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\newcommand{\PMTen}{ PM$_{10}$ }
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\title{eAppendix for: Impact of National Ambient Air Quality Standards nonattainment designations on particulate pollution and health }
\date{}
%\author{
%Corwin M. Zigler\thanks{Department of Biostatistics, Harvard TH Chan School of Public Health, Boston, MA, 02115} 
%\and Christine Choirat \thanks{Department of Biostatistics, Harvard TH Chan School of Public Health, Boston, MA, 02115}
%\and  Francesca Dominici\thanks{Department of Biostatistics, Harvard TH Chan School of Public Health, Boston, MA, 02115}
%}
\IfFileExists{upquote.sty}{\usepackage{upquote}}{}
\begin{document}


\maketitle

\appendix

\section{Details and Tools For Constructing the Analysis Data
  Set}\label{app:dataconstruction}
\subsection{Data Sources}

\begin{table}
  \begin{tabular}{p{4cm}p{5cm}p{3cm}c} \hline \hline

    Measure & Data Source & temporal, spatial resolution & web link \\

    \hline

    Ambient \PMTwo & EPA Air Quality System & annual, monitoring site & \url{https://www.epa.gov/aqs}\\

    Nonattainment designations & EPA Green Book &annual, county & \url{https://www.epa.gov/green-book} \\

    Medicare Mortality and Hospitalization & Center for Medicare and Medicaid Services & annual, zip code & \url{https://www.cms.gov/}\\

    Population Demographics & US Census and MCDC Data Archive
                          & year 2000, zip code
                            tabulation area & \specialcell[t]{\url{https://www.census.gov/}\\\url{http://mcdc.missouri.edu/}}\\

    Smoking Rates & CDC and \cite[Additional file
                    3]{dwyer-lindgren_cigarette_2014} & annual, county
                                                         &
                                                           \specialcell[t]{\url{https://www.cdc.gov/}
    \\ \url{https://goo.gl/tNbpsS}}\\

    Climate & Automated Surface Observing System & annual, monitoring site & \url{http://www.nws.noaa.gov/asos/}\\

  \end{tabular}
\end{table}

%% https://goo.gl/tNbpsS short version of
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The tools to construct the analysis data include:
\begin{enumerate}
\item The AREPA R Package (available at:
  \url{https://github.com/czigler/arepa})
\item Several R scripts (available at: \url{https://github.com/czigler/PM2.5-Nonattainment})
\item Several raw data files (available at: \url{https://dataverse.harvard.edu/dataverse/pm97naaqs}).  
Note that Medicare data are not provided, rather, we provide simulated data of the same format used for linking the data sets as described below.
\end{enumerate}

To assemble the analysis data set, we first obtained the locations of
all ambient \PMTwo monitors in the AQS in operation between 1997 and
2012, and enumerated which were located in areas designated as
nonattainment for \PMTwo in 2005.  To help ensure that monitors had
data spanning all seasons, monitors were only included in a given year
if the annual percentage of valid measurements for that year was at
least 67\%. For each monitoring location, demographic information for
the surrounding general population in the year 2000 was obtained by
aggregating Census variables among all zip codes with centroids
located within 6 miles of the monitoring location.  Climate variables
for each monitoring location were obtained by averaging readings taken
from all ASOS monitors located within 150km of the AQS monitoring
station.  Medicare health outcomes for each monitoring location were
obtained by aggregating mortality and hospitalization outcomes among
all Medicare beneficiaries residing in a zipcode with a centroid
located within 6 miles of the monitoring location.  Medicare
beneficiaries residing within 6 miles of more than one monitoring
station were uniquely assigned to the closest monitoring station, and
zip codes with fewer than 15 beneficiaries were excluded.
Individual-level health data are aggregated to the level of of the
monitoring location to yield outcome rates (mortality rate,
hospitalization rates) for all beneficiaries residing near that
location.

Finally, any monitoring location with missing \PMTwo measurements
during the baseline and follow up period or with fewer than 20
Medicare beneficiaries residing in the zip codes located within 6
miles in 2012 are excluded to yield the initial analysis data set of
829 monitoring locations.

