The STDRATE Procedure

Indirect Standardization and Standardized Morbidity/Mortality Ratio

Indirect standardization compares the rates of the study and reference populations by applying the stratum-specific rates in the reference population to the study population, where the stratum-specific rates might not be reliable.

The expected number of events in the study population is

\[  {\mathcal E} = \sum _{j} \;  {\mathcal T}_{sj} \;  {\hat\lambda }_{rj}  \]

where ${\mathcal T}_{sj}$ is the population-time in the jth stratum of the study population and ${\hat\lambda }_{rj}$ is the rate in the jth stratum of the reference population.

With the expected number of events, ${\mathcal E}$, the standardized morbidity ratio or standardized mortality ratio can be expressed as

\[  {\mathcal R}_{sm} = \;  \frac{\,  {\mathcal D} \, }{\mathcal E}  \]

where ${\mathcal D}$ is the observed number of events (Breslow and Day, 1987, p. 65).

The ratio ${\mathcal R}_{sm} > 1$ indicates that the mortality rate or risk in the study population is larger than the estimate in the reference population, and ${\mathcal R}_{sm} < 1$ indicates that the mortality rate or risk in the study population is smaller than the estimate in the reference population.

With the ratio ${\mathcal R}_{sm}$, an indirectly standardized rate for the study population is computed as

\[  {\hat\lambda }_{is} = {\mathcal R}_{sm} \;  {\hat\lambda }_ r  \]

where ${\hat\lambda }_ r$ is the overall crude rate in the reference population.

Similarly, to compare the risks of the study and reference populations, the stratum-specific risks in the reference population are used to compute the expected number of events in the study population

\[  {\mathcal E} = \sum _{j} \;  {\mathcal N}_{sj} \;  {\hat\gamma }_{rj}  \]

where ${\mathcal N}_{sj}$ is the number of observations in the jth stratum of the study population and ${\hat\gamma }_{rj}$ is the risk in the jth stratum of the reference population.

Also, with the standardized morbidity ratio ${\mathcal R}_{sm} = {\mathcal D} / {\mathcal E}$, an indirectly standardized risk for the study population is computed as

\[  {\hat\gamma }_{is} = {\mathcal R}_{sm} \;  {\hat\gamma }_ r  \]

where ${\hat\gamma }_ r$ is the overall crude risk in the reference population.

The observed number of events in the study population is ${\mathcal D} = \sum _{j} d_{sj}$, where $d_{sj}$ is the number of events in the jth stratum of the population. For the rate estimate, if $d_{sj}$ has a Poisson distribution, then the variance of the standardized mortality ratio ${\mathcal R}_{sm} = {\mathcal D} \,  / {\mathcal E}$ is

\[  V({\mathcal R}_{sm}) = \frac{\,  \, 1}{{\mathcal E}^{2}} \;  \sum _{j} \,  V(d_{sj}) = \frac{\,  1 \, }{{\mathcal E}^{2}} \;  \sum _{j} \,  d_{sj} = \frac{\,  {\mathcal D} \, }{{\mathcal E}^{2}} = \frac{ {\mathcal R}_{sm} }{\mathcal E}  \]

For the risk estimate, if $d_{sj}$ has a binomial distribution, then the variance of ${\mathcal R}_{sm} = {\mathcal D} \,  / {\mathcal E}$ is

\[  V({\mathcal R}_{sm}) = V \left( \frac{1}{\mathcal E} \;  \sum _{j} d_{sj} \right) = \frac{\,  \, 1}{{\mathcal E}^{2}} \;  \sum _{j} \,  V(d_{sj}) = \frac{\,  \, 1}{{\mathcal E}^{2}} \;  \sum _{j} \,  {\mathcal N}_{sj}^{2} V({\hat\gamma }_{sj})  \]

where

\[  V({\hat\gamma }_{sj}) = \frac{ {\hat\gamma }_{sj} (1-{\hat\gamma }_{sj}) }{{\mathcal N}_{sj}}  \]

By using the method of statistical differentials (Elandt-Johnson and Johnson, 1980, pp. 70–71), the variance of the logarithm of ${\mathcal R}_{sm}$ can be estimated by

\[  V( \mbox{log}( {\mathcal R}_{sm} ) ) = \frac{1}{ {\mathcal R}_{sm}^{2} } \,  V({\mathcal R}_{sm})  \]

For the rate estimate,

\[  V( \mbox{log}( {\mathcal R}_{sm} ) ) = \frac{1}{ {\mathcal R}_{sm}^{2} } \,  V({\mathcal R}_{sm}) = \frac{1}{ {\mathcal R}_{sm}^{2} } \,  \frac{ {\mathcal R}_{sm} }{\mathcal E} = \frac{1}{ {\mathcal R}_{sm} } \,  \frac{1}{\mathcal E} = \frac{1}{\mathcal D}  \]

The confidence intervals for ${\mathcal R}_{sm}$ can be constructed based on normal, lognormal, and Poisson distributions.

