Quantitative Characterization of Uncertainty in the Concentration-Response Relationship Between Long-term PM_2.5 Exposure and Mortality at Low Concentrations

Abstract

Extensive epidemiologic evidence supports a linear, no-threshold concentration-response (C-R) relationship between long-term exposure to fine particles (PM2.5) and mortality in the United States. While examinations of the C-R relationship are designed to assess the shape of the C-R curve, they do not provide the information needed to quantitatively characterize uncertainty at specific PM_2.5 concentrations, which is often needed in the context of risk assessments and benefits analyses. We developed a novel approach, using information that is typically available in published epidemiologic studies, to quantitatively characterize uncertainty at different concentrations along the PM_2.5 concentration distribution. Our approach utilizes the annual mean PM_2.5 concentration and corresponding standard deviation from a published epidemiologic study to estimate the standard deviation of hypothetical PM_2.5 concentration distributions defined at 0.1 μg/m³ increments. The hypothetical distributions are then used to derive adjusted uncertainty estimates in the reported effect estimate at low concentrations. We demonstrate the application of this method in six individual epidemiologic studies that examined the relationship between long-term PM_2.5 exposure and mortality and were conducted in different geographic locations worldwide and at different PM_2.5 concentrations. This new method allows for a more comprehensive quantitative evaluation of uncertainty in the shape of the C-R relationship between long-term PM_2.5 exposure and mortality at concentrations below the mean annual concentrations observed in current studies.

1. INTRODUCTION

Epidemiologic studies encompassing populations representative of diverse geographic locations (i.e., within the U.S. and globally) and demographic characteristics (e.g., various lifestages and occupations) provide a large body of evidence indicating a relationship between long-term exposure to fine particles (PM_2.5: particulate matter with a mean nominal aerodynamic diameter less than 2.5 micrometers) and mortality (e.g., ^1–6). The current body of evidence describing the relationship between long-term PM_2.5 exposure and mortality has examined four key categories of uncertainty, which has provided additional support for a robust PM2.5-mortality relationship: 1) model specification (i.e., the covariates included in statistical models to control for confounding; e.g., 7), 2) exposure error (i.e., the difference between a concentration metric and a true exposure; e.g., 8), 3) heterogeneity across studies/cohorts (i.e., observed differences in the PM_2.5-mortality relationship across studies; e.g., 9), and 4) the shape of the concentration-response (C-R) relationship (e.g., ^10–11). However, a key remaining area of uncertainty that has been less thoroughly explored is uncertainty in the PM_2.5-mortality relationship at low concentrations where data are sparse, specifically as it relates to quantifying uncertainty at a given concentration.

Many cohort studies have conducted extensive analyses using a variety of statistical methods to examine the shape of the concentration-response (C-R) relationship and investigate whether a threshold exists below which mortality effects are not observed. These analyses suggest that there is not strong evidence to reject the hypothesis that the C-R relationship is linear. Furthermore, a concentration below which no association exists has not yet been clearly identified. Uncertainty in the C-R shape tends to be least near the study mean and most in concentration ranges with the least data support, such as near the lowest observed exposures (5, 12–13). C-R analyses typically consist of fitting various types of smoothing splines, with results reported graphically by mean predictions over the concentration range and their corresponding uncertainty intervals (5, 12–13). Although these analyses are informative in assessing the general shape of the curve and identifying a concentration below which there is increased uncertainty in the shape of the relationship, they are generally not designed to provide information that allows for the quantitative characterization of risk at specific PM_2.5 concentrations. For example, the predicted associations using splines are often not monotonically increasing. As a result, in risk assessments and analyses that attempt to quantitatively characterize risk or the potential public health implications of improving air quality, it is often assumed that risk and standard error remain constant across the full distribution of PM_2.5 concentrations, reflecting the linear, no-threshold relationship demonstrated in previous C-R analyses. However, as depicted in most C-R analyses it is evident there is less certainty in the shape of the C-R curve at both high and low ends of the PM_2.5 concentration distribution because of lower data density (e.g., ^14–15), such that the use of a constant standard error may result in an underestimation of uncertainty as concentrations deviate from the mean. Conversely, the use of a constant standard error may result in an overestimation of uncertainty at concentrations near the mean.

