# An Argument in Support of Ordinary Factor Analysis of Dichotomous Variables

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Larry Percy (1976) ,"An Argument in Support of Ordinary Factor Analysis of Dichotomous Variables", in NA - Advances in Consumer Research Volume 03, eds. Beverlee B. Anderson, Cincinnati, OH : Association for Consumer Research, Pages: 143-148.

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http://acrwebsite.org/volumes/9189/volumes/v03/NA-03

The number of scale points necessary to effect a cogent factor analytic solution is explored. An argument is offered for the use of ordinary factor analysis in dichotomous variables by comparing the factor solutions generated from 2-point Likert scale and 5-point Likert scale representations of the same items. The results are discussed in terms of congruence and possible implication.

INTRODUCTION

One is frequently confronted in designing a research study with the disquieting dilemma of whether or not to use a multi-point scaling of a continuous variable and feel reasonably comfortable with the reliability and validity of the measurement, along with its applicability to most multivariate models, or to opt for a more convenient dichotomous variable and wonder if in fact one is somehow "losing" information, and quite possibly violating the assumptions of many multivariate models. Evidence is accumulating that would seem to arrest any continuing anxiety over this question.

On the problems of reliability or validity of measurement, neither are dependent on the number of scale intervals. Matell and Jacoby (1971), in a study based on a series of eighteen different scale alternatives varying from 2 through 19 scale intervals, report no significant difference evidenced in either reliability or validity, regardless of the number of scale intervals used. Similar results were obtained when higher ordered scales were collapsed. They conclude that the typical Likert-type scale measures primarily direction, and only marginally intensity. The overwhelming majority of the variance contained in a measurement derived from Likert-type composite rating scales is accounted for by the directional component; so increasing the number of scale intervals does not increase the precision of the instrument. One is not "losing" information, accordingly, by employing a dichotomous scaling

The second question is more widely debated: what is the effect on a multivariate model of reducing a continuous measurement scale to a dichotomy, particularly when the Pearson product-moment correlation is used to represent the relationship between the variables? Addressing a specific interest in factor analysis and the so-called problem of the appropriate correlation measure, one may find support for any of his favorite coefficients: Carroll (1961) feels the tetrachoric is perhaps more suited when dealing with dichotomous variables, while Horst (1965) states that its use is unwarranted; Cattell (1952) favors phi-over-phi-max. Henrysson and Thunbert (1965) discuss the tetrachoric vs. phi coefficients in factor analysis. In one of the few empirical studies (as reported in Rummel, 1970) addressing itself to this question, Comrey and Levonian (1958) concluded that the "same" factors result from all three, and thus recommended phi (or the usual Pearson product moment correlation associated with most factor analysis programs) as the most reasonable choice. Their reasoning followed because phi-over-phi-max and the tetrachoric lead to overly high communality estimates.

After coming 'round the bush, so to speak, in determining the appropriateness of actually using the Pearson product moment correlation with dichotomous data, what actual differences in factor solutions are to be expected as the number of scale intervals of a continuous variable is reduced to two? Martin et al. (1974) hypothesized that the reduction in the number of scale intervals would have a substantial effect on factor analysis results. They were wrong. Utilizing seven different numbers of scale intervals down to a dichotomy, a general decrease in size of the eigenvalues, commonalities, and factor loadings was noted as the number of scale divisions was reduced; however, the patterns of the rotated factor loadings were not appreciably affected, and remained consistent over the entire range of scale intervals. So much for the effect of the number of scale intervals on the results of a factor analysis.

Yet in the typical application of factor analysis in marketing research, the magic number of scale intervals remains five, along with severe warnings to those foolish enough to suggest only two. Guertin and Bailey (1970), for example, have stated that it is highly undesirable to use less than five interval scale points, and that to go beyond five is not particularly useful in providing more information. It is the point of this paper to debunk such notions.

Specifically, this paper investigates the effect on a factor solution of collapsing a typical 5-point agreement scale into a dichotomous measure. From a pragmatic standpoint we wish to see if the dichotomous scale provides sufficient information to uncover the latent dimensionality of a series of attitude questions as efficiently as a 5-point scale. If such is the case, from a marketing research viewpoint the saving in questionnaire administration time and the increase in respondent motivation to complete a long list of scales occasioned by the reduction in the number of scale intervals, would suggest that dichotomous scales substitute for the traditional 5-point scale, particularly in the development phase of an inquiry where intensity of response is conceptually less compelling than direction.

