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The 'no-choice' alternative in conjoint choice experiments.

Rinus Haaijer [*]

Wagner Kamakura [+]

Michel Wedel [*]

Conjoint choice designs are frequently applied in practice, and often a base alternative is added to the design. When such a 'no-choice' base alternative is present in conjoint choice experiments a constant term should be added to the design ('X'-) matrix with attribute dummies when effects type and/or linear coding is used for the attribute levels. Not including such a constant may result in a much lower model and predictive fit and even biased estimates for the (linear) attributes.

Introduction

Choice experiments have become prevalent as a mode of data collection in conjoint analysis in applied research. Their popularity seems due to their ability to mimic real market decisions. Experimental choice analysis has a number of advantages over traditional conjoint analysis, including that there are no differences in scale usage across respondents, that the choice tasks have higher external validity being more akin to the behaviour respondents display in reality, and that assumptions made in designing choice simulators can be avoided. The availability of new computer-assisted data-collection methods, in particular CBC from Sawtooth Software, has also greatly contributed to its popularity. In conjoint choice experiments respondents make choices from several sets of alternatives. Respondents choose one profile from each of several choice sets. In order to make the choice more realistic, in many conjoint experiments one of the alternatives in the choice sets is a 'no-choice' or 'none' option. This option can entail a real no-choice alternative ('None of the above') or an 'own-choice' alternative ('I keep my own product'). This base alternative, however, presents the problems of how to include it in the design of the choice experiment and how to accommodate it in the choice model. Regular choice alternatives are most often coded in the data matrix with effects-type or dummy coding. Since the no-choice alternative does not possess any of the attributes in the design, one may be tempted to code it simply as a series of zeros, which makes the fixed part of its utility zero in each choice set. In this paper we investigate several models that can be used to accommodate the no-choice option and we show that it is best to add a constant to the design or X-matrix of the conjoint experiment for the no-choice alternative when effects-type coding and/or linear coding is used for all attributes. Not including this constant leads to reduced fit and biased estimates of linear attributes.

The base alternative

In conjoint choice experiments a base alternative is included in the design of the experiment, among others, to scale the utilities between the various choice sets. A base alternative can be specified in several ways. First, it can be a regular profile that is held constant over all choice sets. Second, it can be specified as 'your current brand', and third, as a 'none', 'other' or 'no-choice' alternative (e.g. Louviere & Woodworth 1983; Batsell & Louviere 1991; Carson et al. 1994). Additional advantages of including a 'no-choice' or 'own' base alternative that are mentioned in the literature are that it would make the choice decision more realistic and would lead to better predictions of market penetrations. A disadvantage of a no-choice alternative is that it may lead respondents to avoid difficult choices, which detracts from the validity using the no-choice probability to estimate market shares. However, Johnson & Orme (1996) claim that this seems not to happen in conjoint choice experiments. In addition, the no-choice alternative gives no information about preferences for attributes of the choice alternatives, which is the main reason for doing a conjoint choice experiment. Dhar (1997) gives an overview of why and when respondents may choose a no-choice option in general. He states that respondents may choose the no-choice when none of the alternatives appears to be attractive, or when the decision-maker expects to find better alternatives by continuing to search. Furthermore, when subjects are uncertain about the range of potential alternatives they may continue to look for better alternatives and choose the 'none' in the early stages of a choice process. Dhar (1997) also shows that adding an attractive alternative to an already attractive choice set increases the preference of the no-choice option and adding an unattractive alternative to that choice set decreases the preference of the no-choice. This implies that when alternatives are close to each other in preference, people will choose the no-choice mor e often than when there is a clearly dominant or unattractive profile in the choice set. Huber & Zwerina (1996) stated that conjoint choice sets that are utility balanced are more informative and efficient compared to sets with dominant alternatives. However, according to the above, adding a no-choice alternative to such a choice set may influence this negatively, because people may avoid making the difficult choice. This is related to findings in the psychological literature, where it has been found that people prefer consequences that arise from inaction over those arising from action since the decision to stay within a status quo has certain psychological advantages (Baron & Ritov 1994; Dhar 1997). This may make people choose 'none' if there is no dominant alternative present in the choice set. The above shows that the reasons to choose the no-choice may be different from choosing any of the other 'real' profiles in a conjoint choice experiment. In this view, 'no-choice' cannot be seen as just another choi ce alternative, leading to potential violations of IIA, among others.

In this paper we investigate the no-choice option from a modelling point of view. [1] We begin by discussing a number of alternative model formulations. First, simply having a series of zeros describing the attribute values of the no-choice alternative seems a straightforward option, but this formulation may produce misleading results. When no-choice is coded as a series of zeros, its fixed part of utility is equal to zero by definition. However, when there are linear attributes present, these zero values of the no-choice alternative act as real levels of the linear attributes. For instance, when price is a linear attribute in the design, the zero value for no-choice will correspond to a zero price. We hypothesise that this can lead to a biased estimate of the parameter of the linear attribute.

