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?
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$\begingroup$ If the regressions are truly independent, then what use is this? $\endgroup$– RepmatCommented Jul 3, 2015 at 14:00
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$\begingroup$ 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. $\endgroup$– whuber ♦Commented Jul 3, 2015 at 14:38
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$\begingroup$ The point estimate is for a future observable, so I have a prediction interval. The 13 regressions are from 13 independent datasets. $\endgroup$– earlyriser70Commented Jul 3, 2015 at 14:53
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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).
answered Jul 3, 2015 at 14:14
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$\begingroup$ 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? $\endgroup$ Commented Jul 3, 2015 at 15:03
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$\begingroup$ Yes! (Using $t$ distributions with appropriate degrees of freedom, of course, not just uniform draws from p% prediction intervals.) $\endgroup$ Commented Jul 3, 2015 at 15:05
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$\begingroup$ Ah, this makes sense to me now and will work fine for my needs. Thanks! $\endgroup$ Commented Jul 3, 2015 at 15:15