# Multiple variable prediction interval

Hey!

I'm working with some regression related stuff at the moment and i'd need some help with multiple variable prediction interval. Prediction interval for a single variable can be calculated using

$$PI = \hat{\beta_0}+\hat{\beta_1}x_i \pm t^* s_e \sqrt{1+\frac{1}{n} + \frac{(x_i-mean(x))^2}{S_{xx}}}$$

where x can be thought as a 1 dimensional vector (or matrix/set) which holds the values x_0, x_1, x_2 and so on. Also, $\hat{\beta_0}+\hat{\beta_1}x_i$ is a linear regression line $\hat{y}$. Finally, $t^*$ is the t-percentile, $s_e$ is standard deviation, $n$ is the amount of points in the sample and $S_{xx} = \sum{(x_i-mean(x))^2}$ from 1 --> n.

Now what does the equation look like for multiple variable regression?

I'd suppose $\hat{\beta_0}+\hat{\beta_1}x_i$ is easily changed to
$$\hat{\beta_0}+\hat{\beta_1}x_{0i}+\hat{\beta_2}x_{1i}+\hat{\beta_3}x_{2i}+...$$
but what do i do with
$$\frac{(x_i-mean(x))^2}{S_{xx}}$$
?

I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?

FactChecker
Gold Member
Now what does the equation look like for multiple variable regression?

I'd suppose $\hat{\beta_0}+\hat{\beta_1}x_i$ is easily changed to
$$\hat{\beta_0}+\hat{\beta_1}x_{0i}+\hat{\beta_2}x_{1i}+\hat{\beta_3}x_{2i}+...$$
but what do i do with
$$\frac{(x_i-mean(x))^2}{S_{xx}}$$
?
Off the top of my head, I would say that $s_e$ would be replaced by a cross-covariance matrix of the $x_{j}$s and that the square root would be replaced by a vector where each element is calculated with the square root equation.

PS. Your equations should drop the i subscript where x is now an arbitrary input rather than the sample data point i.

PPS. I don't know which sign of the square root to pick. I think that an authoritative answer to your OP will take more expertise than I have.

Last edited:
1 person
Homework Helper
You'll find formulae if you look in a book on multiple regression, linear models, or basic multivariate analysis. Essentially you replace the quantity you ask about with the matrix equivalent. If $\widehat y$ is the fitted value from the equation, and $\mathbf{x}_0$ is the specified value of the predictor, the interval estimate for the mean value of the response is

$$\widehat y \pm t \sqrt{\, \hat{\sigma}^2 \mathbf{x}'_0 \left(X' X\right)^{-1} \mathbf{x}_0 }$$

If you want the interval for the particular value it is

$$\widehat y \pm t \sqrt{\, \hat{\sigma}^2 \left(1 + \mathbf{x}'_0 \left(X' X\right)^{-1} \mathbf{x}_0 \right) }$$

1 person
Thanks alot for the replies. I looked through a couple of books but they only talked about multiple variable regression in too vague manner. This'll help me get forward. Thanks again.