 Research article
 Open Access
 Published:
Using logic regression to characterize extreme heat exposures and their health associations: a timeseries study of emergency department visits in Atlanta
BMC Medical Research Methodology volume 21, Article number: 87 (2021)
Abstract
Background
Shortterm associations between extreme heat events and adverse health outcomes are wellestablished in epidemiologic studies. However, the use of different exposure definitions across studies has limited our understanding of extreme heat characteristics that are most important for specific health outcomes or subpopulations.
Methods
Logic regression is a statistical learning method for constructing decision trees based on Boolean combinations of binary predictors. We describe how logic regression can be utilized as a datadriven approach to identify extreme heat exposure definitions using health outcome data. We evaluated the performance of the proposed algorithm in a simulation study, as well as in a 20year timeseries analysis of extreme heat and emergency department visits for 12 outcomes in the Atlanta metropolitan area.
Results
For the Atlanta case study, our novel application of logic regression identified extreme heat exposure definitions that were associated with several heatsensitive disease outcomes (e.g., fluid and electrolyte imbalance, renal diseases, ischemic stroke, and hypertension). Exposures were often characterized by extreme apparent minimum temperature or maximum temperature over multiple days. The simulation study also demonstrated that logic regression can successfully identify exposures of different lags and duration structures when statistical power is sufficient.
Conclusion
Logic regression is a useful tool for identifying important characteristics of extreme heat exposures for adverse health outcomes, which may help improve future heat warning systems and response plans.
Introduction
Extreme heat events have significant public health impacts as demonstrated, for example, by historical heat waves in Chicago [1, 2] and Europe [3, 4]. Recent epidemiology studies have also found consistent shortterm associations between extreme heat events and various causespecific mortality [5,6,7,8,9] and morbidity outcomes [10,11,12]. However, synthesizing existing evidence has been challenging because of the use of various exposure definitions in previous studies [13, 14]. How extreme temperatures become hazardous can vary across locations due to differences in societal and biological adaptations [15], as well as across outcomes due to differences in physiological mechanisms [16]. Identifying extreme heat characteristics that are most important for specific health outcomes or vulnerable subpopulations may help improve heat warning systems and response plans.
Extreme heat events are typically characterized by their exceedance over an intensity threshold and their sustained duration. For example, previous studies have defined heat waves as a period where temperature exceeds the 98th percentile over two or more consecutive days [17,18,19]. Many other relative (or absolute) thresholds and durations have been used to define extreme heat in health studies [20]. Furthermore, the choice of heat metric represents another source of variation across studies. Daily maximum and average temperatures have been the most commonly used heat metrics. But there is increasing interest in investigating apparent temperature or wetbulb temperature [21] that may better reflect human discomfort, and minimum temperature that reflects nighttime exposure [22,23,24].
The current approach of assessing effect heterogeneity due to exposure definitions within a study involves examining different heat metrics, different extreme thresholds, and different durations oneatatime. Studies often also need to take into account statistical power in exposure definition due to the low frequency of extreme heat days [25]. There has been limited work in leveraging health data directly to develop exposure definitions for extreme heat. Recently, machine learning methods have been applied to predict adverse health outcomes using meteorological variables [26]. However, the resulting algorithms can be difficult to interpret in terms of the two key characteristics of duration and intensity threshold. These approaches may also suffer from the lack of rigorous control for confounders, making results more difficult to translate into causal effects for subsequent intervention and impact analysis.