\section{Details of the Propensity Score ``Design'' Stage}\label{app:propensityscores}
Propensity scores are estimated from a logistic regression of the
form:
\begin{equation}
  log(\frac{p(A_i=1)}{1-p(A_i=1)}) = X_i\gamma \label{psmod}
\end{equation}
where $A_i=1$ denotes that the $i^{th}$ monitoring location lies in a
nonattainment area, $X_i$ is a vector of covariates measured at the
$i^{th}$ location, including an intercept, main effects for the
variables denoted in Table \ref{datatab} of the main text, and two
interaction terms: one between average temperature and average
relative humidity and another between average temperature and baseline
ambient \PMTwo during 2002-2004. The unknown parameter $\gamma$ is
estimated with maximum likelihood to construct predicted values from
model (\ref{psmod}), which are estimated propensity scores.

Propensity scores are estimated across the range 0 to 1, with
locations in nonattainment areas tending to have higher estimated
propensity scores.  Importantly, the 109 locations with estimated
propensity scores greater than 0.97 were exclusively nonattainment
locations, and the 316 attainment locations with estimated propensity
scores less than 0.019 were exclusively in attainment areas. These
observations with propensity scores that do not ``overlap'' with the
comparison group are ``pruned'' from the analysis data set to prevent
extrapolation of causal comparisons beyond what can be supported by
the observed data \citep{king_dangers_2006, ho_matching_2007,
  crump_dealing_2009}.  Note that the ability of the pruning strategy
to ensure comparisons are confined to locations that are actually
comparable is underscored by Figure \ref{maps} of the main text.  Even
though geography is not explicitly considered in the propensity score
pruning, the resulting analysis data set is geographically more
tightly clustered around the locations of the nonattianment areas to
prevent, for example, outcomes in western Iowa or northern Maine from
being construed as part of the ``control group'' for nonattainment
locations that are almost certainly not comparable.  After pruning,
the remaining 404 monitoring locations are grouped into four groups
based on quartiles of the estimated propensity score distribution.
eTable 1 lists the number of attainment and nonattainment
areas in each of the four subgroups.
\begin{table}
  \centering
  \caption*{eTable 1: Number of attainment and nonattainment areas in each of the four propensity score subclasses used for confounding adjustment}\label{psgrps}
  \begin{tabular}{lcccc} \hline \hline
    & \multicolumn{4}{c}{\underline{Propensity Score Quartile}}  \\
    & 1$^{st}$ & 2$^{nd}$ & 3$^{rd}$ & $4^{th}$\\ \hline
    Attainment &94&80&40&7\\ 
    Nonattainment &7&21&61&94\\  \hline
    Total &101&101&101&101\\

  \end{tabular}
\end{table}




\begin{table}[p]
\caption*{eTable 2: Covariate and outcome comparison summary between locations retained after propensity score pruning and locations pruned due to non-overlapping propensity scores:  Variables marked with $^a$ are those included in the model that estimates the propensity score, and those marked with $^b$ are those included for additional covariate adjustment in models for pollution and health outcomes.  Medicare health outcomes are listed as rates per 1000 beneficiaries for mortality and per 1000 person-years for hospitalizations. }\label{datatab_pruned_v_retained}
\scriptsize
\begin{tabular}{lcccc|cccc} \hline \hline
 & \multicolumn{4}{c}{\underline{Attainment Areas ($n=537$)}} & \multicolumn{4}{c}{\underline{Nonattainment Areas ($n=292$)}} \\
 & \multicolumn{2}{c}{Retained ($n=221$)} & \multicolumn{2}{c}{Pruned ($n=316$)}  & \multicolumn{2}{c}{Retained ($n=183$)} & \multicolumn{2}{c}{Pruned ($n=109$)} \\
  & Mean & SD & Mean & SD & Mean & SD & Mean & SD\\ \hline
  \underline{Pollution Monitoring Data} \\
Ambient PM2.5 2002-2004 ($\mu g/m^3$)$^a$$^b$&12.94&1.25&10.65&1.65&13.93&1.08&15.41&1.36\\ 
\% Ozone nonattainment 2005$^a$&61.99&48.65&8.54&28&89.07&31.29&98.17&13.48\\ 

