Factor Analysis Using Sas Pdf Bookmark

Factor analysis using sas pdf bookmark

This example uses the data presented in Example Unlike Example The scree plot helps you determine the number of factors, and the loading plots help you visualize the patterns of factor loadings during various stages of analyses.

Factor analysis using sas pdf bookmark

Output If the data are appropriate for the common factor model, the partial correlations controlling all other variables should be small compared to the original correlations. For example, the partial correlation between the variables School and HouseValue is , slightly less than the original correlation of see Output The partial correlation between Population and School is , which is much larger in absolute value than the original correlation; this is an indication of trouble.

Values of or are considered good, while MSAs below are unacceptable.


Only the Services variable has a good MSA. The overall MSA of is sufficiently poor that additional variables should be included in the analysis to better define the common factors. A commonly used rule is that there should be at least three variables per factor.

In the following analysis, you determine that there are two common factors in these data.

Factor analysis using sas pdf bookmark

Therefore, more variables are needed for a reliable analysis. The square multiple correlations are shown as prior communality estimates in Output Because the square multiple correlations are usually less than one, the resulting correlation matrix for factoring is called the reduced correlation matrix.

In the current example, the SMCs are all fairly large; hence, you expect the results of the principal factor analysis to be similar to those in the principal component analysis.

Factor analysis using sas pdf bookmark

The first two largest positive eigenvalues of the reduced correlation matrix account for of the common variance.

This is possible because the reduced correlation matrix, in general, is not necessarily positive definite, and negative eigenvalues for the matrix are possible. A pattern like this suggests that you might not need more than two common factors.

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The scree and variance explained plots of Output Showing in the left panel of Output A sharp bend occurs at the third eigenvalue, reinforcing the conclusion that two common factors are present. These cumulative proportions of common variance explained by factors are plotted in the right panel of Output This decision agrees with the conclusion drawn by inspecting the scree plot.

The principal factor pattern with the two factors is displayed in Output This factor pattern is similar to the principal component pattern seen in Output For example, the variable Services has the largest loading on the first factor, and the Population variable has the smallest. The variables Population and Employment have large positive loadings on the second factor, and the HouseValue and School variables have large negative loadings.

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Comparing the current factor loading matrix in Output The advantage of using this option might not be very obvious in Output The final communality estimates are all fairly close to the priors shown in Output Only the communality for the variable HouseValue increased appreciably, from to.

Therefore, you are sure that all the common variance is accounted for. The partial correlations are not quite as impressive, since the uniqueness values are also rather small.

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These results indicate that the squared multiple correlations are good but not quite optimal communality estimates. As displayed in Output The Services variable is in between but closer to the HouseValue and School variables. A good rotation would place the axes so that most variables would have zero loadings on most factors.

As a result, the axes would appear as though they are put through the variable clusters. In Output To yield the varimax-rotated factor loading pattern , the initial factor loading matrix is postmultiplied by an orthogonal transformation matrix. This orthogonal transformation matrix is shown in Output This rotation or transformation leads to small loadings of Population and Employment on the first factor and small loadings of HouseValue and School on the second factor.

Services appears to have a larger loading on the first factor than it has on the second factor, although both loadings are substantial. Hence, Services appears to be factorially complex.

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The first factor is associated more with the first three variables first three rows of variables : HouseValue , School , and Services.

The second factor is associated more with the last two variables last two rows of variables : Population and Employment. For orthogonal factor solutions such as the current varimax-rotated solution, you can also interpret the values in the factor loading pattern matrix as correlations. For example, HouseValue and Factor 1 have a high correlation at , while Population and Factor 1 have a low correlation at.

Factor analysis using sas pdf bookmark

The variance explained by the factors are more evenly distributed in the varimax-rotated solution, as compared with that of the unrotated solution. Indeed, this is a typical fact for any kinds of factor rotation. In the current example, before the varimax rotation the two factors explain and , respectively, of the common variance see Output After the varimax rotation the two rotated factors explain and , respectively, of the common variance.

