# Resolving Factor Structure Distortions in Consumer Research

##### Citation:

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Robert H. Taylor, E. Laird Landon, Jr., and Jerome E. Scott (1972) ,"Resolving Factor Structure Distortions in Consumer Research", in SV - Proceedings of the Third Annual Conference of the Association for Consumer Research, eds. M. Venkatesan, Chicago, IL : Association for Consumer Research, Pages: 213-219.
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[The authors would like to thank the University of Colorado Computing Center for providing computing time.]

[The authors are all Assistant Professors of Marketing at the University of Colorado.]

Factor analysis has long been one of the principle tools of behavioral research. Its uses range from the reduction of large amounts of data to manageable proportions, to lending insights and understanding to complex behavioral phenomena. Because of this long history of use and frequent appearance in both psychological and marketing literature, there is a feeling of confidence in the use of the technique, when in fact there are many hidden and unresolved problems associated with the use of factor analysis.

The purpose of this paper is to investigate one potential source of trouble relevant to consumer research and to suggest some possible solution procedures. Consideration of this problem should lead to more valid inferences being drawn from behavioral research.

One of the major uses of factor analysis is to derive the underlying structure, or factors, of attitudes or behaviors by analyzing the interrelationships between variables (test scores, overt behavior, etc.) measured on a sample of individuals. An implicit assumption in this approach is that all individuals are guided by the same latent relationships: that is, there is an assumption of a 'universal' factor structure where all individuals perceive the same underlying domain.

Unfortunately, there may be no a priori reason for making this assumption. In fact, the opposite may be more tenable and may be exactly what the researcher is trying to find out. Therefore, to use pooled data (a factor analysis conducted across the total sample) when there are fundamental differences between subsets of subjects can lead to misinterpretations concerning the underlying factors. A factor analysis performed on the total group may differ radically from the analysis performed on the sub-groups of which it is comprised.

The distortions which can result are illustrated in Figure I. The sample consists of two well defined sub-groups, but factoring the total data set would produce one set of axes, while factoring of the two sub-groups separately would produce two entirely different sets of axes (the primed factors).

If the purpose of the analysis is data reduction, then the factor scores are of interest. In this case the analysis would be performed on the total group in order to maintain a common reference point for the factor scores. However, if one is interested in the underlying structure of behavior, the groups should be analyzed separately. To pool the data might lead to a non-meaningful factor matrix.

The technical problem is one of defining modes in the data set, and the subsequent examination of the sub-structures derived from factoring. This problem has been recognized in the clustering literature (Tryon and Bailey, 1970) and in scaling (Tucker and Messick, 1963) as well as the literature dealing with factoring (Harman, 1967), but has not generally been applied to empirical studies dealing with consumer behavior.

The first step in the analysis is to separate the data base into subgroups which are of interest to the researcher. Such groupings are made from hypotheses concerning differential response patterns. Typical examples would be examination of males versus females, blacks versus whites, city versus country residents, etc. Each sub-group would then be separately factor analyzed, in addition to a factor analysis on the total group (to be used for comparative purposes). The second step required in the analysis is to compare and analyze the resulting factor solutions to determine the degree of similarity between them.

POOLED VERSUS SEPARATE GROUP FACTORING

METHODS FOR COMPARING FACTOR SOLUTIONS

The problem of factor matching involves two aspects of factor structure: the analysis of patterns of loadings, and the analysis of the level of loadings. Three approaches have been proposed for comparing factor structures: visual inspection, vector by vector methods, and total matrix comparisons.

Visual Approaches. For the researcher with a good understanding of the problem being investigated, a simple visual inspection of the factor structure might be sufficient to determine the degree of similarity or difference. In addition to level and pattern, the complexity of variables and the communalities can also be examined. The personal impressions formed by such inspection can often lead to subtle insights which raise questions about the comparability of solutions. A visual inspection is generally recommended as a preliminary step in the analysis and may be followed by one or more of the mathematical techniques available.

