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Mean-response and prediction intervals from one Gaussian regression fit

Example

Take $n = 4$ , $p = 2$ and $X = (\begin{array}{cc} 1 & 0 \\ 1 & 1 \\ 1 & 2 \\ 1 & 3 \end{array}), Y = (1, 2, 2, 4)^{T} .$ Under the exact Gaussian regression model, compare the 95% confidence interval for the slope $β_{1}$ with the 95% mean-response and future-response prediction intervals at $x_{0} = (1, 2)^{T}$ . Also distinguish the Gaussian maximum-likelihood estimate of $σ^{2}$ from the residual mean square.

Facts & Assumptions

[F1]

The exact coefficient, mean-response, and future-response intervals use the formulas in Exact confidence and prediction intervals in Gaussian regression.

Verification

Given: The displayed fixed design and response, confidence level $0.95$ , and the interval theorem [F1].

1.1givenalgebra

Direct multiplication gives $X^{T} X = (\begin{array}{cc} 4 & 6 \\ 6 & 14 \end{array})$ , $(X^{T} X)^{- 1} = \frac{1}{20} (\begin{array}{cc} 14 & - 6 \\ - 6 & 4 \end{array})$ , and $X^{T} Y = (9, 18)^{T}$ . Hence $\hat{β} = (0.9, 0.9)^{T}$ , the fitted values are $(0.9, 1.8, 2.7, 3.6)^{T}$ , and $R S S_{F} = {0.1}^{2} + {0.2}^{2} + (- 0.7)^{2} + {0.4}^{2} = 0.70$ . Thus $ν = 2$ , $s^{2} = 0.35$ , while ${\hat{σ}}_{M L}^{2} = R S S_{F} / n = 0.175$ .

2.1F1step 1.1algebra∎

For $x_{0} = (1, 2)^{T}$ , multiplication by $(X^{T} X)^{- 1}$ gives $h_{0} = x_{0}^{T} (X^{T} X)^{- 1} x_{0} = 0.30$ and the fitted mean $x_{0}^{T} \hat{β} = 2.7$ . With $q = t_{2, 0.975}$ , [F1] gives the slope interval $0.9 \pm q \sqrt{0.35 \cdot 0.20}$ , the mean-response interval $2.7 \pm q \sqrt{0.35 \cdot 0.30}$ , and the prediction interval $2.7 \pm q \sqrt{0.35 \cdot 1.30}$ . The prediction half-width is $\sqrt{13 / 3}$ times the mean-response half-width, so it is wider because it includes the future response error.

Depends on

Exact confidence and prediction intervals in Gaussian regression

Used by

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Sources

Dmitry Panchenko, MIT 18.443 Statistics for Applications, Fall 2006, Section 15: Multiple Linear Regression, PDF pp. 4–7 (standard reference, not scraped)
Stanford University, Stats 203, Inference for Linear Regression, Lecture 4, sections on mean-response and prediction intervals (standard reference, not scraped)