\( \newcommand{\logit}{\mathrm{logit}} \newcommand{\alogit}{\logit^{-1}} \newcommand{\sfrac}{l} \newcommand{\tfrac}{Y} \newcommand{\osfrac}{\tilde{\sfrac}} \newcommand{\ssfrac}{\hat{\sfrac}} \newcommand{\otfrac}{\tilde{\tfrac}} \newcommand{\stfrac}{\hat{\tfrac}} \newcommand{\err}{\epsilon} \newcommand{\nobs}{n} \newcommand{\smorthandle}{\texttt{"}\textit{SMortHandle}\texttt{"}} \)

Brass

This build method is an implementation of Brass' logit relational model. Given a set of observed survival fractions \(\osfrac_i\) at ages \(x_i\) , \(i=0,\ldots,\nobs-1\) and a standard curve \(\ssfrac\) calculate logits \begin{equation} \otfrac_i=\logit(\osfrac_i),\quad \stfrac_i=\logit(\ssfrac_i) \end{equation} and solve the linear regression \begin{equation} \otfrac_i = \alpha + \beta\,\stfrac_i+ \err_i \end{equation} for \(\alpha\) and \(\beta\). The smoothed survival fraction at any age \(x\) is then \begin{equation} \sfrac_x=\alogit(\tfrac_x) \end{equation} where \begin{equation} \tfrac_x = \alpha + \beta\,\stfrac_x . \end{equation} If the mortality curve is given in terms of death rates or probabilities then they are converted to survival fractions using the CONSTANT_FORCE build method before applying BRASS. If the curve is specified in terms of death rates, probabilities or survival fractions then the build method property must be of the form BRASS:SMortHandle where SMortHandle is the name of the standard mortality object enclosed in quotes. If \(\alpha\) and \(\beta\) are already known then the curve can be defined using a Parameters subtable with columns Alpha, Beta and Standard.