In this paper, we examine the use of logic regression [27,28,29], a machine learning method, to help identify characteristics of extreme heat events that are associated with adverse health outcomes. Logic regression estimates a decision tree constructed using Boolean combinations of binary predictors. Logic regression has been utilized extensively in genetic association studies for identifying highdimensional interactions [30, 31], and has recently been extended to other exposures [32,33,34]. We show how logic regression provides a datadriven approach to construct a daily extreme heat exposure indicator that is binary (i.e., presence versus absence of the exposure) and can capture impacts of sustained extreme exposure over several days (i.e., heat waves). We evaluated the performance of the approach in simulation studies and applied the method to a 20year timeseries analysis of daily emergency department (ED) visits in Atlanta, Georgia.
Materials and methods
Atlanta emergency department visit and meteorology data, 1993–2012
Patientlevel ED visits data were obtained directly from hospitals within the 20county Atlanta metropolitan area from 1993 to 2004 and then from the Georgia Hospital Association from 2005 to 2012. For some outcomes, secondary diagnoses were also included because they showed stronger associations with temperature in a previous Atlanta analysis [35]. The selected health outcomes are internal causes (INTERN), heat illness (HEAT), ischemic stroke (STK), fluid and electrolyte imbalances (FLEL), all renal disease (RENAL), acute renal failure (ARF), all circulatory system disease (CIRC), hypertension (HT), myocardial infarction (MI), congestive heart failure (CHF), ischemic heart disease (IHD), and diabetes (DIA). Table 1 provides summary statistics of daily ED visits and ICD9 codes for each outcome. Only admissions during the warm seasons (May 1st to September 30th) were used in this analysis because of our interest in comparing extreme heat events versus nonevent warm days.
Hourly ambient air (drybulb) temperature, dewpoint temperature, and apparent temperature were obtained at the Atlanta Hartsfield International Airport weather station from the National Climatic Data Center from 1993 to 2012. We used airport monitor due to its highquality, complete temporal observations and central location in the study area, which has little variation in elevation. Apparent temperature in °C was defined as − 1.3 + 0.92 T + 2.2e, where T is ambient air temperature (°C) and e is water vapor pressure (kPa) [36]. We considered six heat metrics: daily maximum (MX), minimum (MN), and average (Avg) of either drybulb temperature or apparent temperature (AT). For each daily temperature variable, we created binary extreme indicators at the 95th, the 98th or the 99th percentile thresholds based on observations over the 20year study period. Specifically, the extreme indicator takes the value 1 when the temperature value exceeds the percentile threshold.
Timeseries model for ED visits and temperature
We first describe the quasiPoisson loglinear model used to estimate shortterm associations between daily ED visit counts and extreme temperature [36]. Following our previous analysis of heat waves and ED visits in Atlanta, [37], denote \( {\mu}_t^a \) the mean ED visit count for adverse health outcome a on day t. The timeseries model is given by
Our parameter of interest β is the log relative risk (RR) associated with a binary extreme heat exposure H(X_{t}). The use of logic regression to define H(X_{t}) is described in detail later in Section 2.3. The above timeseries model adjusts for nonextreme continuous sameday temperature (T_{t}), average temperature over the last 3 days (lag 1, lag 2 and lag 3) (\( {\overline{T}}_t \)), and sameday maximum dewpoint temperature (DPT_{t}) to reflect the discomfort level due to the humidity. Specifically, in primary analyses, we defined the continuous temperature T_{t} and \( {\overline{T}}_t \) by truncating the value at the extreme heat threshold (the 95th, 98th, or 99th percentile), i.e., setting any daily temperature value above the threshold to be the threshold value. In a sensitivity analysis, we also examined the use of the entire range of observed (i.e., nontruncated) temperature; in this case, H(X_{t}) can be interpreted as the “added” impact of extreme heat beyond the continuous temperature effect. The use of truncated continuous exposure is to provide better interpretation of β because H(X_{t}) only reflects the temperature effect beyond the threshold. Moreover, the tail of exposureresponse function is often difficult to estimate due to sparse data and can be more sensitivity to influential observations. Hence, adjusting for nontruncated temperature may result in βH(X_{t}) accommodating for misspecification of the exposureresponse functions ns(T_{t}) and \( ns\left({\overline{T}}_t\right) \) at the extreme tail. When apparent temperature is the exposure of interest, we did not include dewpoint temperature in the model. To model the possible nonlinear effect of meteorology, natural cubic splines, denoted by ns (.), were used for \( {T}_t,{\overline{T}}_t, \) and DPT_{t} with 2 equidistant internal knots.
