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Atmospheric Environment 45 (2011) 4728e4734

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Atmospheric Environment

journal homepage: www.elsevier .com/locate/atmosenv

Can mortality displacement mask thresholds in the concentration-responserelation between particulate matter air pollution and mortality?

Steven Roberts*

School of Finance, Actuarial Studies and Applied Statistics, College of Business and Economics, Australian National University, Canberra ACT 0200, Australia

a r t i c l e i n f o

Article history:Received 1 May 2010Received in revised form19 April 2011Accepted 19 April 2011

Keywords:Air pollutionConcentration-responseHarvestingMortalityMortality displacementParticulate matterTime-series

* Tel.: þ61 2 61253470; fax: þ61 2 61 50087.E-mail address: steven.roberts@anu.edu.au.

1352-2310/$ e see front matter � 2011 Elsevier Ltd.doi:10.1016/j.atmosenv.2011.04.050

a b s t r a c t

Two critical issues forpublic health assessment and regulationof exposure toparticulatematter air pollution(PM) arewhether there is a threshold in the concentration-response relation between PMandmortality andwhether there is mortality displacement. In this vein, a number of time-series studies have concluded thata linear relation without a threshold is appropriate for describing the concentration-response relation, anda number of other studies have concluded that themortality effects of PM exposure cannot be attributed tomortality displacement alone. Using three-state (healthy, frail, dead) population models that incorporateactual time series data from Cook County, Illinois for the period 1987e2000, the author investigates theshape of the concentration-response relation between PM and mortality, as observable from time-seriesdata, in the presence of mortality displacement. It is found that thresholds in the concentration-responserelation can be masked, or hidden, by linear concentration-response relations if some of the effect of PMexposure is attributable to mortality displacement. This is an important finding that has implications forstudies that have found a linear concentration-response relation appropriate, particularly given the mark-edly different implications for public health assessment and regulation of a linear versus thresholdconcentration-response relation.

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1. Introduction

The shape of the concentration-response relation (Daniels et al.,2000; Koop and Tole, 2006; Samoli et al., 2005; Schwartz et al.,2008; Vedal et al., 2003) and mortality displacement (Dominiciet al., 2003; Fung et al., 2005a,b; Murray and Nelson, 2000;Roberts and Switzer, 2004; Schwartz, 2000; Smith et al., 1999;Zanobetti et al., 2000; Zeger et al., 1999) are two issues that havereceived extensive attention in investigations of the mortality effectof exposure to ambient particulate matter air pollution (PM).Knowledge of the shape of the concentration-response relationbetween ambient PM exposure and mortality is vital for publichealth assessment and the setting of appropriate regulatory stan-dards for PM (Anderson, 2009; Ren and Tong, 2008; Schwartz et al.,2008). Numerous studies have investigated the shape of theconcentration-response relation with many concluding that therelation is approximately linear; that is, that the incremental effect ofexposure to PM onmortality is constant (Daniels et al., 2000; Samoliet al., 2005; Schwartz et al., 2008; Schwartz and Zanobetti, 2000). Inmany studies a hypothesis of particular interest has been whether

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there is a threshold, that is a PM concentration below which expo-sure has no impact on mortality, in the concentration-responserelation (Daniels et al., 2000; Koop and Tole, 2006; Samoli et al.,2008; Vedal et al., 2003). A threshold in the concentration-response relation has important consequences for regulators as itimplies that a tightening of PM standards that results in a standardbelow the threshold will have zero, or less than anticipated, benefitsin terms of reduced mortality. Conversely, a linear concentration-response relation implies that any tightening of PM standards willhave resulting benefits in terms of reduced mortality. Mortalitydisplacement, or harvesting, is another issue that is critical for publichealth assessment of the mortality effect of ambient PM exposure(Roberts, 2011; Roberts and Switzer, 2004; Samet et al., 2000;Schwartz, 2000; Zeger et al., 1999). Mortality displacementdescribes the scenario where the only effect of PM exposure is tohasten the deaths of a frail subset of the population that havea relatively short expected future lifetime, irrespective of exposure toPM. Public health assessment of PM exposure would differsubstantially if it were known that the mortality effect of PM wasmortality displacement alone, rather than the deaths of otherwisehealthy individuals. A number of studies, utilizing different meth-odologies, have investigated the issue of mortality displacement(Dominici et al., 2003;Murray and Lipfert, 2010;Murray andNelson,2000; Schwartz, 2000; Smith et al., 1999; Zanobetti et al., 2000;