Second, when all attributes are modelled with effects-type coding the bias discussed above does not arise, because all part-worths are now specified relative to the zero utility of the no-choice alternative. However, even when all attributes are coded with effects-type dummies, adding such a constant may improve model fit. This can be explained because the no-choice option in fact adds one level to all attributes. If there are, for example, S attributes each with L levels, then there are [L.sup.S] possible alternatives and L part-worths to be estimates for each of the S attributes in the analysis. The inclusion of a no-choice option would produce a design with S attributes each with L + 1 levels. However, the resulting full [(L + 1).sup.S] set of profiles cannot be constructed, since the (L + 1)th level of all attributes is confounded with the no-choice alternative. Thus, instead of [(L + l).sup.S] alternatives only [L.sup.S] + 1 alternatives are possible. This is accommodated in the Multinomial Logit analys is by estimating the L part-worths (through L - 1 effects-type dummies) for each of the S attributes plus one part-worth for the no-choice option. Thus adding the no-choice constant is hypothesised to increase model fit. This change in the design of the choice experiment is hypothesised to reduce the potential bias in the estimates for linear attributes. Although this additional constant increases the number of parameters by one, it sets the utility level of the no-choice alternative, as explained above, and therefore compensates for the bias in the estimates for linear attributes in the design. [2]

Finally, another way to model the presence of a no-choice option is by specifying a Nested Logit model. When two nests are specified, one containing the no-choice and the other the real profiles, the no-choice alternative is no longer treated as just another alternative. The idea is that respondents first decide whether or not to choose and only when they decide to choose a real profile do they select one of them, leading to a nested choice decision. This way of modelling the no-choice potentially also removes the effects of linear attributes because the zeros of the no-choice are no longer treated as real levels, because they are now captured in a different nest. The Nested Logit model can also be tested for the situation with or without the presence of linear attributes next to effects-type coded attributes.

No-choice models

We use three models: (1) the Multinomial Logit model (MNL), (2) the Nested Multinomial Logit model (NMNL), and (3) the No-choice Multinomial Logit model. The difference between the Logit model and the No-choice Logit model is the extra constant ([c.sub.nc]) added for the no-choice option in the design, but both models fall within the standard multinomial Logit context for conjoint experiments. In the Nested Logit model there is one extra parameter ([lambda]) called the dissimilarity coefficient (Borsch-Supan 1990). When its value is equal to 1, the Logit and Nested Logit model are equal. In the Nested Logit model we assume that there are two nests, one containing the no-choice alternative, the other containing the remaining real alternatives.

In a conjoint choice model each respondent has to choose one alternative from each of several choice sets. These choice sets are constructed by dividing the total set of profiles over K choice sets. We assume that each choice set contains the same number of alternatives without losing generality. The utility of alternative m in choice set k for individual j is defined as:

[U.sub.jkm] = [X.sub.km][beta] + [e.sub.jkm] (1)

where [X.sub.km] is a (1 x S) vector of variables representing characteristics of the mth choice alternative in choice set k. In most conjoint choice experiments no individual characteristics are present, so X does not depend on j. Note, however, that when an individualised design is used, X does depend on j, but we omit this index here for convenience. [beta] is a (S x 1) vector of unknown parameters, and [e.sub.jkm] is the error term. The MNL model treats observations coming from the same respondent as independent observations, and falls within the standard random utility approach. The choice probabilities [p.sub.km] that alternative m is chosen from set k in the conjoint MNL approach can be obtained as (see for example Ben-Akiva & Lerman 1985):

[p.sub.km] = exp([X.sub.km][beta])/[[[sigma].sup.M].sub.n=1]exp([X.sub.kn][beta] (2)

The log-likelihood in the conjoint context is extended by adding a sum over choice sets to the standard random utility log-likelihood and is equal to:

l = [[[sigma].sup.J].sub.j=1] [[[sigma].sup.K].sub.k=1] [[[sigma].sup.M].sub.m=1] [y.sub.jkm] ln ([p.sub.km]) (3)

The probabilities in a Nested Logit model are different from the probabilities in the standard MNL model. Assume that there are N nests and each nest n contains [M.sub.n] alternatives (the index representing the choice sets is suppressed for the moment), then the probability that alternative m is chosen from nest n is calculated as (cf. e.g. McFadden 1981):

