Comparison of Stated and Inferred Parameter Values in Additive Models: an Illustration of a Paradigm

James B. Wiley, Wayne State University
Douglas L. MacLachlan, University of Washington
Reza Moinpour, University of Washington
ABSTRACT - A procedure for examining inconsistencies between stated and inferred parameters of additive multiattribute models is presented and illustrated with two product classes, soft drinks and tea drinks. In this example, subjects were found to be inconsistent in their expressed and inferred evaluations of (1) preferences for the various attribute combinations, (2) the importance of attributes, and (3) ideal points on the attributes.
[ to cite ]:
James B. Wiley, Douglas L. MacLachlan, and Reza Moinpour (1977) ,"Comparison of Stated and Inferred Parameter Values in Additive Models: an Illustration of a Paradigm", in NA - Advances in Consumer Research Volume 04, eds. William D. Perreault, Jr., Atlanta, GA : Association for Consumer Research, Pages: 98-105.

Advances in Consumer Research Volume 4, 1977   Pages 98-105


James B. Wiley, Wayne State University

Douglas L. MacLachlan, University of Washington

Reza Moinpour, University of Washington


A procedure for examining inconsistencies between stated and inferred parameters of additive multiattribute models is presented and illustrated with two product classes, soft drinks and tea drinks. In this example, subjects were found to be inconsistent in their expressed and inferred evaluations of (1) preferences for the various attribute combinations, (2) the importance of attributes, and (3) ideal points on the attributes.


Two primary objectives of consumer marketing research are to predict consumer behavior, especially buying behavior, and to control, that is to relate purchase decision making to variables that are controllable by the firm. One of the main problems in achieving these objectives is understanding how consumers develop preferences; for example, preferences for brands in a product class. Recently, a group of marketing researchers have proposed that consumers' preferences among a set of brands can be accounted for by functions of individuals' evaluative beliefs concerning product attributes (Sheth, 1971; Cohen and Ahtola, 1971; Sheth and Talarzyk, 1972; Lehmann, 1971), that is, by multi-attribute models.

A general form of additive multi-attribute models utilized in marketing can be represented as follows (Ginter, 1972, p. 4):



"j = generalized attitude toward brand j, typically operationalized as scaled preference,

Wi = importance of attribute i,

Bij = perceived amount of attribute i believed to characterize brand j,

Ii = ideal amount of attribute i,

n = number of attributes

k = parameter specifying Minkowski metric, normally assume d equal to one (1.00).

The belief component, Bij, allows discriminations between brands on the product attributes. It permits differences in individual preferences for brands to be related to judged differences between brands on product attributes. The weighting component, Wi allows idiosyncratic differences in the "importance" individuals attach to the various product attributes, an intuitively appealing aspect of the model. Likewise the ideal point, Ii, allows idiosyncratic differences in specifying the most desirable level of each attribute. By "standard model," we refer to the special case with k = 1.0 and I= 0.0. Ginter (1972, pp. 38-49) provides a general model linking values, beliefs, and ideals to attitude; attitude to preference; and preference to choice.

At least two approaches have been adopted by those utilizing additive models such as (1) to account for individual preferences (Moinpour, Talarzyk, Wiley, 1976). The composition approach--adopted by Rosenberg and Fishbein as well as numerous marketers (Wilkie and Pessemier, 1973)--builds toward preference or behavior by linking attitude to self-designated judgments on specified product/service attributes for values. The decomposition approach starts with preference or behavior and builds towards underlying cognitive structure. It is clear that both approaches assume a model and that these models may in specific instances be identical. In such cases the decomposition approach represents a scaling method in the sense of Anderson (1962). The estimating procedure, in other words, can be used as a scaling method.

This paper investigates the correspondence between estimated scale values and individual's self-designated judgments of presumably corresponding values. Three areas of correspondence are at issue. The first is the correspondence between inferred and stated "preference"--the dependent variable. Second is the correspondence between inferred and stated "importance." Third is the correspondence between inferred and stated position of the "ideal point."


"Soft drinks" and "tea drinks" are the product classes selected for this study. Both are product classes with which a large number of respondents are likely to be familiar and both have been the object of previous research. [Especially as a result of this latter fact, these product classes offer two important advantages. First, there is some agreement among those experienced in the area (e.g., Ahtola, personal communication; Carroll, 1972) that relatively complex composition functions underlie many people's preferences for "brands" in these product classes. The following comments by Carroll (1972) are illustrative of the complexity of the models necessary to account for "tea data": "As suggested earlier, however, even this fairly complex model [a weighted unfolding model] apparently fails to provide an adequate fit to the present data. Something approaching a cubic seems to be required for the temperature dimension." (pp. 67-68).

