Applied Statistics and Probability for Engineers

The variance components ^2 , ^2 , and ^2 may be estimated by eliminating the first equa-

tion from Equations S14-8, leaving three equations in three unknowns, whose solution is

and

This general approach can be used to estimate the variance components in anymixed model.

After eliminating the mean squares containing fixed factors, there will always be a set of equa-

tions remaining that can be solved for the variance components. Table S14-3 summarizes the

analysis of variance for the two-factor mixed model.

Computer Output

Some software packages have the capability to handle random factors in the analysis of vari-

ance. Table S14-4 is the computer solution to Example S14-1 from Minitab. Notice that the

F-tests on the main effects have been constructed using MSABin the denominator. The results

closely match those in Table S14-2.

Table S14-5 is the Minitab output for the breaking strength data assuming a mixed model,

where operators are fixed and machines are random. Minitab allowed us to assume that

restricted form of the mixed model [not all software packages make this easy, and some auto-

matically use an unrestricted model, that is, the termgia 1 ()ij 0 in general]. Notice that

ˆ^2 MSE

ˆ^2 ^ 

MSAB MSE

n

ˆ^2 ^ 

MSB MSE

an

14-8

Table S14-3 Analysis of Variance for the Two-Factor Mixed Model (Afixed, Brandom)

Source of Sum of Degrees of Expected Mean Mean

Variation Squares Freedom Square Square F 0

Rows (A) SSA a 

1 MSA MSAMSAB

Columns (B) SSB b 

1 ^2 an^2  MSB MSBMSE

Interaction SSAB (a 1)(b 1) ^2 n^2  MSAB MSABMSE

Error SSE ab(n 1) ^2 MSE

Total SST abn 

1

^2 n^2 bng ^2 i 1 a
12

Table S14-4 Minitab Output for Example S14-1

Analysis of Variance (Balanced Designs)

Factor Type Levels Values

Operator random 3 A B C

Machine random 4 1 2 3 4

Analysis of Variance for Strength

Source DF SS MS F P

Operator 2 26.333 13.167 2.21 0.190

Machine 3 24.333 8.111 1.36 0.340

Operator*Machine 6 35.667 5.944 1.59 0.234

Error 12 45.000 3.750

Total 23 131.333

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