The quantitative characterization of the risk of health effects attributed to specific air pollution concentrations is often challenging because there is a time delay of multiple years between the air quality data used in large cohort studies, such as those that examine the relationship between long-term PM_2.5 exposure and mortality, and the more recent, often improved, air quality data used in risk assessments. As a result, the risk estimates and corresponding standard errors for the studies included in risk assessments are representative of mean annual PM_2.5 concentrations that can be much higher than current air quality. For example, U.S.-based cohort studies published prior to 2012 often had mean annual PM_2.5 concentrations > 14 μg/m³, although in 2012 mean annual PM_2.5 concentrations across the U.S. were primarily < 14 μg/m³ (16). Although more recent epidemiologic studies of long-term PM_2.5 exposure and mortality are conducted at lower mean annual PM_2.5 concentrations (i.e., < 12 μg/m³), which can inform the C-R relationship at even lower annual PM_2.5 concentrations, the same question remains:

“Are there approaches that can be instituted to quantitatively characterize uncertainty in the risk of mortality due to long-term PM_2.5 exposure at low PM_2.5 concentrations?”

In order to reduce uncertainty at low concentrations, ideally the epidemiologic literature base would include new studies where the mean of the annual PM_2.5 concentration distribution across the study area was near the lower range where we would like to characterize uncertainty. However, new cohort studies, specifically ones conducted in the U.S., do not exist with low mean annual PM_2.5 concentrations, i.e., ≤ 8 μg/m³ (15, 17–18).

In this paper we propose an innovative method to quantitatively characterize uncertainty at low PM_2.5 concentrations by using information typically reported in epidemiologic studies of long-term PM_2.5 exposure and mortality. Specifically, the effect estimate, often referred to as $\widehat{\beta },$ which represents the linear rate of change between the exposure and the log of the relative hazard of the health effect of interest from the study, its corresponding standard error ($\widehat{\epsilon })$, and the mean and standard deviation of the study-specific PM_2.5 concentration distribution. Using this study-specific information, a hypothetical new study can be postulated such that its mean annual PM_2.5 concentration is lower than those reported in the currently available epidemiologic literature. Statistical properties of the mean and variance of a given PM_2.5 concentration distribution can be used to generate alternative PM_2.5 concentration distributions and subsequently characterize how uncertainty in $\widehat{\beta }$ changes as the mean of the PM_2.5 concentration distribution changes, specifically at low PM_2.5 concentrations, assuming the sample size remains constant. We illustrate this new approach to characterizing uncertainty using data from six individual epidemiologic studies of long-term PM_2.5 exposure and mortality that were conducted in different geographic locations worldwide and at different PM_2.5 concentrations. The resulting uncertainty characterization has the potential to be quantitatively applied in the context of both a risk assessment and benefits analysis.

2. METHODS

2.1. Derivation of PM_2.5 Concentration Distributions to Estimate Uncertainty.

Our method relies on the fact that measured PM_2.5 concentrations are positive, and as such their distributions are non-normal, often skewed to the left. In such cases the concentration distribution has the property that its mean and variance are related, with the variance decreasing as the mean decreases due to the range of the concentration distribution shrinking in size. In addition, the uncertainty or standard error, $\widehat{\epsilon }$, of $\widehat{\beta }$ from an epidemiologic study is proportional to the inverse of the standard deviation of the PM_2.5 concentration distribution (19), indicating that the uncertainty in $\widehat{\beta }$ increases as the variation in observed PM_2.5 concentrations in a study decreases. Based off this property it is possible to predict the uncertainty in $\widehat{\beta }$ under the scenario of concentration distributions where mean annual PM_2.5 concentrations are lower than the mean reported in a study. Within this approach, the assumption is that both $\widehat{\beta }$ and the population size of the study of interest remain constant.

Because PM_2.5 concentrations represent positive values, their distribution within a study (z) can be modelled as a distribution with positive support, such as the log-normal distribution. In this case, the logarithm of $z$ has a normal distribution with mean $\mu$ and variance ${\sigma }^2$. Therefore, the distribution of $z$ has mean $e^{\mu +{\sigma }^2/2}$ and variance $\left(e^{{\sigma }^2}-1\right)e^{2\mu +{\sigma }^2}$ (20). Further suppose the observed mean of $z$ in the available study is $m_{obs}$ with variance $v_{obs}^2$. This results in:

$$m_{\text{obs}}=e^{\mu +{\sigma }^2/2}$$

and

$$v_{\text{obs}}^2=\left(e^{{\sigma }^2}-1\right)e^{2\mu +{\sigma }^2}$$

We can then calculate the mean, ${\mu }_{obs}$, and variance, ${\sigma }_{obs}^2$, of the log-normal distribution from the observed mean and variance of $z$ as:

$${\mu }_{\text{obs}}=-\frac{1}{2}\log \left(\frac{v_{\text{obs}}^2+{m_{\text{obs}}}^2}{{m_{\text{obs}}}^4}\right)$$

and

$${\sigma }_{\text{obs}}^2=2\left(\log \left(m_{\text{obs }}\right)-\mu \right)$$