STUDY DESIGN

The design of this study involves the utilization of data already in hand measuring homemakers' attitudes toward the meal along 20 variables. These variables resulted from a prior factor analysis of a much greater number of variables. Each of the 20 attitude variables are presented to subjects as a randomly shuffled deck of cards, each card containing a single variable. The subjects are then asked to sort the cards into appropriate piles on a sorting board reflecting a typical 5-point Likert-type agreement scale: five blocks labeled Agree Strongly, Agree Slightly, Neither Agree nor Disagree, Disagree Slightly, and Disagree Strongly. These data represented a national probability sampling of homemakers.

Plan of Analysis

The data collected are treated as a traditional 5-point Likert scaling, and are then collapsed as follows to dichotomous variables:

1. Subjects indicating they Agree Strongly or Agree Slightly with a variable are rescored as Agree for that item.

2. Subjects indicating they Disagree Slightly or Disagree Strongly with a variable are rescored as Disagree for that item.

2. Subjects indicating they Neither Agree nor Disagree with a variable are independently introduced to a program which randomly assigns them as Agree or Disagree for that item.

Both sets of data are then submitted to an R-mode factor analysis and the resulting solutions compared for congruence. This provides a comparative analysis of a 5-point Likert-type scale measurement with a dichotomous scale measurement over a common set of variables. The assumption is that both scales measure the same information, are submitted to the same factor analytic procedure, and should thus yield comparable factor solutions.

RESULTS OF THE ANALYSIS

The factor solution for the variables as measured by the 5-point Likert scale revealed eight latent dimensions of attitude (restricting rotation to those factors associated with eigenvalues greater than unity).

The resulting rotated factor loadings are detailed in Table 1. An interpretable solution of the eight dimensions according to appropriate criteria is shown in Table 2.

ROTATED FACTOR MATRIX OF THE EIGHT DIMENSIONS MEASURED BY THE 5-POINT LIKERT SCALE

FACTOR INTERPRETATION OF THE EIGHT DIMENSIONS MEASURED BY THE 4-POINT LIKERT SCALE

Each variable contributing approximately one-quarter of the variance in any one factor (i.e. a rotated loading with an absolute value greater than .5) was considered in interpreting the meaning of that factor. Additionally, any marginal variable with a rotated loading of absolute value greater then .4 but less than .5 was considered if the sense of the variable was compatible with those loading more strongly. One will notice in checking the rotated loadings with those making their way into the solution that certain variables which meet the criteria outlined above nevertheless are not listed (e.g. variable 12 which loads -0.58496 on factor 4 but is not shown in the interpretation). These occasional stray variables find their way into the solution space because of a mathematical congruence with the factor rather than any sense of psychological homogeneity and must be dismissed by the analyst as noise in an otherwise condign system.

A second factor solution for the variables, this time as measured by the dichotomous scale, was sought in a manner identical to that of the initial factoring of the 5-point Likert scale items. This solution also revealed eight latent dimensions of attitude. The resulting rotated factor loadings for this second solution are detailed in Table 3, with the corresponding interpretable solution of the eight dimensions shown in Table 4. Again, the same procedure was used in determining which variables describe the latent attitude dimensions.

From a visual inspection of the two factor patterns, as decocted in Table 2 and Table 4, it appears quite obvious that a great deal of similarity exists between the two solutions. One may conclude that factor one, as derived from the first solution, is all but "identical" to factor one as derived in the second solution. Similar conclusions may be drawn for all but one factor in each solution; factor two matches factor two; factor three matches factor six; factor :four matches factor seven; factor five matches factor four; factor six matches factor three; factor seven matches factor five. Only factor eight in each solution is unique.

ROTATED FACTOR MATRIX OF THE EIGHT DIMENSIONS MEASURED BY THE 2-POINT LIKERT SCALE

FACTOR INTERPRETATION OF THE EIGHT DIMENSIONS MEASURED BY THE 2-POINT LIKERT SCALE

In the first solution, factor eight represents a single variable, which barely increases the accounted for variance; the same is true of the two variable factor eight in the second solution. It is probable that a seven factor-solution was in fact the most parsimonious. Otherwise, the solutions appear congruent regardless of whether the original input resulted from a 5-point or dichotomous scale (see Table 5).

