# Insight into Bias in Time-Stratified Case-Crossover Studies

Xiaoming Wang
Health Services Statistical and Analytic Methods
Alberta Health Services
Edmonton, AB T6G 2J3
<EMAIL_ADDRESS>
Sukun Wang
Financial Crime Unit
Bank of Montreal
Toronto, ON, Canada
<EMAIL_ADDRESS>
Corresponding author. Health Services Statistical and Analytic Methods,
Alberta Health Services, #7235, 2nd Floor, West Wing, Aberhart Centre, 11402
University Avenue, Edmonton, AB T6G 2J3.

###### Abstract

The use of case-crossover designs has become widespread in epidemiological and
medical investigations of transient associations. However, the most popular
reference-select strategy, the time-stratified schema, is not a suitable
solution for controlling bias in case-crossover studies. To prove this, we
conducted a time series decomposition for daily ozone (O3) records;
scrutinized the ability of the time-stratified schema on controlling the
yearly, monthly and weekly time trends; and found it failed on controlling the
weekly time trend. Based on this finding, we proposed a new logistic
regression approach in which we did adjustment for the weekly time trend. A
comparison between the traditional model and the proposed method was done by
simulation. An empirical study was conducted to explore potential associations
between air pollutants and AMI hospitalizations. In summary, time-stratified
schema provide effective control on yearly and monthly time trends but not on
weekly time trend. Therefore, the estimation from the traditional logistical
regression basically reveals the effect of weekly time trend, instead of the
transient effect. In contrast, the proposed logistic regression with
adjustment for weekly time trend can effectively eliminate system bias in
case-crossover studies.

_K_ eywords Case-crossover design $\cdot$ Time-stratified schema $\cdot$
Overlap bias $\cdot$ Permutation $\cdot$ Bias elimination

##  1 Introduction

Since the initial work by Malcolm Maclure in 1991 [1], the use of case-
crossover designs has become widespread in epidemiological and medical
investigations of transient associations between risk factors and adverse
health events[2, 3, 4], notably in the area of ambient air pollution
research[5, 6, 7, 8, 9, 10, 11, 12, 13]. With a case-crossover design, an
investigator samples only cases and compares each individual’s exposure during
a short time period (hazard period) just before onset of a case event with
exposures at other times (referent periods) in a referent window[14].

Comparing with general case-control study designs and cohort study designs,
these self-matching designs have an obvious advantage of controlling by design
all time-invariant confounders, either measured or unmeasured. The challenge
to the case-only design comes from how to control time-varying factors. Being
similar to a cohort or a case-control study, a confounding issue in a self-
matching design arises when there exists unbalanced matching in determinants
between hazard periods and referent periods, leading to various sources of
systematic bias. These sources were systematically reviewed by Mittleman and
Mostofsky[15].

To control time-varying factors, various reference-select strategies have been
proposed in literature. Janes et al.[2] provided a comprehensive review of
these strategies and assessed potential bias associated with them. These
strategies aim to limit the reference-select window to a short period that
restricts time-varying confounders to be nearly constant across reference-
select windows[14]. The ambi-directional[16] and the time-stratified[17], were
among the top two reference-select strategies in literature. And the latter
was claimed to be unbiased[2] and optimal[18], avoiding bias resulting from
time trend in the exposure series as well as specific time-varying
confounders.

As pointed by Wang et al.[4] in a case-crossover study, there are two kinds of
biases. One is point estimate bias from imperfectly controlling of time-
varying factors (confounding issue). The other is the estimation bias of
standard error (SE), the so-called overlap bias[19, 20, 21] from correlations
among observations. We could reduce bias in point estimates by using a shorter
reference-select window which results in higher overlap bias because shorter
reference-select window results in higher correlation among cases and their
matched references. In that paper, the authors pointed out the time-stratified
case-crossover design is by no means a final solution to the overlap bias. As
a remedy, a calibration strategy based on permutation[22, 23, 24, 25] was
proposed to overcome the overlap bias issue. One pitfall of this strategy is
its computational burden.

