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% Make title bold and 14 pt font (Latex default is non-bold, 16pt)
\title{Stat 324: Lecture 20\\
Inference on Regression Parameters}

% For single author (just remove % characters)
\author{Moo K. Chung\\
mchung@stat.wisc.edu}
\date{April 19, 2005}
\maketitle \thispagestyle{empty}


\begin{enumerate}


\item {\em Coefficient of determination} measures the proportion of the
total variations in $y$ that can be explained by the linear model
and it is defined as
$$r^2 = 1 - \frac{SSE}{SST}$$
Since $SST = S_{yy}$ and $SSE = S_{yy}-S_{xy}^2/S_{xx}$, we can see
that $r^2 = S_{xy}^2/(S_{xx}S_{yy}).$ $r$ is called the {\em sample
correlation coefficient}.


\item {\em Example.} Hooke's law states that the length change $y$
in spring is proportional to applied force $x$. The following
measurements are available.\\
\begin{tabular}{|c|cccccc|}
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
\hline
x (kg) & 29.4 & 39.2 & 49.0 & 58.8& 68.6 & 78.4 \\
\hline
 y (mm) & 4.25 & 5.25  & 6.50  & 7.85 & 8.75 & 10.10  \\
\hline
\end{tabular}\\
We fit the model $Y=\beta_0 + \beta_1 x + \epsilon$.

\begin{verbatim}
>x<-c(29.4,39.2,49.0,58.8,68.6,78.4)
>y<-c(4.25,5.25,6.50,7.85,8.75,10.10)
>summary(lm(y~x))

Call: lm(formula = y ~ x)

Residuals:
       1        2        3        4        5        6
 0.06905 -0.10524 -0.02952  0.14619 -0.12810  0.04762

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.658095   0.163995   4.013   0.0160 *
x           0.119825 0.002906  41.238 2.07e-06 ***
---
Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1

Residual standard error: 0.1191 on 4 degrees of freedom Multiple
R-Squared: 0.9977,     Adjusted R-squared: 0.9971 F-statistic:  1701
on 1 and 4 DF,  p-value: 2.067e-06

\end{verbatim}





\item The least squares estimation for $\beta_1$ are given by
$\hat \beta_1 = \frac{S_{xy}}{S_{xx}}$ where $S_{xy} =
n(\overline{xy} -\bar{x}\bar{y})$. Since $\sum_{j=1}^n (x_j-\bar
x)\bar Y = 0$, $\hat \beta_1 = \frac{S_{xy}}{S_{xx}}=\sum_{j=1}^n
c_jY_j,$ where $c_j = (x_j -\bar x)/S_{xx}.$ From $\sum_{j=1}^n c_j
= 0$, $\sum_{j=1}^n c_jx_j=1$, we can show that $\mathbb{E} \hat
\beta_1 = \beta_1$ showing unbiasness. From $\sum_{j=1}^n c_j^2 =
S_{xx}^{-1}$, $\mathbb{V} \hat \beta_1 = \sigma^2/S_{xx}$. Since
$\hat \beta_1$ is a linear combination of normal distributions,
$\hat \beta_1 \sim N(\beta_1, \sigma^2/S_{xx})$. The test statistic
$$T =
\frac{\hat \beta_1 - \beta_1}{S_{\hat \beta_1}} \sim t_{n-2},$$
where $S_{\hat \beta_1} = \hat \sigma/\sqrt{S_{xx}}$ and $\hat
\sigma = SSE/(n-2)$. It can be shown that $SSE =
S_{yy}-S_{xy}^2/S_{xx}$ and $S_{\hat \beta_1} =
\frac{1}{\sqrt{n-2}}\sqrt{\frac{S_{yy}}{S_{xx}}-(\frac{S_{xy}}{S_{xx}})^2}$.
In a slightly different way, $SSE = \sum_{j=1}^n y_j^2 - \hat
\beta_0 \sum_{j=1}^n y_j - \hat \beta_1 \sum_{j=1}^n x_j y_j$.


\item Based on the test statistic, a $100(1-\alpha)\%$ CI for $\beta_1$ is $\hat \beta_1 \pm
t_{\alpha/2, n-2} S_{\hat \beta_1}$. For hypothesis testing
$$H_0: \beta_1 = 0 \mbox{ vs. } H_1:\beta_1 \neq 0$$
we reject $H_0$ if $|t|>t_{\alpha/2,n-2}$ at $100(1-\alpha)\%$
significance. A hypothesis testing on $\beta_0$ and a CI
construction are similar. $\hat \beta_0 = \bar Y - \hat \beta_1 \bar
x$ is distributed as normal with mean $\beta_0$ and variance
$\sigma^2\overline{x^2}/S_{xx}$. Testing $H_0: \beta_0=0$ is based
on $T=\frac{\hat \beta_0 - \beta_0}{S_{\hat \beta_0}} \sim t_{n-2}$,
where $S_{\hat \beta_0} = \sigma^2\overline{x^2}/S_{xx}$.

\end{enumerate}
{\bf Assigned problems.} Exercise 12.30., 12. 36. HW 6 due April
28th. Solve the assigned problems in Lecture 19-22.
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