Quality-adjusted life expectancy (QALE) loss due to smoking in the United States

Abstract

Purpose

Estimate quality-adjusted life expectancy (QALE) loss due to smoking and examine trends and state differences in smoking-related QALE loss in the U.S.

Methods

Population health-related quality of life (HRQOL) scores were estimated from the Behavioral Risk Factor Surveillance System. This study constructed life tables based on U.S. mortality files and the mortality linked National Health Interview Survey and calculated QALE for smokers, non-smokers, and the total population.

Results

In 2009, an 18-year-old smoker was expected to have 43.5 (SE = 0.2) more years of QALE, and a non-smoker of the same age was expected to have 54.6 (SE = 0.2) more years of QALE. Therefore, smoking contributed 11.0 (SE = 0.2) years of QALE loss for smokers and 4.1 years (37%) of this loss resulted from reductions in HRQOL alone. At the population level, smoking was associated with 1.9 fewer years of QALE for U.S. adults throughout their lifetime, starting at age 18.

Conclusions

This study demonstrates an application of a recently developed QALE estimation methodology. The analyses show good precision and relatively small bias in estimating QALE––especially at the individual level. Although smokers may live longer today than before, they still have a high disease burden due to morbidities associated with poor HRQOL.

Appendix

The QALE at age x is calculated by summarizing QALYs throughout remaining expected life starting at age x over the percent of population surviving to age x [7, 8]:

$$ Q(x) = \frac{{\int_{ \ge x} {S(t)y(t){\text{d}}t} }}{S(x)}, $$

where y(t) are HRQOL scores at age t and S(t) is the survival function. Formulas to calculate QALE and their standard errors were provided by Jia et al. [8]. QALE for those at age x is:

$$ Q_{x} = \frac{{\sum\nolimits_{i \ge x} {D_{i} y_{i} } }}{{A_{x} }} $$

The variance of this QALE estimate is:

$$ \begin{aligned} {\text{VAR}}(Q_{x} ) & = \frac{{\sum\nolimits_{i = x}^{84} {\left[ {A_{i}^{2} \left( {\frac{{n_{i} y_{i} }}{2} + {\text{Q}}_{i + 1} } \right)^{2} {\text{VAR}}(q_{i} ) + A_{i}^{2} \left( {1 - \frac{{q_{i} }}{2}} \right){\text{VAR}}(y_{i} )} \right]} }}{{A_{x}^{2} }} \\ & \quad + \frac{{{\text{VAR}}(L_{85} )y_{85}^{2} + D_{85}^{2} {\text{VAR}}(y_{85} ) + {\text{VAR}}(L_{85} ){\text{VAR}}(y_{85} )}}{{A_{x}^{2} }} \\ \end{aligned} $$

where ${\text{VAR}}(q) = \frac{{q^{2} (1 - q)}}{d}$ for age less than 85 and ${\text{VAR}}(L_{85} ) = \frac{{\left( {e^{{ - \sum\nolimits_{k < 85} {n_{k} m_{k} } }} } \right)^{2} }}{{d_{85} m_{85}^{2} }}A_{18 - 24}^{2}$.

QALE loss was the difference in QALE between two groups: $\Updelta_{x} = Q_{x}^{0} - Q_{x}^{1}$. Here, $Q_{x}^{0}$ is QALE for non-smokers and $Q_{x}^{1}$ is QALE for smokers (for individual QALE loss) or for total population (for population QALE loss). The variance of QALE loss is equal to ${\text{Var}}(\Updelta_{x} ) = {\text{Var}}(Q_{x}^{0} ) + {\text{Var}}(Q_{x}^{1} ) - 2{\text{COV}}(Q_{x}^{0} ,Q_{x}^{1} )$. The covariance term is approximated by

$$ {\text{COV}}\left( {Q_{x}^{0} ,Q_{x}^{1} } \right) \approx \frac{1}{{A_{x}^{0} A_{x}^{1} }}\left[ \begin{gathered} \sum\limits_{i < 85} {\frac{{\partial Q^{0} }}{{\partial q_{0} }}\frac{{\partial Q^{1} }}{{\partial q_{1} }}{\text{COV}}(q_{0} ,q_{1} )} + \sum\limits_{i < 85} {\frac{{\partial Q^{0} }}{{\partial y_{0} }}\frac{{\partial Q^{1} }}{{\partial y_{1} }}{\text{COV}}(y_{0} ,y_{1} )} \hfill \\ + \sqrt {{\text{VAR}}\left( {L_{85}^{0} } \right){\text{VAR}}\left( {L_{85}^{1} } \right)} \rho_{{m_{0} ,m_{1} }} y_{0} y_{1} + D_{85}^{0} D_{85}^{1} \rho_{{y_{0} y_{1} }} SE_{{y_{0} }} SE_{{y_{1} }} \hfill \\ \end{gathered} \right], $$

where $\frac{{\partial Q^{0} }}{{\partial q^{0} }} = A_{i} \left( {\frac{ny}{2} + {\text{Q}}_{i + 1} } \right)$ and $\frac{{\partial Q^{0} }}{{\partial y_{0} }} = A_{i} \left( {1 - \frac{{q_{i} }}{2}} \right)$ for i < 85.

Since the number of deaths is estimated from the proportion at risk and the hazard ratio, additional variation from the unreliability of using the estimated proportion (var(p)) and hazard ratio (var(h)) should be included in the variance estimation for q, the probability of dying:

$$ {\text{var}}(q) \approx \frac{{(q^{2} )(1 - q)}}{d} + n^{2} (1 - q)^{2} \left[ \begin{gathered} \frac{{h^{2} m^{2} (h - 1)^{2} }}{{(hp + 1 - p)^{4} }}\text{var} (p) + \frac{{m^{2} (1 - p)^{2} }}{{(hp + 1 - p)^{4} }}\text{var} (h) \hfill \\ + \left( {\frac{h}{hp + 1 - p}} \right)^{2} \text{var} (m) \hfill \\ \end{gathered} \right] $$

The values of var(p) and var(h) are derived from methods of moments estimation (either using designed based on direct estimates or using model-based estimates). For the years 2007–2009, the death data were estimated from a time-series autoregressive moving average model (ARMA) from the 1993–2006 death rates [8, 22]. The death rate at year t, m _t , is specified as ARMA(1,1) or $m_{t} - \mu = \rho (m_{t - 1} - \mu ) + e_{t} - \beta e_{t - 1}$. The predicted death rates for these 3 years therefore should include additional uncertainty in their estimates, and the variance, var(m), from the ARMA model estimates was used to account for this.