# Question about symmetry of confidence interval

I hope somebody could help me with a quick question. I am reading a medical paper where the researchers did a very bad job at reporting the methodology. They are checking the predictive power of a leg raising test in diagnosing a hernia. It is a dichotomous variable - patient says yes or no.

They report predictive values of the tests like this: 0.83 with confidence interval [0.67,0.92]. How is it possible that 0.83, the reported value, is not in the center of the CI. Normally I would think that the found value of 0.83 will be used as a center to which Z*sigma/sqrt(n) is added/subtracted.

If anyone could help, thank you very much! Here's the article if you wanna take a look: https://www.researchgate.net/publication/5460289_The_Sensitivity_and_Specificity_of_the_Slump_and_the_Straight_Leg_Raising_Tests_in_Patients_With_Lumbar_Disc_Herniation

Kind regards!

## 3 Answers

In general the confidence interval gives a range where the value is expected to be in a certain fraction of repeated experiments. But this can mean different things, e.g. based on the distribution of the underlying data. Few examples:

• Normal Distribution ("your definition of CI"): Upper limits and lower limits $$\pm$$ exactly $$Z \sigma / \sqrt{n}$$ around the mean of a normal distribution.
• Bernoulli Distribution: Can be approximated using normal distribution, but there are other ways to calculate confidence intervals. Some of them are symmetric, but others are not. See https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval

The paper you're referring to is probably using one of the confidence intervals in the article linked above, e.g. Wilson with continuity correction, Clopper-Pearson or Agresti-Coull.

The paper indicates that data was put into Epi Info 2000. The Epi Info manual indicates that they use Wilson 95% Confidence Limits. These are described here. These intervals are asymmetric by design.

• Can you elaborate more on this? In the the thread you linked they also talk about CI = Beta +/- 1.96*Error, it doesn't explain the asymmetry @jvl1996 asked about above. – Denwid Jan 5 '20 at 15:45
• They discuss it in the last paragraph of the answer posted. I added an edit as well. – Todd Burus Jan 5 '20 at 16:34
• If you look at the paper the OP linked to neither of your suggestions is true. – mdewey Jan 5 '20 at 16:36
• I did look at the paper. Why do you say that? It seems to reference all the components of a diagnostic odds ratio – Todd Burus Jan 5 '20 at 16:45

A possible explanation for the lack of symmetry in the prediction interval is that the dependent variable could have subjected to a data transformation.

Classically economic data is subject to percentage errors. As such, a traditional transformation is the log function. In medicine, similarly, drug dosing likely varies as a function of percent differences in body weight. As the original error distribution is Lognormal, the log of Lognormal deviates is now Normal (see Wikipedia on Lognormal distribution).

Regression based confidence (or prediction) intervals based on the ln(Y), for example, can be reversed by applying exp(ln(Y)) to the lower and upper bound points. This results in a probabilistically accurate non-symmetric interval with the same level of confidence in the original data. However, the center of the confidence interval, actually corresponds to the exp(mean), which relates to the median of the original lognormal data, and is actually biased (see this work and cited alternative).