\\ \underline{Population Demographics (Year 2000)} \\
log(population)$^a$$^b$&11.82&1.41&11.14&1.66&12.33&1.38&13.03&1.26\\ 
Completely Rural Area$^a$&0.05&0.23&0.1&0.3&0.01&0.1&0&0\\ 
\% Urban$^a$$^b$&80.61&26.75&73.85&30.45&88.49&19.45&95.6&8.29\\ 
\% Black$^b$&18.98&17.04&15.62&17.85&20.59&18.57&23.56&20.65\\ 
\% Hispanic$^b$&4.51&4.45&4.28&6.67&5.63&7.76&7.97&8.69\\ 
\% HS Grad.$^a$$^b$&30.96&6.72&31.31&6.76&31.45&6.55&29.06&8.15\\ 
Median HH Inc. (\$)$^a$&38397.91&8646.72&36594.68&8948.49&40892.48&11702.77&45457.2&17408.51\\ 
\% Poor$^a$$^b$&14.13&5.38&14.64&6.23&13.5&6.2&13.9&7.18\\ 
\% Female&51.56&1.28&51.44&1.59&51.76&1.69&51.67&1.08\\ 
\% Occupied Housing$^a$$^b$&91.88&4.47&89.8&8.52&92.43&3.21&92.41&3.18\\ 
5-Year Migration Rate$^a$&0.48&0.07&0.49&0.08&0.46&0.07&0.47&0.08\\ 
Median House Value (\$)$^a$&104426.49&45537.69&93905.49&32599.74&115520.8&53009.24&153942.61&103056.83\\ 
Smoking Rate$^a$&0.27&0.03&0.26&0.03&0.27&0.03&0.26&0.03\\ 

\\ \underline{Climate (Years 2004 - 2006)} \\
Avg. Dew Point ($^\circ$F)$^a$&45.79&5.26&46.73&9.37&44.82&3.27&44.32&2.94\\ 
Avg. Temperature ($^\circ$F)$^a$&55.58&5.85&56.54&9.98&54.26&3.77&53.46&3.18\\ 
Avg. Rel. Humidity (\%)$^a$$^b$&71.66&1.85&72.76&1.7&71.87&1.7&72&1.48\\ 

 \\ \underline{Baseline Medicare Characteristics (Year 2004)} \\
Total Medicare Benef. 2004$^a$$^b$&10637.62&9504.98&8373.79&10465.32&15184.84&16590.16&17014.87&17036.36\\ 
Avg. Medicare Age 2004 (years)$^a$$^b$&75.23&0.87&75.16&0.99&75.36&0.96&75.59&1.08\\ 
\% Female Medicare Benef. 2004$^a$$^b$&59.57&2.56&58.75&3.09&59.81&2.35&59.81&2.17\\ 
\% White Medicare Benef. 2004$^a$$^b$&85.98&15.2&87.93&15.56&83.04&21.03&78.77&21.96\\ 
\% Black Medicare Benef. 2004$^a$$^b$&12.09&14.84&9.83&14.39&14.49&20.4&16.56&20.21\\ 
Mortality$^a$&53.79&9.37&51.78&8.95&53.21&6.35&53.65&9.21\\ 
All CVD$^a$&112.53&27.36&105.65&25.3&119.38&24.14&122.32&29.25\\ 
Respiratory$^a$&33.27&13.58&30.78&12.31&34.01&10.67&35.38&14.22\\ 
COPD&11.16&7.4&10.14&5.91&11.03&4.92&11.66&5.9\\ 
CV Stroke&19.73&5.61&18.16&6.02&20.56&5.22&20.09&4.69\\ 
Heart Failure&25.51&8.75&22.5&9.2&27.7&9.3&30.01&11.78\\ 
HRD&15.24&5.11&14.96&4.76&16.28&3.99&15.76&3.47\\ 
Ischemic Heart Disease&30.04&11.44&29.57&9.41&30.92&9.19&30.91&8.86\\ 
Peripheral Vascular Disease&6.9&2.95&6.63&3.29&6.98&2.12&7.77&2.87\\ 
Respiratory Tract Infection&22.1&8.02&20.64&8.13&22.99&6.84&23.72&10.28\\ 

