Testing the product of slopes in related regressions

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.