Confidence Intervals for Regression Coefficients and Odds Ratios

1. Constructing a Confidence Interval for the Coefficient ($\beta$)

Just like in linear regression, we can construct a confidence interval around our estimated coefficient, $\hat{\beta }$. In logistic regression, we typically use a Wald confidence interval. The great news is that you won't need to calculate the standard error by hand; it's a standard part of any statistical software output.

The formula for a confidence interval for a coefficient ${\beta }_i$ is:

Where:

• ${\hat{\beta }}_i$ is the estimated coefficient (the "Coef" or "Estimate" from your output).
• $\text{SE}({\hat{\beta }}_i)$ is the standard error of the coefficient (the "SE Coef" or "Std. Error").
• $z_{1-\alpha /2}$ is the critical value from the standard normal distribution for the desired confidence level. For a 95% confidence interval ($\alpha =0.05$), this value is 1.96.

The following resource provides a clear explanation and shows where to find these values in a typical output table.

2. From the Coefficient's CI to the Odds Ratio's CI

Once you have the confidence interval for the log-odds ratio ($\beta$), converting it to a confidence interval for the odds ratio is straightforward. Because the exponential function is monotonic (it preserves order), you can simply exponentiate the lower and upper bounds of the interval for $\beta$.

The Two-Step Process:

1. Calculate the CI for $\beta$:

   • Lower Bound: $L=\hat{\beta }-1.96\times \text{SE}(\hat{\beta })$
   • Upper Bound: $U=\hat{\beta }+1.96\times \text{SE}(\hat{\beta })$

2. Exponentiate the bounds to get the CI for the Odds Ratio (OR):

   • Lower Bound of OR CI: $e^L$
   • Upper Bound of OR CI: $e^U$

This next resource provides a perfectly clear, worked example of this two-step process.

This image provides another excellent illustration, linking the raw data in a 2x2 table, the manual odds ratio calculation, and the corresponding logistic regression output from STATA, including the coefficient and its 95% CI.

3. Interpreting the Odds Ratio Confidence Interval

Now that you have the confidence interval for the odds ratio, how do you interpret it? The key is to check whether the interval contains the value 1. An odds ratio of 1 means there is no association between the predictor and the outcome (the odds are the same in both groups).

Here are the decision rules:

• If the entire CI is above 1: The association is statistically significant. The predictor is associated with an increased odds of the outcome.
• If the entire CI is below 1: The association is statistically significant. The predictor is associated with a decreased odds of the outcome (i.e., it's a "protective" factor).
• If the CI contains 1: The association is not statistically significant. You cannot conclude that there is an association between the predictor and the outcome based on your data.

The following video provides an excellent walkthrough of these three scenarios.

Here is an example of what a final odds ratio table might look like in a report. Notice how the CIs for 'Conservativeness' are entirely above 1 (significant risk), while the CIs for 'Establishmentness' are entirely below 1 (significant protection).

4. Advanced Application: CI for an Interaction Effect

For your exam, you might encounter a question that involves an interaction term. As we briefly mentioned in the last lesson, when a model includes an interaction (e.g., $X_2\times X_3$), the log-odds ratio for one variable depends on the level of the other.

For example, the log-odds ratio for a one-unit increase in $X_2$ when $X_3=1$ might be ${\beta }_2+{\beta }_5$. To find the confidence interval for this combined effect, you must first find its standard error, which requires using the variance-covariance matrix of the model's coefficients.

While this is more complex, the logic is an extension of what you've learned. The following resource shows exactly how to execute these steps.

Conclusion

You have now learned how to quantify the uncertainty around your effect estimates in logistic regression. This is a crucial step in moving from "what is the effect?" to "how sure are we about this effect?".

Key Takeaways:

• The 95% Wald CI for a coefficient $\beta$ is calculated as $\hat{\beta }\pm 1.96\times \text{SE}(\hat{\beta })$.
• To get the CI for the odds ratio, you simply exponentiate the lower and upper bounds of the CI for $\beta$.
• The interpretation of an odds ratio's CI hinges on whether it contains the value 1.
  • CI > 1: Significant positive association (risk factor).
  • CI < 1: Significant negative association (protective factor).
  • CI contains 1: Not a statistically significant association.

In our next lesson, we will formalize the process of testing significance by exploring the Wald and Likelihood Ratio tests. You'll see how the conclusions from these formal hypothesis tests align perfectly with the interpretation of confidence intervals you learned today.