# Combining confidence intervals from several regression point estimates

I have 13 point predictions from 13 independent linear regressions, each prediction with a 95% confidence interval. I want to sum the 13 predictions and calculate the 95%CI for the summed value. How, or should I, combine the 13 CIs to get the CI for the summed value?

• If the regressions are truly independent, then what use is this? – Repmat Jul 3 '15 at 14:00
• Could you please explain what you mean by a "point prediction"? I also wonder about your intended meaning of "independent." Most people would understand that as involving independent datasets, but it occurs to me that you might intend it in a different sense, such as "regressions independently conducted by 13 different people based on the same data." Please clarify. – whuber Jul 3 '15 at 14:38
• The point estimate is for a future observable, so I have a prediction interval. The 13 regressions are from 13 independent datasets. – earlyriser70 Jul 3 '15 at 14:53

Either way, you have a sum of 13 independent $t$ distributed random variables. Unless you have some specific information on your 13 variables, like common variances, the sum does not have a closed form solution.
You can either simulate many realizations and look at the empirical distribution of the sums, or (if you have sufficiently high degrees of freedom) approximate your $t$ distributions by independent normals and hope for the best. The sum of independent normals is normal, with mean equal to the sum of the component means (same for variances).
• Yes! (Using $t$ distributions with appropriate degrees of freedom, of course, not just uniform draws from p% prediction intervals.) – Stephan Kolassa Jul 3 '15 at 15:05