On the Design and Analysis of Correlated Conjoint Experiments Using Difference Designs

ABSTRACT - This paper describes and illustrates an approach that can be used to design correlated conjoint judgment and choice experiments involving numerical attributes. The approach involves the use of fractional factorial designs to create orthogonal difference vectors for numerical attributes. To implement the approach one i) develops a "base" set of absolute attribute level vectors that contain a particular correlational structure of interest, and ii) operates upon these base vectors with the designed orthogonal difference vectors to generate absolute numerical attribute levels of one or more other alternatives. The attribute profiles generated by difference designs are evaluated by respondents in paired or multiple comparison experiments. A variety of response modes and analytical techniques can be used in conjunction with the proposed designs. The paper illustrates some of the design possibilities, discusses their limitations, and describes some academic and applied advantages.

Citation:

Jordan J. Louviere and George G. Woodworth (1988) ,"On the Design and Analysis of Correlated Conjoint Experiments Using Difference Designs", in NA - Advances in Consumer Research Volume 15, eds. Micheal J. Houston, Provo, UT : Association for Consumer Research, Pages: 510-517.

Advances in Consumer Research Volume 15, 1988      Pages 510-517

ON THE DESIGN AND ANALYSIS OF CORRELATED CONJOINT EXPERIMENTS USING DIFFERENCE DESIGNS

Jordan J. Louviere, University of Alberta

George G. Woodworth, University of Iowa

[The authors would like to acknowledge the helpful comments and suggestions on earlier drafts of this paper made by Donald A. Anderson, Norman H. Anderson, Richard D. Johnson, Robert Meyer and Joel Steckel. Errors and deficiencies remaining are the fault of the authors.]

ABSTRACT -

This paper describes and illustrates an approach that can be used to design correlated conjoint judgment and choice experiments involving numerical attributes. The approach involves the use of fractional factorial designs to create orthogonal difference vectors for numerical attributes. To implement the approach one i) develops a "base" set of absolute attribute level vectors that contain a particular correlational structure of interest, and ii) operates upon these base vectors with the designed orthogonal difference vectors to generate absolute numerical attribute levels of one or more other alternatives. The attribute profiles generated by difference designs are evaluated by respondents in paired or multiple comparison experiments. A variety of response modes and analytical techniques can be used in conjunction with the proposed designs. The paper illustrates some of the design possibilities, discusses their limitations, and describes some academic and applied advantages.

INTRODUCTION

Conjoint analysis techniques have been used routinely in commercial marketing research (Cattin and Wittinck, 1982, 1985), and have received considerable academic attention (e.g., Green and Srinivasan, 1978; Rao, 1977; Lynch, 1985). Most published studies have used various types of orthogonal fractional factorial experimental designs originally suggested by Green (1974). Orthogonal designs are useful from the standpoint of identification (i.e., various utility functions can be specified) and precision (i.e., the precision of the parameter estimates of particular specifications can be controlled a priori). However, concern has been expressed recently in both applied (e.g., Riedesel, 1985) and academic (e.g., DeSarbo, Mahajan and Steckel, 1986) papers that orthogonal arrays can fail to capture environmental correlations that exist among attributes in various product markets and/or produce unrealistic or impossible profiles.

Several studies have demonstrated that conjoint results based on uncorrelated attributes differ from those using correlated attributes (See, e.g., Hoffman, 1968; Huber and McCann, 1982; Johnson, Meyer and Ghose, 1986; Levin and Johnson, 1984, 1986). Such findings seem to have been ignored in practice, possibly because of anecdotal reports that consumers are usually ignorant of the feasibility or likelihood of treatment combinations. While possibly true for many consumer product studies, it may be less true for many industrial products. Thus, although orthogonal arrays may be satisfactory in many conjoint analysis studies, it would be useful to be able to design conjoint experiments that accommodate correlated attribute arrays, and retain desirable identification and precision properties.

