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Interpretation of Sample Output
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The output is divided into three sections.
- The first section prints the number of observations (100),
the number of factors (10), and the number of levels for each
factor (10 levels for factor 1). It also prints some overall
summary statistics. In particular, the residual standard
deviation is 0.0059. The smaller the residual standard
deviation, the more we have accounted for the variance
in the data.
- The second section prints an ANOVA table. The ANOVA
table decomposes the variance into the following
component sum of
squares:
- Total sum of squares. The degrees of freedom for
this entry is the number of observations minus one.
- Sum of squares for the factor. The degrees of freedom
for this entry is the number of levels minus one.
The mean square is the sum of squares divided
by the number of degrees of freedom.
- Residual sum of squares. The degrees of freedom is
the total degrees of freedom minus the factor degrees
of freedom. The mean square is the sum of
squares divided by the number of degrees of freedom.
That is, it summarizes how much of the variance in the
data (total sum of squares) is accounted for by the
factor effect (factor sum of squares) and how much is
random error (residual sum of squares). Ideally, we would
like most of the variance to be explained by the
factor effect. The ANOVA table provides a formal
F test for the factor effect. The F-statistic is the
mean square for the factor divided by the mean square
for the error. This statistic follows an
F distribution
with (k-1) and (N-k) degrees of freedom.
If the F CDF column for the factor effect is greater than
95%, then the factor is significant at the 5% level.
- The third section prints an estimation section.
It prints an overall mean and overall standard deviation.
Then for each level of each factor, it prints the number of
observations, the mean for the observations of each
cell (
 in
the above terminology), the factor effect
(
in the above terminology), and the standard deviation of the
factor effect. Finally, it prints the residual standard
deviation for the various possible models. For the
one-way ANOVA, the two models are the constant model, i.e.,
and the model with a factor effect
For these data, including the factor effect reduces the
residual standard deviation from 0.00623 to 0.0059.
That is, although the factor is statistically significant,
it has minimal improvement over a simple constant model.
This is because the factor is just barely significant.
Output from other statistical software may look somewhat different
from the above output.
In addition to the quantitative ANOVA output, it is recommended
that any analysis of variance be complemented with
model validation.
At a minimum, this should include
- A run sequence plot of the
residuals.
- A normal probability plot of the
residuals.
- A scatter plot of the predicted
values against the residuals.
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