### Abstract

Choosing the largest of several means can be a demanding problem, especially in the presence of a covariate. A hierarchical Bayesian approach to ranking and selection, as well as estimation of related means in the presence of a covariate, is considered. For the multiple slopes model we compute, in addition to the posterior means and standard deviations of the parameters, the posterior probabilities that each mean, at a given value of the covariate, is the largest. The vector of posterior probabilities thus obtained provides an easily understandable answer to the selection problem. Although calculation of the posterior probabilities may involve four-dimensional numerical integration in the difficult unbalanced design and unknown variance case, an efficient Monte Carlo method of evaluation has been developed and is given in the article. By reanalyzing a well-known data set on the breaking strength and thickness of starch films, we illustrate how our Bayesian approach produces meaningful conclusions, some of which would perhaps be difficult to obtain otherwise. For the starch film example, we found that it took only 1.4 seconds to compute the quantities of interest using an IBM 3090–600S machine. Because the computation time is quite small, it is apparent that the Bayesian procedure can be implemented for everyday use.

Original language | English (US) |
---|---|

Pages (from-to) | 1128-1136 |

Number of pages | 9 |

Journal | Journal of the American Statistical Association |

Volume | 87 |

Issue number | 420 |

DOIs | |

State | Published - Jan 1 1992 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

}

**Ranking and estimation of related means in the presence of a covariate—a Bayesian approach.** / Fong, Duncan.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Ranking and estimation of related means in the presence of a covariate—a Bayesian approach

AU - Fong, Duncan

PY - 1992/1/1

Y1 - 1992/1/1

N2 - Choosing the largest of several means can be a demanding problem, especially in the presence of a covariate. A hierarchical Bayesian approach to ranking and selection, as well as estimation of related means in the presence of a covariate, is considered. For the multiple slopes model we compute, in addition to the posterior means and standard deviations of the parameters, the posterior probabilities that each mean, at a given value of the covariate, is the largest. The vector of posterior probabilities thus obtained provides an easily understandable answer to the selection problem. Although calculation of the posterior probabilities may involve four-dimensional numerical integration in the difficult unbalanced design and unknown variance case, an efficient Monte Carlo method of evaluation has been developed and is given in the article. By reanalyzing a well-known data set on the breaking strength and thickness of starch films, we illustrate how our Bayesian approach produces meaningful conclusions, some of which would perhaps be difficult to obtain otherwise. For the starch film example, we found that it took only 1.4 seconds to compute the quantities of interest using an IBM 3090–600S machine. Because the computation time is quite small, it is apparent that the Bayesian procedure can be implemented for everyday use.

AB - Choosing the largest of several means can be a demanding problem, especially in the presence of a covariate. A hierarchical Bayesian approach to ranking and selection, as well as estimation of related means in the presence of a covariate, is considered. For the multiple slopes model we compute, in addition to the posterior means and standard deviations of the parameters, the posterior probabilities that each mean, at a given value of the covariate, is the largest. The vector of posterior probabilities thus obtained provides an easily understandable answer to the selection problem. Although calculation of the posterior probabilities may involve four-dimensional numerical integration in the difficult unbalanced design and unknown variance case, an efficient Monte Carlo method of evaluation has been developed and is given in the article. By reanalyzing a well-known data set on the breaking strength and thickness of starch films, we illustrate how our Bayesian approach produces meaningful conclusions, some of which would perhaps be difficult to obtain otherwise. For the starch film example, we found that it took only 1.4 seconds to compute the quantities of interest using an IBM 3090–600S machine. Because the computation time is quite small, it is apparent that the Bayesian procedure can be implemented for everyday use.

UR - http://www.scopus.com/inward/record.url?scp=21144468831&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21144468831&partnerID=8YFLogxK

U2 - 10.1080/01621459.1992.10476269

DO - 10.1080/01621459.1992.10476269

M3 - Article

AN - SCOPUS:21144468831

VL - 87

SP - 1128

EP - 1136

JO - Journal of the American Statistical Association

JF - Journal of the American Statistical Association

SN - 0162-1459

IS - 420

ER -