Epidemiology • Volume 25, Number 5, September 2014
Letters
on A and R recovers cross-world counterfactual independencies for the mediator and outcome,7 that is for all a and a* Y ( a, m) ⊥⊥ M ( a*) A = a, R( a) = r (5) Tchetgen Tchetgen and VanderWeele7 establish that PDE ( a, a*) becomes identified in the NPSEM-IE for Figure 1B provided an additional assumption also holds, either 1. the treatment and confounder R are binary and the effect of treatment on R is monotone at the individual level, that is, R( a*) ≤ R(a) for a* < a , or, 2. there is no average additive interaction between R and M in their joint effects on Y, that is, E (Y a, m, r) − E (Y a, m*, r)
− E (Y a, m*, r) + E (Y a, m*, r *) = 0 Alternative identifying assumptions under the NPSEM-IE were also considered by Robins and Richardson.6 Note that Tchetgen Tchetgen and VanderWeele7 thus continue to make a cross-world counterfactual independence assumption now given by (5). Here, we extend the Robins–Richardson bounds for PDE ( a, a*) so that they may be used in the presence of exposure-induced confounding as depicted in Figure 1B upon interpreting the causal diagram strictly as encoding a ssumptions (1) and Y (a, m) ⊥⊥ M (a) A = a, R = r (6) In the eAppendix (http://links. lww.com/EDE/A824), we establish that for binary A and M, and a = 1, a* = 0, under assumptions (1) and (6), L* ≤ PDE (1, 0) ≤ U * (7) L* = max{0, Pr(M (0) = 0) + E[Y (1, 0)] − 1} + max{0, Pr(M (0) = 1) + E[Y (1, 1)] − 1} − E[Y (a = 0)]
U * = min{Pr(M (0) = 0), E[Y (1, 0)]}
Pr(M (0) = 1), E[Y (1, 1)] + min − E[Y (a = 0)]
776 | www.epidem.com
Therefore, the bounds L* and U* are identified provided E [Y (a,m)], E [Y (a)], and E [M(a)] are themselves identified. Under the usual no unobserved confounding assumptions (1) and (6), we indeed have E[Y (a, m)] = ∑ E[Y a, m, r] f (r a), r
E[Y (a)] = E[Y a], E[M (a)] = E[M a]
ACKNOWLEDGMENTS We thank James Robins and Thomas Richardson for their insightful comments on a previous version of the manuscript. Eric J. Tchetgen Tchetgen Department of Biostatistics Department of Epidemiology Harvard School of Public Health Boston, MA [email protected]
Kelesitse Phiri Department of Epidemiology Harvard School of Public Health Boston, MA
REFERENCES 1. Robins JM, Greenland S. Identiability and exchangeability for direct and indirect effects. Epidemiology. 1992;3:143–155. 2. Pearl J. Direct and Indirect Effects. In: Proceedings of the 17th Annual Conference on Uncertainty in Artificial Intelligence (UAI-01). San Francisco, CA: Morgan Kaufmann; 2001:411–442. 3. VanderWeele TJ, Vansteelandt S. Odds ratios for mediation analysis for a dichotomous outcome - with discussion. Am J Epidemiol. 2010;172:1339–1348. 4. Imai K, Keele L, Yamamoto T. Identification, inference and sensitivity analysis for causal mediation effects. Stat Sci. 2010;25:51–71. 5. Tchetgen Tchetgen EJ, Shpitser I. Semiparametric theory for causal mediation analysis: efficiency bounds, multiple robustness, and sensitivity analysis. Ann Stat. 2012;40:1816–1845. 6. Robins JM, Richardson TS. Alternative graphical causal models and the identication of direct effects. In: Keyes KM, Ornstein K, Shrout PE, eds. Causality and Psychopathology: Finding the Determinants of Disorders and Their Cures. Oxford, New York: Oxford University Press; 2011:103–158. 7. Tchetgen Tchetgen EJ, VanderWeele T. On identification of natural direct effects when a confounder of the mediator is directly affected by exposure. Epidemiology. 2013;25:282–291.