We then want to know how the variance of $z$ would change as the mean of the concentration distribution decreases. We can predict how the variance changes if we relate the mean of $\mathrm{l}\mathrm{o}\mathrm{g}\left(z\right)$ to its variance by $\frac{{\sigma }_{\text{obs }}^{\text{z}}}{{\mu }_{\text{obs }}}={\alpha }_{obs}$. In this case we can write both the mean, $e^{\mu \left(1+\frac{a_{obs}}{z}\right)}$ and variance, $\left(e^{{\alpha }_{obs}\mu }-1\right)e^{\mu \left(2+{\alpha }_{obs}\right)}$, of the distribution of $z$ as functions of $\mu$ and ${\alpha }_{obs}.$ Then for any new hypothesised mean, $m_{new}$, we can calculate the corresponding variance of $z$, as

$$v_{new}^2=\left(e^{{\alpha }_{obs}\tilde{\mu }}-1\right)e^{\tilde{\mu }\left(2+{\alpha }_{obs}\right)}$$

where $\tilde{\mu }=\log \left(m_{new}\right)/\left(1+{\alpha }_{obs}/2\right)$. The variance of a new PM_2.5 concentration distribution can then be predicted given a new mean value using an estimate of the ratio of the log-normal variance to the mean annual PM_2.5 concentration (${\alpha }_{obs}$) from any epidemiologic study.

2.2. Calculation of Uncertainty at Alternative Mean Annual PM_2.5 Concentrations.

After estimating the variance of a PM_2.5 concentration distribution with a given new mean value, it is then possible to estimate the new standard error, ${\widehat{ϵ}}_{new}$, of $\widehat{\beta }$ under the condition of the new mean $\left(m_{new}\right)$:

$${\widehat{ϵ}}_{new}\left(m_{new}\right)=\widehat{\epsilon }\times \frac{v_{obs}}{v_{\text{new }}}\equiv \widehat{\epsilon }\times ADJ\left(m_{\text{new }}\right)$$

with ${\widehat{ϵ}}_{new}\left(m_{\text{obs }}\right)=\widehat{\epsilon }$ and the standard error adjustment factor $ADJ\left(m_{obs}\right)=1$. That is, the new estimate of uncertainty is equal to the estimate of uncertainty from any single epidemiologic study when the mean annual PM_2.5 concentration of the new concentration distribution is equal to the mean annual PM_2.5 concentration of the study of interest. This equation is derived from the standard error of $\widehat{\beta }$ in a Cox PH model, which is a function of the inverse of the variation in the underlying distribution (e.g., PM concentrations; ²¹). In a linear model, this relationship is exact (19); however, because the variation in the underlying distribution is not readily isolable in the standard error equation for a non-linear Cox PH model, this adjustment represents a relative approximation. Based on this approach, the adjustment factor would be less than unity for mean exposures above the observed study mean, resulting in narrower uncertainty intervals compared to what would be expected using the study specific standard error. As a result, because the focus of this new approach is in characterizing uncertainty at exposures less than the observed study mean, we instituted a hybrid uncertainty method. Specifically, we set ${\widehat{ϵ}}_{new}$ equal to $\widehat{\epsilon }$ from the epidemiologic study of interest above the study mean, resulting in a constant standard error above the mean annual concentration of the study, and allowing for increasing uncertainty as PM_2.5 concentrations decline from the study mean.

2.3. Evaluation of Uncertainty for Individual Studies

The proposed method can be implemented using parameters from an individual cohort study, even if there has been no formal assessment of shape or uncertainty in the C-R relationship. To demonstrate the applicability of this new method in characterizing uncertainty in the C-R relationship between long-term PM_2.5 exposure and mortality at low PM_2.5 concentrations, we apply this method using results extracted from six cohort studies selected to represent different geographic locations and different PM_2.5 concentration distributions. These large cohort studies include an extended-analysis of the American Cancer Society’s Cancer Prevention Study II (ACS CPS II; 22), a nationwide study in Canada – the Canadian Census Health and the Environment Cohort (CanCHEC) (23), a subset of the Nurses’ Health Study (NHS) consisting of participants in the Northeastern and Midwestern U.S. (8), the National Cohort of Chinese Men (NCCM) (24), the Rome Longitudinal Study (RoLS) (25), and a nationwide study of all Medicare beneficiaries (5). As detailed previously in Sections 2.1 and 2.2, four parameters are required to quantitatively characterize uncertainty at various PM_2.5 concentrations below the annual mean: $\widehat{\beta },{\widehat{\epsilon }}^2$, and the study-specific mean ${(m}_{obs})$ and standard deviation ${(v}_{obs})$ of the PM_2.5 concentration distribution. In the case of a single epidemiologic study, this information can generally be extracted directly from the study results. All analyses were conducted using R Statistical Software (26). Example R code using the equations defined in Sections 2.1 and 2.2 to calculate hypothetical PM_2.5 distributions and hybrid uncertainty estimates for an individual cohort study can be found in Appendices 1 and 2 of the supplemental file.

