 Research article
 Open access
 Published:
Alternative adjustment for seasonality and longterm timetrend in timeseries analysis for longterm environmental exposures and disease counts
BMC Medical Research Methodology volume 21, Article number: 2 (2021)
Abstract
Background
Timeseries analysis with caseonly data is a prominent method for the effect of environmental determinants on disease events in environmental epidemiology. In this analysis, adjustment for seasonality and longterm timetrend is crucial to obtain valid findings. When applying this analysis for longterm exposure (e.g., months, years) of which effects are usually studied via survival analysis with individuallevel longitudinal data, unlike its application for shortterm exposure (e.g., days, weeks), a standard adjustment method for seasonality and longterm timetrend can extremely inflate standard error of coefficient estimates of the effects. Given that individuallevel longitudinal data are difficult to construct and often available to limited populations, if this inflation of standard error can be solved, rich caseonly data over regions and countries would be very useful to test a variety of research hypotheses considering unique local contexts.
Methods
We discuss adjustment methods for seasonality and timetrend used in timeseries analysis in environmental epidemiology and explain why standard errors can be inflated. We suggest alternative methods to solve this problem. We conduct simulation analyses based on real data for Seoul, South Korea, 2002–2013, and timeseries analysis using real data for seven major South Korean cities, 2006–2013 to identify whether the association between longterm exposure and health outcomes can be estimated via timeseries analysis with alternative adjustment methods.
Results
Simulation analyses and realdata analysis confirmed that frequently used adjustment methods such as a spline function of a variable representing time extremely inflate standard errors of estimates for associations between longterm exposure and health outcomes. Instead, alternative methods such as a combination of functions of variables representing time can make sufficient adjustment with efficiency.
Conclusions
Our findings suggest that timeseries analysis with caseonly data can be applied for estimating longterm exposure effects. Rich caseonly data such as death certificates and hospitalization records combined with repeated measurements of environmental determinants across countries would have high potentials for investigating the effects of longterm exposure on health outcomes allowing for unique contexts of local populations.
Background
Timeseries analysis is a prominent method to estimate the effect of shortterm (e.g., days, weeks) environmental exposures (e.g., air pollution, extreme weather) on health outcomes [1, 2]. Timeseries studies form the basis for our understanding of environmentally susceptible populations and the number of adverse health outcomes that can be potentially avoided if an adverse exposure level were to be reduced. Timeseries data can consist of caseonly records such as death certificates, hospitalization records, and insurance claims. So, timeseries analysis with rich caseonly data worldwide can contribute to not only the identification of the relationships between environmental exposures and a variety of health outcomes but also the improvement of generalizability of scientific findings, given that it is common that the health effects of environmental exposure are heterogeneous across regions and countries due to numerous modifiers such as demographics, socioeconomic positions and health behaviors [3,4,5,6]. For example, a recent study collected timeseries data for mortality and air pollution over 652 cities across 24 countries [4] and found that the mortality effect of shortterm exposure to air pollution may differ by countries.
In addition to shortterm exposure effects, some environmental exposures, such as air pollution, have longterm (e.g., several months, years) exposure effects on health. These effects that are generally stronger than shortterm exposure effects are usually investigated via cohort data. In cohort studies, exposure is estimated over long time periods, often accounting for mobility of study participants [1]. Cohort data with location information over time allow for more accurate assessment of exposure over longtime frames as participants are followed throughout time and changes in residence, which could affect exposure. However, cohort data are costly and difficult to construct, available for limited populations, and sometimes not representative for the general population so that generalizability for the magnitude of exposure effects is often limited: for example, most of the published air pollution cohort studies for Western countries, several for East Asian countries, and none for the rest of the world [6]. In contrast, registry data, such as death certificates and hospitalization records, are rich in many countries, but usually do not have information for mobility. This leads to higher potential exposure misclassification in studies of longterm exposure compared to studies of shortterm exposure, which assume that participants had the same exposure for a few days prior to the event. Nevertheless, for general populations, such exposure misclassification is likely to result in underestimation [7, 8], which would be a basis of the conservative view if caseonly data are used for inferring longterm exposure effects (i.e., understating the effects rather than overstating them).