In this paper, we were seeking a better solution to the overlap bias in case-
crossover studies. For this purpose, we scrutinized the ability of the time-
stratified schema on controlling time-varying factors and found its failure on
controlling weekly time trend. Based on this finding, we further propose a
logistic regression model in which we do adjustment for weekly time trend. As
we hypothesized, this model can effectively eliminate the overlap bias as well
as potential bias in point estimate. This implies that the overlap bias issue
intrinsically roots from unbalanced matching between hazard periods and
referent periods of the time-series under investigate. It allows us to have a
deep insight into the collection between bias in point estimate and bias in SE
estimate.

##  2 Insight into Bias in Case-Crossover Studies

###  2.1 Overlap bias

The term “overlap bias” was first introduced to case-crossover studies by
Lumley and Levy[17], who observed that this bias is similar to the “friend
control bias” in matched case-control designs[20, 21]. In the time-stratified
design in our simulation study, for example, a hazard day could be viewed as a
random sample from a study period, and 3-4 referent days matched are days in
the same year, same month and matching weekday. Selecting of referent days is
conditioned on hazard days. This reference-select scheme makes the hazard days
and the referent days to be selected from different strata in the study
period, causing a positive correlation between exposure levels on hazard and
referent periods. Austin et al.[20] showed that choosing friends or siblings
as the matched controls for each case can lead to the so-called overlap bias.

The deep reason for the overlap bias roots from the correlation among matched
cases and controls and leads to SE estimation bias and therefore a biased
significance test of the exposure effect. To answer of where the correlations
among cases and their matched references come from, we investigated it using
an air pollution time-series data.

###  2.2 Limitation of time-stratified schema

We use the same air pollution data used by Wang et al.[4]. Briefly, daily
records for five criteria air pollutants (in urban area of Edmonton, Canada)
spanning a period of 10 fiscal years (from April 1, 2000 to March 31, 2010)
were obtained from National Ambient Pollution Surveillance (NAPS)
Database[26]. The five pollutants were carbon monoxide (CO), nitrogen monoxide
(NO), nitrogen dioxide (NO2), ozone (O3), and particulate matter with an
aerodynamic diameter $\leq 2.5$ (PM2.5). Because different pollutants have
different units, we standardized each of the five pollutant measures by its
interquartile range (IQR, difference between the 75th and the 25th
percentile).

To see the limitation of the time-stratified reference-select schema, we
conducted a time-series decomposition for the daily records of O3. From the
daily time-series of O3 we calculated its yearly average (O${}_{3}^{y}$),
monthly average after removing the yearly time trend (O${}_{3}^{m}$), weekly
average after removing the yearly and monthly time trends (O${}_{3}^{w}$) and
daily average after removing the yearly, monthly and weekly time trends
(O${}_{3}^{d}$). The above process is an usual time-series decomposition, such
that $O_{3}=O_{3}^{y}+O_{3}^{m}+O_{3}^{w}+O_{3}^{d}$. We have depicted the
time-series and its components in Figure 1.

(Figure 1 here)

From Figure 1, we see that (a) the time-stratified schema can control the
yearly and monthly time trends of O3, because a case and its matched
references (with 3 to 4 references) are always in the same year and the same
month, in which yearly and monthly average values of O3 are constant; and (b)
the time-stratified schema can do nothing on controlling weekly time trend of
O3.

The transient effect of an air pollutant on a health outcome is a within-week
impact of the air pollutant. We believe that a lack of controlling weekly time
trend is the main source of potential point estimation bias as well as overlap
bias. Correlations among a case and its matched references could not be
ignorable if the weekly trend time-series is highly correlated.

##  3 Correct Bias in Case-Crossover Studies

Suppose $Y$ is the response variable where $Y=0/1$ indicates exposure to
hazard/referent period in a case-crossover study. The following traditional
logistic regression model is the typical analytic tool being used in
literature to assess transient association between adverse events and exposure
to a pollutant.

$\mbox{logit}(p)=\alpha+\beta O_{3},$ (1)

here $\mbox{logit(p)}=\log(p/(1-p)$ with $p=Pr(Y=1)$, and $\beta$ is the
transient effect of O3 under estimation. Again, we use O3 as example for an
interested pollutant. For illustration simplicity, we consider a model without
covariates.