P(n, m) = P(m\n) * P(n) (4)

where

P(m\n) = exp ([X.sub.nm][beta]) / [[[sigma].sup.[M.sub.n]].sub.m'=1] exp ([X.sub.n'm][beta]) (5)

is the standard MNL probability within a nest, and

P(n) = exp ([lambda][V.sub.n]) / [[[sigma].sup.N].sub.n'=1] exp ([lambda][V.sub.n']) (6)

is the probability of choosing nest n, [lambda] is the above-mentioned dissimilarity coefficient, and [V.sub.n] = ln [exp ([[[sigma].sup.[M.sub.n]].sub.m=1] [X.sub.nm][beta]) is the inclusive value for nest n. When [lambda] = 1 it is easy to show that the Nested Logit is equal to the Logit model. When there is only one nest (N = 1) (5) reduces to the standard Logit formula (2) for conjoint experiments. In the case of the Nested Logit model, when the no-choice is coded as a series of zeros for the attributes, the inclusive value ([V.sub.n]) of the nest containing only the no-choice is equal to zero in the Nested Logit model. Furthermore, P(m\n) = 1 for the nest containing the no-choice because it contains only one alternative.

Empirical investigations of modelling options

The following provides two applications to commercial conjoint choice data to illustrate the relative fits of the alternative models and coding of the attributes.

First application

We describe the results of the models outlined in the previous section, using a conjoint choice experiment on a technological product with six attributes: [3] brand (six levels), speed (four levels), technology type (six levels), digitising option (no and two yes levels), facsimile capable (y/n), and price (four levels). For all models we use three versions: in the first situation the price and speed attributes are coded linear with {1, 2, 3, 4} for the four levels respectively (speed ascending, price descending) and the other attributes are coded using effects-type coding. In the second situation we use as linear codes {-3, -1, 1, 3} for the linear levels instead, to investigate whether mean centring the linear levels solves (part of) the problem. In the third situation all attributes are coded with effects-type coding. We used 200 respondents who each had to choose from 20 choice sets with four alternatives, where the last alternative is the 'no-choice' option, which is defined as none of the above alternat ives. We used the first 12 choice sets for estimation and the last eight for prediction purposes. Each respondent had to choose from individualised choice sets. We compare the results of the models on the log-likelihood value, AIC (Akaike 1973) and BIC statistics (Schwarz 1978) and the Pseudo [R.sup.2] value (e.g. McFadden 1976) relative to a null-model in which all probabilities in a choice set are equal to 1/M. The AIC criterion is defined as: AIC = -2 ln L + 2n, where n is the total number of estimated parameters in the model and the BIG criterion is defined as: BIC = -2 ln L + n ln (O), where O is the number of independent observations in the conjoint choice experiment. We test differences in the likelihood values for models that are nested with the likelihood ratio (LR) test.

In Table 1 the estimation results are listed for all models with linear attributes; the left-hand side gives the results for the models with linear levels {1, 2, 3, 4}, the right-hand side with linear levels {-3, -1, 1, 3} (the results for the models where all attributes are coded with effects-type dummies are not shown). Note that when effects-type coding is used for an attribute, the part-worth for the last level of that attribute can be obtained by taking the sum of the estimates of the other levels of that attribute and changing the sign. Note also that the Nested Logit and the No-choice Logit models are not nested, but both are nested within the standard Logit model. In the nested Logit model we do not estimate [lambda] itself but estimate (1 -- [lambda]) instead, to have a direct test on [lambda] = 1.

The first conclusion that can be drawn from Table 1 is that the No-choice Logit model gives the best overall fit, in both situations, and converged to the same point. The likelihood is significantly better than the standard MNL model (LR(1 df) tests, p [less than] 0.01). The No-choice Logit model and the Nested Logit model are not nested, so these models cannot be compared with an LR test. The AIC and BIC values show, however, that the No-choice Logit model fits better than the Nested Logit model, which itself is significantly better than the Logit model (LR(1) tests, p [less than] 0.01) again in both situations. Table 1 also shows that the estimates for the dissimilarity coefficients ([lambda]) are significantly different from 1 for the Nested Logit model, hence the Nested Logit differs significantly from the MNL model.

