Applied Statistics and Probability for Engineers

14-4 TWO-FACTOR FACTORIAL EXPERIMENTS 513

interaction, and error:

Assuming that factors Aand Bare fixed factors, it is not difficult to show that the expected

valuesof these mean squares are

From examining these expected mean squares, it is clear that if the null hypotheses about main

effects H 0 : i0, H 0 : j0, and the interaction hypothesis H 0 : ()ij0 are all true, all four

mean squares are unbiased estimates of 

2.

To test that the row factor effects are all equal to zero (H 0 : i0), we would use the ratio

E 1 MSE 2 E (^) a
SSE
ab 1 n 12
b
^2
E 1 MSAB 2 E (^) a
SSAB
1 a 121 b 12
b
^2 
n^ g
a
i 1
(^) g
b
j 1
1 ^22 ij
1 a 121 b 12
E 1 MSB 2 E a
SSB
b 1
b
^2 
an^ g
b
j 1
^2 j
b 1
E 1 MSA 2 E a
SSA
a 1
b
^2 
bn^ g
a
i 1
i^2
a 1
MSA
SSA
a 1

MSB

SSB

b 1

MSAB

SSAB

1 a 121 b 12

MSE

SSE

ab 1 n 12

F 0 

MSA

MSE

F 0 

MSB

MSE

F 0 

MSAB

MSE

which has an F-distribution with a1 and ab(n1) degrees of freedom if H 0 : i0 is true.

This null hypothesis is rejected at the level of significance if f 0

 f ,a1,ab(n1). Similarly, to

test the hypothesis that all the column factor effects are equal to zero (H 0 : j0), we would

use the ratio

which has an F-distribution with b1 and ab(n1) degrees of freedom if H 0 : j0 is true.

This null hypothesis is rejected at the level of significance if f 0

 f ,b1,ab(n1). Finally, to test

the hypothesis H 0 : ()ij0, which is the hypothesis that all interaction effects are zero, we use

the ratio

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