2.4. Application of Uncertainty Characterization

Once the four necessary parameters are extracted from an individual study, we can then employ the hybrid uncertainty method to characterize uncertainty across the PM_2.5 concentration distribution. For the purposes of demonstrating this method, we generated hypothetical PM_2.5 concentration distributions with mean annual PM_2.5 concentrations at 0.1 μg/m³ increments above and below the mean annual PM_2.5 concentration reported in each of the individual studies. As an illustrative example, we focus on Turner et al. 2016 (22). Hypothetical distribution means ranged from a counterfactual concentration of 2.0 μg/m³, which represents an approximation of the lower limit of PM_2.5 concentrations in the U.S. based on the lowest measured level in recent studies (e.g., ^{22, 27}), to a defined concentration above the mean annual PM_2.5 concentration to depict how the uncertainty calculation changes at PM_2.5 concentrations below the study mean. After generating hypothetical PM_2.5 distributions based off the mean and variance of the PM_2.5 concentrations from each individual study, we calculated the standard error of $\widehat{\beta }$ at each 0.1 μg/m³ deviation from the individual cohort mean using the equations described in Sections 2.1 and 2.2. This application demonstrates the degree to which the standard error of $\widehat{\beta }$ increases as the mean annual PM_2.5 concentration decreases. Notably, this hybrid uncertainty method holds $\widehat{\epsilon }$ constant at concentrations above the mean.

For estimating the potential health impacts and economic benefits associated with strategies that lower population exposures to air pollutants, analysts require knowledge of the mathematical relationship between air pollution concentrations and the probability of the occurrence of a health outcome, known as the Health Impact Function (HIF) (28). The HIF, which provides an estimate of the excess number of events attributable to exposure, ΔY, is given by:

$$\text{Δ}Y=\left(1-e^{-\widehat{\beta }\times \text{Δ}AQ}\right)\times Y_0\times Pop$$

where $\widehat{\beta }$ is the effect estimate of interest, ΔAQ is the defined change in air quality (from baseline air quality to a control scenario air quality), Y₀ is the baseline incidence of the given health effect, and Pop is the exposed population (29).

Uncertainty in the HIF is defined by the statistical uncertainty in $\widehat{\beta }$, which is given by its standard error, $\widehat{\epsilon }$. Typically, a 95% confidence interval (CI) in the HIF can be defined by:

$$95\text{\%}C{I}_{\text{Δ}Y}=\left(1-e^{-(\stackrel{⌢}{\beta }\pm 1.96\times \widehat{ϵ})\times \text{Δ}AQ}\right)\times Y_0\times Pop$$

Notably, this formula is dependent on a constant standard error. In the case of the hybrid standard errors (${\widehat{ϵ}}_{\text{new }}$) calculations detailed in Section 2.2, the standard error is a function of concentration. As such, a single ${\widehat{ϵ}}_{new}$ cannot be assigned to the CI equation above, because the standard error varies across $\Delta AQ$.

Though we do not estimate health impacts in this paper, we suggest an alternative method to calculating uncertainty in the HIF to account for the concentration-dependent standard errors estimated using the approach described in this paper. Specifically, an average ${\widehat{ϵ}}_{\text{new }}$ can be calculated for the range of values from the baseline air quality concentration to the control scenario air quality concentration using a defined increment (i.e., calculate ${\widehat{\epsilon }}_{new(i)}$ for the baseline air quality concentration and then at each defined increment between the baseline and control air quality concentrations and subsequently calculate the average). The resulting ${\stackrel{⌢}{\mathcal{X}}}_{{\widehat{ϵ}}_{new}}$. can then be applied to the HIF CI function. The defined increment can take any value less than $\Delta AQ$, though smaller increments will result in more accurate estimates due to the non-linear relationship between concentration and ${\widehat{ϵ}}_{\text{new }}$.

