How do you estimate the specific risk a smoking has on the probability of being hospitalized. If smokers on average have lower income and less educational achievement, is smoking truly causing the increase in hospitalization or could the covariates fully or partially explain the increased hospitalization rates?
A paper by Kleinman and Norton suggests using adjusted risk ratios with logistic regressions. The formula for this procedure is as follows:
ARR = [n -1 Σ i=1 to N risk i (X i |as if exposed)] ÷ [n -1 Σ i=1 to N risk i (X i |as if unexposed)] (1)
ARD = [n-1Σi=1 to N riski(Xi|as if exposed)] - [n-1Σi=1 to N riski(Xi|as if unexposed)] (2)
The authors explain the first equation as follows:
“The denominator of equation (1) is the mean of this calculated risk for each observation when the exposure variable is assumed to be unexposed and represents an MLE of the unexposed (baseline) risk for a population whose covariates are distributed as for the observed covariates for the entire study population. The numerator in equation (1) represents an MLE of the adjusted risk among the exposed. This approach is a specific example of using what are called “recycled predictions.”
Standard errors can be calculated using either bootstrapping or the Delta Method. However, the authors wisely recommend bootstrapping the standard errors since it reduces the computations resources needed and can also allow for asymmetric confidence intervals.