The purpose of this paper, therefore, is to propose an approach to the design of correlated attribute conjoint experiments, which applies to the following types of problems: 1) all attributes are numerical (except for brand or Stow name), and at least one pair of attributes is correlated; 2) there is a mix of qualitative and numerical attributes, but only correlations among the numerical attributes are of interest; and 3) there is a mix of numerical and qualitative attributes, and correlations among numerical and (uncorrelated) qualitative attributes are of interest. Towards this end, the paper is organized as follows: we first define difference designs and explain the logic of their application to conjoint problems, next we describe and illustrate their construction and analysis, and finally we discuss some of their limitations, and make suggestions for additional research extensions.

INTRODUCTION TO DIFFERENCE DESIGNS

Difference designs can be thought of as a series of possibilities between totally orthogonal and totally correlated designs. The logic and construction of orthogonal designs for conjoint analysis problems is well-understood. Similarly, combining correlated attributes into orthogonal designs as "composite" attributes has been proposed and demonstrated (e.g., Green, 1974; Green and Srinivasan, 1978; DeSarbo, Mahajan and SteckeL 1986). However, we are unaware of efficient designs with intermediate correlation levels.

The difference designs that we propose allow one to have more control over the degree of correlation in the absolute values of numerical attributes than is now possible with other design methods. Furthermore, difference designs produce a large number of different levels of numerical attributes, rather than the two or three levels commonly used in applied conjoint studies. ID particular, the proposed design approach permits one to control the identification and precision properties of the attribute arrays based on the following model of the response process:

U[d(ij)l, U(i) - U(j),      (1)

where,

U[d(ij)] is the utility of the difference in the utilities of alternatives i and j (i=1,...,I; j=1,...,J)

U(i), U(j) are, respectively, the separate utilities of alternatives i and j.

U(i) and U(j) are defined as follows:

U(i) = f[X(ik)]

U(j) = f[X(jk)],      (2)

where

X(i), X(j) are vectors of absolute levels of the K-th (k=1,...,K) attributes for the ith and jth alternatives, respectively.

Thus,

U[d(ij)] = g{f[X(i)] - f[X(j)]}.      (3)

If the functional specifications g and f can be represented as linear-in-the-parameters-and-variables forms, equation (3) can be expressed as follows:

U[d(ij)] = b'[X(i) - X(j)],       (4)

where

b' is a K dimensional vector of parameters associated with the K attribute difference vectors to be estimated from responses to the experimental difference treatments.

Decision models based on differences in the attributes of alternatives are not new. For example, the Thurstone (1927) paired comparisons model allows one to estimate the overall utility values of a set of alternatives as differences from an arbitrarily chosen origin stimulus. Difference models also have been proposed for preference (e.g., Scheffe, 1952; Anderson, 1981, pp. 29-31) and personality impression formation judgments (e.g., Schmidt and Levin, 1972). What is new in our approach is the way in which one uses orthogonal, fractional factorial design techniques to develop correlated conjoint designs for both judgment and choice problems.

THE LOGIC OR DIFFERENCE DESIGNS

To construct difference designs, one uses orthogonal arrays to represent differences among quantitative variables. The orthogonal difference vectors correspond to and operate upon an array of absolute attribute levels which one designs so as to incorporate interattribute correlations of experimental interest. This designed array of correlated attribute levels consists of profiles that describe a "base" set of alternatives. The difference-in-utility model of Equation (4) can be estimated efficiently if the difference vectors are orthogonal; hence, one is free to choose the absolute numerical attribute levels for one alternative -- the "base" alternativeCso long as the differences between the absolute levels of the attributes of the arbitrarily chosen alternative and the absolute levels of the attributes of all other alternatives are orthogonal. In general, the absolute attribute levels and their correlational structure should span a range of possibilities representative of the product market of interest.

A difference design is therefore an orthogonal array of difference vectors which is applied to the arbitrarily correlated vectors of absolute attribute levels of a "base" alternative to create one or more additional alternatives. Respondents must compare alternatives in such a way that one can estimate the differences between the attribute levels of the "base" alternative and one or more other alternatives (profiles of absolute attribute levels). Thus, paired or multiple comparison tasks are obvious ways to implement difference designs.

