# Confidence interval for superiority/inferiority tests

I am looking into superiority/non-inferiority tests and my understanding is that the null hypothesis for a superiority test is

$$H_{0}: \epsilon \leq \delta$$

where $$\delta \geq 0$$ and $$\epsilon$$ is the true effect size.

I feel that a lot of articles I saw report two-sided confidence intervals for this type of test (e.g. here or here). Is this correct? Is the choice of the confidence interval approach (two-sided vs one-sided with a bound on the left-side in this case) separate from whether we are interested in $$H_{0}: \epsilon = 0$$ vs $$H_{0}: \epsilon \leq 0$$?

The superiority hypotheses $$\text{(for example:}\text{ }\text{ }\text{ } H_0: \mu_1 - \mu_0 \leq 0 \text{ }\text{ }\text{ }\text{ }\text{ } \text{VS}\text{ }\text{ }\text{ }\text{ }\text{ } H_1: \mu_1 - \mu_0 > 0\text{)}$$ and non-inferiority hypotheses $$\text{(for example:}\text{ }\text{ }\text{ }H_0: \mu_1 - \mu_0 \leq -M \text{ }\text{ }\text{ }\text{ }\text{ } \text{VS}\text{ }\text{ }\text{ }\text{ }\text{ } H_1: \mu_1 - \mu_0> -M \text{)}$$ are both one-sided tests.
In the superiority example above, you would consider the 1-sided 97.5% confidence interval of $$\mu_1 - \mu_2$$ or the lower bound of the 2-sided 95% confidence interval and compare it to 0 (for some reason superiority is usually tested at the 2.5% significance level in the pharma industry).