An Introductory Lecture to Environmental Epidemiology Part 2. Time Series Studies

1. An Introductory Lecture to Environmental Epidemiology Part 2. Time Series Studies Mark S. Goldberg INRS-Institut Armand-Frappier, University of Quebec, and McGill University July 2000

2. The first example in Part 1 (the landfill site study) dealt with analyses of airborne air pollutants using geographic regions to represent presumptive levels of exposure. Standard methods for ecological studies (comparison of rates by region) as well as more complex methods using higher quality data (population-based case-control study) were compared.

3. The second study (the Harvard Six Cities Study) used the city as the geographical unit, and a complex longitudinal cohort analysis compared adjusted mortality rates between cities. Because of the relative homogeneity of ambient fine particles, it is likely that there was less exposure misclassification than in the landfill site study.

4. The following example shows an analysis of air pollution over a circumscribed geographic region across time. The purpose is to determine whether daily counts of death increase if air pollution on the same day or on previous days increases. Thus, the study is used to investigate acute effects rather than chronic effects.

5. Example: Time Series Studies • Objective: To determine whether the daily number of deaths increases when air pollution increases on that day or on preceding days. • Method: Juxtapose a time series of deaths with a time series of air pollution.

6. Target population consists of all persons living in a well-circumscribed geographical area • Study is not entirely ecological in that there are no comparisons by place. There are, however, comparisons by time. • There are no denominators. • Exposures: Daily measurements from fixed-site monitors

8. Confounding factors: Any factor that varies on short time scales and is associated with daily mortality (e.g., weather patterns, influenza epidemics). Smoking can not be a confounding variable unless patterns of consumption change on the scale of days.

9. Time Series Plot of All Nonaccidental Mortality • The following graph shows the time series of daily counts of nonaccidental deaths by day among residents of Montreal, Quebec, who died in the city during the 10-year period 1984-1993. There are peaks in the winter. The solid line is a running smooth showing the slow increase in mortality by time.

10. 80 Number of deaths per day 60 40 20 500 1500 2500 3500 Days from January 1, 1984

11. Time Series Plot of the Coefficient of Haze • The following graph shows the time series of the coefficient of haze (a measure of ambient carbon particles), by day. Daily means from about 11 monitoring stations in the city are combined in the plot. As with mortality, there are peaks in the winter. The solid line is a LOESS smooth showing the secular trend in air pollution in Montreal from 1984-1993.

12. 15 Coefficient of haze (linear meters) 10 5 0 500 1500 2500 3500 Number of days since January 1, 1984

13. Analysis: Must account for • non-independence of daily counts of death (serial autocorrelation) • overdispersion from a Poisson process (variance > mean number of daily deaths) • Method of Analysis: Poisson regression (using quasi-likelihood; see Hastie and Tibshirani, 1990).

14. Statistical model: • E(log(Yi)) =  + f1(timei) + f2(meteorologyi) + f3(pollutioni) + …. • Covariance corrected for non-Poisson variation

15. This is a complex statistical model that makes use of quasi-likelihood within the context of the Generalized Additive Models (Hastie & Tibshirani, 1990). The Yi variable are counts of deaths that, because of clustering in time, do not follow a Poisson distribution (mean  variance). The quasi-likelihood method allows the estimation of a dispersion parameter that corrects the estimated covariance matrix.

16. The second term is referred to as a temporal filter. It is an arbitrary function of the data, and its functional form is governed by the data itself and by the amount of smoothing that is required. This term is used to adjust out the seasonal and sub-seasonal patterns in the mortality time series. In addition, it will remove serial autocorrelation, which can be substantial, thereby making the Yi independent.

17. The smoothing parameter can be determined by appealing to the Bartlett statistic to determine whether the residual mortality time series is consistent with a white noise process (Priestly, 1980). • The third term is also a smooth nonparametric function of the data, and is used to adjust for the effects of weather and other variables that vary over short time scales.

18. Filtered Mortality by Time • The first goal of the statistical analysis is to remove seasonal and sub-seasonal variations in mortality that are unrelated to air pollution. By judiciously choosing a filter in the statistical model (the f1 term in the model) using Bartlett’s test, we are able to produce a residual time series that is consistent with a white noise process.

19. 2.0 Observed/fitted number of deaths 1.5 1.0 0.5 500 1500 2500 3500 Time in days

20. The regression then adjusts for the effects of the weather variables (the f2 term) and the resultant estimate for air pollution is unbiased.

21. Exposure-response function for nonaccidental mortality and COH • The following graph shows the residual effect of coefficient of haze on nonaccidental mortality. This represents the fourth term in the statistical model (the f3 term). The y-axis is on a natural logarithmic scale. The solid line represents the nonparametric smooth function and the broken lines are pointwise 95% confidence intervals.

22. 0.20 0.15 0.10 0.05 0.0 -0.05 15 0 5 10 Coefficient of haze Natural logarithm of daily counts of death

23. Nonaccidental Deaths by Age Group • The following graph shows the estimated increase in daily mortality (in percent) for an increase in air pollution equal to the inter-quartile range for a variety of indices of ambient air particles, by age group. The top and lower horizontal bars represent 95% confidence limits, and vertical bars not crossing zero are statistically significant.

24. 4 > 65 < 65 3 2 Mean Percent Change in Daily Mortality 1 0 -1 Predicted Predicted Sutton PM2.5 COH Extinction PM2.5 sulfate sulfate -2

25. The increase in mortality is small, in the order of a few percent. However, all persons are exposed, so the attributable risk could be theoretically high. For a discussion of some of the more technical issues in interpreting these results, see Schwartz, 1994; Goldberg et al., 2000.

26. References • Environmental Epidemiology • Hertz-Piccioto, I. “Environmental Epidemiology”, in Rothman and Greenland: Modern Epidemiology, Second edition, Lippincott-Raven Publishers, 1998, Philadelphia, Chapter 28, pages 555-583.

27. Generalized additive models • Hastie, T., and Tibshirani, R. (1990). London: Chapman and Hall. These models are implemented in Splus (http://www.mathsoft.com). • Statistical considerations in time series studies Priestly, MB. Spectral Analysis of Time Series. 1981, Academic Press, New York. • Schwartz J. Nonparametric smoothing in the analysis of air pollution and respiratory illness.Cdn J Stat 1994; 22:471-87

28. Identifying subgroups of the general population that may be susceptible to short-term increases in particulate air pollution: A time series study in Montreal, Quebec. • Goldberg, M.S., Bailar, J.C. III, Burnett, R., Brook, J., Tamblyn, R., Bonvalot, Y., Ernst, P., Flegel, K.M., Singh, R., and Valois, M.-F. (2000) Health Effects Institute, Cambridge, MA (http://www.healtheffects.org).