Published on November 2, 2007
Choice modelling and Conjoint Analysis: Choice modelling and Conjoint Analysis Dr. Anil Markandya Department of Economics and International Development University of Bath [email protected] tel. +44 1225 386954 Choice Modelling: Choice Modelling CM is a non-market valuation technique that is becoming increasingly popular in environmental economics, but also in other fields, such as management of cultural goods, planning, etc. Stated-preference technique—elicits preferences and places a value on a good by asking individuals what they would do under hypothetical circumstances, rather than observing actual behaviors on marketplaces. Survey-based technique. Contingent valuation is a special case of choice modeling 3 main approaches to elicit preferences with choice modeling: ranking (choose the most preferred, then the second most preferred, etc.) rating (give to each alternative a number from 1 to X to indicate strength of preference) choice (choose the most preferred“conjoint choice”) Contingent Ranking : Contingent Ranking Respondents are asked to rank a set of alternative representations of the good from the most preferred to the least preferred. Limitations of ranking approach: Limitations of ranking approach Heavy cognitive burden It is probably easy to identify the most preferred and the least preferred options, but it might be not so easy to rank the options in the middle “noise” Contingent Rating: Contingent Rating Respondents are shown different representations of the good and are asked to rank each representation on a numeric or semantic scale. Limitations of Rating: Limitations of Rating One of the major drawbacks of this technique is the strong assumptions that must be made in order to transform ratings into utilities. For example, the same representation of a good might receive the same rate by two different respondents, but this does not necessarily mean that the two answers are identical: a rate of “8” by a respondent might be completely different by the same “8” given by another respondent. Conjoint Analysis: Conjoint Analysis Conjoint Analysis (conjoint choice analysis, choice experiments, conjoint choice experiments): Conjoint Analysis (conjoint choice analysis, choice experiments, conjoint choice experiments) In a conjoint choice exercise, respondents are shown a set of alternative representations of a good and are asked to pick their most preferred. Similar to real market situations, where consumers face two or more goods characterized by similar attributes, but different levels of these attributes, and are asked to choose whether to buy one of the goods or none of them. Alternatives are described by attributes—the alternatives shown to the respondent differ in the levels taken by two or more of the attributes. The choice tasks do not require as much effort by the respondent as in rating or ranking alternatives. Slide9: If we want to use conjoint analysis techniques for valuation purposes, one of the attributes must be the “price” of the alternative or the cost of a public program to the respondent. If the “do nothing” (or “status quo” option—i.e., pay nothing and get nothing) is included in the choice set, the experiments can be used to compute the value (WTP) of each alternative. Note that we only learn which alternative is the most preferred, but we do not know anything about the preferences for the options that have not been chosen the exercise does not offer a complete preference ordering. Example of conjoint choice question from Boxall et al. (1996). : Example of conjoint choice question from Boxall et al. (1996). Conjoint choice question from Hanley et al. (2001) : Conjoint choice question from Hanley et al. (2001) Example of conjoint choice question from San Miguel et al. (2000).: Example of conjoint choice question from San Miguel et al. (2000). Example of conjoint question from Alberini et al. 2005: Example of conjoint question from Alberini et al. 2005 1) Land use 2) Moorings 3) New Buildings 4) Fast connections with other parts of the city 5) New jobs created 6) Cost (regional tax for year 2004) No connections Yes connections 350 new jobs 350 new jobs No new moorings No new moorings No new buildings Yes new buildings English Why is conjoint analysis useful?: Why is conjoint analysis useful? Useful in non-market valuation, because it places a value on goods that are not traded in regular marketplaces. It can also be used to value products, or improvements over existing products—popular technique in marketing research. Allows one to estimate WTP for a good that does not exist yet, or under conditions that do not exist yet—for example, a lake after water pollution has been reduced, but people have always seen the lake as a polluted body of water. Allows one to elicit preferences and WTP for many different variants of goods or public programs, and so it can help make decisions about environmental programs where the scope of the program has not been decided upon yet (e.g., EPA’s arsenic in groundwater rule—should it be 50 ppb, 25 ppb, 10ppb?) An advantage of conjoint choice is that researchers usually obtain multiple observations per interview, one for each choice task from each respondent. This increases the total sample size for statistical modeling purposes, holding the number of respondents the same. Designing a Conjoint Analysis Study: Designing a Conjoint Analysis Study 1st task: select the attributes