A critical challenge for applying timeseries analysis with caseonly data into estimating the effect of longterm exposure on health outcomes is to adjust for seasonality and longterm timetrend. Unlike its application for shortterm exposure, adjustment for seasonality and longterm timetrend may mask a real signal of the effect of longterm exposure on an health outcome [9, 10] and can greatly reduce temporal variability of residual timeseries of longterm exposure that is necessary to detect the signal [11]. So, the adjustment can inflate standard error of estimates, which make it difficult to make an inference of the effect of interest. In this regard, timeseries studies in environmental epidemiology have been generally focused on shortterm exposure, usually of a few days, at most about 40–60 days of an exposure timewindow [10, 12,13,14,15,16,17]. In this paper, we address issues of adjustment for seasonality and longterm timetrend in estimating the effects of longterm exposure on health outcomes.
In the following sections, we discuss adjustment methods frequently used in timeseries studies and identify possible problems in estimating the effect of longterm exposure when such methods are used. To solve these problems, we suggest alterative adjustment methods. We present statistical simulations and real data analysis to demonstrate that the effects of longterm exposure can be estimated via timeseries analysis with alternative adjustment methods.
Methods
Model formulation
Suppose that an individual’s hazard is estimated by a multiplicative hazard model with exposure variables, and a study population is a general population that can be seen as an open cohort. For shared environmental exposure over all individuals in a study population at time t and X_{t}, populationaveraged effect can be estimated using Poisson regression model with aggregatedlevel timeseries data as follows [18]:
where Y is the number of events and Z is a set of measured potential confounders such as temperature (for air pollution studies), relative humidity, and influenza epidemic. e(X_{t},…,X_{tL}) is a function of the populationaveraged effect of X from lag0 (same day) to lagL (i.e., cumulative exposure for L+ 1 days) on Y. In the distributed lag model framework [19], this function is generalized to \( {\sum}_{l=0}^L{\beta}_l{X}_{tl} \). The overall cumulative coefficient (i.e., the logarithm of the overall cumulative relative rate/risk) is \( \overline{\beta}={\sum}_{l=0}^L{\beta}_l \). In practice, an observable e(X_{t},…,X_{tL}) may not be equal to e(X_{t},…,X_{tL}) due to mortality displacement and/or an exposure measurement error [20, 21]. g(t) is a function of the populationaveraged effect of unmeasured risk factors that shape seasonality and timetrend in a timeseries of Y [18]. Since X has seasonality and timetrend, confounding could arise. To control for this confounding, several techniques have been used in the timeseries literature. We discuss them in the context of estimating the effect of longterm X (i.e., X_{t},…,X_{tL}, L>months or years) on Y.
Adjustment for seasonality and longterm timetrend
Detrending prior to regression
Prior to a regression analysis, seasonality and timetrend in an exposure series and an outcome series can be decomposed or removed by algorithms [14, 22, 23]. Detrended timeseries can be used to estimate the effect of shortterm X on Y.
For estimating the effect of longterm X on Y, detrending prior to regression is more challenging because a real signal of the effect of longterm X on Y may also be removed. For example, detrending a timeseries of Y will remove some longterm fluctuation of Y that is formed by \( {\sum}_{l=0}^L{\beta}_l{X}_{tl} \). Consequently, an estimate of the effect of longterm X on Y would be biased toward the null.
