Beyond Burden: A Health Capability Gap Metric as the Foundation for a GBD+ Framework

Appendix A.1. Conceptual Accounting Framework

The Health Capability Gap, denoted $HCG$, is defined as the aggregate life-course shortfall between full health capability and realised health capability. It comprises three components: complete health capability exclusion, partial health capability loss, and health resource constraint loss. In the paper’s conceptual sketch, these correspond respectively to $YLL\left(K\right)$, $YLD\left(K\right)$, and $YLS\left(K\right)$.

$$HCG=YLL\left(K\right)+YLD\left(K\right)+YLS\left(K\right)$$

where:

$$YLL\left(K\right)$$

denotes years of life lost interpreted as complete health capability exclusion due to premature mortality;

$$YLD\left(K\right)$$

denotes years lived with disability interpreted as partial health capability loss due to morbidity, disability, or functional limitation;

and:

$$YLS\left(K\right)$$

denotes years of health capability lost due to structural constraints, including deficits in the social, economic, institutional, and environmental conditions required to realise health capability.

The standard DALY identity is:

$$DALY=YLL+YLD$$

In the HCG framework:

$$YLL\left(K\right)\equiv YLL$$

and:

$$YLD\left(K\right)\equiv YLD$$

Therefore:

$$DALY=YLL\left(K\right)+YLD\left(K\right)$$

and:

$$HCG=DALY+YLS\left(K\right)$$

provided that $YLS\left(K\right)$is estimated net of structural deprivation already expressed through morbidity or mortality outcomes captured in DALYs.

Appendix A.2. Estimating YLL (K)

Let:

$$d_{a,s,c,t}$$

denote deaths in age group $a$, sex $s$, cause $c$, and time period $t$, and let:

$$L_a$$

denote the standard remaining life expectancy at age $a$.

Then:

$$YLL(K{)}_{a,s,c,t}=d_{a,s,c,t}{L}_a$$

and:

$$YLL\left(K\right)=\sum _a\sum _s\sum _c\sum _t{d}_{a,s,c,t}{L}_a$$

This is numerically identical to standard $YLL$. The difference is interpretive: in the HCG framework, $YLL\left(K\right)$measures the complete loss of the opportunity to exercise health-related freedoms over the foregone life course.

Appendix A.3. Estimating YLD (K)

Let:

$$P{Y}_{a,s,r,t}$$

denote person-years lived in age group $a$, sex $s$, place $r$, and time period $t$.

Let:

$$p_{a,s,r,t,j}$$

denote the prevalence of health state or sequela $j$in that population stratum.

Let:

$$D{W}_j$$

denote the disability weight for health state $j$, where:

$$0\le D{W}_j\le 1$$

Then, for a single health state:

$$YLD(K{)}_{a,s,r,t,j}=P{Y}_{a,s,r,t}{p}_{a,s,r,t,j}D{W}_j$$

Aggregating over population strata and health states:

$$YLD\left(K\right)=\sum _a\sum _s\sum _r\sum _t\sum _{j}P{Y}_{a,s,r,t}{p}_{a,s,r,t,j}D{W}_j$$

where individual-level data are available, let $H_{i,j}$denote whether individual $i$experiences health state $j$:

$$H_{i,j}\in \left\{\mathrm{0,1}\right\}$$

For multiple overlapping health states, the combined disability weight may be estimated using the multiplicative function:

$$D{W}_i^{*}=1-{\displaystyle \prod _{j=1}^J}(1-H_{i,j}D{W}_j)$$

Then:

$$YLD\left(K\right)=\sum _{i}P{Y}_{i}D{W}_i^{*}$$

This is numerically equivalent to standard $YLD$, but it is interpreted as partial health capability loss.

Appendix A.4. Estimating Raw Structural Capability Loss, $\mathit{Y}\mathit{L}\mathit{S}(\mathit{K}{)}^{\mathit{r}\mathit{a}\mathit{w}}$

Let $m=1,\dots ,M$index structural deprivation states. These may include lack of essential healthcare access, food insecurity, unsafe water or sanitation, inadequate housing, climate vulnerability, displacement, or environmental insecurity. The manuscript treats these as structural capability-deprivation states rather than disease sequelae or ordinary GBD risk factors.