###  3.1 Control weekly time trend

We propose the following analytic methods. Given a suitable model adjustment
for weekly, monthly and yearly time trends, we seek to obtain unbiased point
estimate and unbiased significance test. Our first logistic regression model
is the one adjusted for the yearly, monthly and weekly time trends of O3,
i.e.,

$\mbox{logit}(p)=\alpha+\beta
O_{3}+\gamma_{1}O_{3}^{y}+\gamma_{2}O_{3}^{m}+\gamma_{3}O_{3}^{w}.$ (2)

The adjustment for the yearly and monthly time trends is mainly because some
cases matched with 3 references while the others matched with 4 references,
leading to possible unbalanced matching of the yearly and monthly time trends
in hazard and reference periods.

The second logistic regression model is designed to estimate directly the
impact of the daily average (O${}_{3}^{d}$) and adjusted for the yearly,
monthly and weakly time trends.

$\mbox{logit}(p)=\alpha+\beta
O_{3}^{d}+\gamma_{1}O_{3}^{y}+\gamma_{2}O_{3}^{m}+\gamma_{3}O_{3}^{w}.$ (3)

The transient impact from O3 is considered to be the effect of O${}_{3}^{d}$.
It is worth to point out that model 2 and 3 are equivalent with each other.

If model 2 and 3 are correct, i.e., there is no other unobserved time-varying
factors confounding the association under estimation, we can conclude that
both of them can provide unbiased point estimate (regardless of whether the
estimate of SE is biased or not) and unbiased significance test (if estimate
of SE is unbiased).

Generally speaking, if we cannot control the weekly time trends, correlations
among weekly average records will enter into the errors, leading to relatively
higher correlations among them. However, after controlling weekly time trend,
correlations among daily average records could be low for days in the same
month but with one week or multiple weeks apart. If we can assume that the
daily average records in a hazard day and its matched reference days are
independent with each other, then we can conclude the significance test from
model 2 or 3 is unbiased.

###  3.2 Calibrate bias using permutation

From statistical point of view, all of the proposed models are only for
“univariate analysis". However, bias could also come from other potential
time-varying confounders[13]. An idea strategy is to observer those time-
varying confounders and do suitable model adjustment for them. In this way,
model (2) can be rewritten as

$\mbox{logit}(p)=\alpha+\beta
O_{3}+\gamma_{1}O_{3}^{y}+\gamma_{2}O_{3}^{m}+\gamma_{3}O_{3}^{w}+\mu Z,$ (4)

here the item of $Z$ represents suitable adjustment for some observed
covariates. However, a big challenge is it is impossible for an investigator
to observer all the potential time-varying confounders.

Another possible (but not perfect) solution to the challenge is using the
calibration method suggested by Wang et al.[4] (see detail of the calibration
algorithm therein). The basic assumption of the calibration method is that the
estimated exposure effect ($\hat{\beta}$) can be always divided into design
bias($b$), true exposure effect ($\beta$), and random error ($e$), i.e.

$\hat{\beta}=\beta+b+e,$ (5)

which means regardless of whether the true exposure effect $\beta$ is zero or
not, the design bias $b$ will always be a constant.

Under this assumption, we can correctly estimate the design bias $b$ by
permutation and completely eliminate it from point estimate $\hat{\beta}$.
Although this assumption may be questionable in practice, we still believe
this non-parametric technique can gain benefit for us on bias reduction in
both point estimate and significance test. We will show performance of our
calibration strategy by simulation.

##  4 Simulation Study

We use O3 as an example to describe our simulation study design. The same
design applies to all the five pollutants. Based on the 3,652-day study period
(from April 1, 2000 to March 31, 2010), we simulated “adverse event days” to
create different scenarios to show the performance of the following modeling
strategies.

  * •

Traditional logistic model 1;

  * •

Proposed logistic model 2;

  * •

Traditional model 1 with calibration;

  * •

Proposed model 2 with calibration.

For each scenario, we replicate 1,000 experiments. In each of these
experiments: 5,000 event days are randomly sampled with replacement from the
study period with probability proportional to $\exp(\beta
O_{3}^{d}+\gamma_{1}O_{3}^{y}+\gamma_{2}O_{3}^{m}+\gamma_{3}O_{3}^{w})$;
references are selected using the time-stratified schema; data are analyzed
using the above four modeling strategies. All analyses were conducted in R
(version 3.6.1, 2019-07-05).