S. Roberts / Atmospheric Environment 45 (2011) 4728e4734 4729

Zeger et al., 1999). For the most part, these studies have concludedthat the mortality effects of PM exposure cannot be attributed tomortality displacement alone. If the effect of PM exposure is notmortality displacement alone this means that PM, apart from killingfrail individuals, may be killing otherwise healthy individuals and/orcausing otherwise healthy individuals to transition into a frail state.

Studies investigating thresholds in the concentration-responserelation have typically proceeded by directly modeling thresholdrelations and statistically comparing these relations to a linear relationand/or allowing the concentration-response relation to bemodeled bya smooth function and visually determining whether a linear orthreshold relation appears appropriate (Daniels et al., 2000; Samoliet al., 2008; Schwartz and Zanobetti, 2000; Smith et al., 2000). Asnoted above, the general consensus from investigations of the shape ofconcentration-response relation is that a linear relation is appropriatefor modeling the mortality effects of PM (Daniels et al., 2000; Samoliet al., 2005; Schwartz et al., 2008; Schwartz and Zanobetti, 2000).However, these results should be tempered somewhat by the findingsof Roberts and Martin (2006) who concluded that Akaike’s Informa-tion Criterion (AIC) was not always successful in detecting nonline-arities in the concentration-response relation between PM andmortality. The shape of the concentration-response relationship is animportant issue for other ambient air pollutants including ozone (Bellet al., 2006; Kim et al., 2004).

Mortality displacement in the association between PM andmortality has been investigated using a range of methods in anattempt to obtain estimates of the effect of PM that exclude short-termmortality displacement or to explicitly estimate the size of thefrail population and the effects of PM on this frail population(Dominici et al., 2003; Murray and Nelson, 2000; Schwartz, 2000;Smith et al., 1999; Zanobetti et al., 2000; Zeger et al., 1999). Asdiscussed above, these studies have generally concluded that themortality effects of PM exposure cannot be attributed to mortalitydisplacement alone. A common feature of a number of these studiesis the use of a three-state population model (healthy, frail, dead) tomotivate mortality displacement (Murray and Nelson, 2000; Smithet al., 1999; Zanobetti et al., 2000; Zeger et al., 1999). Under thismodel, individuals transition from the healthy population into thefrail population and deaths only occur from the frail population;mortality displacement corresponds to the situation where themean lifetime in the frail population is short and PM only affectstransitions from the frail population to death. Previous studies haveutilized the three-state model to simulate mortality for exploringthe properties of methods used to investigate mortality displace-ment (Fung et al., 2005a,b; Murray and Nelson, 2000; Roberts andSwitzer, 2004; Smith, 2003; Smith et al., 1999; Zeger et al., 1999).Some of these simulation studies have concluded that methodsdeveloped for investigatingmortality displacement may not alwaysbe successful (Fung et al., 2005b; Roberts and Switzer, 2004; Smith,2003). In a recent study the three-state population model wasextended to allow for multiple healthy and frail populations(Roberts, 2011). Mortality displacement is also an important issuefor other ambient air pollutants including ozone (Zanobetti andSchwartz, 2008).

