Applied Statistics and Probability for Engineers

F-ratio for blocks may not provide reliable information about block effects. For more discus-

sion see Montgomery (2001, Chapter 4).

13-4.2 Multiple Comparisons

When the ANOVA indicates that a difference exists between the treatment means, we may

need to perform some follow-up tests to isolate the specific differences. Any multiple com-

parison method, such as Fisher’s LSD method, could be used for this purpose.

We will illustrate Fisher’s LSD method. The four chemical type averages from

Example 13-5 are:

Each treatment average uses b5 observations (one from each block). We will use 

0.05, so t0.025,122.179. Therefore the value of the LSD is

Any pair of treatment averages that differ by 0.39 or more indicates that this pair of treatment

means is significantly different. The comparisons are shown below:

3 vs. 1y 3 .y 1 .1.381.140.240.39

2 vs. 3y 2 .y 3 .1.761.380.380.39

2 vs. 1y 2 .y 1 .1.761.140.620.39

4 vs. 2y 4 .y 2 .3.561.761.800.39

4 vs. 3y 4 .y 3 .3.561.382.180.39

4 vs. 1y 4 .y 1 .3.561.142.420.39

LSDt0.025,12

B

2 MSE

b

2.179

B

21 0.08 2

5

0.39

y 1 .1.14 y 2 .1.76 y 3 .1.38 y 4 .3.56

13-4 RANDOMIZED COMPLETE BLOCK DESIGN 497

Table 13-14 Minitab Analysis of Variance for the Randomized Complete

Block Design in Example 13-5

Analysis of Variance (Balanced Designs)

Factor Type Levels Values

Chemical fixed 4 1 2 3 4

Fabric S fixed 5 1 2 3 4 5

Analysis of Variance for strength

Source DF SS MS F P

Chemical 3 18.0440 6.0147 75.89 0.000

Fabric S 4 6.6930 1.6733 21.11 0.000

Error 12 0.9510 0.0792

Total 19 25.6880

F-test with denominator: Error

Denominator MS0.079250 with 12 degrees of freedom

Numerator DF MS F P

Chemical 3 6.015 75.89 0.000

Fabric S 4 1.673 21.11 0.000

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