## Link function selection in stochastic multicriteria decision making models (2006)

Venue: | European Journal of Operational Research |

Citations: | 2 - 1 self |

### BibTeX

@ARTICLE{Hahn06linkfunction,

author = {Eugene D. Hahn},

title = {Link function selection in stochastic multicriteria decision making models},

journal = {European Journal of Operational Research},

year = {2006}

}

### OpenURL

### Abstract

A stochastic formulation of the Analytic Hierarchy Process (AHP) using an approach based on Bayesian categorical data models has been developed. However, in categorical data models it is known that the selection of the link function may have an impact on the model estimates. In particular, the selection of the probit link implies an assumption that model error terms are normally distributed and this normality assumption is regularly utilized in other related methods such as the multiplicative AHP. We examine model performance with respect to the choice of two model link functions. With regard to point estimates, it is found that the logit formulation is better able to replicate the estimates obtained by the eigenvector decomposition associated with the original formulation of the AHP. By contrast, the probit link produces priorities which are consistently more moderate than those of the AHP. The results suggest that the logit formulation will be preferred by decision makers who wish to replicate the AHP priorities as closely as possible. The results also suggest that the unexamined use of the normality assumption in other stochastic AHP methods may have an impact on priority estimates and thus is worthy of further attention.

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Citation Context ... a powerful tool for decision makers. The approach assumes that the underlying weights have a multinomial sampling distribution such that the priorities may be estimated via multinomial logit models (=-=McFadden, 1973-=-). Inference on the priority vector is obtained by the adoption of a Bayesian approach and the use of Markov chain Monte Carlo methods. 2 Model Formulation A brief review of the methodology is as foll... |

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Citation Context ...omially distributed conditional on the priorities. In (9)-(11) the differenced linear predictors are used to obtain the priorities by way of the bivariate normal link function. It can be shown (e.g., =-=Hausman and Wise, 1978-=-; Terza, 1998) that in an independent multinomial probit model Ω after standardization has entries of 1 on the diagonal and 1/2 elsewhere. Note that the off-diagonal elements of 1/2 result from the di... |

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Citation Context ...ivariate statistics; Simulation. 1 Introduction The Analytic Hierarchy Process (AHP; Saaty, 1977) is a methodology useful for multicriteria decision making which has received substantial application (=-=Vaidya and Kumar, 2004-=-). In the AHP, a decision maker provides his or her relative preferences for the various alternatives by way of a series of pairwise comparisons. From these judgments, a matrix of pairwise comparisons... |

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Citation Context ... one to make inferential statements on the elements of the priority vector. As such, the development of probabilistic or statistical formulations of the AHP has attracted considerable interest (e.g., =-=Arbel, 1989-=-; Basak, 1991, 1998; Crawford and Williams, 1985; de Jong, 1984; Genest and Rivest, 1994; Haines, 1998; Jensen, 1984; Lipovetsky and Tishler, 1999; Ramanathan, 1997; Saaty and Vargas, 1987; Stam and D... |

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Citation Context ... be instances where the choice of link function impacts the analysis. Indeed, the performance of a categorical data model is known to depend at least in part on the choice of the link function (e.g., =-=Pregibon, 1980-=-). Aside from the estimation considerations, the development of a probit model in the context of stochastic multicriteria decision methods would be interesting as in the literature several authors hav... |

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Citation Context ...nts on the elements of the priority vector. As such, the development of probabilistic or statistical formulations of the AHP has attracted considerable interest (e.g., Arbel, 1989; Basak, 1991, 1998; =-=Crawford and Williams, 1985-=-; de Jong, 1984; Genest and Rivest, 1994; Haines, 1998; Jensen, 1984; Lipovetsky and Tishler, 1999; Ramanathan, 1997; Saaty and Vargas, 1987; Stam and Duarte Silva, 1997). Some authors have obtained s... |

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Citation Context ...31 σ32 σ 2 3 The probability that, say, the first alternative is chosen can be written as Pr(u1 > u2, u1 > u3). The dimensionality of the problem can be reduced from J to J − 1 by differencing (e.g., =-=Alvarez and Nagler, 1995-=-). We may therefore write the probability of the first alternative being chosen as Pr(η1 − η2 > ɛ2 − ɛ1, η1 − η3 > ɛ3 − ɛ1). Let ¯η1j = η1 − ηj and ¯ɛj1 = ɛj − ɛ1. The joint covariance matrix for the ... |

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Citation Context ...of probabilistic or statistical formulations of the AHP has attracted considerable interest (e.g., Arbel, 1989; Basak, 1991, 1998; Crawford and Williams, 1985; de Jong, 1984; Genest and Rivest, 1994; =-=Haines, 1998-=-; Jensen, 1984; Lipovetsky and Tishler, 1999; Ramanathan, 1997; Saaty and Vargas, 1987; Stam and Duarte Silva, 1997). Some authors have obtained stochastic judgments by eliciting intervals for pairwis... |

