Correction: exposure measurement error in time-series air pollution studies. (Correspondence).

David Mage pointed out an error in the first complete paragraph in the second column of page 423 in our paper "Exposure Measurement Error in Time-Series Studies of Air Pollution: Concepts and Consequences" (1). This section contains a brief analysis of the role of pollution originating from indoor and outdoor sources that is incorrect in its derivation but correct in its basic finding that average personal exposure is roughly proportional to ambient concentration. Hence, regression models that use ambient measurements to predict mortality can give different estimates of pollution relative risks than would be obtained if average personal exposure were available. As we show (1), however, the corresponding coefficient for personal exposure can be obtained from the coefficient for ambient concentration by a simple rescaling.

Below is a corrected analysis that depends on the following definitions: [z.sup.*.sub.t] = ambient concentration on day t; [I.sub.it] = concentration from indoor sources for person i on day t; [[delta].sub.it] = proportion of pollutant of ambient origin that penetrates indoors for person i on day t; and [p.sub.it] = proportion of time spent outdoors by person i on day t.

The personal exposure for person i on day t is given by

[x.sub.it] = [p.sub.it][z.sup.*.sub.t] + (1 - [p.sub.it]) {[I.sub.it] + [[delta].sub.it] [z.sup.*.sub.t]} = [q.sub.it][z.sup.*.sub.t] + [J.sub.it'],

where [q.sub.it] = [p.sub.it] + (1 - [p.sub.it])[[delta].sub.it] is the fraction of ambient concentration to which a person is exposed on a given day by either being outdoors or by being indoors and being exposed to ambient pollution which has penetrated indoors; and [J.sub.it] = (1 - [p.sub.it])[[I.sub.it] is the effective concentration of pollution originating from indoor sources to which person i is exposed on day t.

If we average the equation above across all people in a given region, we have that

[[bar]x.sub.t] = [[bar]q.sub.t][z.sup.*.sub.t] + [[bar]J.sub.t].

Thus, the average personal exposure is linearly related to the true ambient concentration with slope coefficient [[bar]q.sub.t], the average of [q.sub.it] across people. Here [[bar]J.sub.t] is the average concentration of pollution from indoor sources to which the population is exposed on day t.

If we further assume that conditional on weather, season and other adjustment variables in the time-series models, [[bar]J.sub.t] is roughly independent of the ambient level [z.sup.*.sub.t], this equation shows that using ambient concentration [z.sup.*.sub.t] to predict daily mortality will produce a regression coefficient that differs from what would have been obtained using mean personal exposure [bar]x.sub.t] by a multiplicative factor that is roughly [[bar]q], the average of the [[bar]q.sub.t]s over time. Note that [[bar]q]is the fraction of outdoor pollution to which the population is on average exposed, either outdoors or via penetration indoors. There is no further bias introduced by [[bar]J.sub.t] because this is an example of Berkson rather than classical measurement error as described above.

In the United States, the average proportion of time spent outdoors tends to be small, so that [[bar]q.sub.t] is to first approximation, equal to the average percentage of ambient concentration that penetrates indoors [[bar][delta].sub.t]. If nearly all small particles penetrate indoors, then [[bar][delta].sub.t] [approximately equal to] 1 and average personal exposure will equal the ambient level plus the contribution of indoor sources. Again [[bar]J.sub.t] it is roughly independent of the ambient level [z.sup.*.sub.t], then regressing on ambient levels will give similar results to regressing on average personal exposure for small particles most of which penetrate indoors.

In the original paper, the equation above mistook [[bar]q.sub.t] to be the average fraction of total exposure that originates outdoors; the correct analysis here shows that it is the average ambient fraction of ambient pollution concentration to which a person is exposed while outdoors (100%) and indoors (100[delta]%). In addition, the original article identified [[bar]J.sub.t] as the average concentration of particles originating indoors. It is in fact the average of the concentration originating indoors to which persons are exposed and therefore includes a term representing the fraction of time persons spend indoors as well as the pollutant level there.

Scott L. Zeger
Peter J. Diggle
Johns Hopkins University
Bloomberg School of Public Health
Baltimore, Maryland
E-mail: [e-mail]

REFERENCES AND NOTES

(1.) Zeger SL, Thomas D, Dominici F, Samet JM, Schwarz J, Dockery D, Cohen A. Exposure Measurement Error in Time-Series Studies of Air Pollution: Concepts and Consequences. Environ Health Perspect 108:419-426 (2000).