To the best of the author’s knowledge this paper provides thefirst joint investigation of the issues of mortality displacement andthresholds in the concentration-response relation. The goal of theinvestigation is to determine the shape of concentration-responserelation between PM and mortality, as observable from time-series data, in the presence of mortality displacement and/ora frail population. In the presence of mortality displacement,a recent study developed a relationship between the sum of thecoefficients of a distributed lag model and the size of the frailpopulation (Roberts, 2011). Here our interest is not in the devel-opment of such a relationship for concentration-response models,

but rather in empirically observing the shape of the concentration-response relation in the presence of mortality displacement. It isillustrated that thresholds in the concentration-response relationcan be masked if some of the mortality effect of PM exposure isattributable to mortality displacement. This observation meansthat in some investigations a finding of a linear concentration-response relation may be an artifact of mortality displacementmasking an underlying threshold concentration-response relation.This is an important finding that has implications for studies thathave found a linear concentration-response relation appropriate,particularly given the markedly different implications for publichealth assessment of a linear versus threshold relation.

2. Data

The data for this study were obtained from the NationalMortality, Morbidity, and Air Pollution Study (NMMAPS) database.The NMMAPS database is a freely available database that containsdaily mortality, weather and air pollution data for selected areas inthe United States (US), for the period 1987e2000. The NMMAPSdatabase has been extensively used to investigate the associationbetween air pollution and mortality (Martin and Roberts, 2008;Peng et al., 2005; Ren et al., 2008). Further information on thisdatabase and access to this database can be obtained at http://www.ihapss.jhsph.edu/. For this study relevant data for CookCounty, Illinois, for the period 1987e2000 was extracted from thedatabase. The daily data extracted were: the number of non-accidental deaths of individuals aged over 65 years; averageconcentration of ambient particulate matter of less than 10 mm indiameter, measured in units of mgm�3; and average measures oftemperature and dew point temperature. The average ambientparticulate matter (denoted PM10) concentrations were con-structed by adding back a trend to a daily time-series of detrendedparticulate matter concentrations. Both the trend and detrendedtime-series are available in the NMMAPS database.

For the purposes of the simulations, missing PM10 concentra-tions (251 days) were imputed as the average of the one-day’sprevious and one-day’s subsequent PM10 concentrations. Twooutlying PM10 concentrations, four negative PM10 concentrationsand four outlying mortality counts had their values replaced usingthis scheme. In the investigations that follow the previous day’s(lag-1) PM10 concentration is used as this has been the focus ofmost attention in large multi-city studies conducted in the US(Peng et al., 2005; Welty and Zeger, 2005). Hereafter, PM10 willrepresent the lag-1 PM10 concentration.

3. Statistical methods

A three-state (healthy, frail, dead) population model is used togenerate mortality time-series that incorporate mortalitydisplacement. This model assumes that individuals start in thehealthy population and, at some point in time, transfer into the frailpopulation and that deaths occur from the frail population. Togenerate mortality time-series from the three-state model theauthor closely followed the specifications used by Roberts (2011),Roberts and Switzer (2004), Smith et al. (2000), and Zeger et al.(1999). Under this specification, on day t, it is assumed that etindividuals transition from the effectively infinite healthy pop-ulation into the frail population of size ft, and that among the nowftþ et individuals in the frail population dt of them die on day t. Tocomplete the model, it is assumed that (1) et is Poisson distributedwith mean mt; (2) dt is binomially distributed with probability ftand number of trials ftþ et; (3) the long-run average number ofentrants into the frail population is equal to the long-run averagenumber of deaths (d); and (4) the number of individuals in the frail

S. Roberts / Atmospheric Environment 45 (2011) 4728e47344730

population at the beginning of day 1 (ft) is equal to m� d, wheremis themean lifetime in the frail population. Themean lifetime in thefrail population is a measure of how long, on average, individualsspend in the frail population and is approximately equal to theaverage of the values of ft/dt. This approximation can be seen bynoting that the expected length of time in the frail population canbe approximately computed as 1fþ 2fð1� fÞ þ 3fð1� fÞ2þ4fð1� fÞ3 þ/ ¼ PN

