1. Analysis of time-stratified case-crossover studies in environmental epidemiology using Stata
Aurelio Tobías
Spanish Council for Scientific Research (CSIC), Barcelona, Spain
Ben Armstrong and Antonio Gasparrini
London School of Hygiene and Tropical Medicine, UK
2014 UK Stata Users Group meeting
London, 12th September 2014

2. Background
The time stratified case crossover design is a popular alternative to conventional time series regression for analysing associations between time series of environmental exposures (air pollution, weather) and counts of health outcomes

3. The case-crossover design
Proposed by Maclure (1991) to study transient effects on the risk of acute health events
Have you made ​​any unusual activity …
Health
event
… compared to your usual routine?
Only cases are sampled assessing changes in exposure

4. The case-crossover design
Proposed by Maclure (1991) to study transient effects on the risk of acute health events
Exposed?
Exposed?
Health
event
Control period
Risk period
Only cases are sampled assessing changes in exposure

5. The case-crossover design
Proposed by Maclure (1991) to study transient effects on the risk of acute health events
Analysis likewise a
matched case-control
Exposed?
Exposed?
Health
event
Control period
Risk period
Only cases are sampled assessing changes in exposure
Control
Exp. No
Risk
period a c b d
OR = b/c

6. Application to environmental studies
Firstly adapted to air pollution studies
Philadelphia (Neas et al. 1999) and Barcelona (Sunyer et al. 2000)
comparing exposure levels for a given day (t) when health event occurs vs. levels before (t-7) and after (t+7) the health event
Allows to control for time-trend and seasonality by design, since compares exposure levels between same weekdays within each month of each year

7. Time-stratified case-crossover
There is a large industry on how to choose the referent window (Wt).
It has been shown that this method of analysis sometimes can produce ‘overlap bias’.
Matched case-control studies require the distribution in each matched set to be excangeable.
This exchangeability condition may not be met since samplig the reference window (Wt) determines the exposure.
(Figueiras et al. 2010)

8. Option 1: conditional logistic regression
Dataset needs to be reshaped from time-series format to individual matched case-control by using the new command,
. mkcco vary, date(vardate)
This also creates three new variables,
case case days (=1)
wn weight observations by n events on case day
ccset set num. to match case-control days
Then, use Stata’s clogit command

9. . use madrid, clear
. list date y pm10 temp in 1/31, noobs clean
date y pm10 temp
01jan
02jan
03jan
04jan
05jan
06jan
07jan
08jan
09jan
10jan
11jan
12jan
13jan
14jan
15jan
16jan
17jan
18jan
19jan
20jan
21jan
22jan
23jan
24jan
25jan
26jan
27jan
28jan
29jan
30jan
31jan

10. Sun Mon Tue Wed Thu Fri Sat 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
. generate dow = dow(date)
. list date y pm10 temp in 1/31 if dow==1, noobs clean
date y pm10 temp
06jan
13jan
20jan
27jan
. mkcco y, date(date)
Dataset reshaped from 1096 to 5480 obs. (1096 case days and 4384 control days).
New variables case, wn and ccset have been added to the resaphed dataset.
Sun
Mon
Tue
Wed
Thu
Fri
Sat 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

11. Sun Mon Tue Wed Thu Fri Sat 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
. list date y pm10 temp case ccset wn in 1/36 if dow==1, noobs clean
date y pm10 temp case ccset wn
06jan
13jan
20jan
27jan
...
Sun
Mon
Tue
Wed
Thu
Fri
Sat 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

12. Sun Mon Tue Wed Thu Fri Sat 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
...
date y pm10 temp case ccset wn
06jan
13jan
20jan
27jan
Sun
Mon
Tue
Wed
Thu
Fri
Sat 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

13. Sun Mon Tue Wed Thu Fri Sat 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
...
date y pm10 temp case ccset wn
06jan
13jan
20jan
27jan
Sun
Mon
Tue
Wed
Thu
Fri
Sat 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

14. Sun Mon Tue Wed Thu Fri Sat 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
...
date y pm10 temp case ccset wn
06jan
13jan
20jan
27jan
Sun
Mon
Tue
Wed
Thu
Fri
Sat 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