Vector Comparisons. The vector by vector approaches focus on how congruent any two factors, considered separately, seem to be. Harman (1967) proposed a root mean square approach for determining the extent of agreement. Using his notation:

gives the agreement between Factor p of the first study and Factor q of the second study across the n common variables. The ads represent the factor loadings for the j'th variable on the p'th factor, etc. This measure is proportional to the distance between factors (positioned in the common n-dimensional space) and imposes both pattern and level restrictions on the definition of similarity. In addition, the index may be difficult to interpret since all one knows is that a perfect fit would yield an index of zero.

A second possible index is the coefficient of congruence, defined as:

with the same meaning and variables as the above formula. This coefficient is the cosine of the angle between factors, and varies from zero (noncorrespondence) to + one. It is less restrictive since the lengths of the vectors are normalized prior to comparison. Thus, level differences are partially removed.

Additionally, the simple correlation coefficient between factor loadings could be used to measure similarity. This is least restrictive in its requirements since it is a measure of pattern without regard for the magnitude of the loadings.

Several cautions are important to observe when employing these coefficients. First, the researcher may be comparing factor structures from two different studies where only a sub-set of the variables are common to both. In this case, the indexes are applicable only to the common variables. Cattell (1952) points out that this can lead to mistaken judgements about the similarity of the factors if the elimination of dissimilar variables also removes the variables which load heavily on the factors and are therefore the defining variables. The comparisons may then be using only the variables with modest loadings which can lead to assertions that the factors are similar when they are defining different constructs (or that they are different when in fact they are defining the same construct). To reduce the likelihood of such spurious judgements, Cattell (1952) suggests that only variables which uniquely define a factor be included in the comparison, and that communalities provide the basis for selecting these variables. If one restricts comparisons to only those variables with loadings exceeding .8, then (for orthogonal factors) its loadings must be low on the remaining factors.

A further problem with the coefficient of congruence is that the relative size of the index tends to be quite large, even for factors which are quite different. Penneau and Newhouse (1964) point out that the lowest value the index can take for comparisons involving factors with only positive loadings is .67, and that factors with similar patterns but quite different magnitudes will result in indices of above .9.

Matrix ComParisons. The disadvantage of the vector approaches is that the factor loadings are taken as given by the rotation method. However, exogenous influences may affect independent rotations and confound comparisons. Since factor rotations are arbitrary (an infinite number of solutions exist), a better approach might be to rotate one of the factor solutions to a least squares fit with the other. Ahmavaara (1954) developed such a target matrix approach.

A difficulty with Ahmavaara's approach is that the solution depends upon which matrix is the target and which is the rotated factor loading matrix. In general, there may be no a priori reason for assigning a particular loading matrix as the target. In this situation it is better to rotate both factor solutions simultaneously to a common point of maximum congruence. A procedure was developed by Horst (1965) as a generalization of the two group canonical correlation problem where factor loadings are orthogonally rotated so that the correlation between loadings is maximized. Since this procedure is based on the correlation between factor loadings, it is a measure of the pattern similarity of loadings and does not measure differences in level.

RESULTS OF FACTOR STRUCTURE COMPARISONS

In a recent study, several hypotheses were generated concerning differential behavior of high and low need for achievement subjects in Bayesian probability revision experiments. However, a key to this relationship is that risk preference is considered a major component of need for achievement (Feather, 1959), and this relationship must be demonstrated.

Ninety-seven males were measured on a twenty-six item measure of need for achievement (Mehrabian, 1968) and eight risk preference items. The pooled group was factor analyzed is determine the dimensionality and to find the key definers of the dimensions. [All factor analyses were conducted using the squared multiple correlation coefficient as the communality estimate. Varimax rotations were performed. Since the total group structure included six factors (eigen-values greater than one), all subsequent analyses were carried to six factors.] The results are shown in Table I. The most surprising outcome was the lack of a relationship between the risk items (which emerged as an independent factor) and need for achievement.

