Statistics Review - Mathematics and Statistics for the Environment - page 142

With unequal replication, the F value in Scheffe's test is computed by the equation

ˆ

2

Q

F =

2 2

a
n

a
n

1 2

k ++ … ( Errormean square)

1 2

Selecting the coefficients ( a i ) for such contrasts can be tricky. When testing the hypothesis that there
is no difference between the means of two groups of treatments, the positive coefficients are usually

n
p

i

Positive a =

i

where p is the total number of plots in the group of treatments with positive coefficients.
The negative coefficients are

n
m

j

Negative a =

j

where m is the total number of plots in the group of treatments with negative coefficients.
To illustrate, if we wish to compare the mean of treatments A, B, and C with the mean of treat-
ments D and E and there are two plots of treatment A, three of B, five of C, three of D, and two of
E, then p = 2 + 3 + 5 = 10, m = 3 + 2 = 5, and the contrast would be

2
10

3
10

5
10

3
5

2
5

−

Q =

ABC + + DE +

7.16.3 r andomized b loCK d esign
There are two basic types of the two-factor analysis of variance: completely randomized design
(discussed in the previous section) and randomized block design . In the completely randomized
design, the error mean square is a measure of the variation among plots treated alike. It is in fact
an average of the within-treatment variances, as may easily be verified by computation. If there is
considerable variation among plots treated alike, the error mean square will be large and the F test
for a given set of treatments is less likely to be significant. Only large differences among treatments
will be detected as real and the experiment is said to be insensitive.
Often the error can be reduced (thus giving a more sensitive test) by use of a randomized block
design in place of complete randomization. In this design, similar plots or plots that are close
together are grouped into blocks. Usually the number of plots in each block is the same as the
number of treatments to be compared, though there are variations having two or more plots per
treatment in each block. The blocks are recognized as a source of variation that is isolated in the
analysis. A general rule in randomized block design is to “block what you can, randomize what
you can't.” In other words, blocking is used to remove the effects of nuisance variables or factors.
Nuisance factors are those that may affect the measured result but are not of primary interest. For
example, in applying a treatment, nuisance factors might be the time of day the experiment was run,
the room temperature, or the specific operator who prepared the treatment (Addelman, 1969, 1970).
As an example, a randomized block design with five blocks was used to test the height growth of
cottonwood cuttings from four selected parent trees. The field layout looked like this:

D B B C A D B A C D

C A A D B C C D A B

I II III IV V