k¼1kfð1� fÞk�1 ¼ f�1

, where f is theaverage probability of death, that is, the average of the ft . Togenerate mortality from this model, values of mt and ft for each day,m and d need to be specified. In this study we will set d ¼ 83 toroughly match the average daily mortality observed in CookCounty. The critical parameter in the three-state model, withrespect to PM10 exposure, is the effect of PM10 on entry into the frailpopulation mt (Roberts and Switzer, 2004; Smith et al., 1999). Thisparameter determines whether the mortality effect of PM10 is morethan mortality displacement alone, that is, whether PM10 isresponsible for shortening life by more than a few days, weeks, ormonths. The three-state population model has been used ina number of previous studies to motivate mortality displacementand/or generate mortality with different degrees of mortalitydisplacement for use in simulations (Fung et al., 2005a,b; Robertsand Switzer, 2004; Smith, 2003; Smith et al., 1999; Zeger et al.,1999).

Mortality generated from the three-state model falls into threecategories with respect to PM10 exposure (Roberts and Switzer,2004): (a) enter model: only entry into the frail population ðmtÞdepends on PM10; (b) exit model: only exit from the frail populationðftÞ depends on PM10; (c) enter/exit model: both entry into and exitfrom the frail population depend on PM10. The exit model corre-sponds to the situation where the effect of PM10 is mortality

Fig. 1. Plots of (a) daily values of the mean number of entrants into the frail population ðmtÞ;death ðftÞ; and (d) a simulated series of daily deaths ðdtÞ corresponding to the enter/exit modays.

displacement alone. The enter/exit model is consistent withprevious studies that have concluded that the mortality effects ofPM10 are not mortality displacement alone.

The variables used to generate mortality are the actual valuesobserved for Cook County described above. To investigate thehypothesis of interest, mortality time-series were generated wherethere was a threshold concentration-response relation betweenPM10 and entry into the frail population mt and a linear relationbetween PM10 and exit from the frail population or death ft :

Specifically, the relations were defined such that the percentageincreases in mt and ft above their baseline levels for a givenconcentration of PM10 were expfb1ðPM10 � 30Þþg � 1 andexpfb2PM10g � 1, respectively. ðPM10 � 30Þþ ¼ PM10 � 30 ifðPM10 � 30Þ � 0 and ðPM10 � 30Þþ ¼ 0 if ðPM10 � 30Þ < 0: Thethreshold relation for mt means that PM10 has no effect on healthyindividuals for concentrations below 30 mgm�3. However forconcentrations above 30 mgm�3 each 1 mgm�3 increment in PM10increases the mean number of entrants into the frail population byapproximately 100b1%. Likewise, the linear relation for ft meansthat each 1 mgm�3 increment in PM10 increases the probability ofdeath by approximately100b2%. Similar to a previous study, toincrease the reality of the simulations, both mt and ft also dependedon smooth functions of concurrent and lagged temperature,concurrent and lagged dew point temperature, and a smooth slow-changing time trend (Roberts and Switzer, 2004). For illustration,Fig. 1 contains plots of mt , the size of the frail population ft , ft , andthe number of deaths dt corresponding to a mortality time seriessimulated from the enter/exit model with b1 ¼ 0:00025,b2 ¼ 0:0005, and a mean lifetime in the frail population of 5 days.

To investigate the shape of the resulting PM10-mortalityconcentration-response relation that would be estimated in

(b) daily size of the frail population ðftÞ; (c) daily values of the individual probability ofdel with b1 ¼ 0:00025;b2 ¼ 0:0005 and an average lifetime in the frail population of 5