Difference designs for two alternatives (paired comparisons) can be used with a variety of commonly used response modes that satisfy equation (4), such as graded pair comparisons (ratings), resource allocations and discrete choices, or the preference intensity measures suggested by Hauser and Shugan, 1980). Graded paired comparison tasks require subjects to estimate the degree of dominance (e.g., "preference") of one alternative over a second using a rating scale . In allocation tasks, subjects allocate some fixed set of resources to each alternative in a pair such that the allocations represent how much one dominates the other (e.g., in "attractiveness"). In discrete paired choice tasks subjects choose the alternative that dominates on a particular response dimension (e.g., "purchase likelihood"). In the multiple comparison case, we concentrate on resource allocation or discrete choice responses because they satisfy the comparisons implied by equation (4).

CONSTRUCTION OF DIFFERENCE DESIGNS

The construction of difference designs can be illustrated by the following four cases that cover the possible applications.

Case I: Multiple Numerical Attribute Difference Vectors, Paired Alternatives

Let us consider a problem in which a consumer has comparative ratings information available on several restaurants in a particular city on the attributes of food quality, service and price. Table 1 contains eight hypothetical food quality, service and price ratings on a 1-10 scale. We demonstrate the use of a difference design in Table 1 by using a 2x2x2 factorial design to vary levels of "differences" in the ratings of the three attributes; the levels of the attribute differences are -1 or +1 in rating units. One can use any numerical difference levels, however, one should use difference levels that generate absolute attribute levels (in this example, "ratings" units) of the second alternative that will lie within the range of the absolute levels in the product market of interest (In this case between the values 1 and 10).

The difference design levels are added to the attribute ratings for each of the eight hypothetical restaurant profiles to produce eight additional hypothetical restaurant profiles as shown in Table 1. Correlations among the ratings levels of the "base" set of eight restaurants are also shown in Table 1.

Table 1 also contains the "B" restaurants that are created by applying the difference design to the ratings levels of the "A" restaurants. That is, the ratings levels of the eight "B" restaurants in Table 1 are produced by adding the levels of the difference design to the numerical ratings levels in Table 1. The experiment represented by the eight pairs of restaurant ratings in Table 1 is implemented by asking respondents to compare each pair on some response dimension of interest, such as "preference", "likelihood of choice", etc. For example, one might ask respondents to allocate 100 points to the restaurants in each pair in such a way as to reflect the relative degree of preference for one over the other.

One can choose any arbitrary correlational structure for the absolute attribute levels of a base alternative. In fact, one can create a wide range of correlations among the attributes of a second alternative by systematically interchanging the rows of the base alternative to approximate a desired correlational structure. Correlations among the attributes of a base and second alternative are therefore irrelevant because difference designs are orthogonal.

TABLE 1

A 3-ATTRIBUTE PAIRED DIFFERENCE DESIGN BASED ON A FRACTION OF THE 2X2X2 FACTORIAL DESIGN

TABLE 2

A 3-ATTRIBUTE MULTIPLE COMPARISON EXPERIMENT BASED ON A FRACTION OF ThE 2X2X2X2X2X2 FACTORIAL

Case II: Several Attribute Difference Vectors, Multiple Alternatives

It is a straightforward matter to extend the design in Table 1 to a multiple comparison experiment involving more than two brands. As before, a "base" set of treatments and a set of orthogonal difference vectors is developed. Continuing the example of Table 1, we create a multiple comparison experiment for sets of three different restaurants, "A", "B" and "C": The same difference design in Table 1 is used for the "B" restaurants, and three additional orthogonal difference vectors (shown in Table 3) are used to create the "C" restaurants. That is, the difference design in Table 2 contains six orthogonal difference vectors -- the three already used in Table 1, plus an additional three drawn from the 26 factorial. Combining the ratings for restaurants "A" and "B" in Table 1 with the ratings for restaurant "C" in Table 2 produces eight multiple comparisons (triples). The ratings for "C" are produced in the same manner as the ratings for "B" in Table 1: he difference vectors in Table 2 are added to the corresponding absolute attribute level vectors for restaurant"A" (the "base"). Correlations among the derived absolute attribute rating vectors also are shown in Table 2.

