Cognitive Differentiation Analysis: a New Methodology For Assessing the Validity of Means - End Hierarchies

Citation:

Thomas J. Reynolds and W. Steven Perkins (1987) ,"Cognitive Differentiation Analysis: a New Methodology For Assessing the Validity of Means - End Hierarchies", in NA - Advances in Consumer Research Volume 14, eds. Melanie Wallendorf and Paul Anderson, Provo, UT : Association for Consumer Research, Pages: 109-113.

Advances in Consumer Research Volume 14, 1987      Pages 109-113

COGNITIVE DIFFERENTIATION ANALYSIS: A NEW METHODOLOGY FOR ASSESSING THE VALIDITY OF MEANS - END HIERARCHIES

Thomas J. Reynolds, University of Texas at Dallas

W. Steven Perkins, University of Texas at Dallas

INTRODUCTION

Means-end hierarchies or chains (Gutman 1982) representing attribute - consequence - value linkages that, in theory, represent how consumers give personal meaning to product attributes and thus serve as the basis of how products are differentiated have recently received-considerable attention in the literature (Gutman 1984; Gutman and Alden 1984; Gutman and Reynolds 1983; Gutman, Reynolds and Fiedler 1984; Olson and Reynolds 1983; Reynolds and Gutman 1984a; Reynolds and Gutman 1984b; Reynolds and Jamieson 1984). The primary methodology used to abstract means-end chains is an in-depth interviewing technique, laddering (Gutman and Reynolds 1979), which first takes a basic distinction elicited by the consumer as to how products differ and tries to uncover the personal relevance of the distinction by probing for the higher level meanings with which it is connected.

Once the initial distinction is determined, which typically is at the lowest level of differentiation, the laddering process is initiated by the "why is that important to you question" which produces a higher level meaning. This obtained meaning is then used as the basis for the next level probe. This process continues until both a consequence and a personal value are elicited from the consumer. Thus, the end result of laddering is a combination of the identification of higher level meanings, consequences and values, that stem from the basic attribute level, and the connections or linkages between the various levels.

One key problem that has researchers questioning the validity of the output from laddering has been the issue of demand characteristics of the interviewing process (Morgan 1985; Durgee 1986). Put simply, the interviewing process will virtually always recover the consequence and value levels, thus the obvious question becomes: To what degree are these higher levels of meaning representing different constructs of product differentiation relevant to constructs such as perception and preference? Or, put another way, if we had consumers sort products at each level of the ladder, thereby representing their differentiation at that level, which level would correspond most closely to judgements of either overall product similarity (perceptual difference) or preference?

From a theoretical perspective the assumptions as to the relative relation of the levels of meaning to judgments of both perceptual differences and preference differences are clearly stated. The central tenet of means-end theory is that the higher the level the more personal relevance the distinction is likely to have and thus it should correspond more closely to preference. Also, perceptual differences are thought to be concrete, dealing primarily with the physical aspects of products and thus should operate at the lowest level of abstraction. Attribute level distinctions, then, should have the highest correspondence to perceptual differences.

Several methodological problems emerge when the measurement of the relation of product ratings by level of meaning to differences between products is considered. Basically, judgments between products imply measures of a pairwise nature as would be input into multidimensional scaling analysis. Thus, the relations between a vector of product ratings can be thought of as representing the independent variables and the dependent variable is the pairwise judgments of either psychological or preference distance.

The first measurement problem is how to assess the correspondence of a vector of ratings to a symmetric matrix of distances. This problem is complicated by the fact that only an ordinal assumption is possible when sorting methods are utilized. This problem has been addressed by a pairwise unfolding technique suggested by Reynolds (1983), which only assumes ordinal judgments. The output of this procedure is a set of summary association statistics closely resembling Kendall's tau measures (Kendall 1955).