Normal Distribution Confidence Interval for SMR

A $(1-\alpha )$ confidence interval for ${\mathcal R}_{sm}$ based on a normal distribution is given by

\[  ({\mathcal R}_ l, \;  {\mathcal R}_ u)= \left( \;  {\mathcal R}_{sm} - z \,  \sqrt {V( {\mathcal R}_{sm} )} \,  , \; \;  {\mathcal R}_{sm} + z \,  \sqrt {V( {\mathcal R}_{sm} )} \;  \right)  \]

where $z = \Phi ^{-1} (1-\alpha /2)$ is the $(1-\alpha /2)$ quantile of the standard normal distribution.

A test statistic for the null hypothesis $H_0: \mbox{SMR} = 1$ is then given by

\[  \frac{{\mathcal R}_{sm}-1}{ \sqrt {V({\mathcal R}_{sm}}) }  \]

The test statistic has an approximate standard normal distribution under $H_0$.

Lognormal Distribution Confidence Interval for SMR

A $(1-\alpha )$ confidence interval for $\mbox{log}( {\mathcal R}_{sm})$ based on a normal distribution is given by

\[  \left( \;  \mbox{log}({\mathcal R}_{sm}) - z \,  \sqrt {V( \mbox{log}({\mathcal R}_{sm}) )} \,  , \; \;  \mbox{log}({\mathcal R}_{sm}) + z \,  \sqrt {V( \mbox{log}({\mathcal R}_{sm}) )} \;  \right)  \]

where $z = \Phi ^{-1} (1-\alpha /2)$ is the $(1-\alpha /2)$ quantile of the standard normal distribution.

Thus, a $(1-\alpha )$ confidence interval for ${\mathcal R}_{sm}$ based on a lognormal distribution is given by

\[  \left( \;  {\mathcal R}_{sm} \;  e^{ -z {\sqrt { V( \mbox{log}({\mathcal R}_{sm}) ) }}} \,  , \; \;  {\mathcal R}_{sm} \;  e^{ z {\sqrt { V( \mbox{log}({\mathcal R}_{sm}) ) }}} \;  \right)  \]

A test statistic for the null hypothesis $H_0: \mbox{SMR} = 1$ is then given by

\[  \frac{ \mbox{log}({\mathcal R}_{sm})}{ \sqrt {V( \mbox{log}({\mathcal R}_{sm}})) }  \]

The test statistic has an approximate standard normal distribution under $H_0$.

Poisson Distribution Confidence Interval for Rate SMR

Denote the $(\alpha /2)$ quantile for the $\chi ^{2}$ distribution with $2{\mathcal D}$ degrees of freedom by

\[  q_ l = {( {\chi }_{2 {\mathcal D}}^{2} )}^{-1} \,  (\alpha /2)  \]

Denote the $(1-\alpha /2)$ quantiles for the $\chi ^{2}$ distribution with $2({\mathcal D}+1)$ degrees of freedom by

\[  q_ u = {( {\chi }_{2 ({\mathcal D}+1)}^{2} )}^{-1}\,  (1-\alpha /2)  \]

Then a $(1-\alpha )$ confidence interval for ${\mathcal R}_{sm}$ based on the $\chi ^{2}$ distribution is given by

\[  ({\mathcal R}_ l, \;  {\mathcal R}_ u) = \left( \;  \frac{q_ l}{2 \,  {\mathcal E}} \,  , \; \;  \frac{q_ u}{2 \,  {\mathcal E}} \right)  \]

A p-value for the test of the null hypothesis $H_0: \mbox{SMR} = 1$ is given by

\[  2 \,  \mr {min} \left( \,  \sum _{k=0}^{\mathcal D} \,  \frac{ e^{-{\mathcal E}} {\mathcal E}^{k} }{ k! }, \;  \sum _{k=\mathcal D}^{\infty } \,  \frac{ e^{-{\mathcal E}} {\mathcal E}^{k} }{ k! } \,  \right)  \]

Indirectly Standardized Rate and Its Confidence Interval

With a rate-standardized mortality ratio ${\mathcal R}_{sm}$, an indirectly standardized rate for the study population is computed as

\[  {\hat\lambda }_{is} = {\mathcal R}_{sm} \;  {\hat\lambda }_ r  \]

where ${\hat\lambda }_ r$ is the overall crude rate in the reference population.

The $(1-\alpha /2)$ confidence intervals for ${\hat\lambda }_{is}$ can be constructed as

\[  ( {\mathcal R}_ l \,  {\hat\lambda }_ r, \;  {\mathcal R}_ u \,  {\hat\lambda }_ r )  \]

where $({\mathcal R}_ l, \;  {\mathcal R}_ u)$ is the confidence interval for ${\mathcal R}_{sm}$.

Indirectly Standardized Risk and Its Confidence Interval

With a risk-standardized mortality ratio ${\mathcal R}_{sm}$, an indirectly standardized risk for the study population is computed as

\[  {\hat\gamma }_{is} = {\mathcal R}_{sm} \;  {\hat\gamma }_ r  \]

where ${\hat\gamma }_ r$ is the overall crude risk in the reference population.

The $(1-\alpha /2)$ confidence intervals for ${\hat\gamma }_{is}$ can be constructed as

\[  ( {\mathcal R}_ l \,  {\hat\gamma }_ r, \;  {\mathcal R}_ u \,  {\hat\gamma }_ r )  \]

where $({\mathcal R}_ l, \;  {\mathcal R}_ u)$ is the confidence interval for ${\mathcal R}_{sm}$.