Multiple variable prediction interval

  • Thread starter Uniquebum
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  • #1
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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

[tex] 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}}}[/tex]

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, [itex]\hat{\beta_0}+\hat{\beta_1}x_i[/itex] is a linear regression line [itex]\hat{y}[/itex]. Finally, [itex]t^*[/itex] is the t-percentile, [itex]s_e[/itex] is standard deviation, [itex]n[/itex] is the amount of points in the sample and [itex]S_{xx} = \sum{(x_i-mean(x))^2}[/itex] from 1 --> n.

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

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

Answers and Replies

  • #2
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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?
 
  • #3
FactChecker
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Now what does the equation look like for multiple variable regression?

I'd suppose [itex]\hat{\beta_0}+\hat{\beta_1}x_i[/itex] is easily changed to
[tex]\hat{\beta_0}+\hat{\beta_1}x_{0i}+\hat{\beta_2}x_{1i}+\hat{\beta_3}x_{2i}+...[/tex]
but what do i do with
[tex]\frac{(x_i-mean(x))^2}{S_{xx}}[/tex]
?
Off the top of my head, I would say that [itex]s_e[/itex] would be replaced by a cross-covariance matrix of the [itex]x_{j}[/itex]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.
 
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  • #4
statdad
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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 [itex] \widehat y [/itex] is the fitted value from the equation, and [itex] \mathbf{x}_0 [/itex] is the specified value of the predictor, the interval estimate for the mean value of the response is

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

If you want the interval for the particular value it is

[tex]
\widehat y \pm t \sqrt{\, \hat{\sigma}^2 \left(1 + \mathbf{x}'_0 \left(X' X\right)^{-1} \mathbf{x}_0 \right) }
[/tex]
 
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  • #5
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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.
 

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