We adjusted for longterm temporal trend in the timeseries model as follows. Within a year, seasonal variation (from May to September), denoted by f(DATE_{t}), was modeled smoothly with natural cubic splines and monthly knots. We included yearspecific indicators \( {\alpha}_{1, yea{r}_t} \) and their interactions with the seasonal pattern in our model to allow for betweenyear variation. We also adjusted for day of week effect using indicators DOW_{ti} and for federal or state holiday using indicators HOLIDAY_{tk}. Finally, HOSPITAL_{tk} represents indicator variables for whether hospital k contributes to the ED visits counts on day t; these indicators were used to account for potential temporal drops in ED counts due to missing data from individual hospitals.
Logic regression
Logic regression is an adaptive regression method that attempts to construct predictors as Boolean combinations of binary covariates. It constructs a simple decision tree or a set of decision trees (multiple trees) with binary predictors connected by and (∧), or (⋁), and not (^{c}) operators. For example, consider the following four binary extreme heat indicators based on minimum apparent temperature (ATMN),
and
where we suppress the subscript t for presentation purposes. A simple extreme heat definition may be H(X_{t}) = X_{1} ∧ X_{2} ∧ X_{3}, which describes a period of 3consecutive days with high temperature. Hence, using these set of indicators, we can potentially capture both lagged and sustained effects of extreme heat. In our application, we focus on the use of the single tree model to estimate H(X_{t}). In another example, \( H\left({\boldsymbol{X}}_t\right)={X}_1^c\wedge {X}_2\wedge {X}_3 \) describes a period of 2consecutive days with high temperature, excluding the lag0 day. This is different from X_{2} ∧ X_{3,} which does not place a restriction on whether X_{1} being 0 or 1. We also note that H(X_{t}) is a binary exposure variable and offers better interpretation for risk associations compared to a more naïve approach of including X_{1}, X_{2}, X_{3,} and X_{4}, as well as their interactions, jointly in a model.
Different logic trees may result in the same classification of days. For example, the two trees identified by logic regressions:
give the same classification according to the distributive law (i.e., A ∧ (B ⋁ C) = (A ∧ B) ⋁ (A ∧ C)). To aid in interpretation of logic regression results, we used the following scheme to describe H(X_{t}). The Boolean combinations of heat indicators generated by logic regression can always be expressed by a series of logic statements joined by the ⋁ (or) operators. Hence in the above example, H_{1}(X_{t}) =(X_{1}^{c} ∧ X_{2} ∧ X_{3}) ⋁ (X_{2} ∧ X_{3}∧ X_{4}) is the preferred form with logic statements \( {X}_1^c\wedge {X}_2\wedge {X}_3 \) and X_{2} ∧ X_{3} ∧ X_{4}.
The ‘Logicreg’ package in R was used to fit logic regression models. Estimation was based on simulated annealing as the optimization algorithm to stochastically explore all \( {2}^{2^k} \) Boolean combinations for k predictors. Tenfold cross validation was used to select the size of the tree (i.e., number of leaves) and reduce issues related to overfitting.
A multistage estimation approach
We applied logic regression to timeseries analysis of ED visit data in a threestage approach. In the first stage, for each temperature variable (i.e., ambient air temperature and apparent temperature) and metric (i.e., maximum, minimum, or average), a quasiPoisson loglinear model without H(X_{t}) was fit.
In the second stage, the Pearson residuals from the firststage model were used to identify the Boolean combination of different extreme heat indicators at various lags. The Pearson residuals were calculated as
where \( {Y}_t^a \) and \( {\hat{\mu}}_t^a \) are, respectively, the observed number of daily ED visits and the predicted mean number of daily ED visits for outcome of interest a on day t, and \( V\left({\hat{\mu}}_t^a\right) \) is the product of \( {\hat{\mu}}_t^a \) and the dispersion parameter. In the first stage we removed effects of continuous sameday and lagged temperatures, as well as other temporal trends. Because Pearson residual represents a scaled difference between the observed and expected counts, the logic regression tree H(X_{t}) estimated using Pearson residuals aims to captures additional lagged and sustained associations due to extreme temperature not explained by the base model. Finally, in the third stage, we refit the full timeseries model with H(X_{t}) and all other covariates.
Simulation study
Simulation setup
We performed a simulation study to assess the performance of logic regression a our multistage procedure in detecting the structure of extreme heat exposures and in estimating associations with health outcomes. Let X_{1}, X_{2}, and X_{3} be binary indicators for the minimum apparent temperature exceeding the 98th percentile threshold on lag 0, lag 1, and lag 2 day, respectively. We considered three different true H(X_{t}) exposures:

