Chapter 13.1, Problem 14E

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 $H_0:{\mu }_{Y.x}={\beta }_0+{\beta }_{1}x$ for some values β₀, β₁ (the true regression function is linear)

Versus

H_a: H₀ is not true (the true regression function is not linear)

Suppose observations are made at x₁, x₂, …, x_c. Let Y₁₁, Y₁₂, …, $Y_{1n_1}$ denote the n₁ observations when x = x₁; …; Y_c1, Y_c2, …, $Y_{c{n}_c}$ denote the n_c observations when x = x_c. With n=∑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:

     $\begin{array}{ccc}\text{SSPE}& =& {\displaystyle \underset{i}{\sum }{\displaystyle \underset{j}{\sum }{(Y_{ij}-{\bar{Y}}_{i\cdot })}^2}}\\ & =& {\displaystyle \sum {\displaystyle \sum Y_{ij}^2-{\displaystyle \sum n_i{\bar{Y}}_{i\cdot }{}^2}}}\\ \text{SSLF}& =& \text{SSE-SSPE}\end{array}$

The n_i observations at x_i contribute n_i – 1 df to SSPE, so the number of degrees of freedom for SSPE is ${\displaystyle \underset{}{\sum }(}{n}_i-1)=n-c$, and the degrees of freedom for SSLF is n – 2 – (n – c) = c – 2. Let MSPE = SSPE/(n – c) and MSLF = SSLF/(c – 2). Then it can be shown that whereas E(MSPE) = σ² whether or not H₀ is true, E(MSLF) = σ² if H0 is true and E(MSLF) > σ² if H₀ is false.

The test statistic is F = MSLF/MSPE, and the corresponding P-value is the area under the $F_{c-2,n-c}$ curve to the right of f. 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 = 5 metabolic clearance rate/body weight.

x	110	110	110	230	230	230	360
y	235	198	173	174	149	124	115
x	360	360	360	505	505	505	505
y	130	102	95	122	112	98	96

(So c = 4, n1 = n2 = 3, n3 = n4 = 4.)

a. Test H0 versus Ha at level .05 using the lack-of-fit test just described.

b. Does a scatterplot 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.)