Adjustment in regression models
Many timeseries studies in environmental epidemiology adjust for seasonality and longterm timetrend by adding a variable or a set of variables representing time (t) into regression models [1, 2, 11, 24]. Natural cubic splines (NCS) with some degrees of freedom (df) per year are frequently used, and its performance for adjustment was tested by simulation studies [24,25,26]. We use NCS(t,pdf/year) to denote this in generalized linear models where t represents a variable representing time and p reflects the degrees of freedom. Nonparametric splines in generalized additive models are also frequently used, but for simplicity, we focus on the former as they may provide less biased estimates [11, 24]. In mortality studies, 3–8df/year is often used [24, 26, 27], but the optimal df may decrease depending on whether other variables explain seasonality to some extent (e.g., influenza epidemic, heat wave) are included in models [27].
Often, a combination of some variables is used [28, 29], such as NCS of calendar time of the year called day of the year (or day of the season) and an indicator of year (i.e., dummy variables) or equivalent (e.g., NCS of year). We use NCS(doy,pdf)+I(year) to denote NCS of day of the year (doy) and an indicator of year. But, NCS(doy,pdf) may not allow for seasonality that may vary between years. To address this, we additionally consider an NCS of the order of week throughout a study period, NCS(week,qdf) or an NCS of the order of month throughout a study period NCS(month,rdf). NCS(doy,pdf)+I(year)+NCS(week,qdf)+NCS(month,rdf) can collectively capture seasonality and longterm timetrend.
To present comparison between these functions, we smoothed daily timeseries of PM_{10} and mortality in Seoul, South Korea, 2002–2013 through NCS with various degrees of freedoms (Fig. 1). Hourly measurement of PM_{10} concentration and allcause mortality (International Classification of Diseases10th, A00R99) were obtained from the National Institute of Environmental Research and Statistics Korea, respectively, and hourly measurements from multiple monitors within Seoul on a given day were averaged. Figure 1 shows that NCS(t,4df/year) can capture seasonality and longterm timetrend. NCS(t,10df/year) can further capture fluctuations more aggressively, which in cases of allcause mortality, may be related to other environmental exposures such as heat waves [27]. NCS(doy,10df)+I(year)+ NCS(week,5df)+NCS(month,5df) also captures seasonality and longterm timetrend although there are some differences compared to NCS(t,10df/year): for example, a peak around 3300 time point (i.e., day). Such differences may be captured using additional dummy variables.
In estimating the effect of longterm X on Y, standard errors can become very high, so that unstable estimates or spuriously significant estimates can come out. This is often referred to as overfitting as analogous of collinearity problems. To illustrate overfitting, we regressed a oneyear moving average of PM_{10} concentration in Seoul, 2002–2013 on functions of variables representing time. Table 1 presents variance of residuals of this moving average and R^{2}. NCS(t,4df/year) explains nearly all variability of the moving average (R^{2}=99.3%). In comparison to NCS(t,4df/year) and NCS(t,10df/year), NCS(doy,10df)+I(year) and NCS(doy,10df)+I(year) +NCS(week,5df)+NCS(month,5df) explains to a lesser extent variability of the moving average (R^{2}=96.6 and 97.1% respectively). The perhaps seemingly small difference in residual variances between these methods could make striking difference in standard errors. Assuming that there are no other covariates to be considered, standard errors may be approximately 17 times higher when NCS(t,4df/year) is used than when NCS(doy,10df)+I(year) +NCS(week,5df)+NCS(month,5df) is used, which is based on a ratio of the two standard deviations of the residuals [30].
To maintain variability in residuals of longterm X using NCS(t,pdf/year), df needs to be reduced, but low df results in loss of adjustment for seasonality. For example, NCS(t,1.5df/year) does not adequately capture seasonality (Fig. 1b). Note that this function still captures variability of a oneyear moving average of PM_{10} to a greater extent than NCS(doy,10df)+I(year)+NCS(week, 5df)+NCS(month, 5df) (Table 1). Thus, NCS(t,pdf/year) may not be relevant for estimating the effect of longterm X on Y.