One can implement the multiple comparison experiment represented by the attribute ratings levels in Tables 1 and 2 as a discrete choice or resource allocation task. For example, subjects might be asked to indicate which one of the three restaurants in each of the 8 comparison sets (choice sets) would be the one they would be most likely to purchase if they had to choose from the three brands in that set. This design can be extended to additional alternatives by using a larger design to create their absolute attribute levels (e.g., a 16 treatment fraction of a 215 would allow one to create up to six different restaurants per comparison set.). One can also use one or more constant alternatives in each set, such as "I want to delay purchase" and/or "... not purchase". The sets of alternatives (in this case, "restaurants") also can be branded; i.e., instead of "A", the eight restaurant profiles under "A" could be branded as "French", the eight "B" as "Chinese",etc., or alternatively, "A" might be MacDonalds, "B" might be "Burger King", etc. That is, the attributes are "nested" under the brand or store names.

Case m: A Single Numerical Attribute Correlated With Several Qualitative Attributes, Paired Comparisons

An example of the third case might be paired difference designs for pricing bundles of options added to a base model, which is a common strategic marketing problem in many product classes. Commonly in a bundling situation, the more options bundled on a base alternative, the higher the price. As well, higher prices often are associated with more valued features or options. Table 3 provides an example of a difference design in which price is positively correlated with the number and type of bundled automobile options such as a) air conditioning, b) AM/FM Dolby Stereo, c) sunroof d) turbo charging, and e) mag wheels. For the sake of example, assume that the prices of each feature are $1,000; $500; 5250; $2,000; and $750, respectively. Table 3 contains orthogonal feature vectors for a base brand (Mercury Cougar) and a correlated price vector. To the price vector is added a price difference vector, which is orthogonal to the presence/absence of the features. Thus, the design in Table 3 is a six factor, orthogonal main effects plan for the features and the price differences.

A second alternative (Ford T-Bird) is generated from the base attribute treatments by i) constructing a "foldover" of the orthogonal feature design for Cougar to produce the T-Bird feature combinations, and ii) adding the difference vector levels to the Cougar price levels in Table 3 to produce the T-Bird price levels. A foldover design can be constructed for"yes/no" attributes (like features) by replacing "yes" levels in the original main effects design with "no" levels, creating a second main effects plan. The profiles nested under Cougar and T-Bird that are produced by pairing each treatment in the Cougar main effects plan with its foldover are shown in Table 3. Although the prices are perfect linear combinations of the features, differences in prices are orthogonal to features. Thus, it is possible to estimate the effects of features and differences in price from this design.

TABLE 3

A SINGLE ATTRIBUTE DIFFERENCE DESIGN FOR PRICING OPTION BUNDLES

Case IV: Several Numerical Attributes Correlated With Each Other And With Orthogonal Qualitative Attributes, Paired Comparisons

The last case can be generalized to paired difference designs in which there are several correlated numerical attributes and several orthogonal qualitative attributes. For example, ratings vectors such as those in Tables 1 and 2, or actual levels of attributes such as MPG or warranty might be combined with qualitative features like those in Table 3 to create ratings levels, or MPG and warranty levels for a second or comparison alternative. The salient aspect of this case is that the qualitative attribute vectors must be orthogonal to one another; numerical attributes can be intercorrelated and/or correlated with qualitative attributes.

The preceding four cases permit generalizations to be made about the construction of difference designs. Table 4 indicates that multiple comparison designs can be constructed only if all attributes are numerical. However, paired comparison designs exist for all cases. Having discussed the relevant domain of applications and some limitations, we next discuss the analysis of response data obtained from these designs.