This measurement approach has been applied to means-end data (Reynolds, Gutman and Fiedler 1984; Reynolds and Jamieson 1984) with the following summary findings which lend initial support for the means-end approach:

1. On average, attribute level distinctions correspond more closely to judgments of psychological distance or similarity, followed by consequences and then values.

2. On average, value level distinctions correspond more closely to pairwise judgments of differences in preference, followed by consequences and then attributes (see Reynolds (1985) for a discussion as to why and how pairwise measures of preference are both relevant and necessary in this context.)

The major limitation of this initial research was that no statistical tests were made on the significance of the measures of association, due to the lack of an appropriate set of statistical assumptions in the initial formulation. The research reported was based upon simple averages of the association measures across the sample. Importantly, these averages were across all attributes, consequences and values, without reference to the individual components. Recent statistical work in this area (Reynolds and Sutrick 1986), however, has now made univariate tests of this type for individual elements of ladders possible.

One notable limitation of this statistical extension is the failure to deal with differentially-weighted multiple variables, thereby providing the researcher with a statistical basis to infer the relative contribution of the higher levels of meaning on an individual basis. Thus, effectively an ordinal regression model that provides both the respective weights of the variables representing the levels of meaning and tests of the relative contribution of each to the model is necessary if a true test of means-end hierarchies is to be undertaken.

Recognizing this key research problem, more recent work detailing the specifics of how a regression extension can be developed has been completed (Reynolds and Sutrick, in review; Reynolds, Weeks and Perkins, in review). The purpose of this paper, then, is twofold. The first is to summarize the statistical nature of these recent developments and the second, and most important, is to provide a basis of understanding of these methods by contrasting them to a more traditional methodology, factor analysis. The understanding of these new methods should provide consumer researchers with the means to study means-end hierarchies and their relationship to consumer perceptions and preference as well as to other theoretical constructs.

COGNITIVE DIFFERENTIATION ANALYSIS (CDA)

The (n/2) pairwise measures of distance will be denoted as Y_ij, i < j. The descriptor ratings for the i-th stimulus for a given vector corresponds to Z_i The components of Z_i(t) = (t)th descriptor rating on the i-th stimulus. The goal of CDA is to assess the degree of correspondence of {Yij} either individually, or from a composite of the respective ordinal pairwise relationships of Zi This is accomplished by unfolding (Coombs 1958) both {Yij} and Zi into EQUATION pairs of pairs relations, comparing Yijs i'j' and Zis Zj to Zi w Z for appropriate ij, i'j'. (See Reynolds and Sutrick (1986) for a detailed exposition of the analytical method.)

To demonstrate the analytic procedure define random vectors Zij

EQUATION (1)

where

Z_ij^w(l)= MIN {Z_i^(l),Z_j^(l)}    (2)

Z_ij^U(l) = MIN {Z_i^(l),Z_j^(l)}    (3)

The expansion into the pairs of pairs framework for the independent variables permits the specification of ordinal relationships only when Z_ij^w(l) < Z_i'j'^w(l) < Z_i'j'^u(l) < Z_ij^u(R) with either Z_ij^w(l) = Z_i'j'^{w(l) or} Z_ij^u(l) = Z_i'j'^u(l). Thus a function b which indicates a relation can be defined as follows

                          1 IF Z_ij^w(l) < Z_i'j'^w(l) < Z_i'j'^u(l) < Z_ij^u(l)

b{Z_ij^(l),Z_i'j'^(l)}= -1 IF Z_i'j'^w(l) < Z_ij^w(l) < Z_ij^u(l) < Z_i'j'^u(l)    (4)

The unfolding expansion of {Y_ij} to ij, i'j' involves only comparisons of the respective pairwise elements and can be summarized as follows

                  1 IF Y_ij > Yi'j'

Y_ij, Yi'j' = -1 IF Y_ij < Yi'j'     (5)

                  0 IF Y_ij = Yi'j'

A straight forward set of prediction rules in the univariate case, noting the relations between {Y_ij} and Z_i can then be denoted as

predict Y_ij > Yi'j' IF b{Z_ij^(l),Z_i'j'^(l)} > 0

predict Y_ij < Yi'j' IF b{Z_ij^(l),Z_i'j'^(l)} < 0    (6)

An obvious extension to interval assumptions can be made by simply comparing the absolute differences between the corresponding ij and i',j' pairs. This additional extension offers the further benefit of being able to measure the "metricness" of data by comparing it to the solution that is based upon only ordinal assumptions.