In summary, adjustment for seasonality and longterm timetrend should be made with avoiding reducing variability of X to a great extent. NCS(t,pdf/year) as a standard method in timeseries studies appears inefficient. A candidate we explored is a combination of some variables such as NCS(doy, pdf)+I(year)+NCS(week,qdf)+NCS(month,rdf) with moderate df. Simulations and realdata analyses were conducted to test this method.
Confounding by longterm association
Although our primary interest is to estimate the effect of longterm X on Y, whether the effect of shortterm X on Y (e.g., \( {\sum}_{l=0}^M{\beta}_l \) (M<few days or weeks) can be confounded by longterm association (e.g., \( {\sum}_{l=M}^L{\beta}_l{X}_{tl} \)) merits investigation in that neighboring lagged variables can have an confounding effect [31] and longterm association can be seen as an weighted moving average of X that would contribute to seasonality and longterm timetrend of Y. If conventional adjustment for seasonality and a timetrend does not sufficiently adjust for longterm association, an inclusion of exposure variables may be needed. Simulations and realdata analyses were also conducted to explore confounding by longterm association.
Simulation methods
We investigate performance of different functions to adjust for seasonality and longterm timetrend in estimating the effect of longterm X on Y using simulation analyses.
Generating simulation samples
Figure 2 shows two hypothetical observable lag patterns to generate simulation samples. The assumed overall cumulative coefficients were 0.01 and 0.001 per a 1 μg/m^{3} increase of PM_{10}, for the observable lag patterns in Fig. 2a and b, respectively. We considered the effect of cumulative exposure on an outcome and mortality displacement; we did not consider exposure measurement errors. For example, epidemiologic evidence shows that the logarithm of the relative risk of the exposure to PM_{2.5} on mortality generally increases with the exposure duration (i.e., L increases in the distributed lag model framework) while the highest marginal increase arises at the most proximal time of the exposure [32, 33]. We considered two distributions of decreases in mortality due to mortality displacement (Fig. S1 in Additional file 1). Findings of many timeseries studies focused on shortterm exposure to PM imply that observable lag patterns for shortterm exposure to air pollutants and mortality are nonlinear [10, 12, 13, 15,16,17, 34] suggesting that shortterm mortality displacement may play a role in shaping observable lag patterns.
We generated simulation samples for daily timeseries of PM_{10} and mortality based on real data of Seoul, 2002–2013. For this, we regressed the logarithm of daily average of PM_{10} and the daily number of allcause mortality cases (except for accidental causes) on functions of variables representing time and potential confounders in timeseries studies, including a lagstructure of temperature from lag0–21, relative humidity, influenza epidemic, national holidays, and day of the week. For temperature and relative humidity, hourly measurements at the center of Seoul obtained from the Korean Meteorological Administration were averaged to daily values. Influenza epidemic was based on the number of hospital visits for influenza, which was obtained from the National Health Insurance System. To generate a timeseries of PM_{10}, values were sampled from normal distributions and then were exponentiated. A mean of normal distributions was predicted with values obtained from the regression model for the logarithm of PM_{10}. Their standard deviation was estimated from the model (low concurvitiy) or the value divided by 10 (high concurvity) [24]. To generate timeseries of allcause mortality, the number of events were sampled from Poisson distributions. A mean of Poisson distributions was the sum of predicted values obtained from the Poisson regression model for allcause mortality and a product of a lag pattern and distributed lags of generated PM_{10} series. Technical detail for this section is presented in Additional file 1. Fig. S2 in Additional file 1 shows an example of simulated sample of timeseries.
Testing models
To adjust for seasonality and longterm timetrend in generated timeseries samples, we used NCS(t,4df/year), NCS(t,10df/year), and NCS(doy,10df)+I(year)+NCS(week,5df)+NCS(month, 5df). We used constrained distributed lags of PM_{10} (lag0–730) to estimate the association between PM_{10} and mortality. We also used only twoday moving average of PM_{10} (lag0–1) to see whether the cumulative exposure to PM_{10} and mortality displacement (lag2–730) can have a confounding effect. We adjusted for a lagstructure of temperature, relative humidity, influenza epidemic, national holidays, and day of the week. Technical detail for this section is presented in Additional file 1.