The second advantage is that there is some agreement across the reported studies regarding the salient dimensions of these two product classes. The "temperature" and "sugar" dimensions reported by Carroll for the "tea data" have intuitive appeal and appear to have worked well in practice. Three groups of researchers have worked with soft drinks and each group has arrived at roughly equivalent sets of attributes. The fact that the correlation between predicted and revealed preferences using these attributes have been relatively high (.25 - .625 for groups, to over .80 for individuals) in comparison with results in other product classes suggests the attributes common to these studies may indeed represent a generally salient subset of attributes.] Respondents were selected, on the basis of willingness to participate, from particular business classes at the University of Washington and Wayne State University.

Two types of tasks were required of participants. First, each individual generated choice data by indicating preferences for (a) hypothetical soft drinks and (b) hypothetical tea drinks. Secondly, individuals indicated (stated) in a "traditional" manner their judgments regarding "preference," attribute "importance,'' and "ideal points" for "brands" and attributes of the two product classes. A laboratory setting was adopted. Figure 1 represents the experimental design schematically and summarizes tasks required for completion.



Each subject was given a complete deck of 25 blocks and asked to order the "brands" in each block from most to least preferred. One half of the subjects received a "tea drink" data set in the first session, one half received a "soft drink" data set in the first session. The subjects were asked to rank order the blocks independently. After an individual ordered all 25 blocks, his deck was collected.

Following the first experimental session, each deck was read into a line file, thus preserving order information. Cards were then randomized within blocks and the blocks arranged in arbitrary order within decks.

For the second session, each subject was given the alternative deck to the one sorted during the first session. After the decks were collected the subjects filled out all portions of questionnaire A. The data storage and randomization procedure was repeated. For the third session, participants were given the same deck they sorted during the first session. For the fourth session they were given the same deck they sorted during the second session and they completed questionnaire B.

Choice Data

For both "product classes" the subjects were given a deck of 225 computer cards. The cards were arranged into 25 subsets, or "blocks," of nine cards. Such a design allows 25 unique "brands" to be arranged into blocks such that each "brand" is paired with every other brand in precisely three blocks and no brand appears in the same block with itself.

A hypothetical brand of soft drink was described on the back of each card in the soft drink deck. Likewise, a hypothetical brand of tea drink was described on the back of each card in the tea deck. Soft drinks were characterized in terms of their level on three 7-point attribute scales: calories (1 = 0 calories/8 oz.; 7 = 160 calories/8 oz.), taste (1 = bitter; 7 = sweet), and carbonation (1 = none; 7 = heavy). In order to assure that each subject was exposed to a range of attribute levels and yet avoid extremity effects, brands characterizing all combinations of the second, fourth, and sixth levels of each attribute were selected for study. The brand characterized by the lowest level of each attribute and the brand characterized by the highest level of each attribute were excluded to meet design requirements (the design accommodated only 25 brands).

Tea drinks were characterized in terms of their level on two attributes: temperature and sugar. Drinks characterizing all combinations of five levels on each attribute were included in the design. The five levels of temperature were: ice cold, cold, lukewarm, hot, and steaming hot. The five levels of sugar were: no sugar, 1/2 teaspoon, 1 teaspoon, 2 teaspoons, and 4 teaspoons per cup.

Twenty subjects provided replicated soft drink data. Twelve of twenty subjects who provided soft drink data also provided replicated tea data. Analysis of the choice data is discussed below.

Self-explicated Data

In addition to providing choice data, subjects stated in a "traditional" manner their judgments regarding "preference," attribute "importance," and "ideal points." Measures of the former two appeared in the first of two questionnaires, questionnaire A (Figure 1). "Importance" was measured on a 7-point scale (1-very unimportant, 7-extremely important). The "ideal point" measure replicated Ginter (1972) in requiring the subject to directly indicate his ideal point on the respective attribute scales. A measure of preference appeared on the second questionnaire, questionnaire B (Figure 1). Preference was indicated on a 10-point acceptable-unacceptable scale. Thus, indices of preference, importance, and ideal-points inferred from choices could be compared with self-explicated judgments of presumably corresponding values.

The twenty subjects who provided soft drink choice data also provided self-explicated attribute importance and ideal point judgments. The twelve subjects who provided replicated tea drink choice data also provided a self-explicated attribute importance judgments and (with one exception) self-explicated ideal-point judgments for this data set. The twelve subjects also rated the hypothetical soft drink brands on a 10-point acceptable-unacceptable scale.


