# What if a confidence interval starts at 1.0?

I am new to statistics so this might be an easy question. I know that if a confidence interval includes 1.0 then the result is not statistically significant because it includes the null. But, what if the confidence interval starts at 1.0? Like 95% CI: 1.0 - 1.9? Is that still statistically significant? It includes the null, but it doesn't cross it. Thanks!

• A confidence interval containing $1$ (or any other values) doesn't automatically correspond to non-rejection of the null hypothesis. In what context are you using that rule? What null hypothesis are you testing? (It sounds like a variance ratio.)
– Dave
Commented May 13, 2020 at 19:32
• @Dave primarily interpreting epidemiology study results where the null is 1.0. For example they gave an RR = 1.5 with a 95% CI: 1.0 - 1.9 Commented May 13, 2020 at 19:40
• @Dave Can you provide an example/source where the null would not be rejected? I know that 2 CIs can have some overlap, but you could still reject the null that the point estimates are the same, but I am not sure if the case you bring up makes sense to me. Commented May 13, 2020 at 20:05
• @DimitriyV.Masterov If the null hypothesis is $\theta=0$, then a confidence interval of $(1,1.9)$ would be fairly strong evidence against the null hypothesis.
– Dave
Commented May 13, 2020 at 20:08
• @Dave For epidemiology studies using risk ratio, hazard ratio, and odds ratio, the primary null hypothesis is θ = 1. That is the context my question is in. So the CI of 1.0 - 1.9 includes the null hypothesis but it is also the endpoint, so I'm not sure if this is considered statistically significant or not. Commented May 13, 2020 at 22:51

It is incredibly unlikely that a confidence interval has the null value as an end point. But, let's assume that it did happen. This would mean that the p value for your associated test would be equal to $$\alpha$$, the false positive rate. In the case of a z test

$$0 = \bar{x} - z_{\alpha/2} \sigma/\sqrt{n} \implies z_{\alpha/2} = \bar{x}/\sigma/\sqrt{n}$$

and

$$2 \mathbf{\Phi}^{-1}(z_{\alpha/2}) = \alpha$$

by definition. Here, $$\mathbf{\Phi}^{-1}$$ is the standard normal quantile function. What the investigator would do at this point is not something I am prepared to discuss at the point. Though I will say this: If the CI you've been given (be it from software or otherwise) has only one digit of precision, ask for more. I guarantee you that a CI which includes the null is likely due to rounding.

• You assume the test statistic is continuous. This often is not the case and there may be considerable probability (under the null) attached to an endpoint of a confidence interval.
– whuber
Commented May 13, 2020 at 21:00