Methods for real data analysis
We conducted a twostage timeseries analysis to estimate the effect of longterm exposure to PM_{10} on mortality in seven major cities (Seoul, Busan, Daegu, Incheon, Gwangju, Daejeon, and Ulsan) of South Korea from 2006 to 2013. Datasets are described in Additional file 2. In the first stage, we fitted cityspecific Poisson regression models with overdispersion considered. Constrained distributed lags of PM_{10} (lag0–365) were used as an exposure metric. A lagstructure of temperature (lag0–14 or lag0–21), O_{3} (lag0–45), relative humidity, and influenza epidemic was adjusted. Seasonality and longterm timetrend were also adjusted using (a) NCS(t,pdf/year), (b) NCS(doy,pdf)+I(year), or (c) NCS(doy,pdf)+I(year)+NCS(week,pdf)+ NCS(month,pdf). Technical detail for modelling is provided in Additional file 2. In the second stage, we applied multivariate metaanalysis to pool cityspecific estimates [35]. We compared pooled estimates of the association between adjustments for seasonality and longterm timetrend. This procedure was to check possible overfitting.
As sensitivity analyses, we used different dfs in each NCS. We also additionally adjusted for multiple dummy variables indicating days when deviations of seasonality were identified. For this additional adjustment, we found that days when predicted daily deaths of a cityspecific model with NCS(doy,20df)+I(year)+NCS(week,3df)+NCS(month,3df) and those of a cityspecific model with NCS(t,10df/year) did not coincide one another; if deviations from normal seasonal patterns exist across years, these two models would yield different predicted daily deaths because NCS(t,10df/year) is likely to capture deviations better. We used cutoff values to determine deviations of seasonality: A_{t}B_{t}>a cutoff value where A_{t} is predicted logarithm of daily deaths from one model and B_{t} is predicted logarithm of daily deaths from the other model at time t. We varied a cutoff value from 98th percentile, 96th percentile, …, to 80th percentile of A_{t}B_{t}.
Software
We used R software 3.5.3 (R Foundation for Statistical Computing) for simulations and realdata analyses. We used splines package for NCS [36], dlnm package for distributed lags of PM_{10} and distributed lag nonlinear terms of temperature [37], and mvmeta package for pooling cityspecific estimates [35]. R codes for simulations and realdata analysis are provided in the first author’s website, http://hkimresearch.com, or Github, https://github.com/HonghyokKim/AlternativeAdjustment.
Results
Results of simulations
Table 2 presents bias, standard deviation, and nominal coverage of 95% confidence interval for the overall cumulative association between distributed lags of PM_{10} (lag0–730) and mortality. Models adjusted for NCS(doy,10df)+I(year)+NCS(week,5df)+NCS(month,5df) produced estimates of the overall cumulative coefficient with negligible bias and almost perfect coverage of 95% confidence intervals (Table 2). Lag patterns were estimated accurately (Fig. 3). As expected by very small residual variance of PM_{10} as in Table 1, models adjusted for NCS(t, 4df/year) and NCS(t,10df/year) produced very large standard errors, indicating overfitting. These models also produced lagmortality associations that deviated far from lag patterns used to generate simulation samples (Fig. 3).
Our simulation results suggest that the effect of shortterm exposure to PM_{10} on allcause mortality may be confounded by the effect of longterm exposure to PM_{10} on allcause mortality in some circumstances. For example, for observable lag pattern #2, bias with respect to a moving average of PM_{10} from lag0–1 in its relation with allcause mortality was − 11.3% when NCS(t,10df/year) was used and PM_{10} from lag2–730 was not adjusted (Table S1 in Additional file 1). When PM_{10} from lag2–730 was adjusted, the bias decreased to 2.8%. The bias decreased with higher df. For observable lag pattern #1, the bias decreased to 0.4% from 4df to 10df when PM_{10} from lag2–730 was not adjusted.
