Conjoint Analysis
6.2 Utility Models
In order to properly construct the conjoint experiment, one should have a clear idea of the utility function one intends to estimate. Therefore, the utility function is discussed in this section, while the construction of the experiment is discussed in the next section.
6.2.1 The Utility Function
As already explained in the Introduction, the utility function describes to what extent the attributes influence the overall utility derived from residential alterna-tives. Typically it is assumed that utility consists of a structural part that can be explained by the estimated model and an error component, the part of utility than cannot be explained. Furthermore, it is typically assumed that the structural part of utility is a linear summation of part-worth (or marginal) utility contributions of the attributes, which can be expressed as follows:
0
=1
= + = +
å
I +j j j i ij j
i
U V e b bX e (6.1)
where,
Uj = overall utility attached to alternative j;
Vj = the structural component of utility; that part of the utility that is determined by the model;
ej = an error component or random part of utility; that part of utility that is not determined by the model;
b0 = the utility constant;
bi = the coefficient to be estimated for attribute i;
Xij = the value of attribute i describing alternative j;
iXij
b = part-worth (or marginal) utility contribution of attribute i to the overall utility of alternative j.
6.2.2 Effects Coding
In order to include categorical attributes into the utility function, like, for example, tenure or dwelling type, the attribute levels need to be coded. Different coding schemes can be applied, such as dummy coding, effects coding, and orthogonal coding. These coding schemes differ in the direct interpretation of the estimated coefficients, but do not differ with respect to the resulting part-worth utilities of the attribute levels. As effects coding is applied in the example study presented later in this chapter, only this coding scheme is described in more detail here. An advantage of effects coding is that the estimated utility constant can be interpreted as the average utility attached to the residential alternatives included in the experiment. In the more well-known dummy coding scheme, the estimated constant denotes the utility of the alternative of which all levels are coded by zero. This is only relevant if that alternative serves as a bench-mark alternative for which it is relevant to compare all other alternatives to. As this is often not the case, effects coding usually offers a more attractive interpretation.
Table 6.1 provides the coding scheme for effects coding for two, three and four level attributes. This scheme indicates that L levels are coded by L-1 indicator variables iv. For each indicator variable l, a coefficient ßl. is estimated. Hence, for
a four-level attribute, three parameters ß1, ß2, and ß3 are estimated. The estimated coefficients are used to calculate the part-worth utility (pwu) of each level by mul-tiplying the estimated parameter with its code and summing the results across the indicator variables (coded columns). The resulting part-worth utilities expressed in the estimated coefficient ßl are presented in Table 6.1. From the table it becomes clear that the sum of the part-worth utilities across the levels of a particular attribute is, by definition, zero.
6.2.3 Interpretation of Part-Worth Utilities
The interpretation of the estimated part-worth utilities when effects coding is applied will be demonstrated by a simple example. Assume that in a conjoint experiment respondents evaluated eight residential profiles describing three attri-butes which each are varied in two levels. As respondents evaluated the profiles on a ten-point rating scale, a regression model was estimated that resulted in the part-worth utility contributions presented in Table 6.2. As effects-coding was applied, the estimated intercept or constant is equal to the mean overall utility derived from all the profiles included in the experiment. Hence, the mean utility derived from the residential profiles included in the experiment is equal to 5.
The part-worth utilities can be interpreted as the contribution of the attribute levels to the overall utility expressed as the deviation from the constant, thus from the mean overall utility. Hence, a positive part-worth utility means that the presence of the attri-bute level in a residential alternative increases the total utility derived from that alter-native, and consequently, a negative part-worth utility decreases the overall utility.
Table 6.1 Effects-coding for two-, three- and four-level
attributes Indicator variables (iv)
Part-worth utility Two levels iv1
1 1 ß1
2 −1 −ß1
Parameters: ß1
Three levels iv1 iv2
1 1 0 ß1
2 0 1 ß2
3 −1 −1 −(ß1 + ß2)
Parameters: ß1 ß2
Four levels iv1 iv2 iv3
1 1 0 0 ß1
2 0 1 0 ß2
3 0 0 1 ß3
4 −1 −1 −1 −(ß1 + ß2 + ß3)
Parameters: ß1 ß2 ß3
The part-worth utilities presented in Table 6.2 indicate that owner-occupied houses increase utility, while rental houses decrease utility. Hence, owner-occupied houses are preferred to rental houses. Likewise, Table 6.2 indicates that single-family houses are preferred to multi-family houses and that cheap houses are preferred to expensive houses. Furthermore, considering the size of the esti-mated coefficients of the part-worth utilities, Table 6.2 indicates that of all the attributes, monthly costs has the largest impact on utility, followed by housing type and finally tenure.