S. Roberts / Atmospheric Environment 45 (2011) 4728e4734 4731

a standard time-series analysis the simulated mortality countswere modeled as independent Poisson random variables withmean 6t on day t given by:

logð6tÞ ¼ confounderst þ s�PM10;t ; df ¼ 6

�t ; (1)

where, PM10,t is the lag-1 PM10 concentration on day t,confounderst other time-varying covariates related to dailymortality, and sðÞ a smooth natural cubic spline function with sixdegrees of freedom. The smooth function of PM10 provides enoughflexibility for the shape of the PM10-mortality concentration-response relation, as observable from the simulated mortalitytime-series data, to be investigated. The specification ofconfounderst in model (1) is:

confounderst ¼ sðtempt ;df ¼ 6Þ þ s�tempt;1�3; df ¼ 6

�

þ sðdewt ;df ¼ 3Þ þ s�dewt;1�3;df ¼ 3

�

þ sðt; df ¼ 4�#yearsÞwhere, tempt is the average temperature on day t, tempt;1�3 is theprevious three days’ average temperature and dewt and dewt;1�3are similarly defined quantities for dew point temperature. Theseconfounder adjustments take the same form as the smooth func-tions of temperature, dew point temperature and time that wereused to modulate mt and ft . Models of a similar form to model (1)are utilized extensively in time-series investigations of themortality effects of PM10 (Daniels et al., 2000; Peng et al., 2005;Roberts, 2005).

To conduct the simulations, sets of 3000 mortality time serieswere generated corresponding to different forms of the enter/exit,enter, and exit models by selecting appropriate values of b1, b2, andm. A range of plausible values of b1 and b2 corresponding toapproximately 0% to 1% increases in entry into and exit from the frailpopulation per 10 mgm�3 increment in PM10, and values of the

Fig. 2. Estimates of the PM10-mortality concentration-response relation each based on sets oof b1 and b2 ; and six different values of the mean lifetime in the frail population. Each plot co60, and 120 days) of the mean lifetime in the frail population considered. In each plot, the tPM10 and exit from the frail population (dashed line) are superimposed. For ease of viewin

mean lifetime in the frail population (m) ranging from 5 to 120 dayswere considered. To each set of 3000 mortality time-series, model(1) was fitted and an estimate of the concentration-response rela-tion calculated as the average of the 3000 estimated PM10-mortalityrelations obtained frommodel (1) via the smooth function of PM10.These estimates of the concentration-response relation representwhat an investigator using a single mortality time-series to inves-tigate the shape of the concentration-response relation wouldexpect to observe in practice. The statistical package R was used forall the analyses in this paper (R Development Core Team, 2009).

4. Results

Figs. 2 and 3 contain the results of the simulations. Each indi-vidual plot contains six estimates of the concentration-responserelation (solid lines) each corresponding to a different set of 3000mortality time-series. In each plot, the six sets of mortality timeseries are generated using the same underlying true relationsbetween PM10 and entry into the frail population (dashed-dottedline) and PM10 and exit from the frail population (dashed line) butdifferent values of the mean lifetime in the frail population of 5, 10,15, 30, 60, and 120 days. Ideally, the estimates of the concentration-response relation should match the threshold shape of the relationbetween PM10 and entry into the frail population, as this is thequantity that determines the mortality effects of PM10 net ofmortality displacement. It is worth noting that in each plot the sixestimates of the concentration-response relation are difficult todifferentiate because the six estimates are very similar and closelyfollow the dashed line corresponding to the true relation betweenPM10 and exit from the frail population.

From Fig. 2, we can see that for mortality generated from theenter/exit model that the impact of PM10 on entry into the frailpopulation has negligible impact on the estimates of theconcentration-response relation obtained frommodel (1) across allvalues of the mean lifetime in the frail population and underlying

f 3000 mortality time-series generated using the enter/exit model, the indicated valuesntains six estimated relations (solid lines) corresponding to the six values (5, 10, 15, 30,rue relation between PM10 and entry into the frail population (dashed-dotted line) andg, the x-axes on these plots have been truncated at 80 mgm�3.