Results of realdata analyses
Table 3 presents percentage increases in allcause mortality risk per a 10 μg/m^{3} increase in twoday moving average of PM_{10} (lag0–1) or PM_{10} for oneyear (i.e., lag0–365) in seven major cities of South Korea. When PM_{10} (lag2–365) was adjusted, the percentage increases in allcause mortality were 0.14% (95% confidence interval: − 0.06, 0.34) for NCS(t,10df/year) and 0.14% (− 0.03, 0.31) for NCS(doy, 20df)+I(year)+NCS(week,3df)+NCS(month,3df).
The percentage increase in allcause mortality risk for a 10 μg/m^{3} increase of PM_{10} for 1 year (i.e., lag0–365) was 4.66% (0.14, 9.38) when NCS(doy,20df)+I(year)+NCS(week,3df)+ NCS(month,3df) was used (Table 3). This percentage increase was stable when different df were used for each NCS unless df were too high for NCS(week,qdf) and NCS(month,rdf). For example, estimates were variable with very wide confidence intervals for 6, 10, and 15 df for each NCS (Table 3). When NCS(t,pdf/year) was used, estimates across different df were extremely variable with very wide confidence intervals, indicating overfitting (Table 3). For cardiovascular mortality and respiratory mortality, we found similar patterns of estimates as those found for allcause mortality. Percentage increase in cardiovascular mortality risk and respiratory mortality risk for a 10 μg/m^{3} increase of PM_{10} for 1 year was 10.76% (0.62, 21.93) and 9.34% (− 17.77, 45.40), respectively (Tables S2–3 in Additional file 2).
Figure S3 in Additional file 2 presents an example of differences between timeseries of predicted logarithm of daily deaths from cityspecific models with NCS(doy,20df)+I(year)+NCS(week,3df)+ NCS(month, 3df) and cityspecific models with NCS(t,10df/year). Since a lagstructure of temperature, air pollutants, and other variables such as influenza epidemics were all used, both models predicted daily mortality series similarly. Nevertheless, some deviations of seasonality were also identified: for example, about 1000–1250 time points (i.e., days) and around 1800 time point in Seoul (Fig. S3A). We adjusted for such deviations using multiple dummy variables and found our results robust to these additional adjustments (Fig. S4 in Additional file 2).
Figure 4 presents the percentage increase in allcause mortality risk, cardiovascular mortality risk, and respiratory mortality risk by exposure duration. These increasing patterns reconcile previous observations that the association between exposure to PM and mortality increases with exposure duration: roughly similar to 5–15% increase of mortality per 20 μg/m^{3} increase of oneyear exposure to PM_{10} [32, 33]. Figure 5 presents corresponding lag patterns.
Discussion
We addressed a challenging issue in estimating the effect of longterm exposure on health outcomes using timeseries analysis – adjustment for seasonality and longterm timetrend that do not mask the effect and not reduce substantially variability of residuals of exposure series. NCS(t,pdf/year) that is frequently used in timeseries studies may lead to very high standard errors, which makes it difficult to infer the magnitude of the association between longterm exposure and health outcomes. Our results suggest that a combination of functions can be used to sufficiently adjust for seasonality and longterm timetrend and also allow for enough variability of residuals of exposure series, so that inflated standard errors can be avoided. We also showed that confounding by longterm exposure effects may arise in timeseries studies for estimating shortterm exposure effects; however, we postulate that this confounding might not be an issue for air pollution timeseries studies because adjustment methods for seasonality and longterm timetrend could adjust for this confounding.