ANALYSIS OF RESPONSE DATA OBTAINED FROM DIFFERENCE/CONTRAST DESIGNS

Analysis Of Paired Comparisons Data

The previous section discussed response modes for paired comparison experiments. Graded paired comparison ratings can be analyzed by several methods commonly applied to conjoint ratings data: e.g., O.L.S. regression, constrained regression, and LINMAP (See Green and Srinivasan, 1978, for a review of these and other analytical approaches). Other possibilities include Tobit regression (e.g., Malhotra, 1986) or ordered probit and logit procedures (e.g., Madalla, 1983). Estimation methods used with ranking data (e.g., MONANOVA) are less commonly applied to comparison data. "Partworths" or partial utilities estimated from comparison tasks correspond to the difference or contrast levels, not to attribute levels seen by respondents. Thus, depending upon the choice of coding scheme, results are relative to a "base" or standard stimulus. Thus, one makes inferences about degrees of preference or utility associated with particular difference or contrast levels rather than the absolute levels. However, one can forecast to absolute levels.

TABLE 4

GUIDE TO THE CONSTRUCTION OF DIFFERENCE/CONTRAST DESIGNS

Analysis of Multiple Comparisons Data

We will only discuss discrete choice or resource allocation response modes for multiple comparisons because these are consistent with equation (4). Ranking data also can be used to develop discrete choice models by means of an explosion technique first applied in marketing to conjoint data by Chapman and Staelin (1982). In the case of multiple comparison tasks, subjects must rank all alternatives in each comparison set (choice set), which considerably complicates the task and adds to its length. Nonetheless, one can also use this approach.

Multiple comparison models include the now well-known multinomial logit (MNL) model (Hensher and Johnson, 1981; Louviere and Woodworth, 1983; Corstens and Gautchi, 1983), as well as other less frequently applied models, such as multinomial probit (Daganzo, 1979), and other forms such as hierarchical, sequential and tree models (See, e.g., Tversky, 1973; Tversky and Sattah, 1979; Lehmann and Moore, 1985). Difference designs for both paired and multiple comparison tasks are consistent with MNL models; and because many now are familiar with MNL models, we use them to illustrate the analysis of responses to multiple comparison difference designs.

One can rewrite the MNL model for J (j=1, 2, .... J) brands:

p(a|A) = exp[U(a)]/{exp[U(a)l+...

              +exp[U(j)]}, V j C A      (5)

where

p(a|A) is the probability of choosing alternative a from a choice or comparison set in which a is a member.

U(a), U(j) are unknown utilities of alternatives a and j that are to be estimated from choice or allocation data.

exp is the exponential operator. V j C A means that the operation is defined "for all the j alternatives contained in choice or comparison set A".

Equation (5) can be rewritten to show that choice probabilities are determined by differences in the utilities of the alternatives in each choice or comparison set:

Ln[p(a|A)] = Ln[p(base|A)] + [U(a) - U(base)]       (6)

where

Ln(x) is the natural logarithm of the choice probability of alternative x.

base refers to the choice alternative that will serve as the origin or "base" of the utility scale. It is identical to the "base" brand discussed in the design construction section.

Otherwise, all terms are as previously defined.

Equation (6) indicates that MNL models can be expressed in terms of two components: 1) the choice probabilities of the base alternative, and 2) differences in the utilities of each choice alternative and a base alternative in some particular choice set A. This formulation allows us to outline the domain of application of difference designs (assuming that empirical choice probabilities can be well-approximated by a MNL model). In particular, equation (6) implies that the only specification of the utility function that is consistent with allocations or choices made in response to a difference design is the following:

Ln[p(a|A)] = ln[p(base|A)] + [c(a) - c(base)] +

                     b(k)[(X(a,k) - X(base,k)], (k=1,2,...,K)      (7)

where

c(a), c(base) are "alternative (or brand)-specific" parameters that define the brand weights of brand a and the base brand. If all alternatives are non-branded concepts, all attributes should have identical b parameters.

b(k) is an element of the vector of parameters to be estimated from the choice or allocation data. There are K (k=l, 2,..., K) parameters associated with each attribute difference vector. The K parameters are called "generic' parameters because they are constant across all alternatives. Attribute parameters which vary across alternatives (i.e., are "nested" under alternatives) are called alternative specific.

