Estimating the QALY Output of the Health Sector

Step 1: The Problem

How many quality-adjusted life years (QALYs) does a country's health sector produce each year? If we know this, we can compare the total cost of the health sector to its QALY output, yielding a mean cost per QALY — a summary measure of health sector value.

To answer this, we need two quantities:

  1. Observed QALYs: The actual QALY accrual of the population today.
  2. Counterfactual QALYs: What QALYs would have been without the health sector.

The difference — attributable QALYs — represents the health sector's output: $$\Delta Q = Q_{\text{observed}} - Q_{\text{counterfactual}}$$

Step 2: Observed QALYs — Nearly Trivial

Computing observed QALYs is straightforward. For each demographic cell \(d = (\text{age } x, \text{sex } s)\):

$$Q_{\text{obs}} = \sum_{d} PY_d(t_1) \cdot q_d(t_1)$$

where \(PY_d(t_1)\) are person-years lived (from population counts and life table) and \(q_d(t_1)\) is the EQ-5D utility weight from published population norms.

The inputs are:

  • Population data by single year of age and sex
  • Period life tables providing survival and person-year functions
  • EQ-5D population norms providing mean utility by age and sex

All of these are readily available from national statistics offices and published surveys.

Step 3: Defining the Counterfactual — The Hard Part

We cannot observe a world without the health sector. Instead, we look to the past: survival tables from before the modern health sector was established provide a crude picture of life expectancy without organised healthcare.

However, simply comparing today's population to a historical population is problematic for several reasons:

  1. Different starting population: The historical population was a different size and had a different age-sex structure. We need to ask: what would today's population look like under historical mortality?
  2. Non-health improvements: Much of the improvement in survival since the baseline year is due to factors other than the health sector — improved nutrition, sanitation, hygiene, workplace safety, and environmental regulation.
  3. HRQoL changes: Quality of life has also changed for reasons beyond healthcare, and we need to account for this.
  4. Temporal attribution: Health spending in a given year produces benefits that may extend into the future. Attributing all QALYs to the current year (the ‘snapshot’) is the simplest approach, but a distributed lag model can redistribute attribution over time.

The following steps address each of these challenges.

Step 4: The Starting Population — Back-Calculation

We cannot simply use the historical population, because today's population reflects decades of births, deaths, and migration. Instead, we ask: what hypothetical set of birth cohorts, subject to current mortality, would produce today's observed population?

For each age \(x\), we back-calculate the effective birth cohort size: $$C(x) = \frac{N(t_1, x)}{l_x(t_1) \,/\, l_0}$$ where \(N(t_1, x)\) is today's population at age \(x\) and \(l_x(t_1)/l_0\) is the probability of surviving to age \(x\) under current mortality.

Applying the counterfactual survival schedule to these same cohorts gives us the counterfactual population: $$N_{\text{cf}}(x) = C(x) \cdot \frac{\tilde{l}_x}{l_0} = N(t_1, x) \cdot \frac{\tilde{l}_x}{l_x(t_1)}$$

This procedure sidesteps immigration by treating the evaluation year population as if it arose from cohorts subject to evaluation year mortality, then re-projecting those cohorts under the counterfactual schedule. Differences in health among groups defined by immigration status are not explicitly handled.

Step 5: How Much Longevity Gain Is Due to the Health Sector?

The gap between baseline (historical) and current survival reflects all sources of improvement, not just the health sector. We need an assumption about what proportion of longevity gains are attributable to healthcare.

The parameter \(p_L\) (longevity attribution) controls this. The counterfactual mortality is constructed by blending baseline and current death rates:

$$\tilde{q}_x = (1 - p_L)\, q_x(t_1) + p_L \, q_x(t_0)$$

This rate-level blending is the default method. It preserves the multiplicative structure of the survival process. An alternative is to blend person-years directly (linear method), though this can produce internally inconsistent life tables.

