B.1 Calculation

There are two parts to the calculations preformed in health behaviour attributable scenarios: (A) calculate the risk, and (B) calculate the health outcome: life expectancy or number of deaths.

B.1.1 Part A: Risk calculations

The original multivariable predictive risk algorithm is:

\[ \text{Risk} = \sum_t h_0(t) * e^{\beta_{pred.smoking}*x_{smoking}+\beta_{pred.cancer}*x_{cancer} + \beta_{pred.age}*x_{age} +...} \]

Step 1. Modify the original algorithm to include the external coefficient(s). This means replacing all predictive hazard ratios/betas related to the health behaviour to the causal hazard ratios/betas.

  • Remove the original regression coefficient(s) for the health behaviour.
  • Add the new external coefficient(s) to the algorithm. External coefficients are generated from either: causal models, or from systematic reviews or meta-analysis.

\[ \text{External coefficient risk} = \sum_t h_0(t) * e^{{\beta_\textbf{causal.smoking}}*x_{smoking} + {{\beta_\textbf{causal.cancer}}}*x_{cancer} + \beta_{pred.age}*x_{age} +...} \]

Step 2. Risk is calculated using the modified algorithm created in Step 1 and the respondent’s original profile (e.g. current smoker). This is the “external coefficient risk”.

\[ \text{External coefficient risk} = \sum_t h_0(t) * e^{\beta_{causal.smoking}* {(\textbf{current smoker})} + \beta_{causal.cancer}*x_{cancer} + \beta_{pred.age}*x_{age} +...} \]

Step 3. “Health behaviour-deleted risk’” is calculated by setting an exposure to a reference (non-exposed) value (all other risk exposures remain unchanged).

\[ \text{Health behaviour-deleted risk' } = \sum_t h_0(t) * e^{\beta_{causal.smoking}* {(\textbf{never smoker})} + \beta_{causal.cancer}*x_{cancer} + \beta_{pred.age}*x_{age} +...} \]

Step 4. The “health behaviour attributable risk external” is calculated as “external coefficient risk” (Step 2) minus the “health behaviour-deleted risk’.”(Step 3).

\[\text{Health behaviour attributable risk}_{external} = \text{External coefficient risk} - \text{Health behaviour-deleted risk'}\]

Step 5. Original risk is calculated using the original algorithm and the original respondent’s profile.

\[ \text{Original risk} = \sum_t h_0(t) * e^{{\beta_\textbf{pred.smoking}}*{(\textbf{current smoker})}+{\beta_\textbf{pred.cancer}}*x_{cancer} + \beta_{pred.age}*x_{age} +...} \]

Step 6. The “health behaviour-deleted risk external” is calculated by “original risk” (Step 5) minus the “health behaviour-deleted effect external” (Step 4).

\[\text{Health behaviour-deleted risk}_{ external} = \text{Original risk} - \text{Health behaviour attributable risk}_{external}\]

Risk portion of the health behaviour attribution calculations

Figure B.1: Risk portion of the health behaviour attribution calculations

B.1.2 Part B: Health outcome calculations

Using risks generated above you can then calculate:

  • Health behaviour-deleted life expectancy or health behaviour attributable life expectancy lost
  • Health behaviour-deleted number of deaths or health behaviour attributable number of deaths

Life expectancy calculations

Step I: Calculate the original life expectancy by using the original risk (Step 5 above) in the sex-specific 5-year abridge period life tables.

Step II: Calculate the health behaviour-deleted life expectancy by using the health behaviour-deleted risk external (Step 6 above) in the sex-specific 5-year abridge period life tables.

Step III: Calculate the health behaviour attributable life expectancy lost by: original life expectancy (Step I) minus health behaviour-deleted life expectancy (Step II):

\[ \text{Health behaviour attributable life exectancy lost} = \text{Health behaviour-deleted life expectancy} - \text{Original life expectancy}\]

Number of deaths calculations

Step I: Calculate the number of deaths that would occur using the original risk (Step 5 above).

Step II: Calculate the number of deaths that would occur using the health behaviour-deleted risk external (Step 6 above).

Step III: Calculate the health behaviour attributable deaths by: original number of deaths (Step I) minus health behaviour-deleted number of deaths (Step II):

\[\text{Health behaviour attributable deaths} = \text{Original number of deaths} - \text{Health behaviour-deleted number of deaths}\]