###  4.1 Bias check and size check under null

All scenarios in this subsection are designed under null hypothesis
($\beta=0$) with the settings of $\gamma_{1}=\gamma_{2}=\gamma_{3}=\gamma=0$
and 0.2, respectively. For each of the five pollutants, point estimates from
each of the four analytic methods are depicted in Figure 2 and Figure 3, and
rejection frequencies (size, under the criterion $\alpha_{0}=0.05$) of
significance tests in 1,000 experiments are reported in Table 1.

(Table 1 here)

From the top panel of Table 1, we can see that in scenarios under the setting
of $\gamma=0$, the size of model 1 (before calibration) is far less from the
nominal level 50, especially for CO, NO, and O3. Considering the point
estimates in these situations are almost unbiased (Figure 2-A), we conclude
that the bias in size is from correlation among errors (i.e., overlap bias).
This implies that, without controlling weekly time trend, model 1 provides
overestimate of standard errors, leading to higher overlap bias. On the other
hand, with suitable adjustment on weekly time trend (model 2), the overlap
bias is almost gone and significance tests nearly unbiased. It seems that the
calibration method can always provide reasonable and robust significance
tests.

The bottom panel of Table 1 shows that under the setting of $\gamma=0.2$ there
is a big problem in using model 1 (before calibration). The big size of
significance tests is from the uncontrolled weekly time trend, which leads to
seriously biased point estimates (Figure 3-A) as well as a higher correlation
among errors. To our surprise, in this scenario, the calibration strategy also
can do a great job for overlap bias correction.

The proposed logistic regression model 2 is unbiased in significance test both
before and after calibration. The good performance of model 2 roots from
thoroughly controlling of the weekly time trend. In this case, the calibration
method can do very little on further bias reduction for model 2.

(Figure 2 here)

Figure 2 shows the performance in point estimates of the traditional logistic
model 1 vs the proposed logistic model 2 under the setting of $\gamma=0$.
Because of the null hypothesis, a point estimate is its estimation bias. In
this scenario point estimates from model 1 are nearly unbiased (panel A);
after calibration (panel B) bias in point estimates is eliminated completely.
It is obvious that model 2 provides very similar results on estimation bias
before and after calibration (panel C and D), i.e., unbiased point estimates.

Comparing the top panels (A and B) and the bottom panels (C and D) in Figure
2, we can see a difference in variance of estimation bias: range of top panels
is within $\pm 0.06$, while that of bottom panels is within $\pm 0.15$. This
can be explained by the difference between model 1 and 2. In model 1, the
time-stratified schema is used to control yearly and monthly time trends, so
the estimated effect is from weekly time trends. In contract, with controlling
yearly, monthly and weekly time trends, the estimated effect from model 2 is
that of the daily time trend. In fact, the target estimation of model 1 is the
effect of weekly time trend, instead of the transient effect under
investigation.

(Figure 3 here)

Under the setting of $\gamma=0.2$, model 1 and model 2 perform quite different
in point estimates (Figure 3). It is obvious that (a) without control of
weekly time trend, model 1 could be unacceptably biased in coefficient
estimate (panel A); (b) with thoroughly control of weekly time trend, model 2
can be perfectly unbiased in point estimates (panel C); (c) the calibration
method can eliminate the bias from model 1 by permutation (panel B); and (d)
model 2 and calibrated model 2 perform very similar, reflecting that model 2
is good enough in point estimate and further calibration is unnecessary in the
simulated scenario.

###  4.2 Power check under alternative

In order to compare efficiency, we considered a power check for the above four
analytic strategies. Scenarios are simulated by setting $\gamma=0$ and $\beta$
from ${0,0.1,\cdots,0.4}$. Rejection rates (power, under the criterion
$\alpha_{0}=0.05$) of significance tests in 1,000 experiments under each
scenario are reported in Table 2.

(Table 2 here)

In summary, (a) test power form model 1 is the worst, because of the overlap
bias; (b) model 2 provides competitive best test power with model 2 with
calibration; and (c) model 1 with calibration is much better than model 1 in
test power, but worse than model 2 and model 2 with calibration. The
difference of the two calibrated models in performance can be explained by the
different in their estimation targets (see previous subsection).