Thus, a simple matching of the resulting 1, -1, Os for the respective pairs of pairs relations can be seen to yield measures of fit that correspond to more traditional pairwise measures of fit (i.e. Goodman-Kruskal Gamma (Kruskal 1958), Kendall's Tau (Kendall 1955) and Somer's D (Somers 1962)). Though conceptually similar, however, the degrees of freedom (n/2) for {Yij} and n for Z_i creates a highly complex problem when the statistical properties used in hypothesis testing are of issue. The solution to this central issue will be detailed after the measures of association are presented.

Denoted in simplified fashion, the indices are:

EQUATION (7), (8) and (9)

where C equals the number of concordant relations, (1.1), (-1,-1) and (0,0); D equals the number of discordant pairs (-1,1), (1,-1), (0,1), and (1,0); and I the inconclusive pairs (1,0) and (-1,0).

The statistical properties of the T measures are developed utilizing theories underlying U-statistics (Randles and Wolfe 1979) where an estimate of variance is obtained by the jackknife procedure. Under the null hypothesis, H : f = O. the test statistic e, for all measures of T is

EQUATION (10)

where At is the variance estimated from the jackknife procedure. The jackknife procedure recomputes T successively leaving out stimulus (-s), at a time. Thus, the estimate of variance is

EQUATION (11)

where EQUATION   (12) and

T^-s the statistic T recomputed leaving out the i-th stimulus.

Evaluating the change in the T measure with addition of an independent variable has also been developed. To illustrate, if there are k independent variables and a (k+1)-st variable is added, the new statistic, for each respective measure of T is

EQUATION (13)

where Tk+1, Tk are given by either (7), (8) or (9) ant, following (11)_k,

EQUATION (14)

The extension to the regression case can be readily performed using a standard regression analysis on the 1,-1,0 values, producing beta weights for each of the components of Z. However, a few other considerations need be taken into account before the regression model is applied. First, the specific nature of this data, where a zero for all independent variables implies a zero prediction for the dependent variable, suggests the omission of the intercept term from the standard regression model (Draper and Smith 1966).

Second, the use of regression yields Yi; predictions which do not correspond to the 1,-1,0 values utilized in the univariate measures of T. Thus, for comparative purposes, a transformation must be affected. Under the zero intercept assumption just noted, the solution involves searching by small gradiations for optimal cutpoints equal distant from zero, yielding the necessary trichotomy. The evaluative criterion suggested is T from (8), for the simple reason that it represents a compromise between the other two measures.

Resulting from the regression are the beta weights and, of course R-squared, denoted R²_u, for all ((n/2)/2) pairs. The problem with the directly derived R-squared measure is that it is based upon all of the unfolded pairs of pairs and, as such is not easily interpretable. To permit a more traditional R-squared evaluation with respect to the n/2 pairs of {Yij}, another extension is required. The-statistics derived from the ((n/2)/2) unfolding and subsequent regression analysis need to be transformed into a framework that is better understood and more easily comparable to standard approaches. To accomplish this, the 1,-1,0s resulting from the application-of cutpoints to the {Yij} can be used to "refold" the data into (n2) pairs corresponding to the original dissimilarity judgments. This is done by creating a score vector, {Wij} , as follows:

EQUATION (15)

By summarizing the net wins in this way for all pairs, the vector W.j can be directly related to the original Yi.. A squared Pearson correlation can then be computed between the predictions and the true distances yielding a R-squared "refolded", denoted R2r, that reflects the overall goodness-of-fit of the CDA regression methodology.