Based on the utility function just discussed, one is able to predict the utility for any combination of attribute levels included in the experiment. For example, the utility for an owner-occupied single-family house of 500 euro per month is pre-dicted as: 5 (constant) + 0.5 + 1 + 2 = 8.5. Furthermore, by applying linear interpola-tion the utility contribuinterpola-tion can be predicted for any value of a continuous attribute that falls within the range varied in the experiment. In this example, this would only apply to the attribute monthly costs.
6.2.4 Interaction Effects
The utility function just discussed is a main-effects-only model. A main-effect is the utility contribution of an attribute level to the overall utility irrespective of the presence of any other attribute level in the alternative. For example, our example model predicts that the combined effect of a rental multi-family house is:
−0.5 − 1 = − 1.5 utility points. However, the part-worth utility contribution of an attribute level may not always be independent from other attribute levels in the alternative. This means that a specific combination of two attribute levels may have a different effect on utility than the sum of their associated part-worth utilities (the main-effects). This difference is denoted by an interaction-effect, which can be modeled by estimating a coefficient for the product of the attributes.
Hence, an interaction-effect can be regarded as a correction of the sum of the main-effects. For example, assume that the interaction-effect for tenure and
Table 6.2 Estimated
housing type is equal to 0.25, this indicates that the joint utility of rental and multi-family house is 0.25 larger than the sum of their main effects (since the product of their codes is +1). Likewise, based on this interaction effect, the joint combination of single-family and owner-occupied house is 0.25 larger than pre-dicted by the main-effects, while the joint combination of both rental - single-family house and owner-occupied – multi-single-family house is 0.25 smaller than predicted by the main-effects.
In addition to the two-way interactions just described, higher order interaction effects may also play a role. Hence, the sum of the main-effects may be corrected by the specific combination of three or more attribute levels. However, two-way interaction effects are not often estimated in practice and higher-order interaction-effects are only very rarely estimated as these are hard to interpret.
6.2.5 Estimation of Rating-Based Models
As already discussed in the introduction, at least two different types of responses can be requested from the respondent in conjoint experiments: ratings or choices.
Ratings involve evaluating each residential profile separately and expressing the result as a number on some preference rating scale. Hence, the overall utility Uj for each residential alternative j is directly observed. As the rating observations are assumed to be of interval level measurement, the observed overall profile ratings make up the dependent variable in a regression analysis, and the independent variables are formed by the coded attribute levels. Hence, these data are typically analyzed by applying ordinary least squares regression analysis, which, for example, can be conducted in SPSS.
6.2.6 Estimation of Choice-Based Models
If the conjoint experiment is framed as a choice task then the observed responses indicate whether an alternative was chosen (normally coded 1), whereas the remaining alternatives (coded 0) were not. Hence, the nominal-level data cannot be analyzed by applying ordinary least square regression analysis, but require the application of an appropriate limited dependent analysis technique.
A further difference with rating tasks is that in choice tasks utility is not directly observed, only choices among alternatives. By assuming that respondents choose for the alternative with the highest utility, however, utility can be linked to the prob-ability that an alternative will be chosen. This requires making additional assump-tions about the random component of utility ej. Typically, it is assumed that the errors are independently, identically distributed extreme values (the distribution is also referred to as: type I extreme value, Gumbell or double-exponential), which
results in the well-known multinomial logit (MNL) model (e.g., Ben-Akiva, and Lerman 1985; Train 2003):
j j V
j V
j S
p e
e ¢
¢Î
=
å
(6.2)Where pj is the probability of choosing alternative j; S denotes the choice set of j alternatives; and eVj denotes the exponent of Vj, the structural part of the utility.
MNL models are typically estimated in specialized software packages, such as Nlogit (Limdep), Alogit or Biogeme.
In order to provide a complete introduction to conjoint analysis, both choice-based and rating-based models will be covered in this chapter. However, it should be noted that most academics now prefer choice-based models (e.g. Louviere et al. 2000, 2010) for reasons that will be explained later (see subsection about measurement tasks).