##  5 Empirical study

Using Alberta Health administrative databases, we obtained all historical
records of hospital admissions for acute myocardial infarction (AMI, ICD-10
code I21-I22 or ICD-9 code 410) for Edmonton urban dwellers. In total 9811
admissions for AMI occurred from 1 April 2000 to 31 March 2010 among patients
aged 20 or over and living in the areas of Edmonton.

We explore potential transient associations between AMI hospitalizations and
the five air pollutants (the same data used in the simulation study). Cohort
and 10 sub-cohorts are defined using admission characteristics as Whole (whole
cohort), Male, Female, Age$\geq 65$, Age$<65$, STEMI (ST Segment Elevation
Myocardial Infarction), NSTEMI (Non-ST Segment Elevation Myocardial
Infarction), Diabetes, Hypertension, Dysrhythmia, and PIHD (prehistory of
heart disease); and records in each of them are further divided into subgroups
defined by seasons (All season, Spring, Summer, Autumn and Winter). For each
subgroup: hazard days are selected as $k$th-day pre case events ($k=0-4$);
references are selected using the time-stratified schema; data are analyzed
using the traditional logistic regression model 1 as well as the proposed
logistic regression model 2.

The estimated effects of the 1,375 (11 cohort/sub-cohorts x 5 subgroups x 5
lags x 5 pollutants) candidate associations from the traditional logistic
model 1 and the proposed logistic model 2 are reported in appendix files "Est-
AMI-application-model1.csv" and "Est-AMI-application-model2.csv",
respectively.

From the estimates of model 1, in summary, 793 (582) of them were estimated as
positive (negative) effects; there are 140 (39) candidates with p-value $\leq
0.05$ ($0.01$). From the estimates of model 2, on the other hand, 634 (741) of
them were estimated as positive (negative) effects; there are 75 (10)
candidates with p-value $\leq 0.05$ ($0.01$). Using Bonferroni correction with
criterion $\alpha_{0}=0.05/1375$, none of the associations (neither from model
1 nor from model 2) has a p-value less than the criterion. The correlation is
0.605 between estimates of model 1 and estimates of model 2. From these
numbers we can see an obvious difference in results between the two models.

(Table 3 here)

The 39 associations detected by the model 1 with p-value $\leq 0.01$ are
reported in Table 3. From the table we can see a clear seasonal pattern: CO,
NO, and NO2 have positive effects in spring and negative effects in autumn; O3
has a negative effect in winter or Spring, and positive effect in summer. It
is hard to believe that the same pollutant in one season has a protective
effect while in another season has a harmful effect. This phenomenon implies
that estimate of the traditional logistic model 1 contains some kind of
seasonal trend. We believe it is just because model 1 can not control the
weekly time trend of an exposure time series. On the other hand, model 2 with
adjusting with weekly time trend can be expected to tell a different story.

(Table 4 here)

The 10 associations detected by the model 2 with p-value $\leq 0.01$ are
reported in Table 4. From the table we see there is no seasonal pattern as
shown in Table 3. In detail, all of the 9 estimated effects of CO, NO, O3 or
PM2.5 are positive, and 1 estimated effect of NO2 is negative. This implies
that exposure to a higher concentration of CO, NO, O3 or PM2.5 may associated
with higher risk of AMI hospitalization. For example, STEMI patients could be
sensitive to increased PM2.5 concentration in autumn (OR=1.156; 95% confident
interval(CI): 1.047, 1.276); while Dysrhythmia patients exposed to increased
concentration of PM2.5 in spring could have elevated risk of hospitalization
(OR=1.247; CI: 1.058, 1.468).

It is notable that there are 4 candidate associations with p-value $\leq 0.01$
that were detected by both the two methods. These estimates are marked with an
asterisk in Table 3 and Table 4, respectively. For example, a higher O3 level
in summer is associated with AMI hospitalizations of all patients (OR=1.216;
CI=1.061, 1.392), NSTEMI patients (OR=1.308; CI=1.806, 1.576) and hypertension
patients (OR=1.281; CI=1.072, 1.531) by the proposed model 2.

##  6 Conclusions

Using the decomposition of the ozone time series, we scrutinized the control
ability of the time-stratified reference selecting schema. This schema
provides effective control on yearly and monthly time trends but not on weekly
trend. We believe that a lack of control on weekly trend is the main source of
systematic bias in case-crossover studies.