  • If \(p_L = 0\): no mortality improvement is attributed to the health sector. The counterfactual equals the current population. No longevity QALYs are attributed.
  • If \(p_L = 1\): all mortality improvement is attributed. The counterfactual reverts fully to baseline survival.
  • For values in between: the counterfactual population has the survival properties of a society without the health sector, but with improvements from other factors (nutrition, hygiene, etc.) retained.

Published estimates of \(p_L\) typically range from 0.25 to 0.50, suggesting the health sector accounts for roughly one-quarter to one-half of observed longevity gains. Because the true value is uncertain, the tool allows \(p_L\) to be specified as a range (e.g. 40–60%), producing corresponding ranges for attributable QALYs and cost per QALY.

Step 6: Counterfactual Health-Related Quality of Life

Beyond living longer, the health sector may also improve quality of life. We need to estimate what HRQoL would have been without the health sector.

Default approach: Remaining life span model. This approach assumes that changes in quality of life over time can be approximated by corresponding changes in survival: gaining more remaining life implies similar postponement of HRQoL loss.

We fit a parametric function \(f(k, \theta)\) relating EQ-5D utility to remaining years of life \(k\), using the current population norms:

$$U_{a,s} = \sum_{k=0}^{K} p_{a,k,s} \cdot f_s(k, \theta_s) + \varepsilon_{a,s}$$

where \(p_{a,k,s}\) is the probability that a person of age \(a\) and sex \(s\) will die in exactly \(k\) years (the curtate future lifetime PMF).

When this fitted model is applied to the counterfactual survival schedule (which has shorter remaining life expectancy), it predicts lower utilities — capturing the HRQoL penalty of reduced longevity.

With this default approach (pL-blended counterfactual reference), a separate \(p_H\) assumption is not needed, as the quality channel is directly coupled to the longevity channel via \(p_L\).

Alternative: Simple offset. Assumes a fixed utility reduction \(\delta\) in the counterfactual: \(\tilde{q}_d = q_d(t_1) - p_H \cdot \delta\). This requires a separate assumption about \(p_H\) (the proportion of HRQoL improvement attributable to the health sector). It is simpler but lacks age-specific structure.

Alternative: Baseline reference. The model predicts utilities under pure baseline (\(t_0\)) survival, and \(p_H\) interpolates between the \(t_1\) prediction and the \(t_0\) prediction. Here \(p_L\) and \(p_H\) operate as independent channels.

Step 7: Temporal Attribution (Optional)

The preceding steps compute a snapshot of attributable QALYs: all gains from the health sector are assigned to the current evaluation year. This is the default (lag horizon = 0). Optionally, a distributed lag model can redistribute these QALYs across time, crediting current-year health spending for benefits that accrue in future years.

The lag model projects the evaluation year population forward under assumed future mortality, computes attributable QALYs at each future year, and applies geometrically declining (Koyck) weights to aggregate them. This matters when we think of health spending as an investment that yields returns over time, rather than a purely contemporaneous service.

Implicit assumptions about the world

The choice of counterfactual population method and mortality projection together embody an assumption about how the health sector operates over time:

  • Period counterfactual + constant projection: Models a steady-state world in which the current level of health services has been and will remain constant. Age-by-sex mortality rates are fixed at their evaluation year values for both the backward-looking counterfactual and the forward projection. This is the simpler assumption and is internally consistent: if the health sector’s contribution is constant, mortality rates should not be trending.
  • Cohort-diagonal counterfactual + log-linear projection: Models a world in which health services (and societal conditions more broadly) have been gradually implemented. Each birth cohort experienced a different trajectory of improving mortality as it aged. The log-linear projection extends this gradual improvement trend into the future. This captures the historical reality that mortality has declined year by year, but requires extrapolating that trend forward.

The two pairings are coherent but not mandatory — the controls can be set independently. When the lag horizon is zero (the default), the projection method has no effect and the distinction is moot.

See the Distributed Lag Model tab for the full mathematical framework and illustrative plots.

