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Article Excerpt
The use of adaptive designs in conjoint analysis has been shown to lead to an endogeneity bias in partworth estimates using sampling experiments. In this paper, we re-examine the endogeneity issue in light of the likelihood principle. The likelihood principle asserts that all relevant information in the data about model parameters is contained in the likelihood function. We show that, once the data are collected, adhering to the likelihood principle leads to analysis where endogeneity becomes ignorable for estimation. The likelihood principle is implicit to Bayesian analysis, and discussion is offered for detecting and dealing with endogeneity bias in marketing.
Key words: likelihood principle; adaptive design; Bayes theorem; directed acyclic graphs History: This paper was received January 31, 2006, and was with the authors 2 months for 2 revisions; processed by Eric Bradlow.
1. Introduction
The recent paper by Hauser and Toubia (2005), "The Impact of Utility Balance and Endogeneity in Conjoint Analysis," raises an interesting set of issues related to a wide class of marketing models. In their paper, they present evidence that adaptive designs, where answers to early questions in a conjoint interview are used to select later questions, induce biases in the estimated part-worths. While their paper focuses on analysis associated with Sawtooth Software's popular ACA (Adaptive Conjoint Analysis) software, the implications of their analysis reach beyond conjoint analysis, sequential analysis, and utility balance, touching on important philosophical issues at the core of statistical inference.
In this paper, we reexamine the endogeneity bias identified by Hauser and Toubia (HT) and explain its presence using traditional econometric methods. Bias is an aspect of statistical inference that relies on the notion of a sampling experiment, where hypothetical data sets are used to characterize the performance of an estimator. We argue that sampling experiments are useful to study properties of estimators and other procedures when real data are not available. However, when data are available, analysis should proceed according to the likelihood principle as originally proposed by Fisher (1922).
The likelihood principle asserts that the likelihood contains all the information about model parameters (e.g., conjoint part-worths) in the data. We show that, according to the likelihood principle, the endogeneity created by adaptive questioning is not of concern for estimation, i.e., does not alter the likelihood function of the observed data. Our discussion of the likelihood principle raises a number of philosophical issues at the core of statistical inference that highlight the difference between classical (i.e., frequentist) and Bayesian philosophies.
Our analysis of endogeneity bias in conjoint models uses HT as a springboard for discussion and is not meant to criticize their findings. In fact, our analysis covers some of the same ground as their analysis, restating their findings in terms more familiar to the statistics literature. Our examples are sometimes similar to those examined by HT and sometimes depart from theirs to provide additional insight and analysis. We applaud their interest in analyzing the issue of endogeneity created by adaptive designs because of its importance to both practitioners and academic researchers.
The remainder of the paper is organized as follows. We begin with a review of endogeneity bias in regression models caused by adaptive designs, restating many of the points made by HT. We then introduce the likelihood principle and examine its implication for data analysis. Our analysis shows that, conditional on the dependent variable (y), the way the design is adaptively created is not informative about model parameters. The mechanism employed to pick the design points, and the resulting endogeneity, are "ignorable" for the purpose of likelihood-based inference (Gelman et al. 2004, p. 203; Rubin 1976). (1) Thus, we believe the potential harm from endogeneity created by adaptive designs should be considered when selecting among experimental procedures (e.g., adaptive versus fixed designs), not in estimation. Discussion is provided about the importance of bias in the evaluation of procedures, and tools are offered for detecting when endogeneity created by adaptive questioning and other forms of sample selection will impact conditional inference--i.e., when it alters the likelihood function. Concluding comments are then offered, in which we advocate a Bayesian orientation to conducting analysis in marketing.
2. Endogeneity Bias
Endogeneity bias arises in regression analysis y = X[beta] + [epsilon] when the regressors, X, are not independent of the errors, [epsilon]. When a specific realization of the regressors, [x'.sub.t], is selected based on the outcome of previous choices, [y.sub.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)
The presence of endogenous responses biases the regression estimate because the second term in Equation (1) is not equal to zero. To see this, consider a simple example where there is just one regressor without an intercept (y = x[beta] + [epsilon]), and just two observations. The OLS estimator is given by
[??] = [x.sub.1][y.sub.1] + [x.sub.2][y.sub.2]/[x.sup.2.sub.1] + [x.sup.2.sub.2]. (2)
Then, if [x.sub.2] is determined by the value of [y.sub.1], i.e., [x.sub.2] = f([y.sub.1]) = f([x'.sub.1][beta] + [[epsilon].sub.1]), we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)
The expectation of the first term in the brackets is not equal to zero because [y.sub.1], and hence [[epsilon].sub.1], is used to determine [x.sub.2]. Because [[epsilon].sub.1]...
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