Let:

$$E_{a,s,r,t,m}$$

denote exposure to deprivation state $m$in age group $a$, sex $s$, place $r$, and time period $t$. This may be represented either as an individual-level indicator:

$$E_{i,m}\in \left\{\mathrm{0,1}\right\}$$

or as a group-level prevalence:

$$0\le E_{a,s,r,t,m}\le 1$$

Let:

$${\omega }_m$$

denote the Health Resource Constraint disutility weight for deprivation state $m$, where:

$$0\le {\omega }_m\le 1$$

The weight ${\omega }_m$represents the average proportional reduction in health capability associated with structural deprivation $m$, before accounting for overlap with other structural constraints or with disease outcomes already captured in DALYs. These weights are analogous to disability weights, but they measure capability loss from structural deprivation rather than health loss from disease or injury.

For a single structural deprivation state:

$$YLS(K{)}_{a,s,r,t,m}^{raw}=P{Y}_{a,s,r,t}{E}_{a,s,r,t,m}{\omega }_m$$

For multiple overlapping deprivation states, define the combined raw structural disutility weight as:

$${\omega }_{a,s,r,t}^{raw}=1-{\displaystyle \prod _{m=1}^M}\left(1-E_{a,s,r,t,m}{\omega }_m\right)$$

This multiplicative form ensures that the combined structural disutility remains bounded between 0 and 1, provided that each $E_{a,s,r,t,m}{\omega }_m$lies between 0 and 1.

The raw structural capability loss is therefore:

$$YLS(K{)}^{raw}=\sum _a\sum _s\sum _r\sum _{t}P{Y}_{a,s,r,t}{\omega }_{a,s,r,t}^{raw}$$

Appendix A.5. Adjustment for Double-Counting

Structural deprivation may reduce health capability through two pathways.

First, it may directly reduce the real freedom to achieve health before or apart from any observed disease event:

$$\mathrm{Structural}\mathrm{deprivation}\to YLS\left(K\right)$$

Second, it may indirectly produce morbidity or mortality already captured in DALYs:

$$\mathrm{Structural}\mathrm{deprivation}\to \mathrm{disease}\mathrm{or}\mathrm{death}\to YLD\left(K\right)+YLL\left(K\right)$$

The paper’s limitations section explicitly notes that simply summing $YLS\left(K\right)$and DALYs may double-count the same deprivation, because part of the structural constraint may already be reflected in $YLL\left(K\right)$or $YLD\left(K\right)$. It therefore requires the HRC component to be measured net of outcome effects already counted in DALYs.

Let:

$${\theta }_{a,s,r,t}$$

denote the DALY-mediated share of raw structural capability loss in stratum $\left(a,s,r,t\right)$. More precisely:

$${\theta }_{a,s,r,t}={\displaystyle \frac{\mathrm{raw}\mathrm{structural}\mathrm{capability}\mathrm{loss}\mathrm{already}\mathrm{expressed}\mathrm{through}\mathrm{DALY}-\mathrm{counted}\mathrm{morbidity}\mathrm{or}\mathrm{mortality}}{\mathrm{total}\mathrm{raw}\mathrm{structural}\mathrm{capability}\mathrm{loss}}}$$

with:

$$0\le {\theta }_{a,s,r,t}\le 1$$

Then the double-counting-adjusted structural disutility weight is:

$${\tilde{\omega }}_{a,s,r,t}=(1-{\theta }_{a,s,r,t}){\omega }_{a,s,r,t}^{raw}$$

and the adjusted structural capability loss is:

$$YLS\left(K\right)=\sum _a\sum _s\sum _r\sum _{t}P{Y}_{a,s,r,t}{\tilde{\omega }}_{a,s,r,t}$$

or:

$$YLS\left(K\right)=\sum _a\sum _s\sum _r\sum _{t}P{Y}_{a,s,r,t}(1-{\theta }_{a,s,r,t}){\omega }_{a,s,r,t}^{raw}$$

This adjustment preserves the interpretation of $YLS\left(K\right)$as the component of structural capability loss not already realised as morbidity or mortality.