Step 8: Attributable QALYs and Cost per QALY

Combining counterfactual person-years and counterfactual HRQoL weights, we compute:

$$Q_{\text{cf}} = \sum_{d} \widetilde{PY}_d \cdot \tilde{q}_d$$

The attributable QALYs are the difference:

$$\Delta Q = Q_{\text{obs}} - Q_{\text{cf}} = \sum_{d} \bigl[PY_d(t_1)\, q_d(t_1) - \widetilde{PY}_d\, \tilde{q}_d\bigr]$$

Finally, the mean cost per QALY is:

$$\text{Cost per QALY} = \frac{E}{\Delta Q}$$

where \(E\) is total health expenditure. This can be compared against standard cost-effectiveness thresholds to assess whether the health sector, on average, provides value for money.

Because \(p_L\) (and optionally \(p_H\)) are uncertain, the tool allows specifying ranges for these parameters, producing a range of cost-per-QALY estimates rather than a single point estimate.

If a distributed lag model is used (Step 7), the attributable QALYs \(\Delta Q\) are replaced by their lag-weighted counterpart \(Q_{\text{lag}}\), and the cost per QALY is computed accordingly. The effect is typically modest (±3%) because forward-projected attributable QALYs change slowly over time.



              

                

                
Distributed Lag Model

                

                
Lag Attribution by Year

                
Left: Bars show Koyck-weighted QALYs credited to each projection year
                    (when a pL range is set, dark bars = low estimate, light bars = high estimate).
                    The red line (right axis) shows the total attributable QALYs at each year offset
                    before applying Koyck weights — this is what the lag model redistributes.
                    Right: Running total of weighted QALYs; the dashed line marks the
                    snapshot value (horizon = 0).

                

                  

                    

                  

                  

                    

                  

                

              

              

                

                
QALY Accrual by Age - Males

                

                

                
QALY Accrual by Age - Females

                

              

              

              
Reference Data: Health Expenditure

              

              

              
Reference Data: Longevity Attribution Estimates

              

              

              

            

            

              

              
              
Survival by Age

              

                

                  
Males

                  

                

                

                  
Females

                  

                

              

              

              
Population by Age

              

                

                  
Males

                  

                

                

                  
Females

                  

                

              

              

              
Person-Years (Life Years) by Age

              

                

                  
Males

                  

                

                

                  
Females

                  

                

              

              

              

                Population Pyramids (click to expand)
                

                
Population Composition by Age and Sex

                

                  

                    

                      

                    

                    

                  

                  

                    

                      

                    

                    

                  

                  

                    
Counterfactual

                    

                  

                

                

                
Person-Year Pyramids

                

                  

                    

                      

                    

                    

                  

                  

                    

                      

                    

                    

                  

                  

                    
Counterfactual

                    

                  

                

              

              

              

                
Methods & Sources

                
Data sources: Life tables and population data from the
              Human Mortality Database (HMD)
              for years up to 2022, extended with
              ONS Single Year Life Tables
              for 2023–2024. ONS tables were converted to HMD format; missing columns (\(a_x\), \(L_x\), \(T_x\))
              were computed and ages 101–110 extrapolated via log-linear \(m_x\) fit on ages 85–100.

                

                  Counterfactual survival construction
                

                
Method 1: Rate-level interpolation (recommended). Blends baseline and current
              age-specific death probabilities:
              $$\tilde{q}_x = (1 - p_L)\, q_x(t_1) + p_L\, q_x(t_0)$$
              A complete life table is then rebuilt:
              $$\tilde{l}_0 = 100{,}000, \quad \tilde{l}_{x+1} = \tilde{l}_x (1 - \tilde{q}_x)$$
              $$\tilde{L}_x = (\tilde{l}_x - \tilde{d}_x) + a_x(t_1) \cdot \tilde{d}_x$$
              This preserves the multiplicative structure of the survival process.