Three indices are to be inferred from the choice data: a) an index of preference, b) a measure of "importance,'' and c) an indication of the most preferred level of each attribute (the "ideal point"). In order to develop these numbers we assume the choice data can be represented as the hypothetical data structure shown in Table 1. A cell frequency fjm, represents the frequency with which brand j is assigned to category m. In this instance the number of categories, M is nine and the marginal frequencies, nj., is seven. We also assume:



pjm = fjm/nj. ,

p'j = [pj1 pj2 ... pjM], and

p'1xNM = [p'1 p'2 ... p'N].

That is, for a given design and set of choice objects, it is assumed that an individual's responses can be represented as multinomial. The proportion pjm is an estimate of the probability that object j will be assigned rank m, provided it appears in a block. The multinomial p'j summarizes the individual's response to object j and the supervector p' summarizes the individual's response to the set of objects.

Inferred Preference

The mean rank preference, the measure of inferred preference here utilized, is given by

F(pj) = A.pj

where A is an appropriately defined matrix of weights such that F(pj) is the mean score for the j(th) multinomial distribution and E[F(pj)] = F(pj), where E[F(pj)] is the expectation of F(pj) and F = F(p) is a vector of consistent estimators of F(p), the vector of underlying preferences.

The variance for item j, V(F(pj)), can be represented as,


and the sample variance/covariance matrix for the set, S , is a diagonal matrix with the elements V(F(pj)) (NxN) on the diagonal, under the assumption of independence. [Note that (2) is merely the formula for the variance of grouped data.]

Estimating "Effects"

We now assume that:


where X is a known design matrix of rank v < N, and b is a vector of unknown parameters.

It is well known that a best asymptotic normal (BAN) estimator of b is given by b, where b is the vector which minimizes (F-Xb)/(1xN)' S-1/(NxN) (F-Xb)/(Nx1). The minimum value of this form is a test of the fit of the model C/(dxv) b/(vx1) = 0/(dx1); a test of the null hypothesis C/(dxv) b/(vx1) = 0/(dx1)  is produced by conventional methods of weighted regression, where C is matrix of arbitrary constraints of rank d < v. The test statistic for the fit of the model to the preference measures F(p), is SS[F(p) =  X B = F'/(1xN) S-1/(NxN) F/(Nx1) - b'/(1xv) (X S-1 X)/(vxv) b/(vx1) which has, asymptotically, a central c2 distribution with (N-v) degrees of freedom, if the model fits: where b = (X'S-1X)-1 X'S-1F. Given the fit of the model, the test of the null hypothesis C b = 0 is produced by

SS[C b = 0] = b'C'[C(X'S-1X)-1 C']-1 C b,  (3)

which has asymptotically a central c2 distribution with d degrees of freedom if HO is true.

For example, in the case of a set of "brands" characterized in terms of three attributes a, b, g, we can represent a linear additive model as

E[F(p)] = u + aa + bbgc

where F(p) estimates the vector of preferences, m is the overall mean; aa, a = 1, 2, ..., I, is the differential effect associated with the a(th) level of attribute a; bb, b = 1, 2, ..., J is the differential effect associated with the b(th) level of attribute b; and gc, c - 1, 2, ..., K, is the differential effect associated with the exception of reparameterizing the model so that S aa = S bb = S gc = 0, to provide sufficient constraints for estimation. By way of illustration, for three attributes, two levels per attribute, we have


where the estimate of a2 can be calculated by a2 = -a1; similarly b2 = -b1, g2 = -g1. Under the hypothesis the model is true, the parameters are unbiased estimates which can be used as "scale values" in the tradition of Anderson (1962), Andrews, Morgan, and Sonquist (1967), and Anderson and Messenger (1973). Note the similarity between this model and the conjoint measurement models employing monotonic analysis of variance (Kruskal (1965), Green and Wind (1973)).

The following are parameter estimates and asymptotic covariance matrices for the parameter estimates for the additive linear model.

b = [a1, a2 ... aI, b1, b2 ... bJ, g1, g2 ... gK]   (4a)

   = (X'S X)-1 X'S F(p)

Vb = (X'S-1X)-1   (4b)

Definition of Inferred Importance

Inferred "importance" is defined in this study in terms of the proportion of total variance accounted for by an attribute. This was derived by using (3) to define simultaneous contrasts on the elements of k corresponding to attribute levels (4a). For example, the "importance'' of attributes for a three-attribute, three-level-per-attribute model (reparametized) was provided by three applications of (3), using the following C matrixes:


"Importance measures for a two-attribute, five-level model (reparametized) was provided by two applications of (3) with the following C matrixes:


Proportionate variance, converted to ranks, defines inferred "importance" of the attribute. The greater the proportion of variance "accosted for" by an attribute, the more the inferred "importance" of the attribute.

