Pair Difference (PD)

The Pair Difference in health between two groups is the simple arithmetic difference.  It is calculated as:

$${PD} = \mu_{k} - \mu_{l}$$

where \(\mu_{k}\) and \(\mu_{l}\) are health the average health status in two chosen social groups.  In this case \(\mu_{l}\) serves as the reference group.  \({PD}\) is similar to the Range Difference (RD) but the user selects the reference group and the comparison group.

Point Estimator of \(\widehat{PD}\)

A survey design consistent estimator is

$$\widehat{PD} = \mu_{k} - \mu_{l}$$

where

$$\widehat\mu_{j}=\frac{{\sum}_{h=1}^{H}{\sum}_{\alpha=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}{\delta}^{j}_{h\alpha{i}}{w}_{h\alpha{i}}{y}_{h\alpha{i}}}{{\sum}_{h=1}^{H}{\sum}_{\alpha=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}{\delta}^{j}_{h\alpha{i}}{w}_{h\alpha{i}}}for\epsilon\left(k,l\right)$$

Variance and Confidence Intervals of \(\widehat{PD}\)

The variance for \(\widehat{PD}\) is:

where \(Z_{h\alpha}={\sum}_{i=1}^{n_{h\alpha}}w_{h\alpha{i}}z_{h\alpha{i}},\) \(\overline{Z}_{h}=\frac{{1}}{t_{h}}{\sum}_{\alpha=1}^{t_{h}}Z_{h\alpha},\) and

$${z}_{h\alpha{i}}=\frac{{\partial}\widehat{RD}}{{\partial}{w}_{h\alpha{i}}}={\sum}_{j\epsilon(k,l)}\frac{{\partial}\widehat{RD}}{\partial\widehat{\mu}_{j}}\times\frac{{\partial}\widehat{\mu}_{j}}{\partial{w}_{h\alpha{i}}},$$

with

$$\left(\frac{{\partial}\widehat{PD}}{{\partial}\widehat{\mu}_{k}}, \frac{{\partial}\widehat{PD}}{{\partial}\widehat{\mu}_{l}}\right)=\left(1,-1\right)$$

$$\frac{{\partial}\widehat{\mu}_{j}}{{\partial}{w}_{h\alpha{i}}}=\frac{{\partial}_{h\alpha{i}}^{j}\left({y}_{h\alpha{i}}-\widehat{\mu}_{j}\right)}{{\sum}_{h=1}^{H}{\sum}_{a=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}{\partial}_{h\alpha{i}}{w_{h\alpha{i}}}}forj\epsilon\left(k,l\right).$$

The 95% confidence intervals of \(\widehat{PD}\) are:

$$\widehat{PD}\pm1.96\times\sqrt{\mathrm{var}(\widehat{PD})}.$$