X(a,k), X(base,k) are absolute levels of elements of attribute vectors of alternative a and the base alternative. Their elements correspond to the K attributes whose differences are varied in the difference design.

In equation (7) the parameters of each attribute are the same for all alternatives, although each alternative can have a different "brand weight", which represents a contrast with the base brand. Difference designs permit one to relax the assumption that attribute parameters are the same across brands by specifying "alternative-specific" parameters for one or more of the brands of interest. However, this relaxation may create analytic problems, as we indicate later. Difference designs also permit mixed model specifications in which some vectors of attribute differences have common or generic parameters, while other parameters defined on absolute attribute level vectors may be alternative-specific.

Only attribute differences are orthogonal in difference designs; thus, the absolute levels of attribute values can be highly correlated. This can be a problem if one estimates alternative specific effects for two or more attributes common to all brands because correlations among absolute attribute levels within brands are preserved in alternative- specific effects specifications. That is, if the absolute levels of two or more numerical attributes nested under brands are correlated, collinearity might pose problems for estimating their alternative specific effects.

If only a single attribute is correlated between brands (e.g., the price of each brand), collinearity is not a problem because the vectors of absolute levels are uniquely nested under each brand. For example, Table 5 illustrates a pricing experiment in which there are 11 brands (pizza outlets), one of which is the base alternative (Domino's). The experimental design consists of a set of orthogonal price difference vectors that define the prices of other brands relative to the price of a base brand. The absolute price levels were deliberately designed to span the range of prices that existed in the market at the time of the study. In the example in Table 5 collinearity is not a problem because only a single numerical attribute (price) is involved.

TABLE 5

ONE OF 11 SETS OF CHOICE SETS FROM THE DIFFERENCE DESIGN USED TO CONSTRUCT THE DELIVERED PIZZA PRICING EXPERIMENT

The experiment described in Table 5 was administered to 237 undergraduate business students in a large Midwestern U.S. university who were regular or occasional consumers of delivered pizza. Each comparison set listed the price of a basic cheese pizza as well as the cost of one, two or three additional ingredients (expressed as a linear combination of the price of a cheese pizza) for each restaurant in a particular choice set. Subjects were asked to assume that because of the hour, renovations, bankruptcy or other reasons, the only restaurants from which they could order were those listed in each set. Subjects made discrete choices from all 88 choice sets; the order of appearance of the sets was randomized across subjects. The data of analytical interest are the aggregated choice frequencies of the 237 subjects in the 88 choice sets.

MNL models were fit to the data in Table S using iteratively reweighted least squares techniques to obtain the maximum likelihood estimates (Woodworth and Louviere, 1984). The analytical results (Table 6) can be interpreted as follows: a positive brand-weight parameter indicates that a restaurant was preferred (chosen more often) than the base restaurant (Domino's), while a negative brand weight indicates the reverse. An alternative-specific model was also estimated and compared statistically with the model in Table 6 that assumes a common or generic price parameter for all brands. The statistical results indicated that including alternative-specific price effects did not significantly improve the fit of a generic price parameter model. The generic price parameter in Table 6 can be interpreted to mean that as the price differential between Domino's and a restaurant increases by $1.00, the utility of that restaurant relative to Domino's decreases by 0.80 units. Thus, difference designs for multiple comparison problems involving numerical attributes like price can be developed and analyzed in a straightforward manner.

The final section discusses the design cases described in the paper, indicates some of their limitations, mentions some additional analytical considerations, and describes several advantages of potential interest to academic researchers, concluding on a cautionary note.

