Variance estimation methods for health expectancy by relative socio-economic status

Abstract

In many studies, health expectancies (HE) by relative socio-economic status have been calculated but the estimation of confidence intervals and the performance of tests of significance for differences in HE between sub-populations have been impeded by lack of variance estimation methods. Also in most scenarios, the sampling designs of the surveys from which prevalence of ill-health conditions are obtained have been ignored. This paper aims at presenting variance estimation techniques such as the bootstrap and the delta method taking account of the survey design. The study suggests that using the raw survey data and the Delta method while accounting for the survey design, gives better estimates for the variance compared to the bootstrap method and therefore is a highly recommended method for variance estimation of HE by relative socio-economic status.

Appendix A: Estimation of variance of prevalence of ill health conditions

The following logistic regression model was fit to the data using the prevalence (π) of ill-health conditions as the response variable:

$$ \hbox{logit}(\pi )=\beta _0 +\sum\limits_{j=1}^N {{\beta _j}{\ast} x_j} $$

(4)

where β_j, j =  0,1,2, ...,N represent parameters to be estimated from the data and x _j , j = 1,2,3,,N represent the various predictor variables that can be incorporated into the model. These variables are the relative position on the socio-economic scale, age, sex and other variables depending on the research question.

The fitted model obtained from (4) is used to predict the prevalence of those at the top and the bottom of the socio-economic scale for various combinations of the predictor variables; e.g. for males between 25 and 34 years. Using the predicted prevalence and (1), the HE of persons at the top and bottom of the socio-economic scale is obtained.

The variance of the HE is obtained by first determining the variance of the predicted prevalence. The fitted model from (4) provides estimates for the variance (v) of the log odds (θ): Thus suppose $\hat{\theta}=\hbox{log}(\hat{\pi}/(1-\hat{\pi}))$ and $\hbox{var}(\hat{\theta})=v,$ then $\hat{\pi}=\hbox{exp} (\hat{\theta})/(1+\hbox{exp} (\hat{\theta}))$ and the variance of the predicted prevalence according to the Delta method [17] is given by;

$$ \hbox{Var}(\hat{\pi})= \left[\frac{\partial{\hbox{exp}(\hat{\theta})/(1+\hbox{exp} (\hat{\theta}))}}{\partial \hat{\theta}}\right]^{2} {\ast} v=[\hbox{exp} (\hat{\theta})/(1+\hbox{exp} (\hat{\theta}))^{2}]^{2} {\ast} v $$

(5)