COMPARISON OF THE TWO FACTOR INTERPRETATIONS

Although by inspection one might be satisfied that there was little real difference in the results of the two separate analyses, it would certainly help if there were some analytic way of comparing the goodness of fit between the two solutions. Rummel (1970) has pointed out that in addition to an intuitive mode of comparison two systematic models exist for comparing factors from separate analyses: a matrix comparison technique that transforms the factor solution of one study to a least squares fit to the factors of another; and a vector comparison method such as correlation or congruence coefficients.

Harman (1967) and Cattell (1966) both discuss a __coefficient of congruence__ for determining the extent of similarity or agreement between factors obtained in different solutions. This coefficient, as developed by either Tucker (1951) or Burt (1948) is precisely the same, and is given in matrix notation by Cattell as:

where X and Y are loadings of the same variable, not deviations of loadings. Harman's notation shuns matrix form and is given by:

While this coefficient may appear to be similar in form to the product-moment correlation coefficient, notice that (a) one is not dealing with deviations from loadings, but with actual loadings, and (b) one is summing over common variables not individuals. Any attempt to merely correlate the columns of the factor matrices would be insufficient for one would be ignoring the difference between the origins and the differences of mean loading. Both procedures suffer from a lack of attention to possible differences in level between two patterns, but for purposes of this analysis level is not a crucial consideration. Were level an important consideration, the Cattell et al. (1966] pattern similarity coefficient r_{p} would avoid this difficulty by indicating a high congruence of loadings only when both the shapes of the patterns involved and the level are similar.

Because the n variables are identical for the solutions at hand, a straightforward application of equation (1) or (2) to the corresponding loadings of the hypothesized matching factors as shown in Table 5 is possible. Considering the factors derived from the 5-point Likert scale measurement as the first group X and the factors derived from the dichotomous scale measurement as the second group Y, coefficients of congruence were computed for each of the seven factor pairs. The resulting coefficients are shown in Table 6.

The coefficient of congruence can range in value from +1 for perfect agreement through zero for no agreement at all to -1 for perfect inverse agreement. It differs from a correlation coefficient, however, in not equating means (i.e. in not standardizing the data). The coefficients of congruence computed between these two solutions all indicate almost perfect agreement.

Harman (1967) points out that in situations where factors from two solutions may be matched visually, and the number of variables common to the factors is small, it can be expected that the coefficients will be high. It was assumed in these calculations that only those variables common to the factor interpretations were common to any two factors being compared. Table 6 illustrates the variables and their corresponding loadings which were used in the computation.

This whole notion of factor matching for congruence, is of course, consistent with the model of factors as determiners and with the more subtle idea that a factor either operates fully upon a variable or leaves it entirely alone. Any fall from a unit loading by a variable within a factor is purely a function of the other factors that happen to be operating. When one thinks of a lower factor loading implying a smaller effect on a variable he is tacitly recognizing that for a particular situation other factors are more dominant.

If a factor exists and is central to the domain of study, it will occur regardless of the factor technique or instrumentality of measurement. The degree of efficacy in a factor concept is only reinforced by its appearance under varying sampling situations and forms of analysis. In this case, removal of intensity within the attitude measurement devise did not alter the factor solution.

DISCUSSIONS AND CONCLUSIONS

Matell and Jacoby (1971) have indicated that reliability and validity are found to be independent of the number of scale points contained in a Likert-type rating scale. They additionally demonstrate that regardless of the number of steps originally used in collecting the data, re-scoring of these multi-point response scales to dichotomous measure does not result in any significant decrement in reliability or validity. They contend, as both Peabody (1962) and Cronbach (1950) have, that Likert-type scales measure primarily direction and only to a minor degree intensity.

Since any Likert scale measurement is concerned primarily with direction, utilization of a 2-point Likert scale realizes largely the same information as a multi-point Likert scale. Correlations of these data will reflect this fact because the basic monotone relationship has not been altered. By observing normal cautions with the data, there is no meaningful effect on the correlation matrix transformation as a result of the number of scale points utilized.