\\ \underline{Pollution and Health Outcomes} \\
Ambient PM2.5 2010-2012 ($\mu g/m^3$)&10.01&1.29&8.89&1.74&10.86&1.3&11.53&1.32\\ 
Mortality 2012&49.22&6.26&47.21&8.17&47.96&6.11&48.99&13.41\\ 
COPD 2012&11.36&7.24&9.37&5.75&11.39&4.88&11.54&5.94\\ 
CV Stroke 2012&15.44&4.16&14.13&5.21&15.89&4.05&15.93&4.6\\ 
Heart Failure 2012&17.89&5.85&15.46&6.56&19.11&7.56&19.18&6.6\\ 
HRD 2012&13.96&3.94&13.48&4.86&15.36&4.23&14.26&4.09\\ 
Ischemic Heart Disease 2012&14.9&5.26&16.02&6.26&15.74&4.91&15.59&5.88\\ 
Peripheral Vascular Disease 2012&4.1&1.62&4.27&2.97&4.69&2.4&4.91&2.16\\ 
Respiratory Tract Infection 2012&16.27&6.7&14.52&6.63&16.41&5.35&15.73&5.94\\ 
\end{tabular}
\end{table}

% Investigation of whether this propensity score strategy is
% successful in balancing covariate distributions between attainment
% and nonattainment locations within subgroup appears in Figure
% \ref{balcheck}.  This figure depicts the standardized mean
% difference of each measured covariate between attainment and
% nonattainment locations.  This quantity can be calculated for each
% covariate as a way to measure the threat of confounding for that
% covariate; values far from zero indicate poor balance and strong
% threat of confounding, whereas values closer to zero indicate better
% balance and less susceptibility to confounding.  The red line
% depicts the standardized mean difference for all 829 locations,
% without propensity score adjustment, indicating substantial
% differences between attainment and nonattainment locations and a
% strong threat of confounding bias.  In contrast, the blue line
% indicating differences after propensity score pruning and grouping
% are much closer to zero, indicating that virtually every covariate
% is balanced, on average, among attainment and nonattainment
% locations within propensity score subclass.  Note that while Figure
% \ref{balcheck} indicates balance on average across the five
% subgroups, the following variables exhibit an absolute standardized
% mean difference of at least 0.25 within at least one of the five
% propensity score groups: log(population), \% black, \% high school
% graduates, median household income, \% housing occupancy, average
% dew point, average temperature, average relative humidity, and the
% total number of, average age of, \% female, \%white, and \% black
% Medicare beneficiaries.  These variables are denoted in Table
% \ref{datatab} of the main text, and, as detailed later, will be
% included in analysis models in addition to the propensity score.
% \begin{figure}[h]
%   \center
%   \caption{Plots of average standardized mean difference in each
%   measured covariate between attainment and nonattainment locations.
%   Red line denotes differences before propensity score pruning and
%   subclassification, blue line denotes differences within propensity
%   score subclass.}
%   \includegraphics[width =
%   .5\textwidth]{Figures/balplot.pdf}\label{balcheck}
% \end{figure}







\section{Details of the Models Used in the ``Analysis
  Stage''}\label{app:modelspec}

\subsection{Spatial Hierarchical Model for Air Pollution}
Estimates of effects of the nonattainment designations on average
ambient \PMTwo in 2010-2012 are derived from a model fit to
(log-transformed) potential pollution values of the form:
\begin{align}
  Y_i(A)=X_i\beta^A + W_i^A + \epsilon_i^A, \label{lmcmodelpre}
\end{align}
where $Y_i(A)$ denotes the potential pollution concentration that
would be observed under designation $A$, with $A=0$ denoting
``attainment'' and $A=1$ denoting ``nonattainment.''  In this model,
$X_i$ include dummy variables indicating propensity score subclass
membership as well as main effects for the covariates that showed some
sign of imbalance in the design, as denoted in Section
\ref{app:propensityscores} and in Table \ref{datatab} of the main
text.  $\epsilon_i^A$ are normally-distributed non-spatial ``nugget''
errors, $\epsilon^A_i \sim N(0, \psi^A)$.  $W^A_i$ are spatial random
effects defined with a multivariate gaussian process having an
isotropic and stationary exponential covariance function.  The spatial
decay of covariances between two locations governed by parameters,
$\theta^A$, and the variances of these spatial random effects are
denoted $\tau^A$.  Specifying separate models for the values of $A$
implies the assumption that, conditional on $X_i$, the potential
pollution concentrations under $A=0$ and $A=1$ are conditionally
independent.  Details of this model specification, including the
structure of a sensitivity analysis to this conditional independence
assumption, can be found in \cite{zigler_estimating_2012}.  Further
details of the spatial random effect specification can be found in
\cite{banerjee_gaussian_2008}.


