# Combining Prediction Intervals

#### Rob

##### New Member
I'm having trouble finding how to combine a number of prediction intervals. The best way to explain the problem is:

Imagine a race with 4 relay runners running a 4x100m. I have different bits of information on how fast each runner can complete various distances which I use to predict how quickly they can run 100m. Each runner has a regression model which predicts their expected finish time and an interval in which I am 95% confident they will finish (the predcition interval). How would I calculate the prediction interval for the finish time of the whole race (ie all 4 runners together)? I have the Standard error of the regression for each runner's model and the standard error of the prediction (so I can calculate each individual prediction interval).

Any help greatly appreciated!

#### Rob

##### New Member
Thanks MP83.

I know CIs and PIs are calculated in a very similar way so the Bonferonni test would give a combined PI that was wider than just adding up all the prediction intervals - you're calculating an interval that ensures each separate range is wide enough so that there is only a 5% chance that any of the separate predictions you make from the model are outside your interval.

But - I'm not so interested in the separate predictions, only in the overall prediction, so I want only a 5% chance that my overall prediction is wrong (if some of the separe predictions are outside the 95% range, that doesn't matter).

Intuitively to me I would have thought a combined PI would have a narrower interval than just adding up all the separate intervals - variation in each runner's time would act to 'soften' the impact of any one runner's extremely low (or high) time.

I might have got the logic completely wrong here and happy to be shown the right path!

#### mp83

##### TS Contributor
[In short coz my post got lost!]

If you are looking for a CI for a linear combination [average(mu1+mu2+mu3+mu4) for an estimate of the overall time] then you should use the Scheffe method.

You could use one model instead of 4,simply use indicators variables,then say using R,the Scheffe CI is right in front of you...

#### purpleorchid

##### New Member
Hi,

I have a similar problem. I predicted a bunch of values using a regression equation and calculated the 95% prediction interval for each observation. Now I want to aggregate/average all of them and calculate the combined prediction interval. I am not sure if the methods mentioned above apply to my problem. Any suggestions?