CONTRAST TO FACTOR ANALYSIS

The CDA model can be conceptually viewed as precisely the converse of factor analysis. The goal of factor analysis is to decompose the matrix of intercorrelations into an ordered set of vectors that optimally reproduce the summary projections, in terms of variance accounted for. Thus, factor analysis yields positions or loadings on each of the respective factors for each of the variables. Each factor, then, accounts for a specified amount of the total variance represented in the initial summary correlation matrix. CDA can be thought of as measuring the relation or correspondence of an a priori set of factors (independent variables) to the criterion matrix of summary interrelationships, the correlation matrix (dependent variable).

Given this juxtaposition of the two techniques, the results from a factor analysis can be used to both demonstrate and evaluate the CDA regression model. That is. by taking a factor analysis solution and considering it as "known" and performing CDA using the factor scores as the independent variables and the original correlation as the dependent variable, an evaluation of the stability of this new approach becomes possible. With this type of retrofitting of solutions, and given the metric nature of the factor analysis solution, a comparison can be made between the metric and ordinal assumptions of the CDA decomposition of the independent descriptor vectors. Obviously, it would be assumed that the metric assumption would be most appropriate for reproducing the original correlation matrix.

An eight variable correlation matrix was subjected to factor analysis yielding two varimax rotated factors (Cooley and Lohnes 1971, pp. 134-135). The correlation matrix and the resulting factors input to CDA are presented in Table 1. The first factor accounted for 74% of the total variance with the second representing 24%.

TABLE 1

FACTOR ANALYSIS EXAMPLE FROM COOLEY AND LOHNES (1971)

The univariate tau measures from equations (7), (8) and (9) and the tests of significance from equations (10),(11) and (12) are presented in Table 2. As would be expected the first factor has a higher degree of correspondence than does the second. Also, as expected, the metric assumptions provide a higher degree of correspondence than does the ordinal solution. The measure of significance corresponds to a standard score thus indicating a high degree of correspondence for both factors. It should be noted that the tau values and their corresponding test statistics are negative due to the fact that the correlation matrix represents similarity data as opposed to dissimilarity data as would be the case when distances are involved. Thus, all the signs will be reversed.

TABLE 2

TAU CORRELATIONS AND TEST STATISTICS FOR EACH Zi(l) VECTOR

The regression results are shown in Table 3. The metric CDA analysis recovers the factor analysis results well, with an R-square reasonably close to the 98% variance explained by factor analysis and coefficients in the same proportion as the percent explained by each factor. That is, the 74% and 24% explained by factor analysis produces a ratio of about 3 to 1 and the ratio of the squared coefficients (.5945 to .3477) is also approximately 3 to 1.

Under ordinal assumptions, the R-square is somewhat lower and the second vector is not significant. Obviously, the factor analysis results are not ordinal, and would not be expected to conform as closely to the interval method. To illustrate, a two dimensional non-metric scaling solution (Young, Takane and Lewyckyj 1980) for this matrix produces a Kruskal stress of .009 while a one dimensional configuration increases the stress to only .070, indicating that the space may be adequately explained by a single dimension. Thus the ordinal solution appears to correspond closely to the lower number of dimensions as mirrored by the non-metric multidimensional solution. In sum, the CDA method appears to operate in an expected manner with respect to this simple example with a known solution.

TABLE 3

CDA REGRESSION RESULTS

CONCLUSION

The extension of CDA analysis to a regression format offers a methodology to assess product descriptor ratings to a matrix of pairwise judgments of psychological distance or preference. This new measurement approach has significant potential with respect to small stimulus sets as would typically be the case in marketing research.

Of significant import is the fact that all laddering research can be assessed using this analytical framework thereby providing a straightforward validation paradigm. Tests of significance of the contributions of the higher levels can be incorporated by forcing in the lower levels and then evaluating the contribution of the higher levels in a stepwise manner.

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