e compositedesignsfor fitting second-ordermodelsin k factorsall contain

be portionsof resolutionat leastV, plus axial points,plus centerpoints.Of

urse there must be at least 1/2(k + 1)(k + 2) points in the design, this being th

mber of coefficients to estimate.

Hartley (1959) pointed out that the cube

rtion of the composite design need not be of resolution V.

It could, in fact, be

resolution as low as III, provided that two-factor interactions were not aliased

h two-factor interactions.

(Two-factor interactions could be aliased with main

ects, because the star portion provides additional information on the main

ects.)

This idea permitted much smaller cubes to be used.

Westlake (1965)

k this idea further by finding even smaller cubes for the k

=

5, 7, and 9 case

e following table shows the numbers of points in the various designs

ggested, for 2 ≤ k ≤ 9 .

Factors, k

Coefficients

1

2

(k

+1)(k

+2)

Points inBox-Hunter
(1957)designs

Hartley’s number ofpoints

Westlake’s numberof
points

2

6

8

6

—

3

10

14

10

—

4

15

24

16

--

5

21

26

26

22

6

28

44

28

--

7

36

78

46

40

8

45

80

48

--

9

55

146

82

62

MTB > regress c10 5 c1 c2 c11 c22 c12

The regression equation is
Vtp = 1175 + 16.7 Sub I2 - 70.7 Blnkt I2 - 0.253 C11 - 2.68 C22 + 1.46 C12

PredictorCoefStdevt-ratiop
Constant1175.158.00146.950.000
Sub I216.67480.988916.860.000
Blnkt I2-70.7292.686-26.330.000
C11-0.253340.03424-7.400.005
C22-2.67550.2211-12.100.001
C121.460040.0920315.870.001

s = 3.588R-sq = 100.%R-sq(adj) = 100.%

Analysis of Variance

SOURCEDFSSMSFp
Regression572878814575811324.460.000
Error33913
Total8728826

SOURCEDFSEQ SS
Sub1163238
Blnkt I21558420
C111287
C2213602
C1213240

Student’s t-test to check if
each slope (coefficient) iszero

Alpha risk that each slope
is actuallyzero, & the
non-zero value is due to
chance alone

% of Y variance attributed to
variance of the input variables:

(ðY/ðx1* Sx1)2+ . . . +(ðY/ðxn* Sxn)2
Variance of Y

Test hypothesis that
at least one slope is
not zero.

Attempt to attribute
sum of squares (like
variance) to each input
variable.
Maybe misleading.