The effect of longterm exposure may be more likely to be confounded by seasonality and timetrend than that of shortterm exposure. Higher df of NCS would provide stricter adjustment but reduce variability of residuals of exposure series. A combination of functions allows for higher variability of residuals of exposure series but may require additional adjustments for seasonality, depending on data applied, what covariates are measured and how they are adjusted. For example, deviations of seasonality in an outcome series may still exist even after distributed lags of environmental variables that has strong seasonality such as temperature [38] are adjusted. Other examples may include seasonality in an outcome series attributable to influenza epidemic and longerterm exposure than exposure of interest. Therefore, sensitivity analyses are recommended to determine sufficient but not redundant adjustment for seasonality, timetrend, and possible confounding by longerterm exposure as we did in our realdata analyses.
In line with the merits of sensitivity analyses, overfitting must be considered as it increases variance estimates and can yield highly variable estimates of an association. Sometimes, to our experience, statistically significant but spurious associations such as lag patterns of NCS (4 or 10df/year) in Fig. 3 may come out. To avoid overfitting, identification of residual variance of exposure series may be helpful, as in Table 1. Comparing lag patterns estimated across different lag models may also guide whether overfitting arises or not because highly variable estimates would come out from different lag models when standard errors of coefficient estimates of interest are inflated [13].
Some limitations are noted. We did not extend our focus into longerterm exposure beyond 1 year in realdata analysis. We found that functions of variables representing time explain to a larger extent the variability of longerterm fluctuation (e.g., 2 years) in an exposure series. Thus, there would be smaller variability of residuals of exposure series. This issue may be overcome with collecting more informative data. Second, we did not address exposure misclassification theoretically and analytically. So, we cannot rule out the possibility that exposure misclassification might affect exposure durations to different degrees (such as Figs. 4 and 5), but the overall impact on an estimate of the cumulative coefficient \( \overline{\beta} \) might be toward the null [7, 8].
Nevertheless, an estimate affected by exposure misclassification in the context of personal exposure may be seen as an estimate of the effect of ambient air pollution in a given region that is of public health importance. For example, timeseries analysis could be used to address cessation lag [39] of ambient air pollution exposures in a given population, which is a critical issue in understanding efficiency of ambient air pollution interventions. Changes of loss of life expectancy in that region as a measure of the severity of public health burden [40, 41] may also be estimated considering longterm exposure effect without cohort data [23].
Conclusions
We demonstrated that longterm exposure effects can be estimated using timeseries analysis by addressing confounding by seasonality and longterm timetrend, while maintaining efficiency. Since maintaining efficiency may lead to less strict adjustment for seasonality and longterm timetrend, sensitivity analyses should be conducted to confirm that adjustment is sufficiently made. Rich caseonly data such as death certificates and hospitalization records combined with repeated measurements of environmental determinants across countries would have high potentials for investigating the effects of longterm exposure on health outcomes allowing for unique contexts of local populations.
Availability of data and materials
Daily number of deaths we used are not available without permission of Statistics Korea. Citylevel mortality data are publicly downloadable at the “Download service” section in MicroData Integrated Service (MDIS) by Statistics Korea (https://mdis.kostat.go.kr/index.do, in Korean) and more detailed mortality data can be purchased at the “Use Microdata” section in MDIS. Data for air pollution are publicly downloadable at the “Finally confirmed data” section in Air Korea (https://www.airkorea.or.kr/web, in Korean). Data for meteorological factors are publicly downloadable at the “Data” section in Data Portal by Korean Metrological Administration (https://data.kma.go.kr/, in Korean). R codes for simulations and realdata analysis are provided in the first author’s website, http://hkimresearch.com, or Github, https://github.com/HonghyokKim/AlternativeAdjustment.