When the [beta]-estimates are compared, Table 1 shows that the parameters estimates of the attributes with a dummy coding ([[beta].sub.1],..., [[beta].sub.5], [[beta].sub.7],..., [[beta].sub.14]) are somewhat different, although not dramatically so. However, on the lefthand side of Table 1, the coefficients of the linear attributes ([[beta].sub.6], [[beta].sub.15]) in the standard MNL model differ strongly from the other two models. Whereas the estimate for speed is negative (a high level is unattractive) and significant for the MNL model, it is positive (a high level is attractive) and significant for the other models. The price parameter shows a similar effect; it is negative but not significant in one situation and positive in the other for the MNL model, and positive (lower price is more attractive) and significant in the other two models. Clearly, both estimates for the linear attributes show a strong negative bias. Note, however, that there are also differences in the other part-worth estimates across the models. The righthand side of Table 1 shows that when the speed and price variables are coded with values such that the mean of the levels is zero (the same can of course be obtained by mean-centring the linear levels on the left-hand side of Table 1), the estimates for speed and price no longer show the wrong sign, but are still biased downwards compared to the other models. In the No-choice MNL model the estimates for all parameters are equal in both situations, except for the linear parameters which, on the right-hand side of Table 1, have exactly half the value of those on the left-hand side, which is the result of the doubled step-length of the linear levels. When the attributes price and speed are also coded with effects-type coding (not shown), the P-estimates are more similar across the three models, having the right signs.

The conclusion that can be drawn from the above results is that the presence of a no-choice alternative and linearly coded attributes can give very misleading results, in particular for the parameters of those linear attributes when the conjoint choice data are estimated with a standard Logit model. However, the parameters of attributes coded with effects-type dummies are also affected, be it less severely. When all attributes are coded with effects-type coding the bias seems less strong, but coefficients estimates are still highly attenuated. Overall fit can be improved substantially by specifying a Nested Logit or by adding a No-choice constant to the design. When we compare the Nested Logit and the Nochoice Logit results we see that both compensate for the no-choice zero level for the linear attributes, but there are some differences in the magnitudes of the estimated coefficients -- some in the range 5-10%--which may be important in substantive interpretation. However, the fit of the No-choice Logit mode l is much better than the Nested Logit model. Note that in Table 1 the estimates for the no-choice constant are relatively large and positive. This means that the no-choice has a high overall utility, which is also shown by the number of times the no-choice alternative was actually chosen (in 43.1% of all choice sets).

The estimates in Table 1 were used to predict the eight holdout choice sets. Table 1 also gives the values of the statistics for the predictive fit of the three models for the three different design options considered. The No-choice Logit model gives the best predictions which are significantly better than the Logit model (LR(1) tests, p [less than] 0.01) and which are also better than the Nested Logit model in all situations. The Nested Logit model also predicts significantly better than the standard Logit model (LR(1) tests, p [less than] 0.01). Thus the predictive validity results confirm the results on model fit. Note that although the MNL model with linear levels {1, 2, 3, 4} is clearly misspecified (as could be seen from the speed and price estimates in Table 1), the likelihood, both in estimation and prediction, is better than those of the MNL models with the two other ways of coding. However, in all situations the Nested Logit and No-choice Logit models show superior fit. We consider two further exte nsions.

First, in conjoint studies first-order brand-price interaction effects are often included in the design. The question is whether the above results change when such interactions are included. We also tested what the effects on these interaction effect parameters are for the various models and design specifications. In almost all cases no significant interaction terms were obtained and no (significant) improvement of the likelihood was found.

Second, one could expect combination of the No-choice and Nested MNL models, having two more parameters than the standard MNL model to provide a better fit and predictive validity than the nested and no-choice dummy MNL model specifications. This model matches the data, but converges to the same log-likelihood as the No-choice MNL model and is not identified, as the Hessian matrix of second-order derivatives has a zero eigenvalue. Therefore, we conclude that the No-choice MNL model remains the best-fitting model.

Second application

We describe the results of a second application on a technological product with six attributes: brand (five levels), performance (three levels), way of ordering (three levels), warranty (three levels), service (three levels), and price (four levels). For all models we use two versions; in the first situation the price, warranty and performance attributes are coded linear with warranty and performance: {1, 2, 3} and price: {1.5, 1.8, 2.2, 2.5}, respectively, and the other attributes are coded using effects-type coding. In the second situation we use mean centring so that the linear codes are warranty and performance: {-1, 0, 1} and price: {-0.5, -0.2, 0.2, 0.5}. We used 200 respondents who each had to choose from eight choice sets with four alternatives, where the last alternative is the 'no-choice' option, which is defined as none of the above alternatives. Each respondent had to choose from individualised choice sets. In this case we have no holdout choice sets available. We compare the results of the models on the log-likelihood value, AIC and BIC statistics and the Pseudo [R.sup.2] value relative to a null-model in which all probabilities in a choice set are equal to 1/M, as above.

The estimation results are listed for all models in Table 2. The first conclusion that can be drawn from Table 2 is that the No-choice Logit model again gives the best overall fit in both cases with different coding of the variables. The likelihood is significantly better than the standard MNL model in all cases (LR(1 df) tests, p [less than] 0.01) and the AIC and BIG values show that it fits better than the Nested Logit model, which itself is as good as the Logit model in this application. It is surprising that for this dataset, if the linear attributes are not mean-centred, the nested MNL converges to the same solution as the MNL model. However, the no-choice logit does better on all statistics in both cases. The estimates for the no-choice constant are significant in both cases.