Air Pollution and Life Expectancy To the Editors: ife expectancy is based on current age-specific mortality and represents future overall population health. Correia et al1 are to be commended for contributing to the literature on life expectancy as a measure of pollution abatement benefits, but their analysis suffers from several fundamental flaws that inflate pollution effect estimates, as follows:
L
• Using proxy smoking parameters that greatly underestimate effects. • Estimating life-long environmental exposures with contemporary data. • Neglecting other pollutants and improved medical care. • Ignoring disease incubation periods that delay responses. • Using life expectancy at birth rather than at ages of susceptibility. • Expressing effects for10 μg/m3 instead of the actual change (1.6 μg/m3). The Figure illustrates basic temporal and spatial characteristics of US life expectancies over ~50 years. The close relationship (R = 0.98) between national life expectancy at birth2 and smoking prevalence3 suggests that decreased smoking accounts for most of the observed increased survival. Regressions indicate ~21 years lost per smoker over a range of lags, suggesting additional contributions from associated lifestyles such as alcoholism. Two polluted locations, New York City and the county incorporating Steubenville, Ohio,4 show wide and varying survival differences. They show parallel improvements in the early 1970s, The author reports no conflicts of interest. Supplemental digital content is avail able through direct URL citations in the HTML and PDF versions of this article (www.epidem.com). This content is not peer-reviewed or copy-edited; it is the sole responsibility of the author. Copyright © 2014 by Lippincott Williams & Wilkins ISSN: 1044-3983/14/2505-0776 DOI: 10.1097/EDE.0000000000000140
© 2014 Lippincott Williams & Wilkins
Epidemiology • Volume 25, Number 5, September 2014 Letters
78
80 75
US life expectancy
Jefferson County, OH (Steubenville)
US % nonsmokers
70 65
76
60 74
55 50
72
New York City
45
Percent nonsmokers
Life expectancy at birth, years
80
40
70
35 68 1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
30 2010
reduced to 13 days for life expectancies at age 65; ~6 days if past exposures had also been considered, and ~0 if smoking effects has been properly accounted for. As a result, their claim that reducing PM2.5 would have a significant benefit on public health cannot be supported. See the eAppendix (http://links.lww.com/EDE/A809) for further information.
ACKNOWLEDGMENTS Christian Murray developed the model and provided results for the fourth analysis protocol discussed in the commentary.
Year
Frederick W. Lipfert
FIGURE. Life expectancy and smoking trends for the United States and selected counties.
Independent Consultant Northport, NY [email protected]
possibly because of pollution abatement. By the mid-1980s Steubenville led by ~3 y, in part because of AIDS effects in New York. Survival then leveled off in Steubenville but increased elsewhere, especially in New York. Survival in urban locations now exceeds rural locations nationwide; cross-county comparisons are thus sensitive to timing. The District of Columbia had the highest 1999–2001 white female life expectancy; nearby West Virginia had the lowest.5 Correia et al1 regressed countylevel differences between 2000 and 2007 life expectancies at birth against corresponding fine particulate matter (PM2.5) and ecological variables; the mean longevity gain was 292 days (their Figure shows 241 days). Average PM2.5 decreased 13%; surrogate smoking variables decreased 9%. The PM2.5-related gain in life expectancy was 20 days for a 1.6 μg/m3 decrease. The modest smoking effect was reported as 12 days. According to the Figure, having 9% fewer smokers should increase national longevity by 448 days, leaving no margin for PM2.5 effects. Current life expectancies reflect earlier exposures when urban air quality was much worse; this is also the case when responses lag exposures. Increasing exposure increments decreases © 2014 Lippincott Williams & Wilkins