Fig. 3. Estimates of the PM10-mortality concentration-response relation each based on sets of 3000 mortality time-series generated using the enter model or the exit model, theindicated values of b1 and b2; and six different values of the mean lifetime in the frail population. Each plot contains six estimated relations (solid lines) corresponding to the sixvalues (5, 10, 15, 30, 60, and 120 days) of the mean lifetime in the frail population considered. In each plot, the true relation between PM10 and entry into the frail population(dashed-dotted line) and PM10 and exit from the frail population (dashed line) are superimposed. For ease of viewing, the x-axes on these plots have been truncated at 80 mgm�3.

S. Roberts / Atmospheric Environment 45 (2011) 4728e47344732

true relations considered. This is evidenced by the fact that in eachof the four plots it is difficult to detect the six solid lines repre-senting the estimates of the concentration-response relationbecause they are essentially the same as the true linear-relationbetween PM10 and exit from the frail population. Similarly, fromFig. 3 we can see that for mortality generated from both the entermodel and the exit model that the average relations are, in mostcases, similar to the true linear-relation between PM10 and exitfrom the frail population. The relation between PM10 and entry intothe frail population, again, has negligible influence on the estimatesof the concentration-response relation obtained from model (1).

Only in the situation corresponding to b1 ¼ 0:001; b2 ¼ 0 didthemean lifetime in the frail population appear to havewhat mightapproach a non-negligible impact on the estimates of theconcentration-response relation. In this situation, three of theestimated relations corresponding to mean lifetimes of 5, 10 and 15days deviate from the true relation between PM10 and exit from thefrail population e the amount of the deviation increasing as themean lifetime in the frail population decreases. This phenomenonis due to the fact that the deaths of individuals moved into the frailpopulation by the effect of PM10 are not observed immediately, butrather they are spread out over a period of time with meanapproximately equal to the mean lifetime in the frail population.This means that as the mean lifetime in the frail populationdecreases this “spreading out” of deaths is reduced and, hence,more of the effect of PM10 on the healthy population is observablein the current day’s mortality. The fact that more of the effect ofPM10 on the healthy population is observable for smaller meanlifetimes results in the observed deviation of the estimatedconcentration-response relations for mean lifetimes of 5, 10, and 15days. This issue, termed the “dilution effect”, is discussed in moredetail in the Discussion Section.

The above simulations were repeated (results not reported)using values of 15 mgm�3 and 60 mgm�3 for the threshold. The

results for these additional simulations were qualitatively similar tothose reported above for a threshold of 30 mgm�3, that is, that therelation between PM10 and entry into the frail population hasnegligible influence on the estimates of the concentration-responserelation obtained from model (1).

5. Discussion

The results of these simulations are perhaps not surprising giventhe results of Roberts (2011) and Roberts and Switzer (2004) thatinvestigate the use of distributed lag models (DLM) in the presenceof mortality displacement. The results of the simulations illustratethat the presence of even a very small frail population can mask theshape of the concentration-response relation between PM10 andmortality that is not simply short-term mortality displacement.This masked relation that represents the effect of PM10 on entry ofindividuals into the frail population is of primary importance forpublic health assessment because it represents mortality that isbrought forward by a substantial amount of time due to PM10exposure. The reason for this masking is due to the deaths of theindividuals moved into the frail population, as a result of PM10exposure, not being observed immediately but rather being spreadout (or diluted) over a period of time with average equal to themean lifetime in the frail population. This “dilution” effect wasobserved and discussed in a previous study that investigated theuse of distributed lag models (DLM) as an indicator of mortalitydisplacement (Roberts and Switzer, 2004). This study attributed thedilution effect as the reason why the sum of a DLM’s coefficientswas not always a good measure of the mortality effects of PM10, netof morality displacement. This dilution means that the informationavailable from time-series mortality data on the effect of PM10 onentry into the frail population is severely attenuated and effectivelyswamped by the effects of PM10 on exit from the frail population.This is clearly demonstrated in the enter/exit model where the

S. Roberts / Atmospheric Environment 45 (2011) 4728e4734 4733

estimated concentration-response relations were essentially equalto the shape of the relation between PM10 and exit from the frailpopulation and, additionally, in the results for DLMs of Roberts andSwitzer (2004). Roberts (2011) investigates this issue further bydeveloping a relationship between the sum of the estimatesobtained from a DLM as a function of the number of lags in the DLMand the parameters of the population model including the size ofthe frail population.

