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 for some values , (the true regression function is linear)

versus Ha: H0 is not true (the true regression function is not linear)

Suppose observations are made at . Let denote the n1 observations when denote the nc observations when . With (the total number of observations), SSE has df. We break SSE into two pieces, SSPE (pure error) and SSLF (lack of fit), as follows:

The ni observations at xi contribute df to SSPE, so the number of degrees of freedom for SSPE is

i

(ni 2 1) 5 n 2

c, and the degrees of freedom for SSLF ni 2 1 SSLF 5 SSE 2 SSPE 5 g gYij 22 gni Yi#

2 SSPE 5 g i

g j

(Yij 2 Yi #

)

2 n 2 2 n 5 gni x 5 xc x 5 x 1; c; Yc1, Yc2,

c, Ycnc Y11, Y12,

c, Y1n1 x 1, x 2,

c, xc H0 b0 b1 : mY#

x 5 b0 1 b1x n 5 25 n 2 2 S 5 #SSE/(n 2 2) is . Let and . Then it can be shown that whereas whether or not H0 is true, if H0 is true and if H0 is false.

Test statistic:

Rejection region:

The following data comes from the article “Changes in Growth Hormone Status Related to Body Weight of Growing Cattle” (Growth, 1977: 241–247), with and y 5 metabolic clearance rate/body weight.

x 5 body weight f $ Fa,c22,n2c F 5 MSLF MSPE E(MSLF) . s2 E(MSLF) 5 s2 E(MSPE) 5 s2 MSLF 5 SSLF/(c 2 2)

n 2 2 2 (n 2

c) 5 c 2 2 MSPE 5 SSPE/(n 2 c)

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 .)

a. Test H0 versus Ha 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.)