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 H0: Yx
01x for some values 0, 1 (the true regression function is linear)

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

Suppose observations are made at x1, x2, . . . , xc. Let Y11, Y12, . . . , Y1n1 denote the n1 observations when x
x1; . . . ;

Yc1, Yc2, . . . , Ycnc denote the nc observations when x
xc.

With n

ni (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:

SSPE

i

j

(Yij  Yi)2

Y2 ij 
ni Y2 i

SSLF
SSE  SSPE The ni observations at xi contribute ni  1 df to SSPE, so the number of degrees of freedom for SSPE is

i(ni  1)
n  c and the degrees of freedom for SSLF is n  2  (n 

c)
c  2. Let MSPE
SSPE/(n 

c) and