\subsection{Log-linear model for Medicare Health Outcomes}
Estimates of the effects of the nonattainment designations on Medicare
health outcomes derive from augmenting the spatial pollution model in
(\ref{lmcmodelpre}) with log-linear models for the mortality and
hospitalization rates.  We assume conditional independence of
potential health outcomes, conditional on covariates and air
pollution.  We also assume that potential health outcomes under a
nonattainment designation status are independent of what pollution
would have been under an attainment designation, after conditioning on
covariates and pollution under a nonattainment designation.  As a
result of these assumptions, models for potential health outcomes can
be written as:\begin{align} \log(E[H_i(A)]) = \alpha_0^A +
  X_i\alpha_1^A + Y_i(A)\alpha_2^A + \log(N_i), \label{poismod}
              \end{align}
              where $H_i(A)$ is the potential number of outcomes
              (mortalities or hospitalizations) at location $i$ under
              designation $A$ and $N(s)$ is the total number of
              Medicare beneficiaries (managed care and
              fee-for-service) for the mortality outcome, or
              person-years among fee-for-service beneficiaries for the
              hospitalization outcomes.  $X_i$ is a vector of
              covariates including dummy variables for propensity
              score subclass membership and main effects for the same
              covariates included in (\ref{lmcmodelpre}) and denoted
              in Appendix \ref{app:propensityscores} and in Table
              \ref{datatab} of the main text.

\subsection{Missing Data}\label{app:missingdata}
Our analysis is confronted with two types of missing ambient pollution
data.  Among all locations in the analysis data set, 131 (40 in
nonattainment areas) had missing baseline \PMTwo measurements in
2002-2004.  Missing values of average annual ambient pollution in
2002-2004 were imputed using a single posterior mean prediction from a
spatial hierarchical random effects model of the form
(\ref{lmcmodelpre}) including all baseline demographic and climate
variables listed in Table \ref{datatab} of the main text, as well as
indicators for a nonattainment designation for \PMTwo, PM$_{10}$,
ozone, and sulfur dioxide in 2005.  Among the locations retained after
propensity score pruning, 67 (27 in nonattainment) used this imputed
value of ambient \PMTwo in 2002-2004.

The second type of missing pollution data are missing average annual
ambient concentrations during the follow up period of 2010-2012.  In
the entire set of analysis locations, 263 (74 nonattainment) had
missing follow up pollution.  Among those retained after propensity
score pruning, 118 (52 in nonattainment) had missing follow up
pollution. These follow up pollution measures are used as outcomes in
our analysis, and are multiply imputed from the models described above
as a byproduct of our Bayesian estimation procedure.  No part of the
analysis used locations that had both missing baseline and follow up
pollution.  All other covariates and outcomes listed in Table
\ref{datatab} were fully observed.


\subsection{Bayesian Estimation and Prior Specification}\label{app:pm10:estimation}
Causal effects estimates are derived from the posterior distribution
of the parameters in the above models, estimated with Markov chain
Monte Carlo (MCMC) data augmentation algorithm that iteratively
samples missing potential outcomes conditional on observed data and
parameters, then samples parameters and calculates causal effect
estimates conditional on ``complete'' data.  For example, one
iteration of the MCMC algorithm calculates the average difference
between $Y(A=1)$ and $Y(A=0)$ in the sample of retained nonattainment
locations to estimate the sample average causal effect of the
nonattainment designation on ambient pollution among nonattainment
areas.  For a given location, at most one of these values is observed
and at least one of these values is simulated form its
posterior-predictive distribution.  Health effects are estimated
analogously.  Further details appear in \cite{zigler_estimating_2012}.
For the analyses of the main text, MCMC chains were run for 50,000
iterations, with the first 5,000 discarded as ``burn in'' and every
10$^{th}$ posterior sample retained for posterior inference.