A similar rationale to that used in Roberts (2011) could conceiv-ably be used to determine the parameter estimates that would beobtained from a specific parametric model fitted to mortalitygenerated from the three-statemodel in order to estimate the shapeof the concentration-response relation. However, the goal of thispaper was to investigate the shape of the concentration-responserelation that is observable in mortality time-series generated fromthe three-state model, rather than what would be estimated ifa specific parametric model (such as a piece-wise linear model witha given threshold) was fitted to mortality generated from themodel.Compounding thefindings of this paper are those of a previous paperthat showed that the model selection criterion AIC was not alwayssuccessful in detecting nonlinearities in the concentration-responserelation between PM10 and mortality (Roberts and Martin, 2006).This resultmeans that even if thresholds in the relationship betweenPM10 and entry into the frail population were not masked that itwould still be difficult to detect them.

This study has consequences for investigations that use time-series data to investigate whether there are thresholds in thePM10-mortality concentration-response relation and the shape ofthis relation in general. A caveat for such studies is that if a frailpopulation exists it may only be possible to determine the shape ofthe relation between PM10 and exit from this frail population andnot the more important relation between PM10 and entry into thisfrail population. This means that the conclusions of some studiesthat a linear relation is appropriate may in fact only apply to theeffect of PM10 on a frail population, with the effects of PM10 onentry into this frail population going unexplored. The conclusion ofa linear-relation that only applies to a frail population with a shortexpected lifetime while, for example, a masked threshold relationapplies to the healthy population, could lead to an overstatement ofthe public health consequences of PM10 exposure and possibly anincorrect assessment of the benefits of tightening regulatory stan-dards. Conversely, situations where PM10 only affects entry intoa frail population could lead to an understatement of the publichealth consequences of PM10 exposure.

Disentangling the shape of the concentration-response relationbetween PM10 and movement into the frail population from therelation between PM10 and exit from the frail population mayrequire the use of data in addition to time-series data of dailymortality. For example, if it is believed that there is a frail pop-ulation one potential strategy for investigating the relationbetween PM10 and entry into this population would be to investi-gate the relation between PM10 and counts of morbidity that arebelieved to encompass the frail condition. In this vein, there havebeen studies that have investigated the relationship between airpollutants and end-points other than mortality such as hospitaladmissions (Chan et al., 2006; Halonen et al., 2009). In addition,a few studies have tried to identify subpopulations that are moresusceptible to the effects of air pollution. The results of thesestudies could help informwhat end-points, apart frommortality, toconsider (Berglind et al., 2009; Medina-Ramón and Schwartz,2008; Zeka et al., 2006).

The three-state population model considered in this paper, andutilized in a number of studies of mortality displacement, is clearlya simplification of reality. The three-state populationmodel does notallow for multiple frail populations with different mean lifetimes

corresponding to different levels of PM10 induced frailty or for thepossibility that individualsmay recover from the frail state. However,more general populationmodels incorporating these changeswouldnot alter the underlying issue that the presence of a frail populationeffectively masks the shape of the relation between PM10 andthe healthy population, at least as observable in mortality time-series data. Other generalizations that could be envisaged areconcentration-response relations other than threshold and linearrelations and an investigation of other pollutants such as ozone.Threshold and linear relations and PM10 were the focus of thisinvestigation because they are the shapes of the relation andpollutant that have received most attention in the time-series liter-ature. Thefindings of this studywould remain the same for these twogeneralizations as the masking effect of the frail population wouldstill manifest in a similar manner.

Acknowledgements

This work was supported by the Australian Research Council[DP0878988].

Appendix. Supplementary data

Supplementary data associated with this article can be found inthe online version, at doi:10.1016/j.atmosenv.2011.04.050.

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