Lewin: la primera teoría de campo en ciencias sociales.

To assist in the interpretation of the factors,factor rotationmay be performed. Factor rotation corresponds to a transformation (usually, orthogonal) of the coordinate axes, leading to a different set of factor loadings. We may look upon factor rotation as analogous to a scientist attempting to elicit greater contrast and detail by adjusting the focus of a microscope.

The sharpest focus occurs when each variable has high factor loadings on a single factor, with low to moderate loadings on the other factors. For thehouses example, this sharp focus already occurred on the unrotated factor loadings (e.g.,

TABLE 1.10 Factor Loadingsa

Factor 1 2 age-z 0.590 −0.329 educ-z 0.295 0.424 capnet-z 0.193 0.142 hours-z 0.224 0.193 dem-z −0.115 0.013

a_{Extraction method: principal axis factoring; two factors extracted, 152 iterations required.} Factor loadings are much weaker than for the preceding example.

FACTOR ANALYSIS 21 TABLE 1.11 Communalitiesa Initial Extraction age-z 0.015 0.457 educ-z 0.034 0.267 capnet-z 0.021 0.058 hours-z 0.029 0.087 dem-z 0.008 0.013

a_{Extraction method: principal axis factoring. Communalities are low, reflecting not much} shared correlation.

Table 1.5), so rotation was not necessary. However, Table 1.10 shows that we should perhaps try factor rotation for theadultdata set, to help improve our interpretation of the two factors. Figure 1.6 shows the graphical view of the vectors of factors of loadings for each variable from Table 1.10. Note that most vectors do not closely follow the coordinate axes, which means that there is poor “contrast” among the variables for each factor, thereby reducing interpretability.

Next, avarimaxrotation (discussed shortly) was applied to the matrix of factor loadings, resulting in the new set of factor loadings in Table 1.12. Note that the contrast has been increased for most variables, which is perhaps made clearer by Figure 1.7, the graphical view of the rotated vectors of factor loadings. The figure shows that the factor loadings have been rotated along the axes of maximum variability, represented byfactors 1and2. Often, the first factor extracted represents a “general factor” and accounts for much of the total variability. The effect of factor rotation is to redistribute the variability explained among the second, third, and subsequent factors.

dem_z educ_z hours_z capnet_z age_z −1.0 −1.0 −.5 0.0 .5 1.0 −.5 F actor 2 Factor 1 0.0 .5 1.0

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22 CHAPTER 1 DIMENSION REDUCTION METHODS

TABLE 1.12 Factor Loadings After Varimax Rotationa

Factor 1 2 age-z 0.675 0.041 educ-z 0.020 0.516 capnet-z 0.086 0.224 hours-z 0.084 0.283 dem-z −0.104 −0.051

a_{Extraction method: principal axis factoring; rotation method: varimax with kaiser normaliza-} tion; rotation converged in three iterations.

For example, consider Table 1.13, which shows the percent of variance explained by factors 1and2, for the initial unrotated extraction (left side) and the rotated version (right side).

The sums of squared loadings forfactor 1for the unrotated case is (using Table 1.10 and allowing for rounding, as always) 0.5902+0.2952+0.1932+0.2242+ −0.1152_{=0.536.This represents 10.7% of the total variability and about 61% of the} variance explained by the first two factors. For the rotated case,factor 1’s influence has been partially redistributed tofactor 2in this simplified example, now accounting for 9.6% of the total variability and about 55% of the variance explained by the first two factors.

Next we describe three methods fororthogonal rotation, in which the axes are rigidly maintained at 90◦. When rotating the matrix of factor loadings, the goal is to

dem_z educ_z hours_z capnet_z age_z −1.0 −1.0 −.5 0.0 .5 1.0 −.5 F actor 2 Factor 1 0.0 .5 1.0

