Re: Negative Control Outcomes and the Analysis of Standardized Mortality Ratios

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In a recent issue of this journal, Richardson and colleagues1 suggested to use so-called negative control outcomes to adjust standardized mortality ratios (SMR) for potential confounding bias. SMRs between a cohort and a reference population can be biased if the cohort and population differ not only with respect to the measured exposure(s) of interest but also with respect to other (unmeasured) factors that influence the outcome of interest. The authors suggest using an alternative negative control outcome that is assumed not to be affected by the exposure of interest but by (ideally) all other unmeasured confounders. This negative control outcome is used to adjust the SMR for potential confounding in a Poisson model:
JOURNAL/epide/04.02/00001648-201705000-00032/math_32MM1/v/2017-07-26T080259Z/r/image-tiff
with Y as the observed number of deaths from a disease D1 potentially related to the exposure of interest and Z as the number of deaths from a disease D2, denoted negative control outcome in Richardson et al.,1 in the cohort. I and J are the mortality rates for D1 and D2 in the reference population, respectively. In this model,
JOURNAL/epide/04.02/00001648-201705000-00032/math_32MM2/v/2017-07-26T080259Z/r/image-tiff
is used as an offset to adjust for potential confounding and
JOURNAL/epide/04.02/00001648-201705000-00032/math_32MM3/v/2017-07-26T080259Z/r/image-tiff
is the adjusted SMR. Richardson and colleagues1 suggest using the simple maximum likelihood estimator of
JOURNAL/epide/04.02/00001648-201705000-00032/math_32MM4/v/2017-07-26T080259Z/r/image-tiff
, which is asymptotically normally distributed, that is,
JOURNAL/epide/04.02/00001648-201705000-00032/math_32MM5/v/2017-07-26T080259Z/r/image-tiff
with
JOURNAL/epide/04.02/00001648-201705000-00032/math_32MM6/v/2017-07-26T080259Z/r/image-tiff
. This estimate, however, does not account for the variability in Z, which can be estimated from the cohort data as
JOURNAL/epide/04.02/00001648-201705000-00032/math_32MM7/v/2017-07-26T080259Z/r/image-tiff
with
JOURNAL/epide/04.02/00001648-201705000-00032/math_32MM8/v/2017-07-26T080259Z/r/image-tiff
and
JOURNAL/epide/04.02/00001648-201705000-00032/math_32MM9/v/2017-07-26T080259Z/r/image-tiff
. (1) and (2) is a classical error-in-variables model,2 with the correct estimate of
JOURNAL/epide/04.02/00001648-201705000-00032/math_32MM10/v/2017-07-26T080259Z/r/image-tiff
, denoted as
JOURNAL/epide/04.02/00001648-201705000-00032/math_32MM11/v/2017-07-26T080259Z/r/image-tiff
, as
JOURNAL/epide/04.02/00001648-201705000-00032/math_32MM12/v/2017-07-26T080259Z/r/image-tiff
with
JOURNAL/epide/04.02/00001648-201705000-00032/math_32MM13/v/2017-07-26T080259Z/r/image-tiff
, assuming that
JOURNAL/epide/04.02/00001648-201705000-00032/math_32MM14/v/2017-07-26T080259Z/r/image-tiff
.
The difference between the variances of
JOURNAL/epide/04.02/00001648-201705000-00032/math_32MM15/v/2017-07-26T080259Z/r/image-tiff
and
JOURNAL/epide/04.02/00001648-201705000-00032/math_32MM16/v/2017-07-26T080259Z/r/image-tiff
can be substantial: in the appendix of their article, the authors calculate the confidence interval for an adjusted SMR based on hypothetical data, y = 174 and z = 193, with an adjusted SMR for the outcome of interest of 2.0. Based on
JOURNAL/epide/04.02/00001648-201705000-00032/math_32MM17/v/2017-07-26T080259Z/r/image-tiff
, the authors report a nominal 95% confidence interval for the SMR ranging from 1.72 to 2.32. The correct 95% confidence interval based on
JOURNAL/epide/04.02/00001648-201705000-00032/math_32MM18/v/2017-07-26T080259Z/r/image-tiff
, however, ranges from 1.63 to 2.45. In general, the ratio of the variances of
JOURNAL/epide/04.02/00001648-201705000-00032/math_32MM19/v/2017-07-26T080259Z/r/image-tiff
and
JOURNAL/epide/04.02/00001648-201705000-00032/math_32MM20/v/2017-07-26T080259Z/r/image-tiff
solely depends on the ratio of y and z, as illustrated in the Figure. Thus, depending on y and z, ignoring the variability in z can lead to a severe underestimation of the variance of the adjusted SMR.
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