E1 Sameday effect: H(X_{t}) = X_{1},

E2 Sustained 2day effect: H(X_{t}) = X_{1} ∧ X_{2}, and

E3 Sustained 2day laggedonly effect: \( H\left({\boldsymbol{X}}_t\right)={X}_1^c\wedge {X}_2\wedge {X}_3 \).
We considered three different health outcomes with different sample sizes, temporal patterns, and overdispersion (CIRC, RENAL, and HEAT). For each disease, we first fit the timeseries model with all confounders as described in Section 2.1 with the Atlanta ED and meteorology data during 1993–2012 to obtain the baseline mean daily ED visits. We assumed a true relative risk (RR) of 1.01 or 1.05 for H(X_{t}) and simulated daily ED visit counts from a negativebinomial distribution and observed meteorology data using the timeseries model given in Section 2.2. Regression coefficients and overdispersion were based on estimated values from models fitted with real data. We then applied the threestage algorithm described in Section 2.4 to the simulated data and estimated the log RR of interest.
We ran the simulation 100 times for each scenario. The relative bias and relative root mean squared error (RRMSE) were used to examine the performance of the proposed approach. Relative bias and RRMSE were calculated as
Here, \( {\hat{RR}}_i \) is the estimated RR of \( \hat{H}\left({\boldsymbol{X}}_t\right) \) based on logic regression from the i^{th} simulation and RR_{true} is the true RR for each scenario. We also estimated the sensitivity and specificity by comparing days indicated by \( \hat{H}\left({\boldsymbol{X}}_t\right) \) to be exposed/unexposed to the true exposure status given by E1, E2, or E3.
Simulation study results
Results from the simulation study are summarized in Table 2. We found that performance of the proposed method was better with larger RRs and for outcomes with larger daily event counts (e.g., comparing all renal diseases versus heatrelated illnesses). We also ran the same simulation using the nontruncated continuous temperature and found similar results. Among the three different exposure scenarios, logic regression performed best for scenario E1 (singlelag, sameday) and worst in E3 (sustained twoday lagged consecutive exposure). The frequency of the extreme heat event may explain the different performances; the frequency of occurrence in our Atlanta study for E1, E2, and E3 were 146, 73, and 29 days, respectively. Importantly, we also found that in our Atlanta ED visits application, the average bias was negative, indicating that the effect estimate was attenuated towards the null. This is likely due the presence of exposure misclassification: when sensitivity/specificity is not 100%, some days are classified incorrectly as exposed/unexposed.
Application to the Atlanta ED visit data
Primary analyses
We applied the multistage algorithm described in Section 2.4 separately for each heat metric: MX, MN, and Avg of temperature (T) or apparent temperature (AT), and separately for each threshold (95th, 98th, 99th percentile). Extreme binary indicators were also defined for up to three lagged days. The model with the smallest quasiAIC was selected.
Table 3 summarizes the structure of the extreme heat exposures H(X_{t}) and their associations with ED visits. Supplementary Figure 1 shows the structure of the logic regression tree H(X_{t}) for selected outcomes. Logic trees can be read “bottomup” to construct a corresponding logic statements where the tree split is given by the Boolean statement and/or. Given a logic regression tree, individual days in our 20year study period data were divided into two groups and we defined the reference (i.e., H(X_{t}) = 0) as the type of days more frequently observed.
Overall, we found several positive associations with exposures. For example, for heatrelated diseases (HEAT), extreme heat exposure was defined as days where (1) lag 0 and lag 1 ATMIN are above the 95% percentile or (2) lag 1 and lag 3 ATMIN is above the 95% percentile. This exposure was associated with an increase in mean HEAT ED visits by 40% (95% CI: 27–54%). For acute renal failure (ARF) ED visits, the exposure identified (i.e., recent two days’ ATMIN is above the 95% percentile) was associated with an increase of 5% (95% CI: 2–9%). For all renal diseases, the exposures identified were more complicated and was associated with an increase of ED visits by 2% (95% CI: 1–4%). Associations with circulatory disease were generally null, except for ischemic stroke and hypertension, for which we found a negative association with extreme heat exposure
Sensitivity analyses
We also conducted three additional sensitivity analyses. First, we examined the more conventional heat wave definition where duration is defined as at least two consecutive days exceeding the threshold. Result for HEAT, STK, and RENAL outcomes are given in Supplementary Table S1. We found that the conventional heat wave definitions also indicated positive associations, but the magnitude can be attenuated. Second, because logic regression is a machine learning algorithm that optimizes predictability, the lag structure may contain holes (e.g., exceeding the temperature threshold on lag1 and lag3, but not lag2). Based on the exposure lags identified by logic regression as a guide, we defined alternative exposure metrics by filling in gaps and removing the “not” statement. These alternative metrics are defined over consecutive days and may be more interpretable. Results are given in Table S2. For HEAT and RENAL, we found that using alternate exposures that are more extreme (less frequent) continues to estimate positive associations. However, in the case of STK, where the alternate exposures are less extreme, the associations with ED visits were positive, but confidence intervals included the null.
In a second analysis, we replaced the truncated sameday and 3day movingaverage of temperature heat metric with the original variable without truncation. Here the application of logic regression attempts to identify additional risks beyond that conferred by the continuous exposureresponse function. Results are given in Supplementary Table S2. In general, we found that logic regression identified extreme heat exposure based on similar heat metrics but RR with smaller magnitude. However, the use of truncated continuous temperature, as in our main analysis, was generally associated with better model fit and stronger associations with ED visits. This may be attributed to data sparsity in the extreme tail of the temperature distribution such that when using nontruncated temperature, the tail of the continuous exposureresponse function may be misspecified and has high uncertainty.
Discussion