Proceeding from these conclusions, a study of the congruence of the factor solutions of data as measured in a large scale study by both a 5-point Likert scale and a rescored 2-point Likert scale was undertaken. It was hypothesized that if neither the reliability or validity of one's data was affected by the number of points in a given scale, then the identity of the factor structure of influences that may be unearthed by the factor analysis would be manifest in the data regardless of the manner of collection. Although Nunnally (1967) argues that it is a mistake to compare factors in terms of the patterns of loadings in two analyses because the loadings are not the factors, a coefficient of congruence was none-the-less chosen as the measure of comparison due to its overwhelming prominence in the literature (cf. Cattell, 1966; Harman, 1967; Horst, 1965; Martin et al., 1974; Rummel, 1970). The evidence of the coefficients of congruence in the present study indicates that the ability to extract the underlying factor definition of a space proceeds independently of the number of scale points used for the Likert-type items.

The effect of these findings mean greater flexibility in study design. Data which were collected previously with different rating formats, or data within a given instrument collected with a variety of Likert-type items, may be collapsed into dichotomous measures and factor analyzed because of their now common base measure. It would even be possible to dichotomize open-ended responses along some agreement criteria and include them along with more ordered ordinal or metric measures in the factor model.

These results suggest a much more wide-ranging approach to the traditional uses of factor analysis. The structure of scale imposition no longer seems as critical. At the very least the present findings indicate that for efficiencies in large-scale collection a 2-point Likert scale may be effectively utilized. The resulting factor analysis of this large scale data collection could then form the basis of the critical questioning in a follow-up phase where intensity would be measured along a multi-point scale, but only for the relevant factors.

REFERENCES

Cyril Burt, "The Factorial Study of Temperamental Traits," __British Journal of Psychology (Statistical Section__), 1 (1948), 178-203

Cyril Burt, "Factorial Analysis of Qualitative Data," __British Journal of Psychology (Statistical Section__), 3 (1950), 166-85.

J. B. Carroll, "The Nature of the Data, or How to Choose a Correlation Coefficient," __Psychometrika__, 26 (1961), 347-72.

Raymond B. Cattell, __Factor Analysis: An Introduction and Manual for the Psychologist and Social Scientist__. New York: Harper and Row,1952.

Raymond B. Cattell, "The Meaning and Strategic Use of Factor Analysis," in Raymond B. Cattell, ed., __Handbook of Multivariate Experimental Psychology__. Chicago: Rand McNally & Company, 1966.

Raymond B. Cattell, Malcom A. Coulter and Bien Tsujoka, "The Taxonometric Recognition of Types and Functional Emergents," in Raymond B. Cattell, ed. __Handbook of Multivariate Experimental Psychology__. Chicago: Rand McNally & Company, 1966.

A. L. Comrey and E. Levonian, "A Comparison of Three Point Coefficients in Factor Analysis of MMPI," __Educational and Psychological Measurement__, 18 *1958), 739-55.

L. J. Cronbach, "Further Evidence on Response Sets and Test Design," __Educational and Psychological Measurement__, 10 (1950), 3-31.

Wilson H. Guertin and John P. Bailey, Jr. __Introduction to Modern Factor Analysis__, Ann Arbor: Edwards Brothers, Inc. 1970.

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Jacob Jacoby and Michael S. Matell, "Three-Point Likert Scales Are Good Enough," __Journal of Marketing Research__. 8 [November, 1971).

Warren S. Martin, Benjamin Fruchter and William J. Mathis, "An Investigation of the Effect on the Number of Scale Intervals on Principal Components Factor Analysis," __Educational and Psychological Measurement__, 34 (1974), 537-45.

Michael S. Matell and Jacob Jacoby, "Is There an Optimal Number of Alternatives for Likert Scale Items? Study I: Reliability and Validity," __Educational and Psychological Measurement__, 31 (1971), 657-74.

J. C. Nunnally, __Psychometric Theory__. New York: McGraw Bill, 1967.

D. Peabody, "Two Components in Bipolar Scales: Direction and Extremeness," Psychological Review, 69 (1962), 65-73.

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