Statistics Review - Mathematics and Statistics for the Environment - page 156

The F test for dates is

Date mean square
Subplot errormean square

1 500 017
16 856

, ,

F = = = 88 99 . (significant at 0.01 level)

530 / df

,

And for the date-burning interaction,

= Date-burning interactionmean square
Subplot errormean square

137 277
16 856

,

F = = 8 .14 (significantat 0.01 level)

530 / df

,

Note that the major plot error is used to test the sources above the dashed line in the table, while the
subplot error is used for the sources below the line. Because the subplot error is a measure of random
variation within major plots it will usually be smaller than the major plot error, which is a measure of
the random variation between major plots. In addition to being smaller, the subplot error will gener-
ally have more degrees of freedom than the major plot error, and for these reasons the sources below
the dashed line will usually be tested with greater sensitivity than the sources above the line. This
fact is important; in planning a split-plot experiment the designer should try to get the items of great-
est interest below the line rather than above. Rarely will the major plot error be appreciably smaller
than the subplot error. If it is, the conduct of the study and the computations should be carefully
examined. If desired, the subplots can also be split for a third level of treatment, producing a split-
split-plot design. The calculations follow the same general pattern but are more involved. A split-
split-plot design has three separate error terms. For comparisons among major or subplot treatments,
F tests with a single degree of freedom may be made in the usual manner. Comparisons among major
plot treatments should be tested against the major plot error mean square, while subplot treatment
comparisons are tested against the subplot error. In addition, it is sometimes desirable to compare
the means of two treatment combinations. This can get tricky, for the variation among such means
may contain more than one source of error. A few of the more common cases are discussed below.
In general, the t test for comparing two equally replicated treatment means is

Mean difference
Standard errorofthe mean difference

= D
s D

t =

1. For the difference between two major treatment means:

D = 2(Majorploterrormean square
( ()

s

mR

where R is the number of replications of major treatments, and m is the number of subplots
per major plot; t has degrees of freedom equal to the degree of freedom for the major plot
error.

2. For the difference between two minor treatment means:

D = 2(Subplot errormean square
()()

s

RM

where M is the number of major plot treatments; t has degrees of freedom equal to the
degree of freedom for the subplot error.