that define the good to be valued. The attributes should be selected on the basis of what the goal of the valuation exercise is, prior beliefs of the researcher, and evidence from focus groups. For valuation, one of the attributes must be the “price” of the commodity or the cost to the respondent of the program delivering a change in the provision of a public good. Attributes can be quantitative, and expressed on a continuous scale, such as the gas mileage of a car, or the square footage of a house. The price or cost attribute should be on a continuous scale. Attributes can be of a qualitative nature, such as the style of a house (e.g., Cape Cod, ranch, colonial) or the presence/absence of a specified feature. It is also important to make sure that the provision mechanism, whether private or public, is acceptable to the respondent, and that the payment vehicle is realistic and compatible with the commodity to be valued. Slide16: 2nd step: choose the levels of the attributes. the levels of the attributes should be selected so as to be reasonable and realistic, or else the respondent may reject the scenario and/or the choice exercise. Attributes and levels used in the moose hunting study from Boxall et al. (1996).: Attributes and levels used in the moose hunting study from Boxall et al. (1996). Attributes and levels from San Miguel et al. (2000). : Attributes and levels from San Miguel et al. (2000). Attributes and levels from Alberini et al. (2005).: Attributes and levels from Alberini et al. (2005). Slide20: 3rd task: be mindful of the sample size when choosing attributes and levels. The sample size should be large enough to accommodate all of the possible combinations of attributes and levels of the attributes, i.e., the full factorial design. To illustrate, consider a house described by three attributes: square footage, proximity to the city center, and price. If the square footage can take three different levels (1500, 2000, 2200), proximity to the city center can take two different levels (less than three miles, more than three miles) and price can take 4 different levels (£200,000, £250,000, £300,000, and £350,000), the full factorial design consists of 324=24 alternatives. Fractional designs are available that result in fewer combinations. No of useful observations (no of individualsxchoices per individual) should be at least 1000 Slide21: 4th task: Once the experimental design is created, the researcher needs to construct the choice sets. The choice sets may consist of two or more alternatives, depending on how simple one wishes to keep the choice tasks. The “status quo” should be included in the choice set if one wishes to estimate WTP for a policy package or a scenario. This can be done in a number of different ways. For instance, one can ask the respondent to choose between A and the status quo, then B and the status quo, etc. Alternatively, one can ask the respondent to choose directly between A, B, and the status quo. Or, respondents may first be asked to indicate their preferred option between A and B (the so-called “forced choice”), and then they may be asked which they prefer, A, B or the status quo. When grouping alternatives together to form the choice sets, it is important to exclude alternatives that are dominated by others. For example, if house A and B were compared, and the levels of all attributes were identical, but B were more expensive, A would be a dominating choice. Such pairs should not be proposed to the respondents in the questionnaire, although some researchers believe that this is a way of checking if respondents are paying attention to the attributes of the alternatives they are shown. Complexity: Complexity Should increase with: the number of attributes the number of possible levels for an attribute, how different the alternatives in each choice set are in terms of the level of an attribute, how many attributes differ across alternatives in each choice set, the number of alternatives in a choice set (A and B, or A v. B v. C v. D), the number of choice tasks faced by the respondent in the survey. Fatigue or learning? Model for the Conjoint Analysis : Model for the Conjoint Analysis It is assumed that the choice between the alternatives is driven by the respondent’s underlying utility. The respondent’s indirect utility is broken down into two components. The first component is deterministic, and is a function of the attributes of alternatives, characteristics of the individuals, and a set of unknown parameters, while the second component is an error term. Formally, 1) where the subscript i denotes the respondent, the subscript j denotes the alternative, x is the vector of attributes that vary across alternatives (or across alternatives and individuals), and is an error term that captures individual- and alternative-specific factors that influence utility, but are not observable to the researcher. β is a set of parameters – see next slide. Equation (1) describes the random utility model (RUM). Slide24: We can further assume that the deterministic component of utility is a linear function of the attributes of the alternatives and of the respondent’s residual income, (y - C): 2) where y is income and C is the price of the commodity or the cost of the program to the respondent. The coefficient is the marginal utility of income. Respondents are assumed to choose the alternative in the choice set that results in the highest utility. Because