If there is at least one \(x\) value at which more than one observation has been
made, there is a formal test procedure for testing \(H_{0}:
\mu_{\gamma^{-x}}=\beta_{0}+\beta_{1} x\) for some values \(\beta_{0} .
\beta_{1}\) (the true regression function is linear)
versus
\(H_{\mathrm{a}}: H_{0}\) is not true (the true regression function is not
linear)
Suppose observations are made at \(x_{1}, x_{2}, \ldots, x_{c}\). Let \(Y_{11}\),
\(Y_{12}, \ldots, Y_{1 n_{4}}\) denote the \(n_{1}\) observations when \(x=x_{1} ;
\ldots\); \(Y_{c 1}, Y_{c 2}, \ldots, Y_{c m}\) denote the \(n_{c}\) observations
when \(x=x_{c}\). With \(n=\sum n_{i}\) (the total number of observations), SSE
has \(n-2\) df. We break SSE into two pieces, SSPE (pure error) and SSLF (lack
of fit), as follows:
$$
\begin{aligned}
\text { SSPE } &=\sum_{i} \sum_{i}\left(Y_{i j}-\bar{Y}_{i}\right)^{2} \\
&=\sum \sum Y_{i j}^{2}-\sum n_{i} \bar{Y}_{i \cdot}^{2} \\
\text { SSLF } &=\text { SSE }-\text { SSPE }
\end{aligned}
$$
The \(n_{i}\) observations at \(x_{i}\) contribute \(n_{i}-1\) df to SSPE, so the
number of degrees of freedom for SSPE is \(\Sigma\left(n_{i}-1\right)=n-c\) and
the degrees of freedom for SSLF is \(n-2-(n-c)=c-2\). Let MSPE
\(=\operatorname{SSPE} /(n-c)\) and \(\mathrm{MSLF}=\mathrm{SSLF} /(c-2)\). Then
it can be shown that whereas \(E(\mathrm{MSPE})=\sigma^{2}\) whether or not
\(H_{0}\) is true, \(E(\mathrm{MSLF})=\sigma^{2}\) if \(H_{0}\) is true and
\(E(\mathrm{MSLF})>\sigma^{2}\) if \(H_{0}\) is false.
Test statistic: \(\quad F=\frac{\text { MSLF }}{\text { MSPE }}\)
Rejection region: \(f \geq F_{a c-2, x-c}\)
The following data comes from the article "Changes in Growth Hormone Status
Related to Body Weight of Growing Cattle" (Growth, 1977: 241-247), with \(x=\)
body weight and \(y=\) metabolic clearance rate/body weight.
\begin{tabular}{l|lllllll}
\(x\) & 110 & 110 & 110 & 230 & 230 & 230 & 360 \\
\hline\(y\) & 235 & 198 & 173 & 174 & 149 & 124 & 115
\end{tabular}
\begin{tabular}{l|lllllll}
\(x\) & 360 & 360 & 360 & 505 & 505 & 505 & 505 \\
\hline\(y\) & 130 & 102 & 95 & 122 & 112 & 98 & 96
\end{tabular}
(So \(c=4, n_{1}=n_{2}=3, n_{3}=n_{4}=4\).)
a. Test \(H_{0}\) versus \(H_{2}\) at level .05 using the lack-of-fit test just
described.
b. Does a scatter plot of the data suggest that the relationship between \(x\)
and \(y\) is linear? How does this compare with the result of part (a)? (A
nonlinear regression function was used in the article.)
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