# A New Nonmetric Conjoint Method: Some Preliminary Results

ABSTRACT - A new nonmetric conjoint method is presented. This method is simple and it precludes the inconsistent predictions which can be obtained with the traditional methods, when using fractional designs. Preliminary simulation results are encouraging.

##### Citation:

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Philippe Cattin and Alan Gelfand (1986) ,"A New Nonmetric Conjoint Method: Some Preliminary Results", in NA - Advances in Consumer Research Volume 13, eds. Richard J. Lutz, Provo, UT : Association for Consumer Research, Pages: 460-462.
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A new nonmetric conjoint method is presented. This method is simple and it precludes the inconsistent predictions which can be obtained with the traditional methods, when using fractional designs. Preliminary simulation results are encouraging.

INTRODUCTION

The purpose of this paper is to present a new nonmetric method for estimating conjoint models. The method makes the two following assumptions:

Assumption A - the preference order of the levels of each attribute is known for each respondent;

Assumption B - each respondent is "rational" (i.e., each respondent prefers stimulus A over stimulus B if A dominates B, which means that all the levels of A are the same as or preferred to the levels of B).

Work is being done to adapt the method to cases where either assumption does not bold. Moreover, it can be used only with full profile data. But then "full profile" is the method the most used by practitioners (Cattin and Wittink 1982, Table 4).

POTENTIAL INCONSISTENCIES WITH TRADITIONAL ESTIMATION METHODS

Let R be a (N x 1) vector of observed rankings where the most preferred stimulus receives the highest rank (N) and the least preferred the lowest (1). The design matrix X which defines the stimuli is expressed through indicator variables. An attribute with k_{i} levels can be described by (k_{i-1}) vectors. For instance, for an attribute with 4 levels we have 3 column vectors:

Such an attribute will have (k_{i}-1) parameters associated with it if a model is fitted. In the above notation, if we include an intercept, then X is N x (Sk_{i}-r+1). Let b be the (Sk_{i}-r+1) x 1 vector of parameters, wherein b_{ij} is the parameter associated with level j of attribute i (with the parameter associated with the lowest level, b_{ij}, taken to be O). Under the assumption of ordered levels for each attribute, we have O < b_{i2} <... b_{iki},i=1, 2, ..., r.

Suppose we have 3 attributes, 3 levels each, and that we are using a fractional design of 9 stimuli (Figure 1). For each attribute, level 1 is the least preferred and level 3 the most preferred. One can verify that the rankings (A) shown in Figure 1 are rational. They can be derived from an additive model or from a model with noncrossover interactions (with or without errors.)

If we use regression on rankings (A), we get b_{12} = -1/3, b_{13} = -2/3, b_{22} = 1, b_{23} = 5, b_{32} = 5/3 and b_{33} = 10/3 (b_{il} = O for i = 1, 2, 3). The utility estimates obtained for attribute 1 are contrary to expectations (monotone decreasing instead of increasing). Suppose we use these utility estimates to predict the rankings of stimuli (1, 2, 2) and (2, 2, 2), where (a_{1}, a_{2}, a_{3}) represents a stimulus with levels a_{1}, a_{2}, and a_{3} of attributes 1, 2 and 3, respectively. We get R(1, 2, 2) = 4.33 and R(2, 2, 2) = 4, which is inconsistent (contrary to expectations).

A 3 x 3 x 3 FRACTIONAL DESIGN WITH N = 9

It can be shown analytically that under assumptions (A) and (B) inconsistent utility estimates and predictions can be obtained with fractional designs, but not with full designs (Gelfand and Cattin 1984). There are no analytical results in the context of nonmetric methods, but inconsistent estimates and predictions can also be obtained with such methods. One solution is to constrain parameter estimates to be monotone (increasing or decreasing), which can be done with LINMAP. One can also use the method presented below.

NONMETRIC ESTIMATION OF THE RANKING OF A STIMULUS

The ranking of an unobserved stimulus a = (a_{1}, a_{2} , a_{r}) is estimated using the following formula:

Under rational rankings, (1) precludes the creation of inconsistent rankings. For instance, in the example in Figure (1), R(1, 2, 2) = 3.5 because R(1, 2, 2) = 4 and R(1, 2, 2) = 3, and R(2, 2, 2,) = 3.5, as well.

Note that this method does not estimate any attribute utility. It predicts the ranking of an unobserved stimulus from the rankings provided by a respondent on N stimuli. These predicted rankings can (in turn) be used to estimate preference shares, which is often done by practitioners (Cattin and Wittink, 1982, p. 50).

A BRIEF SIMULATION STUDY

Simulation Design

The purpose of the simulation is to gain insight into the behavior of the nonmetric approach compared to regression. All simulations were carried out with three attributes, each with 4 levels. Two orthogonal arrays of 16 stimuli each were used: one to create estimation samples, the other validation samples. Two main types of data were produced: additive and (non-crossover) interactive, to which varying levels of noise were added. Interactive data were also produced without noise. The rationale is that respondent's unreliability (i.e., noise) and interaction are two major reasons for imperfect fit when estimating additive models.

