MANE 6313
Week 12, Module C
Student Learning Outcome
- Select an appropriate experimental design with one or more factors,
- Select an appropriate model with one or more factors,
- Evaluate statistical analyses of experimental designs,
- Assess the model adequacy of any experimental design, and
- Interpret model results.
Module Learning Outcome
Explaining inference for linear regression models.
Tests on Individual Regression Coefficients
- We can test \(H_0:\beta_j=0\) vs. \(H_a:\beta_j\neq 0\)
\[
\frac{\hat{\beta_j}}{\sqrt{\hat{\sigma}^2C_{jj}}}\sim
t_{n-k-1}
\]
where \(C_{jj}\) is the \((jj)\)th element of \(\mathbf{(X^\prime X)^{-1}}\)
- or equivalently
\[
\frac{\hat{\beta_j}}{se(\hat{\beta}_j)}\sim t_{n-k-1}
\]
- If \(H_0:\beta_j=0\) is not rejected, we can remove the variable \(x_j\) from the model
Example Problem 12.8
C.I. on the Individual Regression Coefficients
- Straightforward to construct
\[
\frac{\hat{\beta}_j-\beta}{\sqrt{\hat{\sigma}^2C_{jj}}}\sim t_{n-p}\;\;\;j=0,1,2,\ldots,k
\]
- A \(100(1-\alpha)\%\) confidence interval for \(\beta_j\) is
\[
\hat{\beta}_j-t_{\alpha/2,n-p}\sqrt{\hat{\sigma}^2C_{jj}}\leq\beta_j\leq
\hat{\beta}_j+t_{\alpha/2,n-p}\sqrt{\hat{\sigma}^2C_{jj}}
\]
- or equivalently
\[
\hat{\beta}_j-t_{\alpha/2,n-p}se(\hat{\beta}_j)\leq\beta_j\leq
\hat{\beta}_j+t_{\alpha/2,n-p}se(\hat{\beta}_j)
\]
R: confint() Function
Source: https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/confint
Example 12.8 Confidence Interval on Parameters
C.I. on the Mean response
-
We can find a confidence interval on the mean response at a point \(x^\prime_0=[1,x_{01},x_{02},\ldots,x_{x0k}]\).
-
Note that
\[
\begin{aligned}
\mu_{y|\mathbf{x_0}}&=\beta_0+\beta_1x_{01}+\beta_2x_{02}+\cdots+\beta_kx_{0k}\\
\hat{y}(\mathbf{x_0})&=\mathbf{X^\prime_0\hat{\beta}}
\end{aligned}
\]
- The variance of \(\hat{y}(\mathbf{x_0})\) is
\[
V\left[\hat{y}(\mathbf{x_0})\right]=\sigma^2\mathbf{x_0^\prime(X^\prime
X)^{-1}x_0}
\]
- A \(100(1-\alpha)\%\) confidence interval for the mean response is
\[
\begin{aligned}
\hat{y}(\mathbf{x_0})&-t_{\alpha/2,n-p}\sqrt{\hat{\sigma}^2\mathbf{x^\prime_0(X^\prime
X)^{-1}x_0}}\\
&\leq\mu_{y|x_0}\leq\hat{y}(\mathbf{x_0})+t_{\alpha/2,n-p}\sqrt{\hat{\sigma}^2\mathbf{x^\prime_0(X^\prime
X)^{-1}x_0}}
\end{aligned}
\]
Prediction Intervals
-
We can use the regression equation to predict values at points other than those in the design matrix. In general, we only want to interpolate; not extrapolate
-
Consider the point \(x^\prime_0=[1,x_{01},x_{02},\ldots,x_{x0k}]\). A point estimate for \(y\) is
\[
\hat{y}(\mathbf{x_0})=\mathbf{x^\prime_0\hat{\beta}}
\]
- A \(100(1-\alpha)\%\) prediction interval for this observation is
\[
\begin{aligned}
\hat{y}(\mathbf{x_0})&-t_{\alpha/2,n-p}\sqrt{\hat{\sigma}^2(1+\mathbf{x^\prime_0(X^\prime
X)^{-1}x_0})}\\
&\leq y_0\leq\hat{y}(\mathbf{x_0})+t_{\alpha/2,n-p}\sqrt{\hat{\sigma}^2(1+\mathbf{x^\prime_0(X^\prime
X)^{-1}x_0})}
\end{aligned}
\]
- Most packages will perform this function for you.
R: predict.lm() Function
Source: https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/predict.lm
Example 12.8 - Confidence Interval
Example 12.8 - Prediction Interval