However, the total variance accounted for by the factors remains unchanged after the varimax rotation.

Lecture 10: Factor analysis (and with regression)

This invariance property is also observed for the communalities of the variables after the rotation, as evidenced by comparing the current communality estimates in Output Clearly, HouseValue and School cluster together on the Factor 1 axis, while Population and Employment cluster together on the Factor 2 axis.

Service is closer to the cluster of HouseValue and School.

Programming the statistical procedures from SAS

An alternative to the scatter plot of factor loadings is the so-called vector plot of loadings, which is shown in Output That is:.

This generates the vector plot of loadings in Output For some researchers, the varimax-rotated factor solution in the preceding section might be good enough to provide them useful and interpretable results. For others who believe that common factors are seldom orthogonal, an obliquely rotated factor solution might be more desirable, or at least should be attempted.

The results of the promax rotation are shown in Output The corresponding plot of factor loadings is shown in Output This is the matrix that transforms the varimax factor pattern so that the rotated pattern is as close as possible to the Procrustean target.

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However, because the variances of factors have to be fixed at 1 during the oblique transformation, a normalized version of the Procrustean transformation matrix is the one that is actually used in the transformation. This normalized transformation matrix is shown at the bottom of Output Using this transformation matrix leads to the promax-rotated factor solution, as shown in Output After the promax rotation, the factors are no longer uncorrelated.

As shown in Output In the initial unrotated and the varimax solutions, the two factors are not correlated. In addition to allowing the factors to be correlated, in an oblique factor solution you seek a pattern of factor loadings that is more "differentiated" referred to as the "simple structures" in the literature.

Example 33.2 Principal Factor Analysis

The more differentiated the loadings, the easier the interpretation of the factors. For example, factor loadings of Services and Population on Factor 2 are and , respectively, in the orthogonal varimax-rotated factor pattern see Output With the oblique promax rotation see Output Overall, however, the factor patterns before and after the promax rotation do not seem to differ too much. This fact is confirmed by comparing the graphical plots of factor loadings.

The plots in Output Unlike the orthogonal factor solutions where you can interpret the factor loadings as correlations between variables and factors, in oblique factor solutions such as the promax solution, you have to turn to the factor structure matrix for examining the correlations between variables and factors.

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Basically, the factor structure matrix shown in Output The critical difference is that you can have the correlation interpretation only by using the factor structure matrix. For example, in the factor structure matrix shown in Output The corresponding value shown in the factor pattern matrix in Output Common variance explained by the promax-rotated factors are 2.


Unlike the orthogonal factor solutions for example, the prerotated varimax solution , variance explained by these promax-rotated factors do not sum up to the total communality estimate 4. In oblique factor solutions, variance explained by oblique factors cannot be partitioned for the factors. Variance explained by a common factor is computed while ignoring the contributions from the other factors.

Factor analysis using sas pdf bookmark

However, the communalities for the variables, as shown in the bottom of Output They are still the same set of communalities in the initial, varimax-rotated, and promax-rotated solutions.

This is a basic fact about factor rotations: they only redistribute the variance explained by the factors; the total variance explained by the factors for any variable that is, the communality of the variable remains unchanged. In the literature of exploratory factor analysis, reference axes had been an important tool in factor rotation.

Factor analysis using sas pdf bookmark

Nowadays, rotations are seldom done through the uses of the reference axes. Despite that, results about reference axes do provide additional information for interpreting factor analysis results. To explain the results in the reference-axis system, some geometric interpretations of the factor axes are needed.

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Consider a single factor in a system of common factors in an oblique factor solution. Taking away the factor under consideration, the remaining factors span a hyperplane in the factor space of dimensions. The vector that is orthogonal to this hyperplane is the reference axis reference vector of the factor under consideration.

Using the same definition for the remaining factors, you have reference vectors for factors. A factor in an oblique factor solution can be considered as the sum of two independent components: its associated reference vector and a component that is overlapped with all other factors.