Variance and Confidence Intervals for Health Disparity Measures

HD*Calc calculates 13 summary measures of health disparity including both relative and absolute measures of disparities and pair comparison methods.  Both analytic methods (the Taylor series linearization method, the quadratic form approximation method, and the weighted least squares method) and Monte Carlo simulation (MCS) based method are used.  HD*Calc allows users to select either type of method to estimate variance and construct confidence intervals.

Let \(\theta\) denote a true health disparity measure estimated by \(\widehat{\theta}=F(Y)\) where \(Y=(y_{1},...,y_{j})\).  Let μ and σ ²  denote the mean and variance of Y. 

Variance of \(\widehat\theta\) based on Taylor Series Linearization Method

To estimate \(Var(\widehat\theta)\), we expand \(F\) using a first-order Taylor series around the mean μ, so

$$Var (\widehat\theta)\approx Var{\left\{{\sum}_{j=1}^{J}{\left(\frac{\partial F}{\partial y_{j}}\right)}_{y_{j}=\mu_{j}}{\left(Y_{j}-\mu_{j}\right)}\right\}}$$,

where the second-order derivative or higher-order terms are assumed to be negligible.  Assume each component is independent, then

$$Var (\widehat\theta)\approx {\sum}_{j=1}^{J}{\left(\frac{\partial F}{\partial y_{j}}\right)}_{y_{j}=\mu_{j}}^{2} \sigma_{j}^{2}$$.

Replacing \(\sigma_{j}^{2}\) with its unbiased estimator \(\hat\sigma_{j}^{2}\) we obtain an estimate of \(Var(\widehat\theta)\) as: 

$$var_{TL} (\widehat\theta)\approx {\sum}_{j=1}^{J}{\left(\frac{\partial F}{\partial y_{j}}\right)}_{y_{j}=\mu_{j}}^{2}\widehat{ \sigma}_{j}^{2}$$.

The corresponding 95% confidence interval for \(\widehat{\theta}\) is: 

$$\widehat{\theta}\pm 1.96\times \sqrt{var_{TL}(\widehat{\theta)}}$$

Variance of \(\widehat{\theta}\) Based on Monte Carlo Simulation (MCS) Method

Randomly generate M age-adjusted rates \(y_{j}^{(m)},m=1,...,M,\) using the distribution: \(y_{j}^{(m)}\sim Gamma(mean=y_{j},var=\widehat{\sigma}_{j}^{2})\) for each social group.  Then calculate \(\widehat{\theta}^{(m)}\) using \(y_{j}^{(m)}\). Thus:

$$var_{MCS}(\widehat{\theta})=(M-1)^{-1}{\sum}_{m=1}^{M} (\widehat{\theta}^{(m)}-\overline{\widehat{\theta}})^{2},$$

where \(\overline{\widehat{\theta}}=M^{-1}{\sum}_{m=1}^{M} \widehat{\theta}^{(m)}\).  \(M=1,000\) is used in HD*Calc. The standard error of \(\widehat{\theta}\) based on the MCS method is:  \(\sqrt {var_{MCS}(\widehat{\theta})}\).

Gamma distribution instead of truncated normal distribution was recommended to simulate age-adjusted rates (1).  The lower and upper bounds of the 95% confidence interval of \(\widehat{\theta}\) based on the MCS approach are the 2.5^th percentile and 97.5^th percentile of the \(1,000 \widehat{\theta}^{(m)}\) values.