3. RESULTS

The individual studies selected within this paper to demonstrate the application of the new method to quantitatively characterize uncertainty at low annual PM_2.5 concentrations consist of studies that examined the association between long-term PM_2.5 exposure and mortality in different geographic locations worldwide and at different PM_2.5 concentrations. The cohort studies included consist of subjects interviewed on their mortality risk factors, such as smoking and diet, and then followed over time to identify the date of death. We identified 6 cohort studies conducted in the U.S., Canada, Italy, and China that examined long-term PM_2.5 exposure and mortality and contained individual-level data on potential confounders (Table 1).

Table 1.

Study-specific details for each of the six cohort studies.

Study	Cohort	Deaths (Population)	Mean Annual PM2.5 Concentration (Standard Deviation)	Hazard Ratio (95% CI)a
Pinault et al. 2017 (23)	CanCHEC	233,300 (2,448,500)	7.4 (2.6)	1.18 (1.15, 1.21)
Di et al. 2017 (5)	Medicare	22,567,924 (60,925,443)	11.0 (2.9)	1.084 (1.081, 1.086)
Hart et al. 2015 (8)	NHS	8,617 (108,767)	12.0 (2.8)	1.13 (1.05, 1.22)
Turner et al. 2016 (22)	ACS	237,201 (669,046)	12.6 (2.9)	1.07 (1.06, 1.09)
Cesaroni et al. 2013 (25)	RoLS	144,441 (1,265,058)	23.0 (4.4)	1.04 (1.03, 1.05)
Yin et al. 2017 (24)	NCCM	50,022 (224,064)	40.6 (19.0)b	1.09 (1.08, 1.09)

ACS = American Cancer Society; CanCHEC = Canadian Census Health and Environment Cohort; CI = confidence interval; NHS = Nurses’ Health Study; RoLS = Rome Longitudinal Study; NCCM = National Cohort of Chinese Men

^a= all HRs standardized to 10 μg/m³ increase in PM_2.5 concentrations

^b= Yin et al. 2017 (24) did not provide an overall mean PM_2.5 concentration and standard deviation, therefore, an overall value was estimated from Table 2 of Yin et al. (24) by simulating PM_2.5 concentrations from the 4 time periods using the mean and standard deviation of each time period.

These cohort studies involve a wide range of subject profiles including only females (8), only males (24), older adults (i.e., > 65 years of age; 5), as well as, more broadly, an entire population (22–23, 25). Across the studies, sample size varies widely from a low of approximately 108,000 subjects (8) to a high of over 60 million subjects (5). Consequently, there is a wide range in mortality risk estimates (i.e., hazard ratios [HRs]) across studies. Table 1 presents each study’s hazard ratio per 10 μg/m³ increase in annual PM_2.5 concentrations (i.e., $\text{exp}(\widehat{\beta }\times 10))$, corresponding 95% confidence interval (CI), study-specific mean annual PM_2.5 concentration, and standard deviation. Across studies, mean annual PM_2.5 concentrations range from a low of 7.4 μg/m³ in the CanCHEC (23) to a high of 40.6 μg/m³ in the National Cohort of Chinese Men (24). HRs also display considerable variability ranging from HR = 1.04 in RoLS (25) to HR = 1.18 in CanCHEC (23).

Using the defined mean and standard deviation values from each of the six cohort studies, we developed study specific new hypothetical PM_2.5 concentration distributions, where the mean varies by 0.1 μg/m³. As an example, Figure 1 depicts hypothetical PM_2.5 concentration distributions at mean annual concentrations below the annual mean reported in Turner et al. 2016 (22), and the resulting reduction in the variance of each concentration distribution (see Table S1 for the variances that correspond to each hypothetical mean annual PM_2.5 concentration for each of the cohorts).

Figure 1.

Log-normal distribution generated from the mean annual PM_2.5 concentration from Turner et al. 2016 (22; blue) and a series of hypothetical annual PM_2.5 concentration distributions with means down to 5.5 μg/m³ (red).

Using the variance of each hypothetical mean annual PM_2.5 concentration distribution we calculated the standard error around the HR at each concentration for each of the six cohort studies. The HR (solid blue lines), standard uncertainty intervals (dashed red lines), and our hybrid uncertainty intervals (grey shaded areas) are displayed for each cohort study in Figure 2. The plots range from a counterfactual concentration of 2.0 μg/m³ to the nearest 5 μg/m³-increment above the mean annual PM_2.5 concentration for each study. A counterfactual PM_2.5 concentration above 1 μg/m³ was selected because as noted previously, this new method requires taking logarithms of mean annual PM_2.5 concentrations, and these are not defined at 1 or below.

Figure 2.

