TY - JOUR

T1 - Bayes factors for testing order constraints on variances of dependent outcomes

AU - Böing-Messing, F.

AU - Mulder, J.

N1 - This research was partly supported by a Rubicon grant which was awarded to the second author by the Netherlands Organization for Scientific Research (NWO).

PY - 2021

Y1 - 2021

N2 - In statistical practice, researchers commonly focus on patterns in the means of multiple dependent outcomes while treating variances as nuisance parameters. However, in fact, there are often substantive reasons to expect certain patterns in the variances of dependent outcomes as well. For example, in a repeated measures study, one may expect the variance of the outcome to increase over time if the difference between subjects becomes more pronounced over time because the subjects respond differently to a given treatment. Such expectations can be formulated as order constrained hypotheses on the variances of the dependent outcomes. Currently, however, no methods exist for testing such hypotheses in a direct manner. To fill this gap, we develop a Bayes factor for this challenging testing problem. Our Bayes factor is based on the multivariate normal distribution with an unstructured covariance matrix, which is often used to model dependent outcomes. Order constrained hypotheses can then be formulated on the variances on the diagonal of the covariance matrix. To compute Bayes factors between multiple order constrained hypotheses, a prior distribution needs to be specified under every hypothesis to be tested. Here, we use the encompassing prior approach in which priors under order constrained hypotheses are truncations of the prior under the unconstrained hypothesis. The resulting Bayes factor is fully automatic in the sense that no subjective priors need to be specified by the user.

AB - In statistical practice, researchers commonly focus on patterns in the means of multiple dependent outcomes while treating variances as nuisance parameters. However, in fact, there are often substantive reasons to expect certain patterns in the variances of dependent outcomes as well. For example, in a repeated measures study, one may expect the variance of the outcome to increase over time if the difference between subjects becomes more pronounced over time because the subjects respond differently to a given treatment. Such expectations can be formulated as order constrained hypotheses on the variances of the dependent outcomes. Currently, however, no methods exist for testing such hypotheses in a direct manner. To fill this gap, we develop a Bayes factor for this challenging testing problem. Our Bayes factor is based on the multivariate normal distribution with an unstructured covariance matrix, which is often used to model dependent outcomes. Order constrained hypotheses can then be formulated on the variances on the diagonal of the covariance matrix. To compute Bayes factors between multiple order constrained hypotheses, a prior distribution needs to be specified under every hypothesis to be tested. Here, we use the encompassing prior approach in which priors under order constrained hypotheses are truncations of the prior under the unconstrained hypothesis. The resulting Bayes factor is fully automatic in the sense that no subjective priors need to be specified by the user.

KW - EQUALITY

KW - Encompassing prior

KW - HYPOTHESES

KW - Heteroscedasticity

KW - INEQUALITY

KW - Inequality constraint

KW - Mixed model

KW - Multivariate regression

KW - Repeated measures

KW - SAMPLE

UR - https://app-eu.readspeaker.com/cgi-bin/rsent?customerid=10118&lang=en_us&readclass=rs_readArea&url=https%3A%2F%2Fwww.tandfonline.com%2Fdoi%2Ffull%2F10.1080%2F00031305.2020.1715257

UR - http://www.scopus.com/inward/record.url?scp=85079428221&partnerID=8YFLogxK

U2 - 10.1080/00031305.2020.1715257

DO - 10.1080/00031305.2020.1715257

M3 - Article

SN - 0003-1305

VL - 75

SP - 152

EP - 161

JO - The American Statistician

JF - The American Statistician

IS - 2

ER -