15. . clogit case pm10 [w=wn], group(ccset)
(frequency weights assumed)
...
Conditional (fixed-effects) logistic regression Number of obs =
LR chi2(1) =
Prob > chi2 =
Log likelihood = Pseudo R =
case | Coef. Std. Err z P>|z| [95% Conf. Interval]
pm10 |

16. Option 1: conditional logistic regression
Advantage
Similar approach to a matched case-control study, being familiar to the epidemiologist
Limitations
Unfriendly data management
Large (huge) data-sets at individual level
Slow convergence when fitting interaction terms for individual factors (sex, age)
Cannot account for overdispersion and residual autocorrelation

17. Option 2: Poisson regression
Lu and Zeger (2007)
“ … the time-stratified case-crossover design leads to the same estimate as obtained from a Poisson regression with dummy variables indicating the strata”
Strata defined by the 3-way interaction between year, month and day of week
Either poisson or glm (jointly with the scale option) Stata’s commands can be used

18. . use madrid, clear
. generate yy = year(date)
. generate mm = month(date)
. generate dow = dow(date)
. poisson y pm10 i.yy##i.mm#i.dow
...
Poisson regression Number of obs =
LR chi2(252) =
Prob > chi2 =
Log likelihood = Pseudo R =
y | Coef. Std. Err z P>|z| [95% Conf. Interval]
pm10 | |
parameters!
_cons |

19. . glm y pm10 i.yy##i.mm#i.dow, family(poisson) link(log) scale(x2)
...
Generalized linear models No. of obs =
Optimization : ML Residual df =
Scale parameter =
Deviance = (1/df) Deviance =
Pearson = (1/df) Pearson =
Variance function: V(u) = u [Poisson]
Link function : g(u) = ln(u) [Log]
AIC =
Log likelihood = BIC =
| OIM
y | Coef. Std. Err z P>|z| [95% Conf. Interval]
pm10 | |
_cons |
(Standard errors scaled using square root of Pearson X2-based dispersion.)

20. Option 2: Poisson regression
Advantages
Avoids reshape of dataset keeping original time-series format
Can account for overdispersion and residual autocorrelation
Limitations
Large number of interaction parameters, slowing the convergence time (even making difficult it with missing data)
Even slower when fitting interaction terms for individual factors (sex, age)

21. Option 3: Conditional Poisson regression
Armstrong and Gasparrini (ISEE 2011)
“ … Poisson models with large numbers of indicator variables can alternatively be fit with conditional Poisson models, conditioning on numbers of events in the time stratum”
Firstly proposed by Farrington (1995) for the self-controlled case-series design for vaccine safety studies
Strata defined by grouping same weekday within each month of each year, and then use Stata’s xtpoisson command jointly with the fe option
But overdispersion needs to accounted for with the new xtpois_ovdp command

22. . egen set = group(yy mm dow)
. xtset set date
...
. xtpoisson y pm10, fe
Conditional fixed-effects Poisson regression Number of obs =
Group variable: set Number of groups =
Obs per group: min =
avg =
max =
Wald chi2(1) =
Log likelihood = Prob > chi =
y | Coef. Std. Err z P>|z| [95% Conf. Interval]
pm10 |
. xtpois_ovdp
Estimate and standard errors corrected for over-dipersion
df: 843 ; pearson x2: ; dispersion:
y | Coef. Std. Err z P>|z| [95% Conf. Interval]
pm10 |

23. CPU time My example (3 years data, 60 events/day) clogit 0.181 sec.
poisson sec.
xtpoisson sec.

24. Option 3: Conditional Poisson regression
Advantages
Avoids reshape of dataset keeping original time-series format
Can account for overdispersion and residual autocorrelation
Easy fit of interaction terms for individual factors (sex, age) with faster convergence
Limitations ?

25. Summary Case-crossover studies can easily be analysed using Stata
Conditional logistic regression requires unfriendly data management, and large (huge) data-sets at individual level
Poisson, 3-way interaction, regression requires to fit large number of interaction parameters, making it difficult convergence
Conditional Poisson seems to solve both

26. Thanks for your attention!

27. Thanks for your attention!