Table 2 shows that the coefficients of the linear attributes, price and performance, in the standard MNL are again affected by the mean-centring operation. The estimates for the linear attributes are biased. Note, however, that there are also differences in the other part-worth estimates across the models, albeit they are smaller. In the No-choice MNL model the estimates for all parameters are equal across the different situations: only the magnitude of the no-choice constant is affected by the different coding of the linear attributes. Again, this is a desirable feature of this model.

The conclusion is that the results for this application replicate those of the first application shown above. The presence of a no-choice alternative and linearly coded attributes can give misleading results, in particular for the parameters of those linear attributes when the conjoint choice data are estimated with a standard Logit model. Fit is substantially improved by adding a no-choice constant to the design matrix.

Conclusions and discussion

Respondents may choose the no-choice alternative for two reasons. First, they may not be interested in the product category under research and for this reason choose the no-choice. In such a situation they would first decide whether or not to choose the offered product profiles and the Nested MNL model may be the most appropriate specification to describe this behaviour, since it puts the no-choice alternative in a different nest from these product profiles. The probability of the no-choice alternative may be an indication of the overall preference of the product in this case, and the model may be used to obtain an estimate of the overall attractiveness of the product category. Thus in this case the no-choice alternative would capture 'real' behaviour of consumers in the marketplace. Second, respondents may choose the no-choice because no real alternative in the choice set is attractive enough or because all alternatives are equally attractive and they do not want to waste more time on making the difficult ch oice. In this case the respondent treats the no-choice as 'just another' alternative, and the no-choice captures an effect specific to the task. If this is the case, the MNL model, with a no-choice constant, is the appropriate model to use, since it treats all alternatives equally. Now the utility of the no-choice option does not have a substantive meaning but serves as an indicator of respondents' involvement with the task. In our applications we saw that the No-choice MNL model produced better results compared to the Nested MNL model. This may be an indication that in these applications the second explanation for choosing the No-choice may have been appropriate. In other words, when a conjoint choice experiment with a no-choice alternative is estimated with both the No-choice MNL model and the Nested MNL model, the (predictive) fit of the models may give an indication of the substantive reasons respondents have to choose the no-choice option. In particular, when the No-choice MNL model provides the best fit it may be inappropriate to interpret the estimates as reflecting the overall attractiveness of the category. It is interesting to note that in our second application the Nested MNL model did not provide any improvement in fit over the standard MNL model, while the No-choice MNL model did fit significantly better. We would like to note that in our applications the effect of the no-choice option itself was significant. If that is not the case, the model converges to the standard MNL solution and may neither fit nor predict better. However, the inclusion of the no-choice constant in the model allows for this test, and at the same time gives an indication of whether the standard MNL would be more appropriate.

Rinus Haaijer is a Research Assistant at MuConsult in Amersfoort working in the field of transportation research. He studied for his Ph.D. at the Department of Marketing and Marketing Research at the University of Groningen, where he wrote a thesis on the estimation of conjoint choice experiments using the Probit model.

Wagner A. Kamakura is Professor of Marketing and holds the Cedar Rapids Business Area Chair at the University of Iowa. He gained his Ph.D. in Marketing from the University of Texas at Austin, and before joining academia he worked in market analysis, forecasting and planning in Brazil. Currently the Editor of the Journal of Marketing Research, his research interests cover market segmentation and market structure, database marketing, and the modelling of customer satisfaction, retention and profitability.

Michel Wedel is Professor of Marketing Research at the Department of Marketing and Marketing Research at the University of Groningen and visiting Professor of Marketing at the University of Michigan Business School. His main research interests are in research methodology and its application to substantive marketing problems. His papers have appeared in a variety of journals and he is Area Editor for Marketing Science. He has also developed GLIMMIX software for estimating mixture and mixture regression models.

(*.) University of Groningen and (+.) University of Iowa

(1.) In the remainder of the paper we only mention the 'no-choice' (or 'none'), but the results also apply to the 'own' alternative when nothing is known about its characteristics to the researcher. Even when the (design) attributes and levels of the 'own' alternatives are known, there may be additional attributes, which have some utility to the consumer but which are not in the design of the experiment. In this case an extra constant in the design may act as some 'threshold' value for the 'own' alternative.

(2.) Note that the same can be accomplished by coding one of the attributes (e.g. brand) with dummy coding instead of effects-type coding. This also increases the number of parameters with one and sets the level of the no-choice. Another possibility is to add a constant for the 'real' alternatives instead and keep a series of zeros for the no-choice; in that case the constant is equal to [-c.sub.nc].

