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.

• A class structure must be strictly hierarchical. No exceptions are allowed.
• The whole range of tested products (rather than attribute ranges) is partitioned into product classes such that all combinations of attribute levels belonging to the class are allowed inside the class. If this is not possible by classifying products in a single hierarchy level, a class should be divided into lower-level sub-classes (nests).
• The ranges of attribute levels in the lowest level class must be narrow enough not to allow generation of extremely attractive or totally unacceptable product profiles. All profiles in a class should be managerially acceptable.
• At the same time, the ranges of attribute levels in the class must be broad enough to allow estimation of their influences.
• Partitioning of the products into classes does not mean partitioning of the attribute ranges. No exclusivity of the attribute ranges in classes is required, i.e., ranges of the respective attribute levels may (and usually do) overlap.
• Classes should reflect different types of anticipated behavior and perceptions of respondents.
• Classes may or may not have common attributes. The latter case requires using ASD - Alternative Specific Design method.
• Each attribute in a class is defined on a given range of values. The range may contain one level (e.g. of a brand attribute) or more levels (e.g. of a price attribute).
• If the range is made up of a single level, the class is a specific alternative.
• In turn, a specific alternative can have only one attribute on a single level. If there were two or more attributes on a single level, their utilities would not be estimable. Either a composite attribute or sub-classes should be created.
• The same number of levels of a common attribute (typically price) should be used in each class. The number should be very reasonable so as not to exceed an acceptable total number of levels in the study.
• The range of level values or items in a class should be broad enough to asses the effect of attribute levels in the class but narrow enough not to exceed managerially acceptable values.
• A conjoint study based on product classes can be understood as a merger of several sub-conjoint studies where there are no prohibitions between attribute levels in any of the sub-conjoint exercises.

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

• Products often fall into classes quite naturally.
• Ranges of attributes levels can be easily defined and modified.
• The profiles shown to respondents are credible.
• As attribute ranges in a class are usually quite narrow interactions between attribute levels can be neglected and omitted from the estimation.
• A blocked design has usually a substantially lower number of parameters than an equivalent model with interactions. In our experience, about the same fit of the models is obtained. The model description is more compact and the precision of a simulation unimpaired.
• Part-worths from a blocked design are usually more stable (have lower variance) than those from a non-blocked design.
• Class attribute is usually hidden from respondents. Its level part-worths may or may not be estimated. The latter case is typical for the classes serving only as an aid in the design.
• If estimated, interpretation of the part-worths related to the classes is usually more straightforward than of those based on interactions. E.g., it is much easier to understand and interpret a single value of price-based elasticity of substitution or conventional share-based price elasticity for a brand than, say, values for several price levels and the same number of values for interaction between the brand and the price levels.
• The spans of the implicit conjoint blocks, i.e. the classes that make them up, usually overlap. This contributes to the overall stability and reliability of the estimates.
• Last but not least advantage is the client can easily see where and why the design might be deficient and can promptly make the proper provisions.
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
• A design made up of classes should not be confused with an alternative specific design analyzed as a nested "mother logit" model.
• A typical use of the class approach is for various types of packages made by bundling products or services. Each package may be of its own class and have ranges of its properties set independently from properties of other packages.
• Representatives of all classes in a conjoint exercise are usually shown to respondent equally often, i.e. with ratio 1 : 1 : 1 : ... . This is in agreement with the null hypothesis that all classes are (a priori) equally attractive to the subject. If relevant the frequency can be set to a ratio of small natural numbers, e.g. 3 : 2 : 1 : ..., e.g. if the class is a brand with many more varieties than other brands offer.
• The cardinal motivation to introduce the product classes based construction of a CBC study was to avoid adaptive methods of interviewing such as ACA - Adaptive Conjoint Analysis. The usual goal in marketing engineering is to asses potential of the concepts embodying the intentions of the vendor as compared to the current products, and the reason for differences. In adaptive approach, respondents tend to exclude undesirable levels in addition to those completely unacceptable. This is in a direct contrast to the fact that customers tolerate minor disadvantages when the important properties are satisfying. An adaptive strategy tends to emphasize the idealized wishes of the customers. The possibility to exclude certain levels of attributes before the actual presentation and evaluation of concepts often leads to exclusion of certain groups of common products from the evaluations, and, consequently, does not provide relevant data to answer the posed problem.
• The recently introduced ACBC - Adaptive Choice-Based Conjoint (Sawtooth, Inc., 2009) is an adaptive method of a new generation. Unfortunately, the author has no direct experience with it.