The cost of frailty in France

The health index combines several measures by means of a data reduction statistical technique (multiple correspondence analysis). The maths behind this synthetic index make it bear the same properties of variance decomposition in the regression model as the set of variables it is made of. In other words, this health index has the advantage of being mathematically equivalent to a set of health measures, without the disadvantage of a larger number of variables could have on multicollinearity. The health index thus does not modify the coefficient estimates in the regression when substituted to a large set of health measures, and provides more accurate standard errors. Because it is a data reduction process, there is no need for external validity of the health index.

GLM offer great flexibility in the choice of the functional form and the variance distribution. Basu and Rathouz [26] suggest the use of extended estimating equations (EEE) in order to estimate the appropriate link and variance functions. EEE rests on a flexible semi-parametric extension of the GLM model using power function (Box-Cox transformation) for the link function. As such, EEE are also referred to as Power-GLM or PGLM. Notice that the log-link function is a special case where the link parameter (i.e., the denominator of the power-function) is zero. Two alternative forms of variance distribution, power or quadratic, are also considered. EEE simultaneously estimates the link and variance parameters from the data along with the regression coefficients. Not only EEE can be used directly to estimate a model for health care costs they can also be used to select the appropriate link and variance functions for GLM. Comparison between PGLM and corresponding GLM based on standard goodness-of-fit tests provides interesting sensitivity analysis. Our results (not displayed here but available upon request) suggest that the value of link parameter λ = 0.205 is significantly equal to zero (p = 0.646) so the link function should be logarithmic, and estimates of variance parameters θ ₁ and θ ₂ are statistically equal to 1 (p = 0.602) and 2 (p = 0.369), respectively. This eventually suggests that the appropriate model should be close to our initial choice of GLM with Log-Gamma specification. Notice that a modified Park test on Model 4B strongly supports the choice of Gamma variance (p = 0.769).