the observed outcome of each choice task is the selection of one out of K alternatives, the appropriate econometric model is a discrete choice model expressing the probability that alternative k is chosen. Formally, 3) where signifies the probability that option k is chosen by individual i. This is very important!!!: This is very important!!! This means that 4) from which follows that 5) Equation (5) shows the probability of selecting an alternative no longer contains terms in (2) that are constant across alternatives, such as the intercept and income. It also shows that the probability of selecting k depends on the differences in the levels of the attributes across alternatives, and that the negative of the marginal utility of income is the coefficient on the difference in cost or price across alternatives. Dataset in LIMDEP: Dataset in LIMDEP 12 obs per respondent because each respondent answers 4 choice questions and each choice question has 3 alternatives (A,B and status quo) Left alternative Right alternative Status quo Status quo is chosen Right alternative is chosen Taxes*castello Castello = dummy =1 if respondent lives in castello nij=3 because in each choice task there are 3 options From Alberini et al 2005 see slide 13 Conditional logit model: Conditional logit model If the error terms are independent and identically distributed and follow a standard type I extreme value distribution, one can derive a closed-form expression for the probability that respondent i picks alternative k out of K alternatives. Since the cdf of the standard type I extreme value distribution is and its pdf is choosing alternative k means that for all jk, which can be written as . The probability of choosing k is, therefore, 6) for all jk Expression (6) follows from the assumption of independence, and the fact that is an error term and not observed, so that it is must be integrated out of Slide28: The product within expression (6) can be re-written as 7) Now write: 8) which allows us to rewrite (6) as 9) where Slide29: The integrand in expression (9) is the pdf of the extreme value distribution and is, clearly, equal to 1. Equation (9) thus simplifies to which by (8) is in turn equal to Recalling (2), the probability that respondent i picks alternative k out of K alternatives is 10) where is the vector of all attributes of alternative j, including cost, and = Slide30: Equation (10) is the contribution to the likelihood in a conditional logit model. The full log likelihood function of the conditional logit model is 11) where yik is a binary indicator that takes on a value of 1 if the respondent selects alternative k, and 0 otherwise. The coefficients are estimated using the method of Maximum Likelihood (MLE). Slide31: We can further examine the expression for in equation (10) to show that depends on the differences in the level of the attributes between alternatives. To see that this is the case, we begin by re-writing (10) as 12) which is equal to 13) and thus to 14) Slide32: For large samples and assuming that the model is correctly specified, the maximum likelihood estimates are normally distributed around the true vector of parameters , and the asymptotic variance-covariance matrix, , is the inverse of the Fisher information matrix. The information matrix is defined as 15) where Marginal Prices and WTP: Marginal Prices and WTP Once model (11) is estimated, the rate of trade off between any two attributes is the ratio of their respective coefficients. The marginal value of attribute l is computed as the negative of the coefficient on that attribute, divided by the coefficient on the price or cost variable: 16) The willingness to pay for a commodity is computed as: 17) where x is the vector of attributes describing the commodity assigned to individual i. It should be kept in mind that a proper WTP can only be computed if the choice set for at least some of the choice sets faced by the individuals contains the “status quo” (in which no commodity is acquired, and the cost is zero). Expression (17) is obtained by equating the indirect utility associated with commodity and residual income with the indirect utility associated to the status quo (no commodity) and the original level of income y, and solving for C. Is conjoint analysis better than contingent valuation?: Is conjoint analysis better than contingent valuation? Several analysts believe that conjoint analysis questions reduce strategic incentives, because individuals are busy trading off the attributes of the alternatives and are less prone to strategic thinking (Adamowicz et al., 1998). The same reasoning and the fact that conjoint choice questions may appear less “stark” than the take-it-or-leave options of contingent valuation has led other researchers to believe that “protest” behaviors are less likely to occur in conjoint analysis surveys. Some valuation researchers (Carson, Hanemann) do not believe in conjoint analysis because they believe that much effort must be spent in stated preference studies to provide a scenario that is fully understood and accepted by the respondent. Changing this scenario from one choice question to the next, they point out, results in a loss of credibility of the scenario and may induce rejection of the choice task. Descriptive statistics from Alberini et al. 2005 : Descriptive statistics from Alberini et al. 2005 Results from Alberini et al. (2005).: Results from Alberini et al. (2005).