regression coefficients. However, life expectancies at age 65 represent more recent lower exposures. Correia et al could have computed and used life expectancies at age 65 years, which are highly correlated with those at birth with a slope of ~0.65. Their PM2.5 effect estimates would have decreased accordingly. They also neglected effects of other pollutant reductions and improved medical care. Other methods of estimating effects of life expectancy have important advantages. Longitudinal studies6 avoid spatial confounding and reflect observable trends, with trivial effects on life expectancy. Time-series studies based on daily mortality fluctuations estimate life expectancies for the most susceptible portions of the population, from first principles.7,8 This model has been validated in 3 US cities; it finds susceptible subpopulations of about 0.1% of those aged 65+ who have remaining lives of ~1 week, of which
Letters
on A and R recovers cross-world counterfactual independencies for the mediator and outcome,7 that is for all a and a* Y ( a, m) ⊥⊥ M ( a*) A = a, R( a) = r (5) Tchetgen Tchetgen and VanderWeele7 establish that PDE ( a, a*) becomes identified in the NPSEM-IE for Figure 1B provided an additional assumption also holds, either 1. the treatment and confounder R are binary and the effect of treatment on R is monotone at the individual level, that is, R( a*) ≤ R(a) for a* < a , or, 2. there is no average additive interaction between R and M in their joint effects on Y, that is, E (Y a, m, r) − E (Y a, m*, r)
− E (Y a, m*, r) + E (Y a, m*, r *) = 0 Alternative identifying assumptions under the NPSEM-IE were also considered by Robins and Richardson.6 Note that Tchetgen Tchetgen and VanderWeele7 thus continue to make a cross-world counterfactual independence assumption now given by (5). Here, we extend the Robins–Richardson bounds for PDE ( a, a*) so that they may be used in the presence of exposure-induced confounding as depicted in Figure 1B upon interpreting the causal diagram strictly as encoding a ssumptions (1) and Y (a, m) ⊥⊥ M (a) A = a, R = r (6) In the eAppendix (http://links. lww.com/EDE/A824), we establish that for binary A and M, and a = 1, a* = 0, under assumptions (1) and (6), L* ≤ PDE (1, 0) ≤ U * (7) L* = max{0, Pr(M (0) = 0) + E[Y (1, 0)] − 1} + max{0, Pr(M (0) = 1) + E[Y (1, 1)] − 1} − E[Y (a = 0)]
U * = min{Pr(M (0) = 0), E[Y (1, 0)]}
Pr(M (0) = 1), E[Y (1, 1)] + min − E[Y (a = 0)]
776 | www.epidem.com
Therefore, the bounds L* and U* are identified provided E [Y (a,m)], E [Y (a)], and E [M(a)] are themselves identified. Under the usual no unobserved confounding assumptions (1) and (6), we indeed have E[Y (a, m)] = ∑ E[Y a, m, r] f (r a), r
E[Y (a)] = E[Y a], E[M (a)] = E[M a]
ACKNOWLEDGMENTS We thank James Robins and Thomas Richardson for their insightful comments on a previous version of the manuscript. Eric J. Tchetgen Tchetgen Department of Biostatistics Department of Epidemiology Harvard School of Public Health Boston, MA [email protected]
Kelesitse Phiri Department of Epidemiology Harvard School of Public Health Boston, MA
REFERENCES 1. Robins JM, Greenland S. Identiability and exchangeability for direct and indirect effects. Epidemiology. 1992;3:143–155. 2. Pearl J. Direct and Indirect Effects. In: Proceedings of the 17th Annual Conference on Uncertainty in Artificial Intelligence (UAI-01). San Francisco, CA: Morgan Kaufmann; 2001:411–442. 3. VanderWeele TJ, Vansteelandt S. Odds ratios for mediation analysis for a dichotomous outcome - with discussion. Am J Epidemiol. 2010;172:1339–1348. 4. Imai K, Keele L, Yamamoto T. Identification, inference and sensitivity analysis for causal mediation effects. Stat Sci. 2010;25:51–71. 5. Tchetgen Tchetgen EJ, Shpitser I. Semiparametric theory for causal mediation analysis: efficiency bounds, multiple robustness, and sensitivity analysis. Ann Stat. 2012;40:1816–1845. 6. Robins JM, Richardson TS. Alternative graphical causal models and the identication of direct effects. In: Keyes KM, Ornstein K, Shrout PE, eds. Causality and Psychopathology: Finding the Determinants of Disorders and Their Cures. Oxford, New York: Oxford University Press; 2011:103–158. 7. Tchetgen Tchetgen EJ, VanderWeele T. On identification of natural direct effects when a confounder of the mediator is directly affected by exposure. Epidemiology. 2013;25:282–291.