                
Method 2: Linear (Lx) interpolation. Directly blends person-years:
              $$\tilde{L}_x = (1 - p_L)\, L_x(t_1) + p_L\, L_x(t_0)$$
              This is simpler but can produce person-years inconsistent with any coherent set of
              underlying \(q_x\) values, because survival to age \(x\) is a multiplicative process
              \(l_x = \prod_{a=0}^{x-1}(1 - q_a)\) and linear mixtures of \(L_x\) do not
              in general correspond to any valid \(q_x\) schedule.

                

                  Counterfactual population method
                

                
Period approach (default): Uses only the baseline (\(t_0\)) and
              evaluation (\(t_1\)) life tables. The counterfactual is a blend of the two endpoint
              schedules. This treats the gap between \(t_0\) and \(t_1\) as if mortality
              changed in a single step.

                
Cohort-diagonal approach: Traces each age cohort backward along
              the actual year-by-year mortality it experienced. For a person aged \(x\) at \(t_1\),
              the diagonal extracts \(q(x-D+k,\, t_1-D+k)\) for \(k=0,\ldots,D-1\) where
              \(D = \min(x,\, t_1-t_0)\). The counterfactual rate at each step is:
              $$\tilde{q}_k = (1 - p_L)\, q_{\text{diag},k} + p_L\, q(\cdot,\, t_0)$$
              This avoids the jump from \(t_0\) to \(t_1\) and produces a less biased estimate
              when the time span is large. When \(t_1 - t_0\) is small, both approaches converge.

                
The tables below illustrate how each method uses the \(q(x,t)\) matrix:

                

                  

                    

                      Period approach
                    

                    

                      
                        

                          Age \ Year
                          t0
                          t0+1
                          t0+2
                          ...
                          t1
                        

                      
                      
                        

                          0
                          q(0,t0)
                          -
                          -
                          ...
                          q(0,t1)
                        

                        

                          1
                          q(1,t0)
                          -
                          -
                          ...
                          q(1,t1)
                        

                        

                          2
                          q(2,t0)
                          -
                          -
                          ...
                          q(2,t1)
                        

                        

                          3
                          q(3,t0)
                          -
                          -
                          ...
                          q(3,t1)
                        

                        

                          4
                          q(4,t0)
                          -
                          -
                          ...
                          q(4,t1)
                        

                      
                    

                    

                      Rust = baseline column, Teal = evaluation column. Only these two columns are used.
                    

                  

                  

                    

                      Cohort-diagonal approach
                    

                    

                      
                        

                          Age \ Year
                          t0
                          t0+1
                          t0+2
                          t0+3
                          t1
                        

                      
                      
                        

                          0
                          q(0,t0)
                          q(0,t0+1)
                          -
                          -
                          -
                        

                        

                          1
                          q(1,t0)
                          -
                          q(1,t0+2)
                          -
                          -
                        

                        

                          2
                          q(2,t0)
                          -
                          -
                          q(2,t0+3)
                          -
                        

                        

                          3
                          q(3,t0)
                          -
                          -
                          -
                          q(3,t1)
                        

                        

                          4
                          q(4,t0)
                          -
                          -
                          -
                          q(4,t1)
                        

                      
                    

                    

                      Rust = baseline column, Teal = diagonal path. Each cohort traces its actual year-by-year path.
                    

                  

                

                

                  Counterfactual population construction
                

                
The counterfactual population by age is derived from the evaluation year population
              by adjusting for the difference between evaluation year and counterfactual survival.
              The procedure avoids complications arising from immigration by back-calculating effective
              birth cohort sizes from the evaluation year population.

