We talk nearly every day with a conjoint method user who has run into a problem of some kind. The problem responsible for the most frequent customer support calls is using too many prohibitions. The system won't be able to produce a good design, and may fail altogether.

Prohibitions, if at all possible, should be avoided. Too many prohibitions, in the best case, can lead to imprecise utility estimation and, in the worst case, unresolvable (confounded) effects and the complete inability to calculate stable utilities. Allegedly, there have been entire data sets with hundreds of respondents going to waste ...


[ Freely by Sawtooth Software, Inc. ]

 

Problem

A frequent requirement in testing products is concerned with prohibitions between levels of two or more attributes. Common is conditional pricing due to presence or absence of some levels such as brand, product alternative, package, size, etc. However, unwisely set prohibitions can be detrimental, and sometimes even fatal, for the design efficiency.  As seeing is believing, a simple example is provided below.

Any experimental plan can be partitioned into smaller design building blocks. Say one of the blocks is a L4 orthogonal array: 

Design plan L4
Profile number Level of A1 Level of A2 Level of A3
1. 1 1 1
2. 1 2 2
3. 2 1 2
4. 2 2 1

The block has this covariance matrix:

Covariances of  design plan L4
Attribute A1 A2 A3
A1 1.0 0 0
A2 0 1.0 0
A3 0 0 1.0

Values in the diagonal elements are variance propagation coefficients. They tell us about the extent by which a random error in the data will be projected into the parameter estimates. Values of non-diagonal elements are the same measure for mutual dependence of errors between the attributes. The design is orthogonal as all non-diagonal elements are zero, and fully balanced as all diagonal elements are equal. 

Now we decide that, for some good reason, we cannot combine all levels and must introduce a prohibition leading to the change of level of attribute A3 from 1 to 2 in profile 4. The covariance matrix changes as follows:

Covariances of  design plan L4 with a prohibition
Attribute A1 A2 A3
A1 2.0 1.0 -2.0
A2 1.0 2.0 -2.0
A3 -2.0 -2.0 4.0

Not only variance of the affected attribute A3 has increased fourfold, but those of the other attributes have doubled. Even worse, a strong dependence between attributes has been introduced which means that errors will not compensate. If there are interactions between attributes, the introduced bias will accumulate over respondents. Since small interactions between attributes are common, a design with prohibitions will lead to the bias that may prevail over the main effects of estimated parameters.

 

Solution

There are many suggestions in the literature how to deal with prohibitions and test their influences. We have adopted the standard method of randomized blocked design made up of "product classes" in combination with ASD -Alternative Specific Design. A class is defined as a complete Cartesian set of profiles from allowed intervals, i.e. a set with no prohibitions among attributes allowed. In plain language, a class is the complete set of products having properties not exceeding the allowed intervals of properties given a fixed values of the other properties and features. Randomized representative subsets of orthogonal profiles obtained for each of the product classes are used in an experiment for each respondent. The method is statistically clean, easy to manage, understandable for client to specify and, as of practical importance, can be converted to the prohibition strategy implemented in commercially available design programs. The class approach guarantees all part-worths to be estimable with a minimal bias.

A class for the purpose of a conjoint study design is composed from the profiles whose certain attributes are limited to fixed intervals. It is desirable that the defined classes can be ordered in some way, typically by price range, assumed quality, way of usage, etc. Nominal classes are usually much harder to be handled with, and, therefore, are often grouped in ordinal hyper-classes. The number of hierarchical levels is, at least in theory, unlimited.

 

Application

In a typical SP (stated preference) experiment, a choice set consists of several alternatives among which the respondent is asked to discriminate. The alternatives should (1) provide sufficient variations of attribute values in the managerially permissible intervals and (2) guarantee estimability of the model parameters without a danger of having introduced correlations caused by improper attribute level prohibitions.

A simple representative of a class is a real or suggested product with the attribute values varied in an allowable range. If some combinations of attribute levels are still detected as not feasible the set of product profiles must be partitioned into 2 (or more) classes with attribute level intervals chosen so that all possible combinations of attribute levels in a class are allowed. A nested class structure is common.

A conjoint exercise should be composed of the product classes relevant for a given respondent, i.e. of the alternatives supposed to be from the respondent's evoked/consideration set. This can be accomplished with use of CBS - Choice Based Sampling that allows to assign a range of classes to a respondent.


Class Properties

Class structure of tested products often comes from the goal of the study quite naturally. E.g., cars can be classified as commercial and personal motor vehicles, these as small, lower middle, upper middle, luxury ones, off-roads, etc., and perhaps further divided into subclasses by make, equipment features, etc. Consumption goods can be divided into categories, these into low-level up to luxury classes, then according to the packaging, etc.The class structure and order of the class levels always depends on the conditions and objectives of the study in a way that does not contradict the rules of DOE - Design Of Experiments.

The approach of composing a conjoint exercise from product classes has appeared as very useful:

To a limited extent, product classes can be given predefined weights to simulate market distribution and availability of the products the classes represent.

In the product classes approach, the overall design may be slightly out of balance in respect to some attribute levels, most often to the outer levels of a value-based attribute such as price. This effect is often recognized as positive because the least frequently shown outer levels are usually the least managerially important. Nevertheless, if the off-balance should be a problem, the design weight (presentation frequency) of the affected levels may be boosted.

As aside