Characterization of uncertainty using both standard and hybrid approaches at annual PM_2.5 concentrations below the mean annual PM_2.5 concentration for six cohort studies (5, 8, 22–25).

Note: HR calculations are anchored to 2.0 μg/m³, such that the increment used to calculate HRs at each mean annual PM_2.5 concentration is equivalent to the mean concentration minus 2.0. If you standardize the HR at each mean concentration to a 10 μg/m³ increase, as is done in most long-term PM_2.5 exposure and mortality epidemiologic studies, the HR is equal to 1.18 for Pinault et al. (2017), 1.084 for Di et al. (2017), 1.13 for Hart et al. (2015), 1.07 for Turner et al. (2016), 1.04 for Cesaroni et al. (2013), or 1.09 for Yin et al. (2017) for all hypothetical mean annual PM_2.5 concentrations.

Figure 2 also depicts that the confidence intervals around the HR using the hybrid uncertainty method are generally much wider at lower PM_2.5 concentrations compared to the traditional approach often employed that assumes the standard error is constant across the full range of the PM_2.5 concentration distribution. The increased uncertainty resulting from the hybrid method is depicted in Table 2, where the hybrid uncertainty 95% CIs for the HR reported in Turner et al. 2016 (22) include unity at mean annual concentrations below 3.7 μg/m³, indicating more uncertainty at lower concentrations. The PM_2.5 concentration at which the hybrid uncertainty intervals first include unity varies by study, depending on the observed mean annual concentration and the standard error. For studies with extremely large sample sizes and/or low mean annual concentrations (e.g., ^{5, 23}), the hybrid 95% CIs never include unity, but are still wider than the standard intervals at concentrations below the respective study means (see Tables S2A-S2C for the HRs and corresponding hybrid uncertainty and standard uncertainty 95% CIs corresponding to each of the cohorts). The increased width of the CIs also suggest a higher probability that there might be a larger HR at low concentrations. In contrast, the traditional approach to estimating uncertainty assumes a constant standard error and the resulting CIs do not include unity at any concentration in the distribution for any of the cohort studies. Across the PM_2.5 concentration distribution, the hybrid uncertainty method results in larger standard errors at lower concentrations. For example, in Turner et al. 2016 (22) the hybrid uncertainty method results in an adjustment to the standard error that was 1.32 times larger at 10 μg/m³; 3.18 times larger at 5 μg/m³, and dramatically increased by a factor of 11.19 just above the counterfactual concentration (i.e., at 2.1 μg/m³) compared to the standard method (see Table S3 for the standard error adjustment factors for each of the cohorts).

Table 2.

Hazard Ratios (HR) and Standard and Hybrid 95% Confidence Intervals (95% CIs) applied to Turner et al. 2016 (22) for annual mean PM_2.5 concentrations ranging from 3.2 – 4.2 μg/m³ at 0.1 μg/m³ increments.

Mean Annual PM2.5 Concentration	Hazard Ratio (95% CI)a	Hazard Ratio (95% CI)b
3.2	1.0082 (0.9983, 1.0181)	1.0082 (1.0065, 1.0098)
3.3	1.0088 (0.9986, 1.0191)	1.0088 (1.0070, 1.0107)
3.4	1.0095 (0.9990, 1.0202)	1.0095 (1.0075, 1.0115)
3.5	1.0102 (0.9994, 1.0212)	1.0102 (1.0081, 1.0123)
3.6	1.0109 (0.9998, 1.0221)	1.0109 (1.0086, 1.0131)
3.7	1.0116 (1.0002, 1.0231)	1.0116 (1.0092, 1.0140)
3.8	1.0123 (1.0006, 1.0240)	1.0123 (1.0097, 1.0148)
3.9	1.0129 (1.0011, 1.0249)	1.0129 (1.0103, 1.0156)
4.0	1.0136 (1.0016, 1.0258)	1.0136 (1.0108, 1.0165)
4.1	1.0143 (1.0021, 1.0267)	1.0143 (1.0113, 1.0173)
4.2	1.0150 (1.0026, 1.0276)	1.0150 (1.0119, 1.0181)

^a= hybrid uncertainty method;

^b= standard uncertainty method

Note: HR calculations are anchored to 2.0 μg/m3, such that the increment used to calculate HRs at each mean annual PM2.5 concentration is equivalent to the mean concentration minus 2.0. If you standardize the HR at each mean concentration to a 10 μg/m3 increase, as is done in Turner et al. 2016 (22) the HR is equal to 1.07 for all hypothetical mean annual PM2.5 concentrations.

