# Can we interpret null-range confidence intervals as likelihoods for the real population value?

This short essay on the NCBI site indicates that:

The most commonly reported interval is the 95% confidence interval. A way to think about the concept of confidence intervals is to imagine that the study was repeated about a thousand times. About 95% of the different possible results obtained will lie in this interval.

So, can we interpret null-valued confidence intervals with proportionally more of the interval in the negative-of-null or positive-of-null range as the likelihood of the population statistic being negative or positive, respectively? For example, If I have a 95% CI from -1 to 0.5, can I state that the true population statistic is more likely negative than positive?

• I don't agree with that interpretation of a CI at all. Over many runs, 95% of them will contain the true mean within the 95% CI. It's not the case that 95% of your mean estimates will fall within a 95% CI - which CI are they even talking about? There isn't one single CI you can compare anything against - the only thing that is fixed is the actual, unknown, true mean. That sentence is a bit sloppy, but it's correct in the following sentence: "We are 95% confident that the true population value of what we are estimating in our study lies within the interval." Commented Feb 4, 2020 at 19:13
• Short answer. NO
– Carl
Commented Feb 4, 2020 at 20:59

## 1 Answer

First, the quote is Wrong!.

If you repeated the study a thousand times then you would get a thousand different intervals and about 95% of those intervals would include the true value. But the only way that 95% of the estimates from the thousand replications would be in the single interval created from the original study is if that interval was based on an estimate that exactly equaled the true parameter, but if that were the case, then we would not need a confidence interval.

Second, the way you want to interpret the interval is closer to the Bayesian interpretation than the traditional frequentist approach. If you really want a Bayesian interpretation, then it is best to do a full Bayesian analysis and look at the posterior distribution. If your interval above was from a Bayesian posterior and the posterior is fairly symmetric, then your statement about the parameter being more likely negative than positive would be true. But it is also possible that the posterior is left skewed (something that you cannot tell from just the interval) enough to make the statement false. If that is what you are interested in, then it would be better to compute the probability of being negative/positive from the posterior.