Without controlling weekly time trend, it will enter into the errors of our
logistic regression model, leading to the so-called overlap bias and a
possible bias in point estimate. Doing model adjustment for weekly time trend
can be an effective way of reducing bias from it. Further model adjustment and
calibration could be necessary if there are other potential time-varying
confounders.

The typical case-crossover study design, i.e., combining the time-stratified
reference-select schema with a traditional logistic regression, leads to an
estimate of weekly time trend effect, instead of the transient effect under
investigation. The difference in results between the traditional approach and
the proposed method could be significant. We strongly recommend researchers to
do controlling (adjustment) for weekly time trend of exposures and covariates
in their case-crossover application studies.

We found, through the empirical study for AMI hospitalization, that higher
concentration of CO, NO, O3 and PM2.5 are associated with higher risk of AMI
hospitalization, while higher concentration of NO2 is associated with lower
risk of AMI hospitalization. A limitation of the empirical study for AMI
hospitalization is that we only did ‘univatiate’ analyses for the five
pollutants without controlling for impact from other potential time-varying
factors. Considering also none of the candidate association has p value less
than the Bonferroni criterion, we believe further confirmation for the
reported “findings" is still needed from other studies.

## Acknowledgements

Ethical approval for the study was granted by the University of Alberta’s
Health Research Ethics Board-Health Panel (IREB Pro00010852). Patient
records/information was anonymized and de-identified with a unique scrambled
ID and released to us in this form by the ministry of health prior to
analysis.

### Funding

The author(s) received no financial support for the research, authorship,
and/or publication of this article.

### Conflicts of Interest

The author(s) declared no potential conflicts of interest with respect to the
research, authorship, and/or publication of this article.

### Supplementary Material

Air pollution data of daily average records in urban Edmonton from April 1,
2000 to March 31, 2010 for carbon monoxide (CO), nitrogen monoxide (NO),
nitrogen dioxide (NO2), ozone (O3), and particulate matter with an aerodynamic
diameter $\leq 2.5$ (PM2.5) is available online. An Excel file reporting
modeling results of the 1375 models for the empirical study is available
online. R code files for the simulation and the empirical study are available
upon request.

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Table 1: Size of significance tests for model 1-2 (before calibration and after calibrations). Model | $\gamma$ | Calibration | CO | NO | NO2 | O3 | PM2.5
---|---|---|---|---|---|---|---
Model 1 | 0.0 | Before | 4 | 9 | 16 | 4 | 42
Model 1 | 0.0 | After | 61 | 50 | 52 | 59 | 63
Model 2 | 0.0 | Before | 48 | 35 | 46 | 59 | 57
Model 2 | 0.0 | After | 49 | 51 | 60 | 46 | 59
Model 1 | 0.2 | Before | 617 | 973 | 619 | 108 | 990
Model 1 | 0.2 | After | 54 | 63 | 58 | 54 | 51
Model 2 | 0.2 | Before | 47 | 38 | 43 | 58 | 51
Model 2 | 0.2 | After | 54 | 55 | 59 | 63 | 63

Note: Model 1=traditional logistic regression model on pollutant without
adjustment; Model 2=proposed logistic regression model on pollutant with
adjustment on its yearly, monthly and weekly time trend; each size (under the
criterion $\alpha_{0}=0.05$) was calculated from 1000 independent replication
of experiments.