USER-DEFINED COMPOSITES 23

TABLE 1.13 Factor Rotation Redistributes the Percentage of Variance Explaineda

Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings Factor Total % of Variance Cumulative % Total % of Variance Cumulative % 1 0.536 10.722 10.722 0.481 9.616 9.616 2 0.346 6.912 17.635 0.401 8.019 17.635 a_{Extraction method: principal axis factoring.}

ease interpretability by simplifying the rows and columns of the column matrix. In the following discussion we assume that the columns in a matrix of factor loadings represent the factors and that the rows represent the variables, just as in Table 1.10, for example. Simplifying the rows of this matrix would entail maximizing the loading of a particular variable on one particular factor and keeping the loadings for this variable on the other factors as low as possible (ideal: row of zeros and ones). Similarly, simplifying the columns of this matrix would entail maximizing the loading of a particular factor on one particular variable and keeping the loadings for this factor on the other variables as low as possible (ideal: column of zeros and ones).

r _{Quartimax rotationseeks to simplify the rows of a matrix of factor loadings.} Quartimax rotation tends to rotate the axes so that the variables have high loadings for the first factor and low loadings thereafter. The difficulty is that it can generate a strong “general” first factor, in which almost every variable has high loadings.

r _{Varimax rotationprefers to simplify the column of the factor loading matrix.} Varimax rotation maximizes the variability in the loadings for the factors, with a goal of working toward the ideal column of zeros and ones for each variable. The rationale for varimax rotation is that we can best interpret the factors when they are strongly associated with some variable and strongly not associated with other variables. Kaiser [7,8] showed that the varimax rotation is more invariant than the quartimax rotation.

r _{Equimax rotationseeks to compromise between simplifying the columns and} the rows.

The researcher may prefer to avoid the requirement that the rotated factors remain orthogonal (independent). In this case,oblique rotationmethods are available in which the factors may be correlated with each other. This rotation method is called oblique because the axes are no longer required to be at 90◦, but may form an oblique angle. For more on oblique rotation methods, see Harmon [9].

USER-DEFINED COMPOSITES

Factor analysis continues to be controversial, in part due to the lack of invariance under transformation and the consequent nonuniqueness of the factor solutions. Ana- lysts may prefer a much more straightforward alternative:user-defined composites. A

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24 CHAPTER 1 DIMENSION REDUCTION METHODS

user-defined composite is simply a linear combination of the variables which com- bines several variables into a single composite measure. In the behavior science literature, user-defined composites are known assummated scales (e.g., Robinson et al. [10]).

User-defined composites take the form

W =aZ =a1Z1+a2Z2+ · · · +akZk

where k_i=1 ai =1,k≤m, and the Zi are the standardized variables. Whichever form the linear combination takes, however, the variables should be standardized first so that one variable with high dispersion does not overwhelm the others. The simplest user-defined composite is simply the mean of the variables. In this case, ai =1/k,i=1,2, . . . ,k. However, if the analyst has prior information or expert knowledge available to indicate that the variables should not all be weighted equally, each coefficientai can be chosen to reflect the relative weight of that variable, with more important variables receiving higher weights.

What are the benefits of utilizing user-defined composites? When compared to the use of individual variables, user-defined composites provide a way to diminish the effect of measurement error.Measurement errorrefers to the disparity between the variable values observed, and the “true” variable value. Such disparity can be due to a variety of reasons, including mistranscription and instrument failure. Measurement error contributes to the background error noise, interfering with the ability of models to accurately process the signal provided by the data, with the result that truly significant relationships may be missed. User-defined composites reduce measurement error by combining multiple variables into a single measure.

Appropriately constructed user-defined composites allow the analyst to rep- resent the manifold aspects of a particular concept using a single measure. Thus, user-defined composites enable the analyst to embrace the range of model character- istics while retaining the benefits of a parsimonious model. Analysts should ensure that the conceptual definition for their user-defined composites lies grounded in prior research or established practice. The conceptual definition of a composite refers to the theoretical foundations for the composite. For example, have other researchers used the same composite, or does this composite follow from best practices in one’s field of business? If the analyst is aware of no such precedent for his or her user-defined composite, a solid rationale should be provided to support the conceptual definition of the composite.

The variables comprising the user-defined composite should be highly cor- related with each other and uncorrelated with other variables used in the analysis. This unidimensionality should be confirmed empirically, perhaps through the use of principal components analysis, with the variables having high loadings on a single component and low-to-moderate loadings on the other components.