Where empirical data permit, ${\theta }_{a,s,r,t}$may be estimated using mediation analysis, multistate life-table methods, sequential g-estimation, inverse probability weighting, or related causal methods designed to separate the direct capability loss of deprivation from disease pathways already counted in DALYs. These approaches are consistent with the manuscript’s proposed treatment of the double-counting problem.

Appendix A.6. Health Capability Gap After Double-Counting Adjustment

Using the adjusted structural component:

$$HCG=YLL\left(K\right)+YLD\left(K\right)+YLS\left(K\right)$$

Since:

$$DALY=YLL\left(K\right)+YLD\left(K\right)$$

it follows that:

$$HCG=DALY+YLS\left(K\right)$$

where:

$$YLS\left(K\right)=\sum _a\sum _s\sum _r\sum _{t}P{Y}_{a,s,r,t}(1-{\theta }_{a,s,r,t}){\omega }_{a,s,r,t}^{raw}$$

Thus:

$$HCG=DALY+\sum _a\sum _s\sum _r\sum _{t}P{Y}_{a,s,r,t}(1-{\theta }_{a,s,r,t}){\omega }_{a,s,r,t}^{raw}$$

If:

$$YLS\left(K\right)=0$$

then:

$$HCG=DALY$$

If:

$$YLS\left(K\right)>0$$

then:

$$HCG>DALY$$

The latter condition applies when structural capability losses remain after excluding the portion already realised through DALY-counted morbidity or mortality.

Appendix A.7. Health-Adjusted Life Expectancy

Let:

$$l_0$$

denote the life-table radix, and let:

$$L_a$$

denote life-table person-years lived in age interval $a$.

Let:

$$D{W}_a$$

denote the average disability weight or average morbidity-related health loss in age interval $a$, with:

$$0\le D{W}_a\le 1$$

Then health-adjusted life expectancy may be written as:

$$HALE={\displaystyle \frac{1}{l_0}}\sum _a{L}_a(1-D{W}_a)$$

Total life expectancy is:

$$LE={\displaystyle \frac{1}{l_0}}\sum _a{L}_a$$

and the aggregate non-fatal health loss is:

$$YLD\left(K\right)=\sum _a{L}_{a}D{W}_a$$

Therefore:

$$HALE=LE-{\displaystyle \frac{YLD\left(K\right)}{l_0}}$$

Appendix A.8. Additive Approximation for Capability-Adjusted Life Expectancy

Capability-Adjusted Life Expectancy, denoted $CALE$, extends HALE by adjusting life expectancy not only for morbidity and disability but also for structural capability loss. The manuscript describes CALE as HALE further adjusted for Health Resource Constraint loss.

Under an additive approximation:

$$CAL{E}^{add}={\displaystyle \frac{1}{l_0}}\sum _a{L}_a\left(1−D{W}_a−{\tilde{\omega }}_a\right)$$

where:

$${\tilde{\omega }}_a$$

is the age-specific double-counting-adjusted structural disutility weight.

This additive specification is valid subject to the constraint:

$$0\le D{W}_a+{\tilde{\omega }}_a\le 1$$

for every age interval $a$. This condition ensures that adjusted person-years do not become negative.

Since:

$$YLD\left(K\right)=\sum _a{L}_{a}D{W}_a$$

and:

$$YLS\left(K\right)=\sum _a{L}_a{\tilde{\omega }}_a$$

the additive specification gives:

$$CAL{E}^{add}=LE-{\displaystyle \frac{YLD\left(K\right)}{l_0}}-{\displaystyle \frac{YLS\left(K\right)}{l_0}}$$

Because:

$$HALE=LE-{\displaystyle \frac{YLD\left(K\right)}{l_0}}$$

it follows that:

$$CAL{E}^{add}=HALE-{\displaystyle \frac{YLS\left(K\right)}{l_0}}$$

This identity holds when $YLS\left(K\right)$is defined over total life-table person-years and when the additive constraint above is satisfied.

Appendix A.9. Multiplicative Specification for Capability-Adjusted Life Expectancy

A multiplicative specification may be preferable where morbidity-related loss and structural capability loss overlap within the same person-years, because it preserves non-negative adjusted person-years without requiring the additive restriction $D{W}_a+{\tilde{\omega }}_a\le 1$.