(3.) We thank Rich Johnson from Sawtooth Software for allowing us to analyse these data. Because the data are confidential we cannot give more details about the attributes and levels.

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            Estimation and prediction results: first application
                              Levels linerar attributes {1,2,3,4}
Model                                         MNL
Parameter                                    Est.
Brand A [[beta].sub.01]                      0.386
Brand B [[beta].sub.02]                     -0.013
Brand C [[beta].sub.03]                      0.052
Brand D [[beta].sub.04]                     -0.150
Brand E [[beta].sub.05]                      0.011
Speed [[beta].sub.06]                       -0.237
Tech. type A [[beta].sub.07]                -0.531
Tech. type B [[beta].sub.08]                 0.505
Tech. type C [[beta].sub.09]                -0.321
Tech. type D [[beta].sub.10]                 0.514
Tech. type E [[beta].sub.11]                -0.132
Dig. opt (n) [[beta].sub.12]                -0.586
Dig. opt (y1) [[beta].sub.13]                0.128
Facsimile [[beta].sub.14]                   -0.445
Price [[beta].sub.15]                       -0.013
Nested Logit 1 - [lambda]                      -
No-choice [c.sub.nc]                           -
Fit statistics
Ln-likelihood
AIC
BIC
Pseudo [R.sup.2]
Prediction statistics
Ln-likelihood
AIC
BIC
Pseudo [R.sup.2]
Model                                        NMNL
Parameter                           s.e      Est.        s.e
Brand A [[beta].sub.01]           0.061 [*]  0.522     0.077 [*]
Brand B [[beta].sub.02]           0.067     -0.009     0.078
Brand C [[beta].sub.03]           0.067     -0.006     0.080
Brand D [[beta].sub.04]           0.071 [*] -0.166     0.087 [*]
Brand E [[beta].sub.05]           0.067     -0.018     0.081
Speed [[beta].sub.06]             0.021      0.118     0.031 [*]
Tech. type A [[beta].sub.07]      0.081 [*] -0.678     0.096 [*]
Tech. type B [[beta].sub.08]      0.060 [*]  0.613     0.074 [*]
Tech. type C [[beta].sub.09]      0.075 [*] -0.366     0.086 [*]
Tech. type D [[beta].sub.10]      0.060 [*]  0.635     0.074 [*]
Tech. type E [[beta].sub.11]      0.071     -0.149     0.084
Dig. opt (n) [[beta].sub.12]      0.049 [*] -0.714     0.055 [*]
Dig. opt (y1) [[beta].sub.13]     0.041 [*]  0.180     0.046 [*]
Facsimile [[beta].sub.14]         0.030 [*] -0.528     0.035 [*]
Price [[beta].sub.15]             0.020      0.385     0.031 [*]
Nested Logit 1 - [lambda]             -      0.924     0.009 [*]
No-choice [c.sub.nc]                  -       -            -
Fit statistics
Ln-likelihood                 -2906.738            -2715.413
AIC                            5843.476             5462.826
BIC                            5930.224             5555.358
Pseudo [R.sup.2]                  0.126                0.184
Prediction statistics
Ln-likelihood                 -1883.069            -1741.723
AIC                            3796.139             3515.448
BIC                            3876.805             3601.492
Pseudo [R.sup.2]                  0.151                0.215
Model                         No-Choice MNL
Parameter                         Est.            s.e
Brand A [[beta].sub.01]           0.472         0.066 [*]
Brand B [[beta].sub.02]          -0.011         0.070
Brand C [[beta].sub.03]           0.037         0.071
Brand D [[beta].sub.04]          -0.187         0.075 [*]
Brand E [[beta].sub.05]           0.020         0.071
Speed [[beta].sub.06]             0.129         0.028 [*]
Tech. type A [[beta].sub.07]     -0.633         0.086 [*]
Tech. type B [[beta].sub.08]      0.575         0.064 [*]
Tech. type C [[beta].sub.09]     -0.368         0.078 [*]
Tech. type D [[beta].sub.10]      0.628         0.064 [*]
Tech. type E [[beta].sub.11]     -0.173         0.075
Dig. opt (n) [[beta].sub.12]     -0.732         0.052 [*]
Dig. opt (y1) [[beta].sub.13]     0.172         0.043 [*]
Facsimile [[beta].sub.14]        -0.543         0.032 [*]
Price [[beta].sub.15]             0.396         0.028 [*]
Nested Logit 1 - [lambda]           -               -
No-choice [c.sub.nc]              2.461         0.121 [*]
Fit statistics
Ln-likelihood                               -2663.017
AIC                                          5358.035
BIC                                          5450.566
Pseudo [R.sup.2]                                0.200
Prediction statistics
Ln-likelihood                               -1706.087
AIC                                          3444.175
BIC                                          3530.219
Pseudo [R.sup.2]
                              Levels linear attributes {-3,-1,1,3}
Model                                         MNL
Parameter                                     Est.
Brand A [[beta].sub.01]                       0.350
Brand B [[beta].sub.02]                      -0.012
Brand C [[beta].sub.03]                       0.024
