(item01,item03,item05,item07,item09) . EXECUTE .

This version of the menu command will allow you to compute a mean from any/all available data. If a participant only responded to one of these items (e.g., let’s say they responded 5 to item 1) then they would receive a value of 5 for the rickles variable.

A slightly different way to compute mean composite scores: COMPUTE rickles = mean.3(item01,item03,item05,item07,item09) . EXECUTE .

This version of the menu commence will only compute a mean composite score for individuals who responded to at least 3 of the 5 items. If a participant only responded to item 1, they will not get a score for the rickles variable.

The following command is identical to the first command (i.e., it will compute a mean composite score for any/all available data): COMPUTE rickles = (item01+item03+item05+item07+item09)/5 . EXECUTE .