                
For each age \(x\), the effective birth cohort size is:
              $$C(x) = N(t_1, x) \cdot \frac{l_0}{l_x(t_1)}$$
              where \(N(t_1, x)\) is the evaluation year population at age \(x\), and \(l_x(t_1)\)
              is survival to age \(x\) in the evaluation year life table. The counterfactual population
              is then:
              $$N_{\text{cf}}(x) = C(x) \cdot \frac{\tilde{l}_x}{l_0}
              = N(t_1, x) \cdot \frac{\tilde{l}_x}{l_x(t_1)}$$
              where \(\tilde{l}_x\) is the counterfactual survival. This sidesteps immigration issues
              by treating the evaluation year population as if it arose from cohorts subject to evaluation
              year mortality, then re-projecting those cohorts under the counterfactual schedule.

                
Counterfactual person-years for demographic cell \(d\):
              $$\widetilde{PY}_d = N_d(t_1) \cdot \tilde{L}_d \,/\, l_d(t_1)$$

              

            

            

              

              
              
Model Comparison (AIC / BIC)

              

              

              
Fitted f(k) vs Remaining Life Years

              

                

                  

                

                

                  

                

              

              

              
Observed vs Fitted Utility by Age

              

                

                  

                

                

                  

                

              

              

              
Predicted Utility by Age: Baseline, Counterfactual, and Evaluation Year

              

                

                  
Males

                  

                

                

                  
Females

                  

                

              

              

              

                Utility Values Table (click to expand)
                

                

              

              

              

                
Alternative Approaches for Counterfactual HRQoL

                
1. Model-based (remaining life) — included: Fits \(f(k, \theta)\) to current norms,
              predicts what utilities would have been under shorter counterfactual life expectancy.
              See ‘Quality Reference Options’ below for the two sub-options.

                
2. Simple offset — included: \(\tilde{q}_d = q_d(t_1) - p_H \cdot \delta\).
              Simple but lacks age-specific structure. The offset \(\delta\) captures all
              assumed HRQoL improvement, including both survival-mediated and direct quality gains.

                
3. Historical EQ-5D survey extrapolation — not included: Fit a time trend in age-sex-specific
              utility using repeated cross-sectional surveys (e.g. HSE 1996–present)
              and back-extrapolate.

                
4. Disease-prevalence decomposition — not included: Model HRQoL as a function of disease
              prevalence and remove health-sector-attributable reductions.

                
5. Cause-deleted life table approach — not included: Remove amenable causes of death and
              compute implied HRQoL from the remaining-life model on the cause-deleted survival.

                
6. International comparator — not included: Use EQ-5D norms from a country with minimal
              health sector provision as a proxy.

              

              

              

                
Model-Based Counterfactual HRQoL: Quality Reference Options

                
When the model-based method is selected, the fitted utility function \(f(k, \theta)\)
              is used to predict what HRQoL would have been under a different survival schedule. The underlying
              assumption is that changes in quality of life over time can be approximated by corresponding changes
              in survival; the model assumes that gaining more remaining life implies similar postponement of HRQoL
              loss. Two options are available for choosing the reference survival schedule:

                
Option A: Baseline year (t0) survival. The model predicts
              utilities under the pure baseline (\(t_0\)) mortality schedule, and \(p_H\) interpolates
              between the \(t_1\) prediction and the \(t_0\) prediction:
              $$\tilde{q}_d = q_d(t_1) - p_H \cdot \bigl[\hat{U}_d(t_1) - \hat{U}_d(t_0)\bigr]$$
              where \(\hat{U}_d(t)\) denotes model-predicted utility under year-\(t\) survival.
              Here \(p_L\) controls the longevity channel (person-years) and \(p_H\) independently
              controls the quality channel (utility weights). The two channels operate independently:
              \(p_H\) has the same effect regardless of \(p_L\).

                
Option B: pL-blended counterfactual survival (default). The model predicts
              utilities under the \(p_L\)-interpolated counterfactual mortality schedule:
              $$\tilde{q}_d = q_d(t_1) - p_H \cdot \bigl[\hat{U}_d(t_1) - \hat{U}_d(\text{cf})\bigr]$$
              where \(\hat{U}_d(\text{cf})\) is predicted under the counterfactual survival that
              depends on \(p_L\). The quality channel is coupled to the longevity channel: when
              \(p_L = 0\) (no mortality improvement attributed), the predicted utility difference is
              zero and \(p_H\) has no effect. This avoids attributing quality gains from mortality
              improvement that has not been attributed to the health sector, but produces a smaller
              \(p_H\) effect overall.

