Exercise 1.3. Exploratory factor analysis

In the previous exercise, it became clear that comparing three different items for 17 countries and three time points can be a daunting task. Rather than working with individual items, we prefer to perform analyses on a single scale that is constructed on the basis of the three items. Working with scales has the additional advantage of higher reliability, because random measurement errors in the different items cancel each other out. However, it needs to be tested whether our three items are sufficiently strongly related to be considered as measuring the same concept. Exploratory factor analysis is one possible way of testing the reliability of the indicators.

Use SPSS to perform an exploratory factor analysis on the three immigration items. Carry out this analysis on the pooled dataset without making distinctions between countries and time points. Use ‘principal axis factoring’ (PAF) as the extraction method and ask SPSS to plot a so-called ‘scree plot’.

SPSS Syntax

*First make sure that the DWEIGHT is on, then run the factor analysis. *This is the end of the exercises in chapter 1. *Save the file to make sure you have an updated version of the file for the exercises in the next chapter. *Please do not forget to change ‘C:\’ to the path where you stored the ESS datasets.

WEIGHT
BY dweight.
FACTOR
/variables IMSMETN IMDFETN IMPCNTR
/print initial correlation extraction univariate
/plot eigen
/extraction paf
/method = correlation.
WEIGHT off.
SAVE OUTFILE='C:\ESS123_immig.sav'
/COMPRESSED.

Questions

  1. Do the three items measure a single dimension?
  2. Can the three items be considered reliable indicators of the intended concept (i.e. anti-immigration attitudes)?

Solution

  1. The number of factors or dimensions measured by the items can be determined in various ways. One possibility is to look at the eigenvalues. The eigenvalues refer to the amount of information that is accounted for by the respective factors, given that every item contributes one unit of information. A common rule of thumb is that only factors with an eigenvalue larger than 1 should be retained. After all, only these factors explain more variance than a single item. The eigenvalue of the first factor is 2.462. Since we have three items, the total variance here equals 3. 2.468, of which three units of variance (i.e. 82.080%) are thus accounted for by factor 1. The second eigenvalue (0.351), on the other hand, is substantially smaller. Since this value is well below 1, the second factor hardly adds anything to the amount of variance explained. Only the first factor should therefore be retained. We conclude that the three items measure a single dimension.
    Table 1.3a. Total variance explained, part 1: Inital Eigenvalues
    Factor Total % of Variance Cumulative %
    1 2.462 82.080 82.080
    2 0.351 11.685 93.765
    3 0.187 6.235 100.000

    Weighted by design weight.

    Table 1.3b. Total variance explained, part 2: Extraction Sums of Squared Loadings
    Factor Total % of Variance Cumulative %
    1 2.212 73.731 73.731
    2
    3

    Weighted by design weight.

    The so-called scree plot confirms the conclusion that the three items are uni-dimensional. This scree plot is a graphical representation of the eigenvalues for the different factors. According to this heuristic, one should look for an ‘elbow’ in the graphic. Only factors on the left hand side of the elbow should be retained, while the rest is ‘scree’ and can be neglected. In this case, the elbow can be found with factor 2. As a result, only the first factor should be retained. By consequence, we can conclude that the three anti-immigration items measure one, single dimension.

  2. In order to judge the reliability of the items, we have to look at the factor matrix containing the factor loadings (note that, since only one factor has been retained, no rotation of the factors was performed). The factor loadings represent how strong the respective items are in relation to the common factor. These factor loadings are essentially correlations, so that they range between -1 and 1. Usually, items with factor loadings higher than .40 (in absolute value) are said to be sufficiently reliable. Here, all three factor loadings are considerably larger, indicating excellent reliability. The first item (IMSMETN) has the weakest factor loading. This is understandable, given that this item mentions a specific immigrant group that is quite different from the groups referred to in the other two items. But even this first item has a very strong factor loading.
    Table 1.4. Factor Matrix
    Variable Factor 1
    Allow many/few immigrants of same race/ethnic group as majority 0.778
    Allow many/few immigrants of different race/ethnic group from majority 0.943
    Allow many/few immigrants from poorer countries outside Europe 0.847

    Weighted by design weight. Extraction Method: Principal Axis Factoring. One (1) factor extracted, 12 iterations required.

    Alternatively, reliability can be judged by looking at the communalities, i.e. the proportions of variance that the indicators share with the common factor. These communalities are the squares of the factor loadings (e.g. for IMSMETN: .778² = .605). All items share at least 60% of their variance with the common factor. Again, this indicates that the items are reliable measurements of the concept of anti-immigration attitudes.

    Table 1.5. Communalities
    Initial Extraction
    Allow many/few immigrants of same race/ethnic group as majority 0.553 0.605
    Allow many/few immigrants of different race/ethnic group from majority 0.715 0.889
    Allow many/few immigrants from poorer countries outside Europe 0.650 0.718

    Weighted by design weight. Extraction Method: Principal Axis Factoring.

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