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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 I like to make fun of others. I make people laugh by making fun of myself. People find me funny when I make jokes about others. I talk about my problems to make people laugh. I frequently make others the target of my jokes . People find me funny when I tell them about my failings. I love to get people to laugh by us ing s arcasm. I am funnies t when I talk about my own weaknesses. I make people laugh by exposing other people's stupidities. I am funny when I tell others about the dumb things I have done. Extraction .147 .148 .424 .403 .475 .700 .462 .395 .395 .472 .404 .401 .248 .282 .339 .343 .209 .199 .397 .312 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 Factor 1 2 3 4 5 6 7 8 9 10 Total 2.510 2.319 1.130 .960 .759 .649 .583 .445 .334 .313 Initial Eigenvalues % of Variance Cumulative % 25.103 25.103 23.186 48.290 11.297 59.587 9.600 69.187 7.585 76.772 6.487 83.259 5.826 89.085 4.447 93.532 3.337 96.869 3.131 100.000 Extraction Sums of Squared Loadings Total % of Variance Cumulative % 1.872 18.724 18.724 1.782 17.822 36.546 Rotation Sums Totalof Squared 1.864a Loadings 1.796 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. Fa ctor Ma trixa Factor 1 I make people laugh by making fun of myself. I talk about my problems to mak e people laugh. I am funniest when I talk about my own weaknesses. People find me funny when I tell them about my failings . I am funny when I tell others about the dumb things I have done. I like to make fun of others. People find me funny when I mak e jokes about others. I frequently mak e ot hers the target of my jok es. I love t o get people to laugh by us ing sarc asm. I make people laugh by ex posing other people's stupidities. 2 .634 .031 .576 .252 .559 .175 .546 .321 .537 .153 -.290 .253 -.201 .812 -.143 .672 -.270 .458 -.153 .420 Ex trac tion Method: Principal Ax is Factoring. a. 2 factors ex trac ted. 11 iterat ions 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 make people laugh by making fun of myself. People find me funny when I make jokes about others. I talk about my problems to make people laugh. .148 -.176 .263 -.103 -.176 .403 -.102 .373 .263 -.102 .700 .089 -.103 .373 .089 .395 .211 -.070 .574 .087 -.077 .356 .151 .395 .194 -.157 .426 -.040 -.117 .359 .030 .366 .150 -.084 .372 .018 -.117 .345 .017 .348 -.042 .005 -.013 .002 .202 I like to make fun of others. Reproduced Correlation Residual a I like to make fun of others. I make people laugh by making fun of myself. People find me funny when I make jokes about others. I talk about my problems to make people laugh. I frequently make others the target of my jokes. People find me funny when I tell them about my failings. I love to get people to laugh by using sarcasm. I am funniest when I talk about my own weaknesses. I make people laugh by exposing other people's stupidities. I am funny when I tell others about the dumb things I have done. I like to make fun of others. I make people laugh by making fun of myself. People find me funny when I make jokes about others. I talk about my problems to make people laugh. I frequently make others the target of my jokes. People find me funny when I tell them about my failings. I love to get people to laugh by using sarcasm. I am funniest when I talk about my own weaknesses. I make people laugh by exposing other people's stupidities. I am funny when I tell others about the dumb things I have done. b b -.042 b b .005 .002 .027 -.013 .202 .027 .010 .046 .017 .076 -.012 -.062 .023 -.134 .039 -.007 -.006 -.017 .093 -.115 .011 .068 -.079 -.046 -.034 -.027 -.019 -.061 -.073 -.166 m t 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 I talk about my problems to make people laugh. People find me funny when I tell them about my failings . I make people laugh by making fun of myself. I am funniest when I talk about my own weaknesses. I am funny when I tell others about the dumb things I have done. People find me funny when I make jokes about others . I frequently make others the target of my jokes. I love to get people to laugh by us ing sarcasm. I make people laugh by exposing other people's stupidities. I like to make fun of others . 2 .630 .061 .630 .136 .598 -.168 .585 -.007 .556 -.021 .129 .837 .129 .685 -.073 .521 .022 .448 -.170 .331 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 I talk about my problems to make people laugh. People find me funny when I tell them about my failings . I make people laugh by making fun of myself. I am funniest when I talk about my own weaknesses. I am funny when I tell others about the dumb things I have done. People find me funny when I make jokes about others . I frequently make others the target of my jokes. I love to get people to laugh by us ing sarcasm. I make people laugh by exposing other people's stupidities. I like to make fun of others . 2 .626 .009 .619 .085 .612 -.217 .586 -.054 .558 -.066 .061 .827 .073 .675 -.115 .527 -.015 .446 -.197 .345 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 twofactor 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 .