Survival Instantaneous Log-odds Ratio from Empirical Functions

The objective of this work is to introduce a new method called the Survivorship Instantaneous Log-odds Ratios (SILOR); to illustrate the creation of SILOR from empirical bivariate survival functions; to also derive standard errors of estimation; to compare results with those derived from logistic regression. Hip fracture, AGE and BMI from the Third National Health and Nutritional Examination Survey (NHANES III) were used to calculate empirical survival functions for the adverse health outcome (AHO) and non-AHO. A stable copula was used to create a parametric bivariate survival function, that was fitted to the empirical bivariate survival function. The bivariate survival function had SILOR contours which are not constant. The proposed method has better advantages than logistic regression by following two reasons. The comparison deals with (i) the shapes of the survival surfaces, S(X1, X2), and (ii) the isobols of the log-odds ratios. When using logistic regression the survival surface is either a hyper plane or at most a conic section. Our approach preserves the shape of the survival surface in two dimensions, and the isobols are observed in every detail instead of being overly smoothed by a regression with no more than a second degree polynomial. The present method is straightforward, and it captures all but random variability of the data.