Air Pollution and Life Expectancy To the Editors: ife expectancy is based on current age-specific mortality and represents future overall population health. Correia et al1 are to be commended for contributing to the literature on life expectancy as a measure of pollution abatement benefits, but their analysis suffers from several fundamental flaws that inflate pollution effect estimates, as follows:
L
• Using proxy smoking parameters that greatly underestimate effects. • Estimating life-long environmental exposures with contemporary data. • Neglecting other pollutants and improved medical care. • Ignoring disease incubation periods that delay responses. • Using life expectancy at birth rather than at ages of susceptibility. • Expressing effects for10 μg/m3 instead of the actual change (1.6 μg/m3). The Figure illustrates basic temporal and spatial characteristics of US life expectancies over ~50 years. The close relationship (R = 0.98) between national life expectancy at birth2 and smoking prevalence3 suggests that decreased smoking accounts for most of the observed increased survival. Regressions indicate ~21 years lost per smoker over a range of lags, suggesting additional contributions from associated lifestyles such as alcoholism. Two polluted locations, New York City and the county incorporating Steubenville, Ohio,4 show wide and varying survival differences. They show parallel improvements in the early 1970s, The author reports no conflicts of interest. Supplemental digital content is avail able through direct URL citations in the HTML and PDF versions of this article (www.epidem.com). This content is not peer-reviewed or copy-edited; it is the sole responsibility of the author. Copyright © 2014 by Lippincott Williams & Wilkins ISSN: 1044-3983/14/2505-0776 DOI: 10.1097/EDE.0000000000000140
© 2014 Lippincott Williams & Wilkins
Epidemiology • Volume 25, Number 5, September 2014 Letters
78
80 75
US life expectancy
Jefferson County, OH (Steubenville)
US % nonsmokers
70 65
76
60 74
55 50
72
New York City
45
Percent nonsmokers
Life expectancy at birth, years
80
40
70
35 68 1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
30 2010
reduced to 13 days for life expectancies at age 65; ~6 days if past exposures had also been considered, and ~0 if smoking effects has been properly accounted for. As a result, their claim that reducing PM2.5 would have a significant benefit on public health cannot be supported. See the eAppendix (http://links.lww.com/EDE/A809) for further information.
ACKNOWLEDGMENTS Christian Murray developed the model and provided results for the fourth analysis protocol discussed in the commentary.
Year
Frederick W. Lipfert
FIGURE. Life expectancy and smoking trends for the United States and selected counties.
Independent Consultant Northport, NY [email protected]
possibly because of pollution abatement. By the mid-1980s Steubenville led by ~3 y, in part because of AIDS effects in New York. Survival then leveled off in Steubenville but increased elsewhere, especially in New York. Survival in urban locations now exceeds rural locations nationwide; cross-county comparisons are thus sensitive to timing. The District of Columbia had the highest 1999–2001 white female life expectancy; nearby West Virginia had the lowest.5 Correia et al1 regressed countylevel differences between 2000 and 2007 life expectancies at birth against corresponding fine particulate matter (PM2.5) and ecological variables; the mean longevity gain was 292 days (their Figure shows 241 days). Average PM2.5 decreased 13%; surrogate smoking variables decreased 9%. The PM2.5-related gain in life expectancy was 20 days for a 1.6 μg/m3 decrease. The modest smoking effect was reported as 12 days. According to the Figure, having 9% fewer smokers should increase national longevity by 448 days, leaving no margin for PM2.5 effects. Current life expectancies reflect earlier exposures when urban air quality was much worse; this is also the case when responses lag exposures. Increasing exposure increments decreases © 2014 Lippincott Williams & Wilkins
regression coefficients. However, life expectancies at age 65 represent more recent lower exposures. Correia et al could have computed and used life expectancies at age 65 years, which are highly correlated with those at birth with a slope of ~0.65. Their PM2.5 effect estimates would have decreased accordingly. They also neglected effects of other pollutant reductions and improved medical care. Other methods of estimating effects of life expectancy have important advantages. Longitudinal studies6 avoid spatial confounding and reflect observable trends, with trivial effects on life expectancy. Time-series studies based on daily mortality fluctuations estimate life expectancies for the most susceptible portions of the population, from first principles.7,8 This model has been validated in 3 US cities; it finds susceptible subpopulations of about 0.1% of those aged 65+ who have remaining lives of ~1 week, of which
