The Delta Method in Arc

This paper describes an Arc add-in for the delta method for regression models. The method works for linear models, generalized linear models, nonlinear models and also for generalized nonlinear models. The delta method is a procedure for finding estimates and approximate standard errors of arbitrary functions of normally distributed random variables. As implemented in Arc, the delta method is used to find estimates and standard errors of nonlinear functions of parameter estimates. 1 Getting the delta method add-on Download the file http://www.stat.umn.edu/arc/lindelta.lsp. Place it in the Extras directory in your Arc directory (if you don’t have such a directory, create one). The file will be automatically loaded every time you start Arc. An additional item called “Delta method” will be added to regression menus, and this item is used to access the delta method code. 2 Usage As an example of the usage of the delta method, consider the Lake Mary data discussed in Cook and Weisberg (1999, Chapter 1), relating Length of fish to Age. It is suggested in the text that a reasonable approximation to the mean length at age over the range of ages in the data is a quadratic curve, E Length Age Age Age An important characteristic of the quadratic curve is the age at which the maximum length is attained (at least according to the quadratic fit). We can find the value of Age that maximizes Length by differentiating the mean function (2) with respect to Age, and set the result to zero. Straightforward calculus, given later in this report, gives AgeMax as the Age with maximum expected length. To estimate AgeMax using Arc, first fit the quadratic regression, given by