\subsubsection{Prior distributions and outline of MCMC strategy}
We treat the parameters
$\beta^0, \beta^1, \psi^A, \psi^1,\tau^A,\tau^1, \theta^0, \theta^1,
\alpha^0$ and $\alpha^1$, as \textit{a priori} independent.  We
specify flat priors for $\beta, \alpha^0$ and $\alpha^1$.  For the
$\psi^A$ and $\tau^A$, we specify independent inverse-gamma
distributions with shape parameters set to 2 and scale parameters set
to 0.5.  For $\theta^A$, we specify uniform prior distributions on the
interval $(0.48,3.61)$.  Parameters for the prior distributions of
$\psi^A$ and $\theta^A$ are meant to reflect diffuse prior information
within the range of plausible parameter values.




\section{Sensitivity Analysis without Pruning}\label{app:sensitivity}
\subsection{Pollution Only Model}
A sensitivity analysis analogous to that described in the main text
was carried out on the entire analysis data set of 829 monitoring
locations (i.e., without the propensity score pruning).  This analysis
uses the same propensity score estimates, but no observation is pruned
due to non-overlapping propensity scores.  Locations were grouped into
four propensity score subclasses according to whether the estimated
propensity score was: 1) $\le 0.25$; 2) $>0.25$ and $\le 0.5$; 3)
$>0.5$ and $\le 0.75$; or 4) $>0.75$.  The number of attainment
(nonattainment) locations in each subclass was 467 (19), 37 (17), 21
(32), 12 (224), respectively.  Figure \ref{balcheck_noprune} shows the
covariate balance when all 829 locations are grouped into a propensity
score subclass, indicating substantial imbalances relative to the
pruned analysis of the main text and pronounced potential for
confounding.  Models of the form (\ref{lmcmodelpre}) and
(\ref{poismod}) were fit, adjusting for propensity score subclass and
main effects for the following covariates: \PMTwo in 2002-2004, ozone
nonattainment designation in 2005, log of population, \% urban
population,\% black, \% high school graduates, \% occupied housing,
average temperature, average relative humidity, and total number of,
average age of, \% female, \% white, and \% black Medicare
beneficiaries in 2004, all of which had average ASMD across all
subclasses of greater than 0.25 or a difference greater than 0.5 in
any subclass.
\begin{figure}[h]
  \center
  \caption*{eFigure 1: Plots of average standardized mean difference in each
    measured covariate between attainment and nonattainment locations
    for the sensitivity analysis without propensity score pruning.
    Red line denotes differences before propensity score
    subclassification, blue line denotes differences within propensity
    score subclass.}
  \includegraphics[width =
  .5\textwidth]{Figures/balplot_noprune.pdf}\label{balcheck_noprune}
\end{figure}

For the unpruned analysis, the estimated causal effect of the
nonattainment designations on ambient \PMTwo in 2010-2012 among the
292 nonattainment areas was -0.459 $\mu g/m^3$, with 95\% posterior
interval (-1.342, 0.289).  dFigure 2 summarizes
posterior distributions for the average causal effects of the
nonattainment desingations on the Medicare mortality rate (per 1000
beneficiaries) and hospitalization rates (per 1000 person-years) among
all nonattainment locations, as well as average dissociative and
associative effects.  This sensitivity analysis indicates significant
overall reductions in rates of all-cause mortality and
hospitalizations for heart failure.  Estimates of dissociative effects
are all very similar to estimates of overall effects.  Associative
effects are more pronounced than dissociative effects for rates of
all-cause mortality and COPD, Heart Failure, IHD, PVD, and RTI
hospitalizations, with significantly negative estimates for all-cause
mortality, COPD, Heart Failure, IHD, and RTI.

\begin{figure}[p]
  \center
  \caption*{eFigure 2: Posterior mean point estimates and 95\% posterior
    probability intervals for overall, associative, and dissociative
    effects in the sensitivity analysis of \PMTwo nonattainment
    designations without propensity score
    pruning.}\label{cehplot_noprune}
  \includegraphics[width=\textwidth]{Figures/20161031_med_noprune.pdf}
\end{figure}

Note that the MCMC chain for the sensitivity analysis of pollution
outcomes was run for 19500 iterations (after discarding 5000 as burn
in) of which every 10$^{th}$ sample was retained for posterior
inference.  Sensitivity analyses of health outcomes were based on MCMC
chains run for between 8000 and 13000 iterations (after discarding
5000 as burn in), depending on the outcome, with every 10$^{th}$
iteration retained for posterior inference.
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\end{document}
