Delta Method for Standard Error Estimates
Abstract
Implement the delta method to report the means and confidence intervals of difference scores from log-transformed longitudinal data, so that results can be reported on the original scale of the outcome.
Delta Method for Log-Transformed Data
Conventions
\(RR\): Rate Ratio
\(lRR\): log-transformed RR (most SAS procedures (e.g. PROC MIXED, PROC GENMOD, ...) will output these statistics in log scale)
Mechanism
\[ lRR=\log (RR); \\ lRR \sim \text{Gaussian} (\mu,\sigma^2); \]
For example, in the output from PROC GENMOD, Z-statistics will be presented for Log-transformed point estimates. Based on these Z-statistics, we can obtain \(\mu\) and \(\sigma\).
Let \(g(\cdot)=\exp(\cdot)\), so that \(g(lRR)=RR\).
We would like to know the Standard Error (SE) of \(RR\), by employing the Delta Method.
First-order Taylor series for \(g(lRR)\): \[ g(lRR)\approx \exp(\mu)+\exp'(\mu)(lRR-\mu)=\exp(\mu)+\exp(\mu)(lRR-\mu). \]
Therefore, the variance of \(g(lRR)=RR\) is: \[ \text{VAR}(\exp(\mu)+\exp(\mu)(lRR-\mu))=\exp^2(\mu)\text{VAR}(lRR)=\exp^2(\mu)\sigma^2. \]
The Standard Error of this RR would thus be: \[ \sqrt{\exp^2(\mu)\text{VAR}(lRR)}=\sqrt{\exp^2(\mu)\sigma^2}=\exp(\mu)\sigma \] From the above formula:
\(\text{exp}(\mu)\) is the mean estimate for RR;
\(\sigma\) is the standard error estimate for lRR.
We can obtain both of these two estimates from the SAS output.
This method is quite useful in Poisson Regression or Negative Binomial Regression since the link function for these two distributions is the log link.