Multiple Linear Regression Deriving the Prediction Interval for a New Response, y

In this video I show the math behind deriving the Prediction Interval for a new response (Y) for the Multiple Linear Regression Model using matrix notation. I discuss more in more detail the reason for finding the standard error of the residual in this video: • Simple Linear Regressi...

[@AhmedThahir2002]

Thanks for the amazing video! Very intuitive and well-explained. How would derive the same for multiple non-linear regression. For eg: Exponential curve-fit: y = e^(ax), where a is the parameter

[@willcluett2493]

Do you have a video and can direct me to a resource that derives the var(y0-y0hat)=var(y0)+var(y0hat)? I believe this relies on y0 and y0hat being independent but how do you know this is the case? Is this true as long as x0 is not contained in X?

[@JR-hl6bt]

Thank you for sharing this amazing video. It really helped me to understand more about PI. BTW, I just want o know, why for Prediction Interval we use Varians of (y0-y0 hat), while for Confident interval, we use Varians of y0hat only.

[@Stats4Everyone]

This is a great question! I recently made a video to discuss this question in more detail: kzbin.info/www/bejne/fICrhYyMgNCXf5Y. I hope that this might be helpful :-)

[@valentinx4126]

Great video! Just one question. Why is it that for y0 we need y 0 = x0tB + e, but for yhat0 we need yhat0 = x0tBhat. Without the e... You wrote it in the variance term without for y0hat but with for y0

[@Stats4Everyone]

Good question. This comes down to the different definitions for y0, and yhat0. y0 is random. yhat0 is the predicted value for y0. We find the predicted value for yhat0 by estimating the expected value of y0. The expected value of e is zero (this is a model assumption for y0 = x0*B + e). Therefore E(y0) = E(x0*B + e) E(y0) = x0*B + E(e) E(y0) = x0*B since B is unknown, we estimate B using Bhat. Therefore an estimate of E(y0) is yhat0 = x0*Bhat I hope this helps...

[@valentinx4126]

Ah yes my bad, thanks alot @@Stats4Everyone

[@willcluett2493]

Do you have a video where you derive the var(b)? You make reference to it being derived elsewhere.

[@Stats4Everyone]

Sure do: kzbin.info/www/bejne/kHfXn6mOobyiZqc I have a MLR Playlist: kzbin.info/aero/PLJDUkOtqDm6U4lY66-a2_nqO3jR9D6DnO

[@willcluett2493]

@@Stats4Everyone Thank you Michelle.

[@picobarco4407]

Hi there, I really like this derivation. I have been looking for the same thing, but the Confidence Interval for Multi-Linear Regression. Hope you have derivation for Confidence Interval for the mean response Y.

[@Stats4Everyone]

Thank you for this comment! Yes, I do have a similar video for confidence interval: kzbin.info/www/bejne/mnawYoZ-nM6AmJI . I hope you find this helpful, or please let me know if you have any follow-up questions.

[@abu-bakrmohamed1707]

Such an amazing explanation , its literally the first time i know why its called y0 🤍🤍🤍

[@Stats4Everyone]

That's great to hear! Thanks for the feedback 😁