Between-Group Variance (BGV)

BGV is defined as the sum of squared differences of average health status in each social group from the average health status in the population weighted by the group population shares as: 

$${BGV} = {\sum}_{j=1}^Jp_j(\mu_j-\mu)^2,$$

where μ_j is the average health status in group j, p_j is the population share in group j, and μ is the average health status in the population.

BGV can be interpreted as the variance that would exist in the population if each individual had the mean health of their social group (i.e., no within-social group variation) (2).  BGV may be a useful indicator of absolute disparity for unordered group data because it weights by population group size and is sensitive to the magnitude of larger deviations from the population average (3).

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

A design consistent estimator is:

$$\widehat{BGV} = {\sum}_{j=1}^J\widehat{p}_{j}(\widehat{\mu}_{j}-\widehat{\mu})^2,$$

where

$$\widehat\mu_{j}=\frac{{\sum}_{h=1}^{H}{\sum}_{\alpha=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}{\delta}^{j}_{h\alpha{i}}{w}_{h\alpha{i}}{y}_{h\alpha{i}}}{{\sum}_{h=1}^{H}{\sum}_{h=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}{\delta}^{j}_{h\alpha{i}}{w}_{h\alpha{i}}}$$

is the estimated average health in group j,

$$\widehat{p}_{j}=\frac{{\sum}_{h=1}^{H}{\sum}_{\alpha=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}{\delta}^{j}_{h\alpha{i}}{w}_{h\alpha{i}}}{{\sum}_{h=1}^{H}{\sum}_{h=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}{w}_{h\alpha{i}}}$$

is the estimated population share of group \(j\), and \(\widehat{\mu}={\sum}_{j=1}^{J}\widehat{p}_{j}\widehat{\mu}_{j}\) is the estimated average health status in the population.

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

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

$$\mathrm{var}(\widehat{BGV})\cong{\sum}_{h=1}^H\frac{{t}_{h}}{{t}_{h}-1}{\sum}_{\alpha=1}^{t_{h}}(Z_{h\alpha}-\overline{Z}_{h})(Z_{h\alpha}-\overline{Z}_{h})^{T},$$

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

$${z}_{h\alpha{i}}=\frac{{\partial}\widehat{BGV}}{{\partial}{w}_{h\alpha{i}}}={\sum}_{j=1}^{J}\left(\frac{{\partial}\widehat{BGV}}{\partial\widehat{p}_{j}}\times\frac{{\partial}\widehat{p}_{j}}{\partial\widehat{w}_{h\alpha{i}}}+\frac{{\partial}\widehat{BGV}}{\partial\widehat{\mu}_{j}}\times\frac{{\partial}\widehat{\mu}_{j}}{\partial\widehat{w}_{h\alpha{i}}}\right)$$

with

$$\frac{{\partial}\widehat{BGV}}{{\partial}\widehat{p}_{j}}=\left(\widehat{\mu}_{j}-\widehat{\mu}\right)^2,$$

$$\frac{{\partial}\widehat{p}_{j}}{{\partial}{w}_{h\alpha{i}}}=\frac{{\delta}_{h\alpha{i}}^{j}-\widehat{p}_{j}}{{\sum}_{h=1}^{H}{\sum}_{a=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}{\partial}_{h\alpha{i}}{w}_{h\alpha{i}}},$$

$$\frac{{\partial}\widehat{BGV}}{{\partial}\widehat{\mu}_{j}}=2\widehat{p}_{j}\left(\widehat{\mu}_{j}-\widehat{\mu}\right),$$

and

$$\frac{{\partial}\widehat{\mu}_{j}}{{\partial}{w}_{h\alpha{i}}}=\frac{{\delta}_{h\alpha{i}}^{j}\left({y_{h\alpha{i}}}-\widehat{\mu_{j}}\right)}{{\sum}_{h=1}^{H}{\sum}_{a=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}{\partial}_{h\alpha{i}}{w_{h\alpha{i}}}}.$$

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

$$\widehat{BGV} \pm 1.96 \times \sqrt{\mathrm{var}(\widehat{BGV})}.$$

For details about the derivations, see Li et al (18).