## How to calculate the confidence interval from a p value

Confidence intervals are widely reported in published research and are usually thought to provide more information than p values from significance tests because confidence intervals indicate how precise an estimate is. Sometimes, however, investigators report an estimate (eg. a mean) and p value, but not the confidence interval about the estimate. In a BMJ statistics note, statisticians Doug Altman and Martin Bland demonstrate how to calculate the confidence interval from an estimate and p value.

Limitations of this method.This method is not correct in studies where sample size is small (less than 60 subjects) where the outcome is continuous and the analysis was done with a t-test or analysis of variance.

Steps to calculate the confidence interval (CI) from the p value (p) and the estimate (Est) for a difference where data are continuous:

- Calculate the test statistic for a normal distribution test (z) from p: z = −0.862 + √[0.743 − 2.404×log(p)]
- Calculate the standard error, ignoring the minus sign: SE = Est/z
- Calculate the 95% CI: Est –1.96×SE to Est + 1.96×SE.

### Worked example

Altman and Bland provide a worked example to demonstrate how these steps are applied. Suppose a randomised controlled trial reports a between-group difference in proportions (of binary outcomes, such as mortality) or means (of continuous outcomes, such as blood pressure). The abstract might state: “patients who received the intervention recovered more than patients in the control group (49% vs. 32%, p=0.032)”. The between-group difference in proportions is Est = 17%. What is the 95% CI about this difference?

We can calculate the 95% CI like so:

- z = –0.862+ √[0.743 – 2.404×log(0.032)] = 2.141
- SE = 17/2.141 = 7.940, so that 1.96×SE = 15.56%
- 95% CI = 17.0 – 15.56 to 17.0 + 15.56 = 1.4 to 32.6%

### Summary

For studies with sample size of at least 60, confidence intervals about an estimate of effect can be calculated using the estimate itself and the p value. Altman and Bland also show how to calculate 95% CI for a ratio, which requires a log transformation.

### Reference

Altman DG and Bland JM (2011) How to obtain the confidence interval of a p value. BMJ 343:d2090.