Carrothers and Evans
to a m ultivariate setting. Although the com putational re-
quirem ents will rise substantially, the m ultivariate solution
will be of the sam e general form as the bivariate solution
shown in eqs 5 and 6.
across a m etropolitan area for a period of m an y m on th s).
With out th ese data, n o defin itive con clusion s can be
m ade regardin g th e possibility of bias due to differen ces
in m easurem en t error am on g correlated pollutan ts.
As ongoing studies in epidem iology and exposure as-
sessment create a larger database of time-series analyses with
concurrent m easurem ents of the fine and coarse particle
exposures, the quantitative tool developed in this paper
will be useful in assessing the possibility of bias from the
interplay of m easurem ent error and correlation. Further
work in quantitative m ethods should be able to better re-
solve the m any other debates regarding the interpretation
of air pollution epidem iology.
The m odel assum ed that the m easurem ent errors for F
and C are uncorrelated. This was based on the assum ption
that the spatial variability error is m uch larger than the
instrum ental error and the assum ption that the day-to-day
changes in the spatial pattern of F and C across the city are
uncorrelated. If either of these assum ptions fails, then the
regression m odel should be adjusted to account for this
additional correlation in the term s.
Finally, errors of the Berkson type can induce bias if
the dose-response is non-linear. In the non-linear case, er-
rors concerning the difference between the average indi-
vidual and the ith individual cannot be ignored. This is a
relatively un explored area of research , an d future work
should specifically address errors in exposure assessm ent
under non-linear dose-response.
ACKNOW LEDGMENTS
Both auth ors were supported by th e Harvard Cen ter for
Risk An alysis an d th e Departm en t of En viron m en tal
Health , Harvard Sch ool for Public Health . Carroth ers was
also supported by a Nation al Scien ce Foun dation Gradu-
ate Research Fellowsh ip. Th e auth ors wish to th an k David
Mage, Joh n Spen gler, an d th e peer reviewers for th eir
h elpful com m en ts.
CONCLUSIONS
This paper introduced a tool for predicting the am ount of
bias present in the ratio of fine and coarse particle regres-
sion coefficients due to the interaction of correlation and
m easurem ent error. The m odel showed that the am ount of
bias depended on several variables: the true correlation of
fine and coarse particle exposures, the m easurem ent errors
for fine and coarse particle exposures, and the underlying
true ratio of the fine particle toxicity to coarse particle tox-
icity. Analysis of the m odel proved that all of these vari-
ables m ust be discussed and analyzed before m aking any
broad conclusions regarding bias. For instance, it is inad-
equate to state that differences in measurement error among
fine and coarse particles will lead to false negative findings
for coarse particles. If the underlying true ratio of the fine
and coarse particle toxicities is large (i.e., greater than 3:1),
fine particle exposure m ust be m easured significantly m ore
precisely in order not to underestimate the ratio of fine par-
ticle toxicity versus coarse particle toxicity.
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72 Journal of the Air & Waste Management Association
Volume 50 January 2000