# One tailed prediction intervals for Multiple Linear regression

After having fit a multiple regression model to my data, I am using it for predicting my dependent variable. I understand how one can predict and compute (using R) two tailed prediction intervals at a certain $$\alpha$$. The requirements of the use case are such that I don’t care about the upper prediction (two-tailed) interval because I need to be able to say that with 100(1-$$\alpha$$)% confidence that the value I have predicted will be above the prediction limit. My questions:

1. Is it statistically valid to compute one tailed prediction interval for linear regression? I searched on CV for questions related to this and could not find anything.
2. If yes, then could one use the predict() function in R to compute the interval? I have read the documentation of the function, it does not seem to have an option to switch between one-tailed and two-tailed. Otherwise, how could one calculate such an interval in R?

## 1 Answer

1. Yes. You simply ignore the upper end of the CI as it is not relevant to you.

2. The trick is to manipulate the level argument to predict. If you specify level=0.9, it will produce a confidence interval where 5 % fall below it, and 5 % end up above it. If you ignore the upper end of that interval, it follows that 95 % is above the lower end.

In other words, if you take a 90 % double sided interval and ignore one side, you get a 95 % one-sided interval.

(Similarly, a 98 % two-sided interval becomes a 99 % one sided interval if you ignore one of the ends. And so on. The thing to realise the double-sided interval is constructed to have half the errors above it, and half below. Ignoring one of those halves means you end up with double the significance for the remaining threshold.)

I found this previous answer which might help make things more graphical: https://math.stackexchange.com/questions/2835809/one-tailed-confidence-interval-1-2-alpha-rationale#2836171