4. DISCUSSION

We describe a novel approach to quantitatively estimate uncertainty at low annual PM_2.5 concentrations by using the ratio of the mean and variance of annual PM_2.5 concentrations from an individual epidemiologic study to estimate the variance for a series of hypothetical mean annual PM_2.5 concentration distributions. Subsequently, it is possible to use the estimated variance of the hypothetical mean annual PM_2.5 concentration distributions to calculate adjusted standard errors of a HR from published epidemiologic studies and construct new 95% CIs at each of the hypothetical mean annual PM_2.5 concentrations. This allows for a more comprehensive characterization of uncertainty below the mean using commonly reported study results without the need for subject-level data. Additionally, this method can be applied to studies of health outcomes other than mortality (e.g., hospital admissions or emergency department visits). Although this new approach allows for the quantitative characterization of uncertainty, it does not empirically estimate the mean annual PM_2.5 concentration where there is less certainty in the shape of the C-R relationship for long-term PM_2.5 exposure and mortality.

Ideally to quantify uncertainty at lower concentrations, one would design and conduct an epidemiologic study where mean annual PM_2.5 concentrations are at the low end of the concentration distribution (i.e., < 8 μg/m³). However, there are not many locations worldwide that fit this requirement. As a result, we are currently relegated to devising alternative approaches that allow for the quantitative characterization of uncertainty at low PM_2.5 concentrations, such as the method presented within this paper.

Uncertainty at low annual PM_2.5 concentrations is just one of many areas of uncertainty that present challenges in quantifying the relationship between long-term exposure to PM_2.5 and mortality. Of the key categories of uncertainty mentioned earlier, namely, model specification, exposure error, heterogeneity in risk estimates across cohorts, and the shape of the C-R relationship, the proposed method can be used in concert with other available data to address some of these issues. First, as depicted in Figure 2, our method can be applied to a variety of cohort studies as long as the study provides a few commonly reported statistics (e.g., $\widehat{\beta },\widehat{\epsilon },$ and the mean and standard deviation of the study-specific PM_2.5 concentration distribution). Differences in results from cohort studies in varying locations and across different populations may be attributable to demographic, exposure, behavioral, and PM_2.5 source/composition differences on the PM_2.5-mortality relationship. Employing the hybrid uncertainty method across studies provides a range of estimates to account for these differences in individual cohort results. In addition to applying the hybrid uncertainty method to different cohorts, the method can be applied to the $\widehat{\beta }$ and $\widehat{\epsilon }$ derived from different model specifications within the same study. Researchers often present results from a variety of models with varying levels of adjustment for potential confounders using both individual-level and ecologic covariates (e.g., ^{3, 5, 30}). Applying the hybrid uncertainty approach to multiple effect estimates from the same study will similarly provide a range of estimates to account for uncertainty due to model specification. All of these estimates have the potential to be used in a risk assessment or benefits analysis and the results could be combined using random effects pooling to account for multiple outcomes within and between studies.

Quantitatively accounting for exposure error is less straightforward. Some epidemiologic studies have attempted to adjust effect estimates to correct for bias due to differences in the exposure surrogate (e.g., fixed-site monitors) and the “true” exposure (e.g., the ambient portion of personal exposure) using risk-set regression calibration (8, 31). While it is possible to apply the proposed method to effect estimates that have been adjusted for exposure error in this manner, this does not account for exposure error associated with the air quality monitoring data or modeled surfaces used in risk assessments or benefits analyses. For example, while Hart et al. (8) calibrate regression models to adjust HRs for differences between measured concentrations and “true” personal ambient exposures, the HR may subsequently be applied in a risk assessment, which often use a spatially-aggregated surrogate metric for ambient exposure. Another study compared health effect estimates from regression models using a variety of ground-based and/or remote-sensing exposure estimates (32). The authors compared Akaike Information Criteria (AIC) values under the assumption that models with better predictive ability would have less measurement error. Though Jerrett et al. (32) were unable to quantitatively estimate the impact of measurement error on effect estimates, they conclude that risk and benefits assessments should utilize effect estimates derived from exposure estimates that are most appropriate given the specific constraints of a study. The authors also propose an ensemble estimate that pools multiple effect estimates using AIC-adjusted weights, another approach that could be used with the method described in this paper.

