The STDRATE Procedure

Risk

Normal Distribution Confidence Interval for Risk
Lognormal Distribution Confidence Interval for Risk
Confidence Interval for Risk Difference Statistic
Confidence Interval for Risk Ratio Statistic
Confidence Interval for Risk SMR

An event risk of a population over a specified time period can be defined as the number of new events in the follow-up time period divided by the event-free population size at the beginning of the time period,

${\hat\gamma } = \frac{d}{\mathcal N}$

where ${\mathcal N}$ is the population size.

For a general population, the subsets (strata) might not be homogeneous enough to have a similar risk. Thus, the risk for each stratum should be computed separately to reflect this discrepancy. For a population that consists of K homogeneous strata (such as different age groups), the stratum-specific risk for the jth stratum in a population is computed as

${\hat\gamma }_ j = \frac{d_{j}}{{\mathcal N}_{j}}$

where ${\mathcal N}_{j}$ is the population size in the jth stratum of the population.

Assuming the number of events, , has a binomial distribution, then a variance estimate of ${\hat\gamma }_ j$ is

$V( {\hat\gamma }_ j ) = \frac{ {\hat\gamma }_ j (1-{\hat\gamma }_ j) }{{\mathcal N}_ j}$

By using the method of statistical differentials (Elandt-Johnson and Johnson, 1980, pp. 70–71), the variance of the logarithm of risk can be estimated by

$V( \mbox{log}( {\hat\gamma }_ j ) ) = \frac{1}{ {\hat\gamma }_{j}^{2} } \, V( {\hat\gamma }_{j} ) = \frac{1}{ {\hat\gamma }_{j}^{2} } \, \frac{ {\hat\gamma _{j}} \, (1-{\hat\gamma _{j}}) }{ {\mathcal N}_{j} } = \frac{ 1-{\hat\gamma _{j}} }{ {\hat\gamma }_{j} \, {\mathcal N}_{j} } = \frac{1}{ d_{j} } - \frac{1}{ {\mathcal N}_{j} }$

Normal Distribution Confidence Interval for Risk

A $(1-\alpha )$ confidence interval for ${\hat\gamma }_{j}$ based on a normal distribution is given by

$\left( \; {\hat\gamma }_{j} - z \, \sqrt {V( {\hat\gamma }_{j} )} \, , \; \; {\hat\gamma }_{j} + z \, \sqrt {V( {\hat\gamma }_{j} )} \; \right)$

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

Lognormal Distribution Confidence Interval for Risk

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

$\left( \; \mbox{log}({\hat\gamma }_{j}) - z \, \sqrt {V( \mbox{log}({\hat\gamma }_{j}) )} \, , \; \; \mbox{log}({\hat\gamma }_{j}) + z \, \sqrt {V( \mbox{log}({\hat\gamma }_{j}) )} \; \right)$

where $z = \Phi ^{-1} (1-\alpha /2)$ is the $(1-\alpha /2)$ quantile of the standard normal distribution and the variance $V( \mbox{log}({\hat\gamma }_{j}) ) = 1 / d_{j} - 1 / {\mathcal N}_{j}$ .

Thus, a $(1-\alpha )$ confidence interval for ${\hat\gamma }_{j}$ based on a lognormal distribution is given by

$\left( \; {\hat\gamma }_{j} \; e^{ -z {\sqrt { \frac{1}{d_{j}} - \frac{1}{{\mathcal N}_{j}} }}} \, , \; \; {\hat\gamma }_{j} \; e^{ z {\sqrt { \frac{1}{d_{j}} - \frac{1}{{\mathcal N}_{j}} }}} \; \right)$

Confidence Interval for Risk Difference Statistic

For rate estimates from two independent samples, ${\hat\gamma }_{1j}$ and ${\hat\gamma }_{2j}$ , a $(1-\alpha )$ confidence interval for the risk difference ${\hat\gamma }_{dj} = {\hat\gamma }_{1j} - {\hat\gamma }_{2j}$ is

$\left( \; {\hat\gamma }_{dj} - z \, \sqrt {V( {\hat\gamma }_{dj} )} \, , \; \; {\hat\gamma }_{dj} + z \, \sqrt {V( {\hat\gamma }_{dj} )} \; \right)$

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

$V({\hat\gamma }_{dj}) = V({\hat\gamma }_{1j}) + V({\hat\gamma }_{2j})$

Confidence Interval for Risk Ratio Statistic

For rate estimates from two independent samples, ${\hat\gamma }_{1j}$ and ${\hat\gamma }_{2j}$ , a $(1-\alpha )$ confidence interval for the log risk ratio statistic $\mbox{log} ({\hat\gamma }_{rj}) = \mbox{log} ({\hat\gamma }_{1j} / {\hat\gamma }_{2j})$ is

$\left( \; \mbox{log} ({\hat\gamma }_{rj}) - z \, \sqrt {V( \mbox{log} ({\hat\gamma }_{rj}) )} \, , \; \; \mbox{log} ({\hat\gamma }_{rj}) + z \, \sqrt {V( \mbox{log} ({\hat\gamma }_{rj}) )} \; \right)$

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

$V( \mbox{log} ({\hat\gamma }_{rj}) = V( \mbox{log} ({\hat\gamma }_{1j}) ) + V( \mbox{log} ({\hat\gamma }_{2j}) )$

Thus, a $(1-\alpha )$ confidence interval for the risk ratio statistic ${\hat\gamma }_{rj}$ is given by

$\left( \; \frac{{\hat\gamma }_{1j}}{{\hat\gamma }_{2j}} \; e^{ -z \sqrt {V( \mbox{log} ({\hat\gamma }_{rj}) ) } } \, , \; \; \frac{{\hat\gamma }_{1j}}{{\hat\gamma }_{2j}} \; e^{ z \sqrt {V( \mbox{log} ({\hat\gamma }_{rj}) ) } } \; \right)$

Confidence Interval for Risk SMR

At stratum j, a stratum-specific standardized morbidity/mortality ratio is

${\mathcal R}_{j} = \; \frac{\, d_ j \, }{{\mathcal E}_ j}$

where ${\mathcal E}_ j$ is the expected number of events.

With the risk

${\hat\gamma }_ j = \frac{d_{j}}{{\mathcal N}_{j}}$

SMR can be expressed as

${\mathcal R}_{j} = \; \frac{\, {\mathcal N}_ j \, }{{\mathcal E}_ j} \; {\hat\gamma }_ j$

Thus, a $(1-\alpha )$ confidence interval for ${\mathcal R}_ j$ is given by

$\left( \; \frac{\, {\mathcal N}_ j \, }{{\mathcal E}_ j} \; {\hat\gamma }_{jl} \, , \; \; \frac{\, {\mathcal N}_ j \, }{{\mathcal E}_ j} \; {\hat\gamma }_{ju} \; \right)$

where $(\, {\hat\gamma }_{jl} \, , \, {\hat\gamma }_{ju} \, )$ is a $(1-\alpha )$ confidence interval for the risk ${\hat\gamma }_ j$ .