The multiplicative form is:

$$CAL{E}^{mult}={\displaystyle \frac{1}{l_0}}\sum _a{L}_a(1-D{W}_a)(1-{\tilde{\omega }}_a)$$

with:

$$0\le D{W}_a\le 1$$

and:

$$0\le {\tilde{\omega }}_a\le 1$$

This guarantees:

$$0\le (1-D{W}_a)(1-{\tilde{\omega }}_a)\le 1$$

The multiplicative form can also be written as:

$$CAL{E}^{mult}={\displaystyle \frac{1}{l_0}}\sum _{a}HALEP{Y}_a(1-{\tilde{\omega }}_a)$$

where:

$$HALEP{Y}_a=L_a(1-D{W}_a)$$

Under this formulation, the structural capability loss component embedded in CALE is:

$$YLS(K{)}^{HALE}=\sum _a{L}_a(1-D{W}_a){\tilde{\omega }}_a$$

Therefore:

$$CAL{E}^{mult}=HALE-{\displaystyle \frac{YLS(K{)}^{HALE}}{l_0}}$$

This differs from the additive identity because the structural loss is applied to HALE-adjusted person-years rather than to total life-table person-years.

Appendix A.10. Equivalence Between HALE and CALE

Both additive and multiplicative specifications show that HALE and CALE are equivalent when structural capability loss is zero.

If:

$${\tilde{\omega }}_a=0\forall a$$

then:

$$YLS\left(K\right)=0$$

Under the additive specification:

$$CAL{E}^{add}={\displaystyle \frac{1}{l_0}}\sum _a{L}_a(1-D{W}_a-0)$$

so:

$$CAL{E}^{add}={\displaystyle \frac{1}{l_0}}\sum _a{L}_a\left(1-D{W}_a\right)=HALE$$

Under the multiplicative specification:

$$CAL{E}^{mult}={\displaystyle \frac{1}{l_0}}\sum _a{L}_a(1-D{W}_a)(1-0)$$

so:

$$CAL{E}^{mult}={\displaystyle \frac{1}{l_0}}\sum _a{L}_a(1-D{W}_a)=HALE$$

Therefore:

$$CALE=HALE$$

when:

$$YLS\left(K\right)=0$$

or equivalently:

$${\tilde{\omega }}_a=0\forall a$$

More generally:

$$CALE\le HALE$$

with equality only when there is no double-counting-adjusted structural capability loss.

Appendix A.11. Summary of Core Identities

The central burden identity is:

$$DALY=YLL+YLD$$

The capability reinterpretation is:

$$YLL\left(K\right)\equiv YLL$$

$$YLD\left(K\right)\equiv YLD$$

(2)

Therefore:

$$DALY=YLL\left(K\right)+YLD\left(K\right)$$

The Health Capability Gap is:

$$HCG=YLL\left(K\right)+YLD\left(K\right)+YLS\left(K\right)$$

or:

$$HCG=DALY+YLS\left(K\right)$$

where:

$$YLS\left(K\right)=\sum _a\sum _s\sum _r\sum _{t}P{Y}_{a,s,r,t}(1-{\theta }_{a,s,r,t}){\omega }_{a,s,r,t}^{raw}$$

and:

$${\omega }_{a,s,r,t}^{raw}=1-{\displaystyle \prod _{m=1}^M}\left(1-E_{a,s,r,t,m}{\omega }_m\right)$$

The additive expectancy identity is:

$$CAL{E}^{add}=HALE-{\displaystyle \frac{YLS\left(K\right)}{l_0}}$$

subject to:

$$0\le D{W}_a+{\tilde{\omega }}_a\le 1$$

The multiplicative expectancy identity is:

$$CAL{E}^{mult}=HALE-{\displaystyle \frac{YLS(K{)}^{HALE}}{l_0}}$$

where:

$$YLS(K{)}^{HALE}=\sum _a{L}_a(1-D{W}_a){\tilde{\omega }}_a$$

Finally:

$$CALE=HALE$$

when:

$$YLS\left(K\right)=0$$

and:

$$CALE<HALE$$

when positive double-counting-adjusted structural capability loss remains.