Another area of uncertainty that this method does not address is the shape of the C-R relationship. In fact, a key assumption of this method is that the $\widehat{\beta }$ remains constant at low concentrations. In contrast, there are several approaches in the literature that are designed to provide quantitative estimates of the shape of the C-R relationship at specific PM_2.5 concentrations. Most of these methods apply function transformations (12, 33) or a combination of function transformations with weighting functions, referred to as a “Shape-Constrained Health Impact Function”, or SCHIF (10–11). The standard error of $\widehat{\beta }$ in a given SCHIF is constant and, as a result, the 95% CIs are always proportional to $\widehat{\beta }$ at a given PM_2.5 concentration. However, rather than selecting an optimal SCHIF, it is possible to incorporate additional adjustment for uncertainty in the shape of the C-R relationship by estimating an ensemble model using Bayesian model averaging of all iterations of the SCHIF, applying likelihood values as weights. The ensemble model CIs incorporate both sampling uncertainty and variation in shapes between models, which tends to generate additional uncertainty over ranges with limited data, as a result of multiple plausible shapes across concentrations where there are fewer data points to inform the flexibility of the model. This demonstrates that the linear C-R assumption of our method is subject to additional uncertainty that is not captured by the standard error of $\widehat{\beta }$. However, while the SCHIF provides a flexible non-linear approach to evaluating C-R relationships that gives greater insight to the magnitude of, and uncertainty in, PM-attributable mortality risk, it is generally only possible to estimate such a model with detailed cohort-level study information. Thus, the SCHIF is more difficult to develop and studies using this, or similar approaches are currently less widely available.

In addition to the assumption of a linear no-threshold C-R relationship, the proposed method also assumes that other parameters of the epidemiologic study being used as the basis for characterizing uncertainty remain constant, including sample size and the baseline mortality rate of the sample population. The assumption of a constant baseline mortality rate is also subject to uncertainty, as recent epidemiologic evidence indicates that health improves (e.g., in the form of reduced death rates) as PM_2.5 concentrations decrease (34). Additionally, while this method will allow for the identification of points along the C-R relationship where CIs include unity, this uncertainty characterization does not negate the observed linearity that exists in the C-R curve above that defined concentration where there is greater certainty. Finally, while the use of a constant standard error may result in an underestimation of uncertainty as concentrations deviate from the mean, it also potentially results in an overestimation of uncertainty at concentrations near the mean. Given that we use the hybrid uncertainty method to adjust the standard error using the mean of a hypothetical PM distribution, it follows that we are likely providing a conservative estimate of uncertainty at any given increment.

Overall, the method outlined within this paper fills a gap that currently exists in the PM literature by providing an innovative method for quantifying uncertainty in the C-R relationship for long-term PM_2.5 exposure and mortality at specific PM_2.5 concentrations. Until newer epidemiologic studies are conducted at lower annual PM_2.5 concentrations using more recent air quality data, this method has the potential to be useful in the process of conducting risk assessments and benefits analyses.

Supplementary Material

Table S1.

The variance for each hypothetical PM_2.5 concentration distribution with mean annual concentrations ranging from the counterfactual (2.0 μg/m³) to the mean reported in each study at 0.1 μg/m³ increments.

Sup1

Table S1. The variance for each hypothetical PM_2.5 concentration distribution with mean annual concentrations ranging from the counterfactual (2.0 μg/m³) to the mean reported in each study at 0.1 μg/m³ increments.

Table S2A. Hazard ratios (HR) and standard and hybrid 95% confidence intervals (CIs) applied to Pinault et al. 2017 and Di et al. 2017 for mean annual PM_2.5 concentrations ranging from the counterfactual (2.0 μg/m³) to the study means at 0.1 μg/m³ increments.

Table S2B. Hazard ratios (HR) and standard and hybrid 95% confidence intervals (CIs) applied to Hart et al. 2015 and Turner et al. 2016 for mean annual PM_2.5 concentrations ranging from the counterfactual (2.0 μg/m³) to the study means at 0.1 μg/m³ increments.

Table S2C. Hazard ratios (HR) and standard and hybrid 95% confidence intervals (CIs) applied to Cesaroni et al. 2013 and Yin et al. 2017 for mean annual PM_2.5 concentrations ranging from the counterfactual (2.0 μg/m³) to the study means at 0.1 μg/m³ increments.

Table S3. Standard error adjustment factors for hypothetical concentration distributions with mean annual PM_2.5 concentrations ranging from the counterfactual (2.0 μg/m³) to the mean reported in each study at 0.1 μg/m³ increments.

Appendix 1. R code for calculating the variance of hypothetical PM_2.5 distributions.

Appendix 2. R code for calculating hybrid uncertainty estimates.

ABBREVIATIONS

C-R

concentration-response

CI

confidence interval

HR

hazard ratio

PM_2.5

fine particulate matter with an aerodynamic diameter less than 2.5 μm