Confidence intervals (CIs) are useful statistical calculations to help get a level of certainty around an estimated effect size. Whenever possible, I advocate to include a CI when reporting an estimated effect size. Sometimes, however, it is of interest to back calculate a p-value from a confidence interval if the p-value is not reported in the manuscript. To do so, we need to remember the basic equations for the confidence interval and the calculation of a p-value. Assuming we are dealing with a 95% CI, we would take the effect size and subtract/add 1.96 times the standard error of the effect size to get our lower and upper bounds of the confidence interval. For the p-value, we just take the effect estimate and divide it by the standard error of the effect estimate to get a z score from which we can calculate the p-value. Therefore, if we are given an effect size and confidence interval all we need to do is back calculate the standard error and combine that with the effect size to get the z score used to calculate the p-value. Below are two examples to illustrate how to do this.
Suppose we have an estimate of a risk difference and a respective 95 percent confidence interval of 3.60 (0.70, 6.50). Here are the steps to follow:
(1) Subtract the lower limit from the upper limit to get the difference and divide by 2: (6.50-0.70)/2=2.9
(2) Divide the difference by 1.96 (for a 95% CI) to get the standard error estimate: 2.9/1.96=1.48
(3) Divide the risk difference estimate by the standard error estimate to get a z score: 3.60/1.48=2.43
(4) Look up the z score using Python, R (ex: 2*pnorm(-abs(z))), Excel (ex: 2*1-normsdist(z score)), or an online calculator to get the p-value. Usually the two-sided p-value is reported: p=0.015 (two-sided)
For an odds ratio, things are a bit trickier because we need to first take the natural log of the estimate and 95% confidence interval before we can carry out the back calculation of the standard error for calculating the p-value. Suppose we have an odds ratio and 95 percent confidence interval of 1.28 (1.05, 1.57). Here are the steps to follow:
(1) Take the natural log (ln) of each value in the 95% CI: 0.25 (0.05, 0.45)
(2) Subtract the lower limit from the upper limit and divide by 2: (0.45-0.05)/2=0.2
(3) Divide the difference by 1.96 (for a 95% CI) to get the standard error estimate: 0.2/1.96=0.10
(4) Divide the log odds ratio by the standard error estimate to get a z score: 0.25/0.10=2.50
(5) Look up the z score using Python, R (ex: 2*pnorm(-abs(z))), Excel (ex: 2*1-normsdist(z score)), or an online calculator to get the p-value. Usually the two-sided p-value is reported: p=0.012 (two-sided)
Hopefully these are good examples to get you started. As you can imagine you can also go from a p-value to a 95% confidence interval by extending these methods in the opposite direction, but in practice it is somewhat unlikely an author would report an effect size and p-value while leaving out the 95% confidence interval.
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