# 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

## 1 Answer

Do you have predictions and prediction intervals (i.e., for future observables), or parameter estimates and confidence intervals (i.e., for unobservable model parameters)? No, it won't make a difference for the answer, but if you have your nomenclature correct, it will be easier for you to find help.

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).

• These are predictions and prediction intervals for a future observable(s). Are you suggesting many simulations of the 13 intervals through random selection from the interval, and then looking at the distribution of simulatio sums? – earlyriser70 Jul 3 '15 at 15:03
• 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
• Ah, this makes sense to me now and will work fine for my needs. Thanks! – earlyriser70 Jul 3 '15 at 15:15