# Can confidence interval be equal to zero?

Good day! I'm doing a study on appropriateness of empiric therapy, and did a logistic regression using the outcome (death or survival) as dependent variable with age and blood culture result as independent variables. The results showed that the odds ratio for age is 2.97E006 CI:0. I asked the technical support of the software I'm using if this was an error result. I only have 107 cases and there were no cell values with zero frequencies in the 2 independent variables.

The tech support reply was, it is an undefined confidence interval. What does it mean to have an undefined CI? Is it safe to report a result of a zero CI, in medical field.

• There is surely a problem here. Hard to say what, given you haven't posted your data, software used, or other details. However, one thing is safe to say: it is NOT safe to report a CI of zero in a medical journal! It would be helpful to post some data and some details of your analysis. – pmgjones Oct 22 '12 at 12:38
• One issue here, orthogonal to your question, is that logistic regression is probably not ideal in your situation. Everyone in your sample will eventually die (without meaning to be morbid), it's just that some haven't yet. You should consider using Survival Analysis, probably the Cox proportional hazards model. – gung - Reinstate Monica Oct 22 '12 at 13:05
• Please note that the confidence interval is practically infinite, not zero! What is zero is the lower confidence limit. It was computed by exponentiating a logarithm that is, therefore, essentially negative infinity. – whuber Oct 22 '12 at 14:19
• On principle, a frequentist confidence interval $C$ on a parameter $\theta$ can be empty or infinite for some realisations of the observed random variable $X$ since the coverage is averaged on $X$: $\mathbb{P}_\theta(C(X)\ni \theta)=\alpha$. (This is a common Bayesian criticism of frequentist confidence intervals.) – Xi'an Oct 23 '12 at 5:31

In a remarkable paper, Gleser and Hwang showed that, for some models, confidence intervals must have infinite expected length for any positive confidence level $\alpha$ to be attained. More precisely, there is a subset of observations with positive measure for which the length of the confidence interval is infinite. Examples include error-in-variable models and the Fieller-Creasy problem, where one estimates the ratio of two normal means.

Conversely, a confidence interval may have zero length for a subset of observations with positive measure without jeopardising the (overall) coverage probability. For instance, in the normal $X\sim\mathcal{N}(\theta,1)$ case, if we pick an empty confidence interval with probability .01 and the interval $(x-2.054,x+2.054)$ with probability .99, we end up with an overall coverage probability of $$\mathbb{P}_\theta(|X-\theta|<2.054)\times .99 = .96\times .99 = .95$$

An odd ratio of 2.97E006 (if you mean 2970000) seems odd to me, since it is way too high. Given your sample size it could be that some categories of the independed variables have a low frequency (e.g. a 1). Based on the information you provided it is indeed not safe to report the CI. I think no journal will expect a CI of zero, if it is zero there is no interval!

The ci for a logit is calculated as $ln(OR)-(Z*SE)\leq\theta \leq ln(OR)+(Z*SE)$

If this results in zero, something is wrong.

• Let's continue your analysis: a near-infinite (not "zero") CI, as reported in the question, therefore indicates the SE was huge. How can you obtain a huge SE in a logistic regression? It's well known (and intuitively obvious) that when the data are too good--that is, when there are linear combinations of the IVs that perfectly predict the outcome--then there is no unique solution. Typically the algorithm will not converge, but often the software will produce results if forced to. (This is how R behaves, for instance.) – whuber Oct 22 '12 at 14:14
• yes, is a agree with Whuber. Karen try, as a ‘litmus test’, take the exp. of 2970000 with the software you are using exp(2970000), it will return an error, nothing (e.g. stata) or infinite (Matlab). – Adam Oct 22 '12 at 14:57
• Adam, I believe you mean "$\log$" instead of "$\exp$" in your comment: the endpoints of the CI are already exponentials and exponentiating them further is meaningless in this situation. However, taking logarithms is not too helpful as any kind of test, because the lower limit of $0$ is just a limited-precision expression of a tiny positive number. – whuber Oct 22 '12 at 15:18
• Yes, the only thing I want to show Karin is that her coefficients are extreme. – Adam Oct 22 '12 at 16:04
• Sounds like a problem of complete or quasicomplete separation. – Peter Flom - Reinstate Monica Oct 22 '12 at 16:55