DISCUSSION AND CONCLUSIONS

This paper proposed a new approach to the construction of conjoint designs that permits one to manipulate the degree of correlation among the absolute levels of numerical attributes. For example, in concept testing problems using unbranded concepts, researchers often wish to correlate attributes within but not necessarily between alternatives. On the other hand, in pricing experiments, researchers often want to correlate the price vectors of various brands across one another, reflecting an existing market structure. The method of design construction described in this paper not only allows one to control correlations among levels of numerical attributes within and between brands, it also enables one to obtain orthogonal estimates of parameters of difference-in-utility models.

TABLE 6

STATISTICAL RESULTS FOR THE PIZZA PRICING EXPERIMENT

The approach proposed in this paper views the design problem in terms of constructing orthogonal difference matrices. A "base alternative is used to set the range, variability and correlation of the attributes. The base alternative and the associated difference design completely define the experimental treatments. Conjoint experiments derived from difference designs require subjects to make comparison judgments, resource allocations or discrete choices; such response data are compatible with difference-in-utility models.

One should be cautious about trying to develop individual-level conjoint models from discrete choice data because such models usually require many more treatments than are used in most applied conjoint problems to achieve asymptotic levels of efficiency comparable to regression models. Chapman (1984) and McFadden (1974) provide tables of efficiencies for logit models with various numbers of parameters and observations. One can use these tables to determine the size of experiment needed to produce reliable and efficient individual-level estimates. Because one can obtain much more choice data from individuals in academic research, one is less limited in such situations. However, one should be aware of the analytical limitations of discrete choice models whether data are experimental or observational. Of course, one can group subjects with similar choice patterns by means of discrete clustering techniques (e.g., Wilkinson, 1986, Chapter 16). Each cluster of subjects can be separately analyzed using their aggregate choice frequency counts as the dependent variable in a logit or probit regression analysis (See Louviere and Woodworth, 1983, for a similar suggestion).

One can specify and estimate interaction effects from cross-products of difference vectors, as in current conjoint models. However, orthogonal polynomials cannot be used to describe non-linearities in such response surfaces, as suggested by Louviere and Woodworth (1983). Rather, if there are non-linear effects, other transformations are appropriate: e.g., a logarithmic transformation applies to cases in which the rate of change in the response depends on the absolute value of the numerical attribute. Other transformations are discussed in Ben-Akiva and Lerman (1985, 174- 182).

Difference designs have several advantages over conventional orthogonal arrays commonly used in conjoint applications: 1) In the case of numerical attributes, subjects see a large number of absolute levels because the absolute levels of the base alternative and the orthogonal differences produce a variety of absolute levels. This may lessen demand characteristics that result when subjects see only a few absolute levels (often only two!), and make the task more reflective of levels in real markets. 2) Difference designs can be used to avoid dominance problems associated with using orthogonal arrays to develop choice sets for non-branded attribute profiles. For example, research into weight shifting in utility functions as a function of the composition of choice sets and the range of the attribute levels requires caution to insure that designs are statistically efficient and there are no dominant alternatives in any choice sets (See, e.g., Meyer and Eagle, 1982; Eagle, 1984). 3) Difference designs can be used to study the performance of MNL or other choice models under different attribute correlational structures within and/or between brands. For example, Johnson, Meyer and Ghose (1986) report that MNL models perform poorly when attributes are negatively correlated, as they often are in real markets (See, e.g., Curry, 1985). However, collinearity in their absolute attribute vectors may have affected the parameter estimates of their MNL models. Orthogonal difference designs therefore can be used to test the generality of such conclusions, as well as to explore the properties of various choice models under controlled conditions.

We would like to end on a cautionary note: the properties of difference designs may vary with different configurations of base alternatives. We have not examined such effects, but we have examined a wide range of different experiments with different base configurations. Our empirical results suggest that variance in base configurations is probably of minor concern because orthogonality of the design matrix seems to the determining factor in the statistical properties of the resulting design. Nonetheless, the question of possible effects on results due to different configurations of base alternatives remains open, and we invite others to assist in illuminating this issue.

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