Brand D [[beta].sub.04]                      -0.142
Brand E [[beta].sub.05]                       0.013
Speed [[beta].sub.06]                         0.046
Tech. type A [[beta].sub.07]                 -0.451
Tech. type B [[beta].sub.08]                  0.432
Tech. type C [[beta].sub.09]                 -0.281
Tech. type D [[beta].sub.10]                  0.471
Tech. type E [[beta].sub.11]                 -0.132
Dig. opt (n) [[beta].sub.12]                 -0.488
Dig. opt (y1) [[beta].sub.13]                 0.099
Facsimile [[beta].sub.14]                    -0.386
Price [[beta].sub.15]                         0.144
Nested Logit 1 - [lambda]                      -
No-choice [c.sub.nc]                           -
Fit statistics
Ln-likelihood
AIC
BIC
Pseudo [R.sup.2]
Prediction statistics
Ln-likelihood
AIC
BIC
Pseudo [R.sup.2]
Model                                        NMNL
Parameter                           s.e      Est.        s.e
Brand A [[beta].sub.01]           0.064 [*]  0.524     0.076 [*]
Brand B [[beta].sub.02]           0.069     -0.010     0.078
Brand C [[beta].sub.03]           0.071      0.001     0.080
Brand D [[beta].sub.04]           0.074 [*] -0.176     0.086 [*]
Brand E [[beta].sub.05]           0.070 [*] -0.011     0.080
Speed [[beta].sub.06]             0.014 [*]  0.067     0.016 [*]
Tech. type A [[beta].sub.07]      0.084 [*] -0.682     0.095 [*]
Tech. type B [[beta].sub.08]      0.062 [*]  0.621     0.073 [*]
Tech. type C [[beta].sub.09]      0.077 [*] -0.377     0.086 [*]
Tech. type D [[beta].sub.10]      0.062 [*]  0.651     0.073 [*]
Tech. type E [[beta].sub.11]      0.073     -0.159     0.083
Dig. opt (n) [[beta].sub.12]      0.050 [*] -0.721     0.055 [*]
Dig. opt (y1) [[beta].sub.13]     0.043 [*]  0.178     0.046 [*]
Facsimile [[beta].sub.14]         0.031 [*] -0.539     0.035 [*]
Price [[beta].sub.15]             0.014 [*]  0.203     0.015 [*]
Nested Logit 1 - [lambda]             -      0.840     0.017 [*]
No-choice [c.sub.nc]                  -       -            -
Fit statistics
Ln-likelihood                 -2947.716            -2701.927
AIC                            5925.431             5435.854
BIC                            6012.180             5528.386
Pseudo [R.sup.2]                  0.114                0.184
Prediction statistics
Ln-likelihood                 -1960.978            -1735.014
AIC                            3951.956             3502.028
BIC                            4032.622             3588.072
Pseudo [R.sup.2]                  0.116                0.218
Model                         No-choice MNL
Parameter                         Est.            s.e
Brand A [[beta].sub.01]           0.472         0.066 [*]
Brand B [[beta].sub.02]          -0.011         0.070
Brand C [[beta].sub.03]           0.037         0.071
Brand D [[beta].sub.04]          -0.187         0.075 [*]
Brand E [[beta].sub.05]           0.021         0.071
Speed [[beta].sub.06]             0.064         0.014 [*]
Tech. type A [[beta].sub.07]     -0.634         0.086 [*]
Tech. type B [[beta].sub.08]      0.575         0.064 [*]
Tech. type C [[beta].sub.09]     -0.368         0.078 [*]
Tech. type D [[beta].sub.10]      0.628         0.064 [*]
Tech. type E [[beta].sub.11]     -0.173         0.075
Dig. opt (n) [[beta].sub.12]     -0.732         0.052 [*]
Dig. opt (y1) [[beta].sub.13]     0.172         0.043 [*]
Facsimile [[beta].sub.14]        -0.543         0.032 [*]
Price [[beta].sub.15]             0.198         0.014 [*]
Nested Logit 1 - [lambda]           -               -
No-choice [c.sub.nc]              1.150         0.048 [*]
Fit statistics
Ln-likelihood                               -2663.017
AIC                                          5358.035
BIC                                          5450.567
Pseudo [R.sup.2]                                0.200
Prediction statistics
Ln-likelihood                               -1706.087
AIC                                          3444.175
BIC                                          3530.219
Pseudo [R.sup.2]                                0.231
(*.)P[less than]0.05
           Estimation and prediction results: second application
Linear levels:                 Performance {1, 2, 3}, warranty
                           {0.5, 1, 5}, price {1.5, 1.8, 2.2, 2.5}
Model                                        MNL
Parameter                                   Est.                        s.e
Brand A                                   0.349 [*]                    0.069
Brand B                                   0.193 [*]                    0.072
Brand C                                   0.149 [*]                    0.069
Brand D                                  -0.165 [*]                    0.072
Performance                               1.157 [*]                    0.045
Order phone                              -0.048                        0.047
Order store                              -0.008                        0.046
Warranty                                  0.173 [*]                    0.016