Table 2: Power of significance tests for model 1-2 (before and after calibrations) with $\gamma=0$. Scenario | Before Calibration | After Calibration
---|---|---
pollutant | $\beta$ | Model 1 | Model 2 | Model 1 | Model 2
CO | 0.0 | 4 | 48 | 61 | 49
CO | 0.1 | 637 | 964 | 819 | 937
CO | 0.2 | 1000 | 1000 | 1000 | 1000
CO | 0.3 | 1000 | 1000 | 1000 | 1000
NO | 0.0 | 9 | 35 | 50 | 51
NO | 0.1 | 979 | 999 | 986 | 999
NO | 0.2 | 1000 | 1000 | 1000 | 1000
NO | 0.3 | 1000 | 1000 | 1000 | 1000
NO2 | 0.0 | 16 | 46 | 52 | 60
NO2 | 0.1 | 420 | 689 | 515 | 663
NO2 | 0.2 | 960 | 1000 | 978 | 1000
NO2 | 0.3 | 1000 | 1000 | 1000 | 1000
O3 | 0.0 | 4 | 59 | 59 | 46
O3 | 0.1 | 45 | 447 | 376 | 557
O3 | 0.2 | 454 | 976 | 898 | 988
O3 | 0.3 | 930 | 1000 | 1000 | 1000
PM2.5 | 0.0 | 42 | 57 | 63 | 59
PM2.5 | 0.1 | 980 | 999 | 979 | 998
PM2.5 | 0.2 | 1000 | 1000 | 1000 | 1000
PM2.5 | 0.3 | 1000 | 1000 | 1000 | 1000

Note: Model 1=traditional logistic regression model on pollutant without
adjustment; Model 2=proposed logistic regression model on pollutant with
adjustment on its yearly, monthly and weekly time trend; each power (under the
criterion $\alpha_{0}=0.05$) was calculated from 1000 independent replication
of experiments.

Table 3: Detected associations with p-value $\leq$ 0.01 between pollutants and AMI hospitalization by the traditional logistic model 1. Sub-cohort | Season | N | Pollutant | Lag (day) | Odds Ratio (95% CI)
---|---|---|---|---|---
Whole | Spring | 2505 | CO | 4 | 1.068 (1.017, 1.121)
Age$<$65 | Spring | 1045 | CO | 4 | 1.121 (1.042, 1.205)
NSTEMI | Spring | 1397 | CO | 4 | 1.093 (1.022, 1.168)
Whole | Autumn | 2408 | CO | 3 | 0.919 (0.865, 0.976)
Age$\geq$65 | Autumn | 1320 | CO | 3 | 0.893 (0.822, 0.971)
Whole | Spring | 2505 | NO | 2 | 1.067 (1.021, 1.115)
Whole | Spring | 2505 | NO | 3 | 1.066 (1.020, 1.114)
Whole | Spring | 2505 | NO | 4 | 1.068 (1.019, 1.119)
Male | Spring | 1631 | NO | 3 | 1.081 (1.024, 1.141)
Male | Spring | 1631 | NO | 4 | 1.078 (1.019, 1.14)
Age$<$65 | Spring | 1045 | NO | 3 | 1.106 (1.034, 1.184)
Age$<$65 | Spring | 1045 | NO | 4 | 1.110 (1.034, 1.191)
STEMI | Spring | 1108 | NO | 3 | 1.091 (1.024, 1.163)
Dysrhythmia* | Spring | 428 | NO | 3 | 1.158 (1.046, 1.282)
NSTEMI | Spring | 1397 | NO | 4 | 1.091 (1.024, 1.162)
Whole | Autumn | 2408 | NO | 3 | 0.915 (0.861, 0.973)
Whole | Autumn | 2408 | NO | 4 | 0.907 (0.852, 0.966)
Male | Autumn | 1607 | NO | 2 | 0.906 (0.841, 0.976)
Male | Autumn | 1607 | NO | 4 | 0.899 (0.832, 0.972)
Age$\geq$65 | Autumn | 1320 | NO | 3 | 0.880 (0.806, 0.96)
Whole | Spring | 2505 | NO2 | 2 | 1.093 (1.030, 1.160)
STEMI | Spring | 1108 | NO2 | 2 | 1.138 (1.04, 1.244)
Whole | Autumn | 2408 | NO2 | 3 | 0.837 (0.761, 0.921)
Whole | Autumn | 2408 | NO2 | 4 | 0.873 (0.794, 0.960)
Male | Autumn | 1607 | NO2 | 2 | 0.857 (0.764, 0.96)
Male | Autumn | 1607 | NO2 | 3 | 0.805 (0.716, 0.905)
Male | Autumn | 1607 | NO2 | 4 | 0.840 (0.748, 0.945)
Age$\geq$65 | Autumn | 1320 | NO2 | 3 | 0.828 (0.727, 0.944)
HTN | Autumn | 1423 | NO2 | 3 | 0.841 (0.743, 0.953)
NSTEMI | Autumn | 1312 | NO2 | 3 | 0.800 (0.703, 0.911)
NSTEMI | Autumn | 1312 | NO2 | 4 | 0.830 (0.729, 0.946)
Whole | Winter | 2475 | O3 | 1 | 0.870 (0.783, 0.967)
Whole | Spring | 2505 | O3 | 4 | 0.882 (0.815, 0.955)
Female | Spring | 874 | O3 | 4 | 0.832 (0.726, 0.954)
NSTEMI | Spring | 1397 | O3 | 4 | 0.870 (0.783, 0.966)
Whole* | Summer | 2423 | O3 | 2 | 1.144 (1.037, 1.261)
NSTEMI* | Summer | 1288 | O3 | 2 | 1.204 (1.052, 1.377)
HTN* | Summer | 1433 | O3 | 2 | 1.183 (1.041, 1.345)
Age$<$65 | Spring | 1045 | PM2.5 | 4 | 1.107 (1.028, 1.191)