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(Show Context)
Citation Context ...ic or statistical formulations of the AHP has attracted considerable interest (e.g., Arbel, 1989; Basak, 1991, 1998; Crawford and Williams, 1985; de Jong, 1984; Genest and Rivest, 1994; Haines, 1998; =-=Jensen, 1984-=-; Lipovetsky and Tishler, 1999; Ramanathan, 1997; Saaty and Vargas, 1987; Stam and Duarte Silva, 1997). Some authors have obtained stochastic judgments by eliciting intervals for pairwise comparisons ... |

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(Show Context)
Citation Context ... Vargas, 1987; Stam and Duarte Silva, 1997). Some authors have obtained stochastic judgments by eliciting intervals for pairwise comparisons as opposed to point values (e.g., Bryson and Joseph, 2000; =-=Sugihara, Ishii, and Tanaka, 2004-=-). While useful, these methods require additional judgments from the decision maker and thereby lack the efficiency of the original formulation of the AHP. Others have obtained a probabilistic formula... |

5 |
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(Show Context)
Citation Context ...ly a stochastic multicriteria decision making method has been developed that permits inference on the elements of the priority vector without requiring interval judgments or multiple decision makers (=-=Hahn, 2003-=-). Moreover, the approach easily allows for dependence among pairwise comparison judgments. This stochastic judgment method utilizes the pairwise comparison matrices of the original AHP and thus probl... |

5 |
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(Show Context)
Citation Context ...ive AHP can also be seen to make use of a link or mapping function. The multiplicative AHP appears to have originally arisen from the line-fitting tradition of minimum least squares estimation (e.g., =-=Lootsma, 1996-=-), thus its stochastic characteristics were originally less explicitly presented. However, these considerations have been made more explicit over time (Van den Honert, 1998; Laininen and Hämäläinen, 2... |

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Citation Context ...ation (e.g., Lootsma, 1996), thus its stochastic characteristics were originally less explicitly presented. However, these considerations have been made more explicit over time (Van den Honert, 1998; =-=Laininen and Hämäläinen, 2003-=-). By contrast the method here has been explicitly derived from a stochastic characterization of the pairwise comparison as the ratio of two kinds of preference outcomes. The pairwise comparison is th... |

4 |
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Citation Context ...ochastic judgment method utilizes the pairwise comparison matrices of the original AHP and thus problems that have been examined with AHP may be examined without modification using this method (e.g., =-=Phillips-Wren, Hahn, and Forgionne, 2004-=-). The value-add of such a two-phase examination is that the latter method allows individuals to make inferential statements regarding the priorities. This can be a powerful tool for decision makers. ... |

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2 |
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(Show Context)
Citation Context ...s attracted considerable interest (e.g., Arbel, 1989; Basak, 1991, 1998; Crawford and Williams, 1985; de Jong, 1984; Genest and Rivest, 1994; Haines, 1998; Jensen, 1984; Lipovetsky and Tishler, 1999; =-=Ramanathan, 1997-=-; Saaty and Vargas, 1987; Stam and Duarte Silva, 1997). Some authors have obtained stochastic judgments by eliciting intervals for pairwise comparisons as opposed to point values (e.g., Bryson and Jos... |

1 |
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(Show Context)
Citation Context ...inferential statements on the elements of the priority vector. As such, the development of probabilistic or statistical formulations of the AHP has attracted considerable interest (e.g., Arbel, 1989; =-=Basak, 1991-=-, 1998; Crawford and Williams, 1985; de Jong, 1984; Genest and Rivest, 1994; Haines, 1998; Jensen, 1984; Lipovetsky and Tishler, 1999; Ramanathan, 1997; Saaty and Vargas, 1987; Stam and Duarte Silva, ... |

1 |
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Citation Context ...e the logged pairwise comparison ratio is taken to be normal. Indeed, the log-normal conceptualization of the pairwise comparison has appeared with some regularity (e.g., Crawford and Williams, 1985; =-=Escobar and Moreno-Jiménez, 2000-=-). Moreover, the multivariate normality of the normalized principal eigenvectors in AHP has been assessed by Stam and Duarte Silva (1997). In the previous section, the case of binary response models w... |

1 |
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(Show Context)
Citation Context ...cal formulations of the AHP has attracted considerable interest (e.g., Arbel, 1989; Basak, 1991, 1998; Crawford and Williams, 1985; de Jong, 1984; Genest and Rivest, 1994; Haines, 1998; Jensen, 1984; =-=Lipovetsky and Tishler, 1999-=-; Ramanathan, 1997; Saaty and Vargas, 1987; Stam and Duarte Silva, 1997). Some authors have obtained stochastic judgments by eliciting intervals for pairwise comparisons as opposed to point values (e.... |

1 |
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(Show Context)
Citation Context ...itional on the priorities. In (9)-(11) the differenced linear predictors are used to obtain the priorities by way of the bivariate normal link function. It can be shown (e.g., Hausman and Wise, 1978; =-=Terza, 1998-=-) that in an independent multinomial probit model Ω after standardization has entries of 1 on the diagonal and 1/2 elsewhere. Note that the off-diagonal elements of 1/2 result from the differencing. M... |

1 | Stochastic group preference modelling in the multiplicative AHP: A model of group consensus - Honert - 1998 |