Further hypothesis testing, confidence intervals and regions

10/07/2020

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Outline

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

Testing one predictor

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

Testing one predictor – continued

  • 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,

    \[ \begin{align} H_0: \boldsymbol{\beta}_i = 0 \end{align} \]

Testing one predictor – continued

  • 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.

Student t-distribution

Image of student t-distributions varying with the number of degrees of freedom 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.

Student t-distribution