Note: N=number of observations; CI=confidence interval; STEMI = ST Segment
Elevation Myocardial Infarction; NSTEMI = Non-ST Segment Elevation Myocardial
Infarction; HTN = Hypertension; PIHD = prehistory of heart disease; *Detected
by both the traditional logistic model 1 and the proposed logistic model 2.

Table 4: Detected associations with p-value $\leq$ 0.01 between pollutants and AMI hospitalization by the proposed logistic model 2. Sub-cohort | Season | N | Pollutant | Lag (day) | Odds Ratio (95% CI)
---|---|---|---|---|---
Dysrhythmia | Spring | 428 | CO | 3 | 1.319 (1.094, 1.591)
Dysrhythmia* | Spring | 428 | NO | 3 | 1.210 (1.048, 1.397)
HTN | All Season | 5848 | NO2 | 0 | 0.912 (0.850, 0.978)
Whole* | Summer | 2423 | O3 | 2 | 1.216 (1.061, 1.393)
NSTEMI* | Summer | 1288 | O3 | 2 | 1.308 (1.086, 1.576)
Diabetes | All Season | 2667 | O3 | 3 | 1.192 (1.047,1.356)
HTN* | Summer | 1433 | O3 | 2 | 1.281 (1.072, 1.531)
PIHD | All Season | 3076 | O3 | 3 | 1.174 (1.040, 1.325)
STEMI | Autumn | 1096 | PM2.5 | 4 | 1.156 (1.047, 1.276)
Dysrhythmia | Spring | 428 | PM2.5 | 3 | 1.247 (1.058, 1.468)

Note: N=number of observations; CI=confidence interval; STEMI = ST Segment
Elevation Myocardial Infarction; NSTEMI = Non-ST Segment Elevation Myocardial
Infarction; HTN = Hypertension; PIHD = prehistory of heart disease;
SE=standard error; *Detected by both the traditional logistic model 1 and the
proposed logistic model 2.

Figure 1: Time series decomposition of daily records of ozone (O3) from April
1, 2000 to March 31, 2010. Daily records of O3 (in Edmonton urban area,
Canada) are depicted in the top panel. Its yearly, monthly, weekly, and daily
components of time trends are depicted in 2nd to 5th panels, respectively.

Figure 2: Boxplot of distribution the estimated biases of effect of one of the
five pollutants using one of the four analytic modeling strategies
respectively. Scenarios are simulated under $\beta=0$ and $\gamma=0$, and
study period is the whole-cycle-period from April 1, 2000 to March 31, 2010.
The five pollutants are carbon monoxide (CO), nitrogen monoxide (NO), nitrogen
dioxide (NO2), ozone (O3), and particulate matter with an aerodynamic diameter
$\leq 2.5$ (PM2.5).

Figure 3: Boxplot of distribution of the estimated biases of effect of one of
the five pollutants using one of the four analytic modeling strategies
respectively. Scenarios are simulated under $\beta=0$ and $\gamma=0.2$, and
study period is the whole-cycle-period from April 1, 2000 to March 31, 2010.
The five pollutants are carbon monoxide (CO), nitrogen monoxide (NO), nitrogen
dioxide (NO2), ozone (O3), and particulate matter with an aerodynamic diameter
$\leq 2.5$ (PM2.5).