Definition of "Ideal Point"

The inferred "ideal point" in this study is defined as the attribute level having the maximum effect, or beta value (4a). An illustrative response surface for tea data is presented in Figure 2. For this subject, the ideal point on sugar is taken to be "one teaspoon" of sugar. The ideal point on temperature is taken to be "steaming hot." Note the appearance of a least preferred level on the temperature dimension. This "negative ideal" appears at the lukewarm level, an intuitively pleasing result.


Three scores were calculated for each subject and product class as a preliminary check on subject consistency. These scores are presented in Tale 2. The first two scores in the table represent the number of choice intransitivities committed by individuals during each trial.

Consistency With and Across Trials

Table 2 indicates that only a few subjects were completely transitive in their judgments. However, there is strong evidence that the subjects were not responding randomly, either. The expected number of intransitivities under the hypothesis of random response is 400 with a standard deviation of 14.4. The hypothesis that subjects are responding randomly can be rejected at beyond the .001 level.





Mean ranks, F(p), were calculated for each individual and each trial. The Rho figures in Table 2 consist of the Spearman rank order correlation between trial one and trial two scores. All of the soft drink correlations are significant at the .05 level. Eleven of 12 tea drink correlations are significant at the .05 level. Some "non-significant" correlations might be expected as a result of "true preference" changes. The fact that subjects were very consistent within sorting tasks in those instances exhibiting relatively lower correlation suggests that the lower intertrial correlations may in fact have been the result of a true change in preference.

In summary, subjects' choices are consistent within trials and accurate predictions of choices within one trial can be made using a function estimated from choices in another trial. These results imply that lack of correspondence between stated and inferred values (if found) is unlikely to be the result of inability to perform the choice task or inconsistency during its completion, nor is it likely the result of preference functions that are unstable over time.

Stated Versus Inferred Preference

Regarding the dependent variables, Craig and Ginter (1974) comment:

Marketing's use of the multi-attribute model has entailed a variety of dependent measures. Variables used as the dependent measure include: actual choice, self-report of purchase, preference and attitude. Most studies select one of these dependent measures and then proceed to examine the model's adequacy in terms of that variable .... Thus, generally, two equally tenable explanations exist for any finding: (1) the model itself did or did not perform well, or (2) the dependent measure selected was appropriate or inappropriate.

Table 3 presents correlations between stated (rated acceptability) and inferred preference for each of 12 subjects. The median correlation is approximately .50 in both trials (notwithstanding the abstract nature of the judgment and the relatively large number of judgments required).

Given the high within-trial correlations between stated and inferred preference exhibited in previous analysis, it is tempting to interpret the median of such correlations (Table 3) as an indicant of the median correlation that may be obtained utilizing a three attribute model and "traditional analysis" (i.e., utilizing rated importance, belief, and criterion measures (Moinpour and Wiley, 1972a; Wilkie and Weinrich, 1972; Craig and Ginter, 1974)). It is tempting in light of current results, in other words, to entertain the notion that the "dependent variable" is a major source of residual error. Only Moinpour and Wiley (1972a) report markedly higher median correlations, utilizing a three attribute model, than those reported here (.75 for "pain relievers," .91 for soft drinks). Wilkie and Weinrich (1972) report a mean correlation of .54 for a three attribute model. Craig and Ginter (1974) report correctly classifying approximately 66 percent of subjects' preference (against an expected 33.9 percent, using proportional priors), a result not inconsistent with Wilkie and Weinrich or with the results reported in Table 3. Thus, at the minimum the results appear to indicate that the second of Craig and Ginter's "tenable explanations" could be true. Perhaps more than previously has been recognized the onus for "less than perfect "results accrues to poor measurement of the dependent variable.



Stated Versus Inferred Importance

Issues related to "importance" are among the most widely investigated in the area. Four of the 42 papers reviewed by Wilkie and Pessemier (1973) evaluated alternative ways of measuring importance. The results suggest item specific factors plus error amount to less than 50 percent of total variance, but are greater than 20 percent. Twenty-four of the 42 papers reviewed investigated the question of whether or not importance weights belong in the formulation. Wilkie and Pessemier's conclusion is that inclusion neither is likely to reduce explanation power of the model, nor is it likely to add strikingly to prediction. An explanation for previous results compatible with current findings, posits "measurement error" in the "I," "B," and "W" components of the model (1). Tables 4 and 5 present data relevant to this issue. [Beckwith and Lehmann (1973) argue against a "measurement error" explanation for the "suppression effect" associated with the "W" component.]

