Multiple Comparison of Several Linear Regression Models

Liu, Wei; Jamshidian, Mortaza; Zhang, Ying

doi:10.1198/016214504000000395

Cited by 70 publications

(67 citation statements)

References 2 publications

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“…Finally, for the comparisons of more than two linear regression models, Spurrier (1999Spurrier ( , 2002, Liu, et al (2004), and Bhargava and Spurrier (2004) proposed various confidence bands. Note however the partial F test for this case does not have a naturally associated set of confidence bands, and so the comparison of more than two linear regression models is characteristically different from the comparison of two linear regression models.…”

Section: Discussionmentioning

confidence: 99%

“…In this appendix, we first provide a geometric representation of T in (3.2) which is central to the method of simulating T in Liu et al (2004). It is then used to show that the partial F test can actually be deduced from the confidence band (2.3).…”

Section: Appendixmentioning

confidence: 99%

“…So the critical value c can be determined if the distribution of T can be found. While it is clearly difficult to derive a closed form formula for the distribution of T in this general setting, the method of Liu, Jamshidian and Zhang (2004) can be used to efficiently simulate a T . So one can simulate a sufficiently large number, R say, replicates of the random variable T , and set the [(1 − α)R]th largest simulated value as the critical constant c. A geometrical representation of T , which is central to the simulation of T and also relevant to establishing some assertions of Section 2, is given in the Appendix.…”

Section: Confidence Bands Over a Restricted Regionmentioning

confidence: 99%

“…In order to find Q one only needs to find the smallest angle between either Z and v or −Z and v while v varies in the cone V . Liu, et al (2004) gave an efficient method for the computation of Q.…”

Section: Appendixmentioning

confidence: 99%

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Some new methods for the comparison of two linear regression models

Liu

Jamshidian

Zhang

et al. 2007

Journal of Statistical Planning and Inference

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The frequently used approach to the comparison of two linear regression models is to use the partial F test. It is pointed out in this paper that the partial F test has in fact a naturally associated two-sided simultaneous confidence band, which is much more informative than the test itself. But this confidence band is over the entire range of all the covariates. As regression models are true or of interest often only over a restricted region of the covariates, the part of this confidence band outside this region is therefore useless and to ensure 1 -α simultaneous coverage probability is therefore wasteful of resources. It is proposed that a narrower and hence more efficient confidence band over a restricted region of the covariates should be used. The critical constant required in the construction of this confidence band can be calculated by Monte Carlo simulation. While this two-sided confidence band is suitable for two-sided comparisons of two linear regression models, a more efficient one-sided confidence band can be constructed in a similar way if one is only interested in assessing whether the mean response of one regression model is higher (or lower) 1 Mortaza Jamshidian's research has been supported in part by the National Science Foundation Grant DMS-0437258. AbstractThe frequently used approach to the comparison of two linear regression models is to use the partial F test. It is pointed out in this paper that the partial F test has in fact a naturally associated two-sided simultaneous confidence band, which is much more informative than the test itself. But this confidence band is over the entire range of all the covariates. As regression models are true or of interest often only over a restricted region of the covariates, the part of this confidence band outside this region is therefore useless and to ensure 1 − α simultaneous coverage probability is therefore wasteful of resources. It is proposed that a narrower and hence more efficient confidence band over a restricted region of the covariates should be used. The critical constant required in the construction of this confidence band can be calculated by Monte Carlo simulation. While this two-sided confidence band is suitable for two-sided comparisons of two linear regression models, a more efficient one-sided confidence band can be constructed in a similar way if one is only interested in assessing whether the mean response of one regression model is higher (or lower) than that of the other in the region. The methodologies are illustrated with two examples.

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Section: Discussionmentioning

confidence: 99%

Section: Appendixmentioning

confidence: 99%

Section: Confidence Bands Over a Restricted Regionmentioning

confidence: 99%

Section: Appendixmentioning

confidence: 99%

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Some new methods for the comparison of two linear regression models

Liu

Jamshidian

Zhang

et al. 2007

Journal of Statistical Planning and Inference

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“…Serfling, 1980). This approach of using sample quantile to approximate the population quantile has been used successfully in solving many otherwise difficult problems; see, for example, Edwards and Berry (1987) and Liu et al (2004Liu et al ( , 2005Liu et al ( , 2010.…”

Section: Exact Sti'smentioning

confidence: 99%

Statistical calibration and exact one-sided simultaneous tolerance intervals for polynomial regression

Han

Liu

Bretz

et al. 2016

Journal of Statistical Planning and Inference

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Please cite this article as: Han, Y., Liu, W., Bretz, F., Wan, F., Yang, P., Statistical calibration and exact one-sided simultaneous tolerance intervals for polynomial regression. J. Statist. Plann. Inference (2015), http://dx.doi.org/10.1016/j.jspi. 2015.07.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Exact Multiple Comparisons of Three or More Regression Lines: Pairwise Comparisons and Comparisons with a Control

Spurrier

2002

Biom. J.

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The problem of finding exact simultaneous confidence bounds for differences in regression models for k groups via the union‐intersection method is considered. The error terms are taken to be iid normal random variables. Under an assumption slightly more general than having identical design matrices for each of the k groups, it is shown that an existing probability point for the multivariate studentized range can be used to find the necessary probability point for pairwise comparisons of regression models. The resulting methods can be used with simple or multiple regression. Under a weaker assumption on the k design matrices that allows more observations to be taken from the control group than from the k‐1 treatment groups, a method is developed for computing exact probability points for comparing the simple linear regression models of the k‐1 groups to that of the control. Within a class of designs, the optimal design for comparisons with a control takes the square root of (k‐1) times as many observations from the control than from each treatment group. The simultaneous confidence bounds for all pairwise differences and for comparisons with a control are much narrower than Spurrier's intervals for all contrasts of k regression lines.

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Multiple Comparison of Several Linear Regression Models

Cited by 70 publications

References 2 publications

Some new methods for the comparison of two linear regression models

Some new methods for the comparison of two linear regression models

Statistical calibration and exact one-sided simultaneous tolerance intervals for polynomial regression

Exact Multiple Comparisons of Three or More Regression Lines: Pairwise Comparisons and Comparisons with a Control

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