title: Testing the product of slopes in related regressions.
abstract: A study was conducted of the relationships among neuroprotective factors and cytokines in brain tissue of mice at different ages that were examined on the effect of dietary restriction on protection after experimentally induced brain stroke. It was of interest to assess whether the cross-product of the slopes of pairs of variables vs. age was positive or negative. To accomplish this, the product of the slopes was estimated and tested to determine if it is significantly different from zero. Since the measurements are taken on the same animals, the models used must account for the non-independence of the measurements within animals. A number of approaches are illustrated. First a multivariate multiple regression model is employed. Since we are interested in a nonlinear function of the parameters (the product) the delta method is used to obtain the standard error of the estimate of the product. Second, a linear mixed-effects model is fit that allows for the specification of an appropriate correlation structure among repeated measurements. The delta method is again used to obtain the standard error. Finally, a non-linear mixed-effects approach is taken to fit the linear-mixed-effects model and conduct the test. A simulation study investigates the properties of the procedure.
authors.affiliation: Mathematics and Statistics Department, Loyola University Maryland, 4501 North Charles St., Baltimore, MD 21210-2699 USA chm@loyola.edu (410)617-2629, ; Laboratory of Cardiovascular Sciences, National Institute on Aging, 5600 Nathan Shock Drive, Baltimore, MD 21224 USA.
authors.forename: Christopher H
authors.initials: CH
authors.lastname: Morrell
authors.affiliation: MedStar Research Institute, Hyattsville, MD USA.
authors.forename: Veena
authors.initials: V
authors.lastname: Shetty
authors.affiliation: Laboratory of Bioengineering & Physical Science, National Institute of Biomedical Imaging & Bioengineering, 9000 Rockville Pike, Bethesda, MD 20892 USA.
authors.forename: Terry
authors.initials: T
authors.lastname: Phillips
authors.affiliation: School of Biomedical Sciences, The University of Queensland, Australia.
authors.forename: Thiruma V
authors.initials: TV
authors.lastname: Arumugam
authors.affiliation: Laboratory of Neurosciences, National Institute on Aging, 251 Bayview Boulevard, Baltimore, MD 21224 USA.
authors.forename: Mark P
authors.initials: MP
authors.lastname: Mattson
authors.affiliation: Laboratory of Neurosciences, National Institute on Aging, 251 Bayview Boulevard, Baltimore, MD 21224 USA.
authors.forename: Ruiqian
authors.initials: R
authors.lastname: Wan
keywords: Multivariate multiple regression; delta method; linear mixed-effects model; non-linear mixed effects model; repeated measures