                
Key distinction: Option A treats longevity and quality attribution
              as independent dimensions. Option B constrains the quality channel to be a subset of the
              longevity channel, so that quality attribution from mortality improvement cannot exceed
              what has been attributed via \(p_L\). Under both options, the model assumes that changes
              in quality of life over time can be approximated by corresponding changes in survival;
              gaining more remaining life implies similar postponement of HRQoL loss. Direct
              quality-of-life improvements (better treatments reducing morbidity, improved medical
              technology) are not captured.

              

              

              

                
Methods & Sources

                

                

                  Remaining-life utility model
                

                
Models are estimated separately for males and females, each using the
              sex-specific survival PMF and EQ-5D norms. Only ages with directly provided
              (published) norm values are used for model fitting; assumed values for child
              ages do not affect the estimated model parameters.

                
$$U_{a,s} = \sum_{k=0}^{K} p_{a,k,s} \cdot f_s(k,\, \theta_s) + \varepsilon_{a,s}$$
              where \(p_{a,k,s} = \frac{l_{a+k,s}}{l_{a,s}} \cdot q_{a+k,s}\) is the sex-specific
              curtate future lifetime PMF.

                

                  
                    

                      Model
                      Functional Form
                      Params
                      Method
                    

                  
                  
                    

                      Linear
                      \(\theta_0 + \theta_1 k\)
                      2
                      OLS
                    

                    

                      Log
                      \(\theta_0 + \theta_1 \log(k+1)\)
                      2
                      OLS
                    

                    

                      Power
                      \(\theta_0 + \theta_1 k^{\theta_2}\)
                      3
                      NLS
                    

                    

                      Exponential
                      \(\theta_2 + \theta_0(1-e^{-\theta_1 k})\)
                      3
                      NLS
                    

                    

                      Gompertz
                      \(\theta_0 - \theta_1 e^{-\theta_2 k}\)
                      3
                      NLS
                    

                  
                

                
Model comparison uses AIC and BIC. ‘Best AIC (auto)’ selects the model
              with the lowest AIC.

              

            

            

              

              
              

                
Overview

                
The distributed lag model redistributes QALY attribution across time.
              The standard ‘snapshot’ approach (lag horizon = 0) assigns all attributable
              QALYs to the current evaluation year. With a positive lag horizon, current-year health
              spending also receives credit for QALYs that accrue in future years, reflecting the
              idea that health investments today produce benefits over time.

                
The lag horizon controls how many future years are included, and the decay
              parameter λ controls how quickly the weight on future years diminishes.
              When horizon = 0, the model reduces to the standard snapshot.

              

              

              

                
Mathematical Framework

                
Forward population projection. Starting from the evaluation year
              population, we project forward \(k\) years:

                
$$N_{\text{obs}}(a\!+\!k,\, t_1\!+\!k) =
              N_{\text{obs}}(a\!+\!k\!-\!1,\, t_1\!+\!k\!-\!1) \cdot
              \bigl(1 - q_{\text{obs}}(a\!+\!k\!-\!1,\, t_1\!+\!k\!-\!1)\bigr)$$

                
The same projection is applied to the counterfactual population. No new births
              enter during the forward projection.

                
Person-years at age \(a\) and year offset \(k\):

                
$$PY(a,k) = N(a,k) \cdot L_x(a) \,/\, l_x(a)$$

                
Attributable QALYs at age \(a\) and year offset \(k\):

                
$$A(a,k) = PY_{\text{obs}}(a,k) \cdot u_{\text{obs}}(a) -
              PY_{\text{cf}}(a,k) \cdot u_{\text{cf}}(a)$$

                
Age-dependent horizon: \(D(a) = \min(a, H)\) —
              young cohorts have truncated horizons (e.g., age 0 gets 100% weight on current year).

