Solved – How to calculate 95% CI for OR for a different reference category without running the SAS logistics again

Question

My question is about calculation of confidence interval (CI) for odds ratio (OR) from a SAS output of a logistic regression model for a different reference category without running the SAS program again. I’ve run SAS logistics modeling dichotomized income level (low coded as 1 versus high coded as 0) with 4 dummy coded race variables as covariates- White, Chinese, Black and Filipinos with White as the reference category. While looking at the OR and CI for Filipinos’ income level against White from a SAS output that only furnished intercept, beta, SE, OR and 95% Wald CI for OR, I’ve become interested to know about the OR and 95% CI for Filipinos against Chinese (instead of White). I know how to calculate the new Beta and OR by calculating the predicted values for Chinese and Filipinos and then taking the difference of the log pie for Chinese and Filipinos to get the new beta and OR, but I’m completely at a loss about calculation of the new CI. Can anyone HELP please?

Answer 1

The log odds ratio ($\beta$) comparing Filipinos with Chinese can be computed as:

$$\beta_{\mathrm{Filipino,Chinese}} = \beta_{\mathrm{Filipino,white}} - \beta_{\mathrm{Chinese,white}}$$

So, the the variance of the sampling distribution of that log odds ratio is:

$$\mathrm{var}(\beta_{\mathrm{Philipino,Chinese}}) = \mathrm{var}(\beta_{\mathrm{Filipino, white}}) + \mathrm{var}(\beta_{\mathrm{Chinese,white}}) - 2\mathrm{cov}(\beta_{\mathrm{Filipino,white}}, \beta_{\mathrm{Chinese,white}})$$

So, you do not only need the standard errors (squaring these will give you the variances) but also the covariance of the sampling distribution of $\beta_{\mathrm{Filipino,white}}$ and $\beta_{\mathrm{Chinese,white}}$. These are typically stored in a variance covariance matrix, but I am not familiar enought with SAS to tell you where SAS leaves this behind.

The standard error ($\mathrm{se}$) is the square root of this variance:

$$ \mathrm{se}(\beta_{\mathrm{Philipino,Chinese}}) = \sqrt{\mathrm{var}(\beta_{\mathrm{Philipino,Chinese}})} $$

The odds ratios is just: $OR_{\mathrm{Filipino,Chinese}} = \exp(\beta_{\mathrm{Filipino,Chinese}})$. You can than use the delta method to approximate the standard error for the odds ratio:

$$\mathrm{se}(OR_{\mathrm{Filipino,Chinese}}) \approx \exp(\beta_{\mathrm{Filipino,Chinese}}) \times \mathrm{se}(\beta_{\mathrm{Philipino,Chinese}})$$

The confidence interval of the odds ratio comparing Filipinos and Chinese is than approximately:

$$ OR_{\mathrm{Filipino,Chinese}} \pm 1.96 \mathrm{se}(OR_{\mathrm{Filipino,Chinese}}) $$