In this paper, we propose the use of logic regression to help identify characteristics of extreme heat exposure that are associated with shortterm adverse health risks. Our 20year timeseries analysis shows that ED visits for various disease outcomes were associated with exposure identified by logic regression using a multistage algorithm. Our motivating hypothesis is that most harmful characteristics of heat exposures are likely to vary between outcomes because of the different vulnerable subpopulations and different pathophysiological mechanisms they may impact. While the strength of association of different exposure definitions varied by outcome, the most healthrelevant exposures were generally those characterized by temperatures exceeding a threshold over multiple lags. We also found evidence that apparent temperature and daily minimum temperature gave better model fit, providing support that humidity and nighttime exposure are important considerations for quantifying adverse health effects of extreme heat events. However, Armstrong et al. [38] found that adding relative humidity or dewpoint temperature to a model with temperature does not improve model fit.
Though studies on heat waves and causespecific ED visits are limited compared to causespecific mortality, our results are consistent with much of the previous research that utilized different definitions. Petitti et al. [39] used three prespecified temperature trigger points (minimum risk temperature, increasing risk temperature, and excess risk temperature) in Maricopa County Arizona. This study found significant associations with all three trigger points and ED visits for heat related diagnosis but found no association with CVD related outcomes and total ED visits. Another study that looked at high ambient temperature and ED visits in California found that same day ambient temperature was positively associated with heat illness and ARF, but was negatively associated with hypertension [40], as were our results.
For hypertension, we found that the exposures estimated from logic regression tended to be negatively associated with ED visits. Previous studies on heat waves and morbidity outcomes have found similar results. Lim et al. [9] and Sherbakov et al. [41] found that hospital admissions due to hypertension and other cardiovascular related outcomes decreased with increased temperatures. Similarly, a study by Michelozzi et al. [42] found that cardiovascular related morbidity was reduced with higher temperatures, but cardiovascular mortality increased. A possible mechanism for these findings is that blood pressure can decrease from vasodilation and sweating in the summer, thus potentially reducing hypertension related hospital admissions [41, 43].
Our extreme heat exposures did not always identify consecutive days of high temperature as being the most harmful. For example, the not logic statement and missing lags were selected for several disease outcomes (Table 3 and S2). This may be due to the lack of statistical power as periods with consecutive days of high temperature were less frequent. It is also possible that the risk of consecutive days of high temperature is less harmful compared to periods with more variable but high temperature due to awareness of extreme heat events.
We found that different specifications of the base model (i.e., adjusting for truncated versus nonuntruncated continuous temperature) can have an impact on the extreme heat exposure identified and its estimated risk ratio. While both types of analyses are common in the literature, associations of extreme heat events from these two approaches should be interpreted differently (i.e., total risks above a certain threshold versus risks in addition to the continuous exposureresponse function). This motivated our multistage estimation approach such that the base model is specified a priori and the datadriven logic regression is only utilized to explore potential excess risks not explained by the base model. We note that when performing risk assessment of high temperature, both extreme temperature events and the tail of the exposureresponse function should be considered.
The main advantage of logic regression is its supervised learning approach for deriving studyspecific and outcomespecific exposure definitions that are flexibly constructed by indicators of different extreme heat characteristics. Study population, geographical region, and the set of exposures being considered may contribute to the observed heterogeneity in extreme heat health effects within and across studies. The algorithm also has a userfriendly software package that can efficiently evaluate a large suite of possible exposure definitions to identify those that are most important for individual health outcomes. Compared to regression tree methods, logic regression has two important advantages: (1) the ability to incorporate “and”, “or” and “not” statement between predictors, and (2) the focus on binary classification of a continuous outcome. In contrast, Classification And Regression Trees (CART) will give multiple terminal nodes and the resulting heat metric will likely be very complex because only “and” statement is allowed as the decision tree is split.
Finally, logic regression has several limitations that warrant further investigations. First, during estimation, logic regression can become trapped in a local minimum when many binary predictors are being considered. Hence, model fitting requires a comprehensive evaluation of tuning parameters (e.g., starting temperature, finishing temperature, and cooling schemes) and initial values, which increases computational burden. In our application, the number of binary predictors is considerably smaller than the typical genetic association studies and we were able to evaluate different control parameters for simulated annealing. Second, we only utilized logic regression to select the exposure lag structure given a heat metric and threshold. We found that the current sample size cannot accommodate including all possible heat exposure indicators that are highly correlated. One future direction is to consider additional penalization within logic regression. Third, our multistage estimation approach, while allowing us to work with established timeseries analysis methodology, does not account for estimation uncertainty associated with the extreme heat metric, which may be important for quantifying associations for rare exposure events. Statistical inference such as pseudolikelihood [44, 45] could be integrated to the current method to account for the ignored uncertainty. Moreover, recent advances in fitting logic regression under a Bayesian framework [46] allows for direct quantification of uncertainties via posterior samples. How to incorporate these uncertainties in a multistage health analysis, similar to an exposure measurement error framework, warrants further investigations.
Availability of data and materials
The daily emergency department data used in this study contain protected health information and cannot be shared due to existing data use agreements. Example R code for carrying out the simulation study and analysis are available from the corresponding author upon request.
Abbreviations
 ARF:

