The STDRATE Procedure

Indirect Standardization and Standardized Morbidity/Mortality Ratio

Normal Distribution Confidence Interval for SMR
Lognormal Distribution Confidence Interval for SMR
Poisson Distribution Confidence Interval for Rate SMR
Indirectly Standardized Rate and Its Confidence Interval
Indirectly Standardized Risk and Its Confidence Interval

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 .

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 .

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 ${\mathcal R}_{sm}$ .