Serv. ship back                          -0.639 [*]                    0.052
Serv. dealer                              0.171 [*]                    0.046
Price                                    -1.220 [*]                    0.067
Nested Logiti 1 - [lambda]                    -                          -
No-choice                                     -                          -
Statistics
Ln-likelihood                                                      -1569.002
AIC                                                                 3160.004
BIC                                                                 3219.159
Pseudo R [2]                                                           0.293
Linear levels:
Model                      Nested MNL           No-choice MNL
Parameter                     Est.         s.e      Est.           s.e
Brand A                     0.349 [*]     0.069   0.342 [*]       0.069
Brand B                     0.193 [*]     0.072   0.192 [*]       0.072
Brand C                     0.149 [*]     0.069   0.147 [*]       0.069
Brand D                    -0.165 [*]     0.072  -0.162 [*]       0.073
Performance                 1.157 [*]     0.050   1.093 [*]       0.049
Order phone                -0.048         0.047  -0.046           0.047
Order store                -0.008         0.046  -0.008           0.046
Warranty                    0.173 [*]     0.016   0.162 [*]       0.016
Serv. ship back            -0.639 [*]     0.052  -0.626 [*]       0.052
Serv. dealer                0.171 [*]     0.046   0.170 [*]       0.047
Price                      -1.220 [*]     0.084  -1.486 [*]       0.097
Nested Logiti 1 - [lambda]  0.000          na     -                 -
No-choice                     -             -    -0.807 [*]       0.215
Statistics
Ln-likelihood                         -1561.606               -1571.630
AIC                                    3162.004                3147.211
BIC                                    3226.537                3211.744
Pseudo R [2]                              0.293                   0.296
Linear levels:             Performance, warranty and
                              price: meancentred
Model                                 MNL                      Nested MNL
Parameter                            Est.                 s.e     Est.
Brand A                            0.360 [*]             0.069  0.360 [*]
Brand B                            0.203 [*]             0.072  0.203 [*]
Brand C                            0.155 [*]             0.069  0.155 [*]
Brand D                           -0.169 [*]             0.072 -1.169 [*]
Performance                        1.170 [*]             0.047  1.170 [*]
Order phone                       -0.051                 0.047 -0.051
Order store                       -0.008                 0.046 -0.008
Warranty                           0.173 [*]             0.016  0.173 [*]
Serv. ship back                   -0.673 [*]             0.052 -0.673 [*]
Serv. dealer                       0.184 [*]             0.046  0.184 [*]
Price                             -1.579 [*]             0.096 -1.579 [*]
Nested Logiti 1 - [lambda]             -                   -    0.000
No-choice                              -                   -     -
Statistics
Ln-likelihood                                        -1571.630
AIC                                                   3165.260
BIC                                                   3224.416
Pseudo R [2]                                             0.291
Linear levels:
Model                                No-choice MNL
Parameter                       s.e      Est.           s.e
Brand A                        0.069   0.342 [*]       0.069
Brand B                        0.072   0.192 [*]       0.072
Brand C                        0.069   0.146 [*]       0.069
Brand D                        0.072  -0.162 [*]       0.073
Performance                    0.050   1.093 [*]       0.049
Order phone                    0.047  -0.046           0.047
Order store                    0.046  -0.008           0.046
Warranty                       0.016   0.162 [*]       0.016
Serv. ship back                0.053  -0.626 [*]       0.052
Serv. dealer                   0.046   0.170 [*]       0.047
Price                          0.098  -1.486 [*]       0.097
Nested Logiti 1 - [lambda]      na     -                 -
No-choice                       -     -0.358 [*]       0.081
Statistics
Ln-likelihood              -1571.630               -1561.606
AIC                         3167.260                3147.211
BIC                         3231.794                3211.744
Pseudo R [2]                   0.291                   0.296
(*.)Significant 5%
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Author:Haaijer, Rinus; Kamakura, Wagner; Wedel, Michel
Publication:International Journal of Market Research
Date:Jan 1, 2001
Words:7016
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