Acute renal failure
 AT:

Apparent temperature
 CHF:

Congestive heart failure
 CIRC:

Circulatory system diseases
 ED:

Emergency department
 DIA:

Diabetes
 FLEL:

Fluid and electrolyte imbalances
 HEAT:

Heat illness
 HT:

Hypertension
 IHD:

Ischemic heart disease
 INTERN:

Internal causes
 MI:

Myocardial infarction
 MN:

Minimum
 MX:

Maximum
 RENAL:

Renal diseases
 RR:

Relative risk
 STK:

Ischemic stroke
 T:

Temperature
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Funding
This research was supported by the National Institute of Environmental Health Sciences (NIEHS) of the National Institutes of Health (NIH) under Award R01 ES027892, R01 ES028346, and R21 ES023763. The content of this manuscript is solely the responsibility of the authors and does not necessarily represent the official views of the NIH and does not endorse the purchase of any commercial products or services mentioned in this manuscript.
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SJ contributed to data analysis, data interpretation and drafting the manuscript. JWL, NS, LAD, MJS, YC, AJN, STE and HHC assisted with the writing and interpretation of findings. STE and SM contributed to health and exposure data acquisition and management. HHC contributed to the conceptual framework of the study. All authors critically reviewed and approved the final manuscript.
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The study is approved by the Emory Institutional Review Board. Informed consent was not obtained because this study involved secondary analysis of administrative databases. All analyses were carried out in accordance with relevant guidelines and regulations.
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Additional file 1: Figure S1.
Structure of logic regression tree of extreme heat exposures for selected warmseason ED visit outcomes in Atlanta, Georgia, 1993–2012. Table S1. Summary of alternative extreme temperature metrics and their shortterm associations with warmseason emergency department visits in Atlanta, 1993 to 2012. Table S2. Summary of alternative extreme temperature metrics with consecutive lags and their shortterm associations with warmseason emergency department visits in Atlanta, 1993 to 2012. Table S3. Summary of extreme heat metrics from truncated continuous versus continuous temperature metric and their shortterm associations with warmseason emergency department visits in Atlanta, 1993 to 2012.
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Jiang, S., Warren, J.L., Scovronick, N. et al. Using logic regression to characterize extreme heat exposures and their health associations: a timeseries study of emergency department visits in Atlanta. BMC Med Res Methodol 21, 87 (2021). https://doi.org/10.1186/s1287402101278x
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