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:

  1. Calculate the test statistic for a normal distribution test (z) from p: z = −0.862 + √[0.743 − 2.404×log(p)]
  2. Calculate the standard error, ignoring the minus sign: SE = Est/z
  3. 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:

  1. z = –0.862+ √[0.743 – 2.404×log(0.032)] = 2.141
  2. SE = 17/2.141 = 7.940, so that 1.96×SE = 15.56%
  3. 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.

 

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