10/07/2020

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- The following topics will be covered in this lecture:
- Testing a single predictor
- The t-test
- Testing a subspace
- Confidence intervals
- Confidence regions

As a general method, we can always use the F-statistic for

**nested models**.Specifically, whenever one model is given by a subspace of another:

- \( \boldsymbol{\omega} \) consists of models over variables \( x_1, \cdots, x_{q-1} \) and corresponds to \( q \) parameters (including the intercept);
- \( \boldsymbol{\Omega} \) consists of models over variables \( x_1, \cdots , x_{p-1} \) and corresponds to \( p \) parameters (including the itercept);
- such that \( q \text{<} p \).

Concretely, the null hypothesis must be \( H_0 : \boldsymbol{\beta}_i = \boldsymbol{0} \) for each \( i=q,\cdots, p-1 \).

The alternative hypothesis is that the larger model holds,

\[ H_1: \boldsymbol{\beta} \neq 0 \]

We compute the F statistic as, \[ \begin{align} F &\triangleq \frac{ \left( RSS_\boldsymbol{\omega} - RSS_\boldsymbol{\Omega}\right)/ (p-1)}{RSS_\boldsymbol{\Omega}/(n-p)} . \end{align} \]

Suppose there is one particular variable that we want to determine the significance of for our model.

Specifically, suppose we have a model,

\[ \begin{align} \mathbf{Y} = \mathbf{X} \boldsymbol{\beta} + \boldsymbol{\epsilon}, \end{align} \] with respect to some choice of variables \( \mathbf{X} \).

Our alternative hypothesis in this case is,

\[ \begin{align} H_1: \boldsymbol{\beta} \neq \boldsymbol{0}. \end{align} \]

**Q:**if we want to determine if \( \boldsymbol{\beta}_i \) specifically gives an appreciable difference in this model, what is our null hypothesis?**A:**our null hypothesis takes the form,\[ \begin{align} H_0: \boldsymbol{\beta}_i = 0 \end{align} \]

We will examine this on the

`gala`

data once again.We define the model

`lmods`

without`area`

as an explanatory variable for the null hypothesis. Then we compute the ANOVA table with the bigger model that contains`area`

```
require("faraway")
lmod <- lm(Species ~ Area + Elevation + Nearest + Scruz + Adjacent, gala)
lmods <- lm(Species ~ Elevation + Nearest + Scruz + Adjacent, gala)
anova(lmods, lmod)
```

```
Analysis of Variance Table
Model 1: Species ~ Elevation + Nearest + Scruz + Adjacent
Model 2: Species ~ Area + Elevation + Nearest + Scruz + Adjacent
Res.Df RSS Df Sum of Sq F Pr(>F)
1 25 93469
2 24 89231 1 4237.7 1.1398 0.2963
```

The result of the \( F \) test is to say, “With probability 29.63%, we will find a value drawn from the F distribution with this value or greater”.

**Q:**do we reject or fail to reject the null hypothesis at \( 5\% \) significance here?**A:**Here we**fail to reject the null hypothesis**because it is reasonable that the difference between the large model and the small model could be due to random variation.**Note:**there may be some statistical relationship, but we haven't detected one that wouldn't be surpising if it was just noise.

Courtesy of Skbkekas CC BY-SA 3.0

- The significance of a
**single variable**can also be found with respect to the student t-test. - Generally, suppose that we have \( n \) independent random variables drawn from a Gaussian distribution \( \left\{Y_i\right\}_{i=1}^n \), with unknown true mean \( \mu_Y \) and standard deviation \( \sigma \).
- As usual, our sample-based estimate of the mean is given by, \[ \begin{align} \overline{Y} = \frac{1}{n}\sum_{i=1}^n Y_i ; \end{align} \]
- and our unbiased, sample-based estimate of the variance is given as, \[ \begin{align} S^2 = \frac{1}{n-1} \sum_{i=1}^n \left(Y_i - \overline{Y}\right)^2. \end{align} \]

- It is a powerful and non-trivial result that, \[ \begin{align} \frac{\overline{Y} - \mu_Y}{S/ \sqrt{n}}, \end{align} \] is distributed according to a student t-distribution, in \( n-1 \) degrees of freedom.