Abbreviations
 PM_{10} :

Particulate matter with aerodynamic diameter ≤10 μm
 NCS:

Natural cubic spline
 df:

Degrees of freedom
 NCS(t,pdf/year):

NCS of time (t) throughout timeseries (or the study period) with pdf per year
 I(year):

An indicator function of the order of year throughout timeseries (or the study period)
 doy:

Day of the year
 NCS(doy,pdf):

NCS of doy with pdf
 NCS(week,pdf):

NCS of the order of week throughout timeseries (or the study period) with pdf
 NCS(month,pdf):

NCS of the order of month throughout timeseries (or the study period) with pdf
 SD:

Standard deviation
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This publication was developed under Assistance Agreement No. RD835871 awarded by the U.S. Environmental Protection Agency to Yale University. It has not been formally reviewed by EPA. The views expressed in this document are solely those of the authors and do not necessarily reflect those of the Agency. EPA does not endorse any products or commercial services mentioned in this publication. Research reported in this publication was also supported by the National Institute On Minority Health And Health Disparities of the National Institutes of Health under Award Number R01MD012769. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
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HK made the conception of the study, designed the study, collected the data, conducted the analysis, interpreted the results, wrote the draft and revised the manuscript. JTL collected data, interpreted the results, and revised the manuscript. KF interpreted the results and revised the manuscript. MB supervised the study, contributed to the conception, interpreted the results, and revised the manuscript. All authors have read and approved the manuscript.
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Supplementary Information
Additional file 1.
Simulation methods and additional results of simulations. Figure S1. Two hypothetical observable lag patterns of the association between 1 μg/m^{3} increase of exposure to PM_{10} and mortality, used to generate simulation samples (the last row of panels). A) and B) Hypothetical actual effect (identical). C) Mortality displacement for a few days to several weeks for Observable lag pattern 1. D) Mortality displacement for a few days to several weeks for Observable lag pattern 2. E) Mortality displacement for few days to two years for Observable lag pattern 1. F) Mortality displacement for few days to two years for Observable lag pattern 2. G) Observable lag patterns 1. H) Observable lag pattern 2. Figure S2. An example of simulated samples. A) allcause death series. B) PM_{10} series. Table S1. Bias and Standard Deviation for the Association between a TwoDay Moving Average of PM_{10} (lag0–1) and AllCause Mortality Estimated by a Model with Different Adjustment Methods for Seasonality and LongTerm TimeTrend over 5000 Samples
Additional file 2.
Real data analysis methods and additional results. Table S2. Percentage increase and 95% confidence intervals of cardiovascular mortality risk per a 10 μg/m^{3} increase in twoday moving average of PM_{10} (lag0–1) or oneyear distributed lags of PM_{10} (lag0–365) in seven major cities of South Korea, 2006–2013. Table S3. Percentage increase and 95% confidence intervals of respiratory mortality risk per a 10 μg/m^{3} increase in twoday moving average of PM_{10} (lag0–1) or oneyear distributed lags of PM_{10} (lag0–365) in seven major cities of South Korea, 2006–2013. Figure S3. Timeseries of logarithm of daily allcause deaths predicted by models with two different adjustments for seasonality and longterm time trend and their absolute differences. A) Timeseries in Seoul, 2006–2013 B) Absolute differences in Seoul. C) Timeseries in Daegu, 2006–2013. D) Absolute differences in Daegu. Grey vertical dotted lines in B) and D) denote 80th, 90th, and 98th percentile of absolute differences. Figure S4. Results of additional sensitivity analyses in seven major cities of South Korea, 2006–2013. A) Allcause mortality. B) Cardiovascular mortality. C) Respiratory mortality. Percentage increases estimated by models with NCS(doy, 20df)+I(year)+NCS(week, 3df)+NCS(month, 3df) in Table 3 (for allcause mortality), Table S2 (for cardiovascular mortality), and Table S3 (for respiratory mortality) were duplicated for comparison (red triangles).
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Kim, H., Lee, JT., Fong, K.C. et al. Alternative adjustment for seasonality and longterm timetrend in timeseries analysis for longterm environmental exposures and disease counts. BMC Med Res Methodol 21, 2 (2021). https://doi.org/10.1186/s12874020011991
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DOI: https://doi.org/10.1186/s12874020011991