The data were separated into two groups on the basis of need for achievement and each group was separately factored. Two problems arise with this procedure. First, the variables used to define the groups were used again in the factor analysis. If the intent were to predict membership, this procedure would surely bias the results. However, since relationships between variables were the focus, the procedure was felt to be valid. Second, the splitting procedure reduced the sample size for each group to forty-eight. Since thirty-four variables were being used, the stability of the correlation coefficients are highly suspect (Cattell (1952) suggests four times as many cases as variables).

The results of the separate factor analyses are shown in Tables II and III. In both solutions, risk preference items loaded on factors containing need for achievement item., indicating a relationship between these two sets of variables within the group. A visual inspection of which variables define the factors for each of the user-groups strongly suggests that the factors appear to be different constructs between groups. Further, both sub-group solutions are different from the total group solution. It would appear that risk preference is related to and interacts with need for achievement, and that this relationship is hidden by the total group factoring.

LOW NEED FOR ACHIEVEMENT GROUP FACTOR STRUCTURE

The sub-group factor solutions were then compared factor by factor using the coefficient of congruence (Table IV). An inspection of the coefficients indicates very low similarity for any pair of factors between sub-groups. These results are consistent with the visual examination.

The sub-group factor solutions were then analyzed using Horst's matrix comparison approach to obtain a simultaneous comparison of the factor structure (Table V). This procedure attempts to maximize the correlation between loadings. The largest correlation is .77 which indicates a relatively low degree of correspondence between the most similar factors of the two structures. The remaining correlations fall quite rapidly with virtually no relationship indicated for the least similar pair. This procedure does indicate somewhat higher correspondence for the pairs of factors than the coefficient of congruence, but this is a result of the rotation of the structure to maximum congruence prior to the calculation of correlations. This has the effect of removing part of the angular separation of the factors, while the use of correlations between factors removes level differences.

HIGH NEED FOR ACHIEVEMENT GROUPS FACTOR STRUCTURE

FACTOR COMPARISONS COEFFICIENT OF CONGRUENCE

The guidance given to the Bayesian research is ta verify that there is a relationship between risk preference and need for achievement, but that the direction and extent of that relationship is a function of the degree of need for achievement. These results also indicate that a simple pooled group factoring may hide the true structure of the system because of an invalid assumption of a universal factor structure.

FACTOR COMPARISONS - HORST'S MAXIMAL CONGRUENCE

REFERENCES

Ahmavaara, Y. The Mathematical Theory of Factorial Invariance Under Selection. Psychometrika, 1954, 19.

Cattell, R. B. Factor Analysis. New York: Harper and Row, 1952.

Harman, H. H. Modern Factor Analysis, 2nd ed., rev. Chicago: The University of Chicago Press, 1967.

Horst, P. Factor Analysis of Data Matrices. New York: Holt, Rinehart, and Winston, Inc., 1965.

Mehrabian, A. Male and Female Scales of the Tendency to Achieve. Educational and Psychological Measurement, 1968, 28. 493-502.

Pinneau, S. R. & Newhouse, A. Measures of Invariance and Comparability in Factor Analysis for Fixed Variables. Psychometrika, 1964, 29, 271-281.

Rummel, R. J. ApPlied Factor Analysis. Evanston, Illinois: Northwestern University Press, 1970.

Tryon, R. C. & Bailey, D. E. Cluster Analysis. New York: McGraw-Hill, 1970.

Tucker, L. R. & Messick, S. An Individual Difference Model for Multi-dimensional Scaling. Psychometrika, 1963, 28, 333-367.

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##### Authors

Robert H. Taylor, University of Colorado

E. Laird Landon, Jr., University of Colorado

Jerome E. Scott, University of Colorado

##### Volume

SV - Proceedings of the Third Annual Conference of the Association for Consumer Research | 1972

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