Only in the case of the "taste" attribute on soft drinks (Table 4) might the possibility of a consistent relationship between stated and inferred values be entertained. In the case of the other two attributes, stated importance fails to correctly classify inferred importance significantly more frequently than would be expected by random assignment with equal priors. Similarly, for the tea data it cannot be said that the distribution of inferred importance for an attribute differs significantly between those who declare the attribute "most" and those who declare the attribute "least" important.





Discrepancy between stated values and values inferred from judgments are well documented. Research in other fields has shown that subjective reports about cognitive processes are often erroneous (for example, Hoffman, 1960; Slovic, 1969). The relationship between stated and inferred values observed in this study is consistent with corresponding relationships observed elsewhere.

Stated Versus Inferred Ideal

At least seven studies requiring stated ("direct") judgments of ideal-points have been reported (Lehmann, 1971; Bass, Pessemier, and Lehmann, 1972; Ginter, 1972; Winter, 1972; Wiley and Richard, 1974; Richard and Wiley, 1974; Wiley, Richard, Becherer, and Johnston, 1975). All found the standard formulation (1)--k = 1, I = 0--to perform slightly better than an "ideal-point" formulation. The respective authors, however, were unwilling to reject the possible utility of ideal-point formulations.

Ginter (1972) suggests the poor performance of the ideal-point formulation stems from subjects' difficulty in understanding the notion of an "ideal-point." None of the seven papers cited, however, provides direct evidence of such "misunderstanding." Tables 6 and 7 provide data relevant to this issue.

Following Ginter (1972), explicit ideal points were obtained from the respondents by having them state, on the same scales adopted for the sorting tasks, where the ideal level of each attribute would lie. Table 6 presents the frequency of inferred ideals, given the stated ideal, for soft drink data (stated categories 1 and 2 are combined, stated categories 3, 4, and 5 are combined, and stated categories 6 and 7 are combined). Comparable results for the tea data are presented in Table 7. For both cases wide variation in inferred ideals is evidenced for given stated ideals. Only in the case of the "temperature" attribute for the tea data is there convincing evidence of consistency between stated and inferred values (and this attribute differs from the others in that the various positive values are generally circumscribed by the presence of a "negative ideal.") Greater consistency, in other words, is to be expected. Thus, the results appear consistent with those previously discussed for preference and importance.


As is usually the case in multi-attribute model research, comparisons between studies are confounded by the number of "brands" and the number of salient product attributes, which vary from product class to product class and from study to study. It is a rare study indeed that incorporates as many as 25 "brands." On the other hand, most previously published studies identify more than three presumably salient attributes characteristic of the product class. Given the high correlations between predicted and criterion preference obtained in this study, one might interpret the median correlation between stated and preferred preference (.50) as an indicant of the upper bound median correlation that may be obtained using three "salient" attributes and "traditional analysis." The correlations between stated and inferred preference reported here are not inconsistent with the correlations between stated preference and predicted preference derived from three-attribute traditional analyses.





Little correspondence is indicated between inferred and stated importance or inferred position for the "ideal point" and stated position for the "ideal point." For soft drink data, knowing the inferred importance of an attribute reduces uncertainty as to the stated importance only in the case of the "taste" attribute. For tea data, the frequency of stated importance does not differ significantly between inferred "important" attributes and inferred "unimportant" attributes. Little consistency between inferred ideals and stated ideals is indicated in either data set. Only in the case of the "temperature" attribute for the tea data is there convincing evidence of consistency between these values. Thus, the results are consistent with those relating to preference and importance.

The present study demonstrates a useful procedure for evaluating inconsistencies between stated and inferred values of model parameters. The procedures are general, however, and promise enhanced power and flexibility in the design of choice experiments. There are at least two ways these procedures can increase the power of choice experiments. First, for a given number of choice objects and replications, it appears these procedures reduce the effort required of subjects compared to that accompanying conventional procedures. Alternatively, the experimental design can be expanded; i.e., for a given effort, more choice objects, more replications, or both can be included. Finally, replications can be unevenly distributed over choices, allowing flexibility in data collection.

A programmatic effort to generalize the substantive results of this study awaits a comprehensive typology specifying similarities between products, attributes, settings, and subjects. However, inconsistency between subjects' stated and inferred values are reported across a sufficiently broad spectrum of academic disciplines and problem types to provide some confidence that the findings reported here are not ultimately to be explained in terms of the methodology, choice objects, subjects, or definitions peculiar to this study.


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