Geoscience Reference
In-Depth Information
7.16
COMPARISON OF TWO OR MORE GROUPS BY ANALYSIS OF VARIANCE
7.16.1 C
omplete
r
andomization
A planter wanted to compare the effects of five site-preparation treatments on the early height
growth of planted pine seedlings. He laid out 25 plots and applied each treatment to 5 randomly
selected plots. The plots were then hand planted and at the end of 5 years the height for all pines was
measured and an average height computed for each plot. The plot averages (in feet) were as follows:
Treatments
A
B
C
D
E
Total
15
16
13
11
14
14
14
12
13
12
12
13
11
10
12
13
15
12
12
10
13
14
10
11
11
Sums
67
72
58
57
59
313
Treatment means
13.4
14.4
11.6
11.4
11.8
12.52
Looking at the data we see that there are differences among the treatment means: A and B
have higher averages than C, D, and E. Soils and planting stock are seldom completely uniform,
however, so we would expect some differences even if every plot had been given exactly the same
site-preparation treatment. The question is, can differences as large as this occur strictly by chance
if there is actually no difference among treatments? If we decide that the observed differences are
larger than might be expected to occur strictly by chance, then the inference is that the treatment
means are not equal. Statistically speaking, we reject the hypothesis of no difference among treat-
ment means.
Problems like this are neatly handled by an analysis of variance. To make this analysis, we need
to fill in a table like the following:
Source of Variation
Degrees of Freedom
Sums of Squares
Mean Squares
Treatments
4
Error
20
Total
24
7.16.1.1 Source of Variation
There are a number of reasons why the height growth of these 25 plots might vary, but only one can
be definitely identified and evaluated—that attributable to treatments. The unidentified variation is
assumed to represent the variation inherent in the experimental material and is labeled error. Thus,
total variation is being divided into two parts: one part attributable to treatments, and the other
unidentified and called error.
7.16.1.2 Degrees of Freedom
Degrees of freedom are difficult to explain in non-statistical language. In the simpler analyses of
variance, however, they are not difficult to determine. For the total, the degrees of freedom are one
less than the number of observations; there are 25 plots, so the total has 24 degrees of freedom.
For the sources, other than error, the degrees of freedom are one less than the number of classes
or groups recognized in the source. Thus, in the source labeled “Treatments,” there are five groups
(five treatments), so there will be four degrees of freedom for treatments. The remaining degrees of
freedom (24 - 4 = 20) are associated with the error term.
Search WWH ::
Custom Search