                
Normalized Koyck weights:

                
$$w_a(k) = \frac{\lambda^k}{\sum_{j=0}^{D(a)} \lambda^j}$$

                
Total lag-weighted QALYs:

                
$$Q_{\text{lag}} = \sum_a \sum_{k=0}^{D(a)} w_a(k) \cdot A(a\!+\!k,\, k)$$

              

              

              

                
Key Properties

                
    
                  
  • Weights sum to 1 per cohort — the lag model
                redistributes attribution across time, it does not multiply it.
    • 
                  
  • Horizon = 0: snapshot (all weight on current year).
                The lag model is disabled and results match the standard approach exactly.
    • 
                  
  • λ = 0: all weight falls on the current year
                regardless of horizon, equivalent to the snapshot.
    • 
                  
  • Increasing λ shifts more weight to future years,
                giving more credit for longer-term benefits of health spending.
    • 
                  
  • Typical effect is small (~±3%) because
                forward-projected attributable QALYs are relatively stable over time —
                the difference between observed and counterfactual populations evolves slowly.
    • 
                

              

              

              
Illustrative Plots

              

                

                  
Normalized Koyck Weights by Age

                  

                

                

                  
Observed vs Counterfactual Population Trajectory

                  

                

              

              

              

                

                  
Unweighted Attributable QALYs by Year Offset

                  

                

                

                  
Weighted (Lag-Attributed) QALYs by Year

                  

                

              

              

              
Lag Attribution Graphs

              
Left: Bars show the Koyck-weighted QALYs credited to each projection year;
                  the red line (right axis) shows the raw attributable QALYs before weighting.
                  Right: Running total of weighted QALYs; the dashed line marks the snapshot value (horizon = 0).

              

                

                  

                

                

                  

                

              

              

              

                
Mortality Projection Methods

                
Constant (period rates): Age-specific mortality rates from the
              evaluation year are held fixed for all forward projection years. Simple and transparent,
              but does not account for ongoing mortality trends.

                
Log-linear trend:
              $$\log q_x \sim \beta_0(a) + \beta_1(a) \cdot t$$
              fitted per age using available year-by-year data, then extrapolated forward. This
              captures ongoing mortality decline and is the coherent pairing with the cohort-diagonal
              counterfactual approach.

                
Note: The projection method only affects forward-projected years (year
              offset > 0). The evaluation year (offset 0) always uses observed mortality.

              

              

              

                
Interpretation

                
Lag-attributed QALYs bar chart: Each bar shows the weighted
              QALYs credited to a future year. The height reflects both the raw attributable QALYs
              at that offset and the Koyck weight. Bars should decline geometrically if attributable
              QALYs are stable over the projection.

                
Cumulative plot: Shows the running total of lag-attributed QALYs
              across projection years. The horizontal dashed line marks the snapshot value. If the
              cumulative line ends above the snapshot, the lag model attributes more QALYs to the
              health sector; if below, fewer.

                
Why the effect is typically modest: The forward-projected
              difference between observed and counterfactual populations changes slowly over time.
              Mortality differences are persistent, so attributable QALYs at year offset 5 or 10
              are similar to those at offset 0. Since the Koyck weights sum to 1, redistributing
              similar quantities across years produces a result close to the snapshot.

              

            

            

              

              
              
Cost per QALY over pL and pH grid

              

              

              

              

              

                
Methods

                
Evaluates the model over a grid of \(p_L\) and \(p_H\) (0.1 to 0.9, step 0.2),
              holding all other parameters fixed.

              

            

            

              

              
              

                
Fixed Parameters for Results Below

                

              

              

              

                

              

              

              

              

              

              

              

              

              

              

            

            

              

              
              
Per-Age-Sex Breakdown