Analysis of Cardinal and Ordinal Assumptions in Conjoint Analysis

By Harrison, R. Wes; Gillespie, Jeffrey et al. | Agricultural and Resource Economics Review, October 2005 | Go to article overview

Analysis of Cardinal and Ordinal Assumptions in Conjoint Analysis


Harrison, R. Wes, Gillespie, Jeffrey, Fields, Deacue, Agricultural and Resource Economics Review


Of twenty-three agricultural economics conjoint analyses conducted between 1990 and 2001, seventeen used interval-rating scales, with estimation procedures varying widely. This study tests cardinality assumptions in conjoint analysis when interval-rating scales are used, and tests whether the ordered probit or two-limit tobit model is the most valid. Results indicate that cardinality assumptions are invalid, but estimates of the underlying utility scale for the two models do not differ. Thus, while the ordered probit model is theoretically more appealing, the two-limit tobit model may be more useful in practice, especially in cases with limited degrees of freedom, such as with individual-level conjoint models.

Key Words: ordered probit, two-limit probit, conjoint analysis, cardinality

Numerous applications of conjoint analysis (CA) have emerged in the agricultural economics literature in recent years. Most of these studies have analyzed consumer preferences for new food products or resource usage and willingness to pay for recreational services. Studies evaluating new food products include Gineo (1990), Prentice and Benell (1992), Halbrendt, Wirth, and Vaughn (1991), Halbrendt, Bacon, and Pesek (1992), Yoo and Ohta (1995), Hobbs (1996), Sylvia and Larkin (1995), Sy et al. (1997), Harrison, Ozayan, and Meyers (1998), Gillespie et al. (1998), and Holland and Wessells (1998). New product acceptance studies typically assume that a respondent's total utility for a hypothetical product is a function of various product attributes. CA is used to estimate "part-worth" utilities, which measure the partial effect of a particular attribute level on the respondent's total utility for hypothetical products. Part-worth estimates are typically used to simulate utility values for products not evaluated by respondents; thus, optimal hypothetical products can be determined.

A second category of CA studies has sought to estimate respondents' willingness to pay for a bundle of attributes associated with a recreational site or activity. Examples include Mackenzie (1990, 1993), Can and Luzar (1993), Lin, Payson, and Wertz (1996), Roe, Boyle, and Teisl (1996), Stevens, Barrett, and Willis (1997), Miquel, Ryan, and Mclntosh (2000), and Boyle et al. (2001). As with the new product acceptance studies, this approach requires respondents to rate or rank attribute bundles as price and other attribute levels are varied (Mackenzie 1990). Willingness to pay is calculated directly from the marginal rates of substitution between price and non-price attributes estimated from conjoint data.

Two commonly used methods for coding respondent preferences are rank-order and intervalrating scales. The rank-order method requires subjects to unambiguously rank all hypothetical product choices. In these cases, the dependent variable is ordinal, and ordered regression models such as ordered probit or logit are most suitable for conjoint estimation. The interval-rating method allows subjects to express order, indifference, and intensity across product choices, a feature allowing both metric and nonmetric properties of utility to be elicited. Model selection becomes less clear if interval-rating scaling is used. As shown in Table 1, a wide range of models have been utilized when ratings have been used. Some studies have used linear regression to estimate part-worth parameters (Halbrendt, Wirth, and Vaughn 1991; Prentice and Benell 1992; Harrison, Ozayan, and Meyers 1998; Stevens, Barrett, and Willis 1997; Roe, Boyle, and Teisl 1996). These studies assume that interval-rating scales are metric and continuous. However, even if one argues that subjects express metric information in their responses, interval-rating scales are limited by upper and lower bounds. Under these circumstances, linear regression models yield truncated residuals and asymptotically biased parameters. The censored nature of the scale can be accounted for with a two-limit tobit (TLT) model, which corrects for censoring and retains metric information between the bounds. …

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