### Excerpt

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.