The additive data were produced using the following model. The rth response, or rating Y_{r}, is such that

The noise (e_{r}) was drawn from a normal distribution with mean zero and variance s^{2}, where s^{2} was set such that the expected percentage of noise in the data took on the values 11.5, 26 and 65 percent. These percentages correspond to high, medium and low predictive validity found in real data (e.g., Scott and Wright, 1976). [See footnote 3 in Cattin and Punj (1984)] The noise was added to the stimuli in the estimation sample, but not in the validation sample. In this manner, it takes fewer replications to obtain significant differences between methods. There were five hundred replications for each level of noise.

The ratings of the stimuli in the estimation sample, obtained using (2), were transformed into rankings. These rankings were used to predict the rankings of the 16 stimuli in the validation sample using the nonmetric approach, i.e., (1). Both the ratings, obtained from (2) and the rankings of the stimuli in the estimation sample were used to estimate a least squares regression model. The estimated models were used to predict the ratings and rankings of the stimuli in the validation sample.

The order of the levels of each attribute is known. Thus, following Srinivasan, et al. (1983) and Cattin and Punj (1984), constrained least squares regressions were also performed to constrain the parameter estimates to be monotone increasing for each attribute.^ This was done as follows. For each attribute, b_{iki} and b_{i,ki-1} were averaged if b_{ik} was found to be less than b_{i,k-1}.

Next, b_{i,ki-1} and b_{i,ki-2} were averaged if b_{i,ki-1} was found to be less than b_{i,ki-2} (and b_{iki}, b_{i,ki-1}, b_{i,ki-2} were averaged if b_{iki} was found to be less than b_{i,ki-2}), and so on. The estimated models thus obtained were used to predict the ratings or rankings of the stimuli in the validation sample.

The predicted ratings or rankings obtained with each method, as well as the true ratings (in the validation sample) were transformed into rankings 1 through 16 (allowing for ties). Spearman rank and Kendall tau correlations were computed between true and predicted rankings and averaged across the 500 replications for each cell of the simulation design.

The interactive data were produced with the following model. Again, the rth rating, Y_{r}, is such that

This is one way to produce interactive data of the noncrossover type. The noise was again drawn from a normal distribution with mean zero and variance s^{2}, where s^{2} was such that the expected percentage of noise in the data was 0% (i.e., no noise), 11.5%, 26% and 65%. Again, the noise (if any) was added to the estimation data only and there were 500 replications for each level of noise. The estimation and prediction procedures used with these interactive data were identical to those used with the additive data.

The b_{ij} and C_{jkl} values in (2) and (3), used to produce the additive and interactive data, were all drawn from the same distribution, thus implying relatively "compensatory" models. To gain further insight into the behavior of the nonmetric approach, another set of additive data was produced in which the b_{3j} values, drawn from the same distribution as before, were multiplied by 10, thus producing data in which the third attribute tends to be more important and implying a model with a "dominant attribute" (similar to a lexicographic model). Data were thus produced, followed by estimation and prediction, as described above.

Simulation Results

The mean Spearman rank and Kendall tau correlations obtained for each cell of the simulation design are shown in Tables 1 and 2. Not surprisingly, the best regression results are obtained when using constrained least squares on ratings. The improvement (compared to nonconstrained least squares) is greater when the data are additive than when they are interactive.

Comparing the results obtained with the nonmetric method to those obtained with constrained least squares on ratings shows that the constrained least squares method outperforms the nonmetric method only with additive data when the noise level is 11.5% (i.e., high predictive validity). (Note that most of the standard errors of the mean correlations shown in Tables 1 and 2 are between .002 and .005.) The predictive validities of the constrained least square and nonmetric methods are about the same when looking at the results obtained with 11.5% noise under additive data with a dominant attribute. The predictive validity of the nonmetric method is higher everywhere else: i.e., with interactive data (whatever the noise level), and with additive data with medium to high noise levels.

MEAN SPEARMAN RANK CORRELATIONS

CONCLUSION

The intent of this paper was to present a new nonmetric method which precludes the inconsistent predictions which can be obtained with traditional methods when using fractional designs. The method is quite simple and preliminary simulation results are encouraging. Additional simulations and empirical work are needed to gain more insight on the behavior of this method, compared to the traditional methods.

The method can be used only with full profile data and makes two important assumptions (given at the beginning Of the paper). This limits its use. However, work is being done to infer the preference order of the levels Of an attribute (when unknown) from full profile data, and to adapt the method to inconsistent (full profile) rankings. This will broaden the potential use of the method.

REFERENCES

Cattin, P. and D. R. Wittink (1982), "Commercial Use of Conjoint Analysis: A Survey," Journal of Marketing, 46, 44-53

Cattin, P. and G. Punj (1984), "Factors Influencing the Selection of Preference Model Form for Continuous Utility Functions in Conjoint Analysis," Marketing Science, 3, 73-82.

Gelfand A. and P. Cattin (1984), "A Nonmetric Alternative to Traditional Model-R>^ed Approaches to Conjoint Analysis: Part I," working paper, University Of Connecticut.

Scott, J. E. and P. Wright (1976), "Modeling an Organizational Buyer's Product Evaluation Strategy: Validity and Procedural Considerations, n Journal of Marketing Research, 13, 211-224.

Srinivasan, V., A. K. Jain and N. K. Malhotra (1983), "Improving Predictive Power of ConJoint Analysis by Constrained Parameter Estimation," Journal of Marketing Research, 20, 433-438.

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##### Authors

Philippe Cattin, University of Connecticut

Alan Gelfand, University of Connecticut

##### Volume

NA - Advances in Consumer Research Volume 13 | 1986

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