Designed experiments are conducted to demonstrate a cause-andeffect relationship between one or more explanatory factors (or
predictors) and a response variable.
A cause-and-effect relationship is demonstrated by altering the
Typical ANOVA models have fixed factor effects. Consider studying
the effects that gender and drug (Tylenol, Advil, control) have on
level of headache.
These are fixed factor levels. Our interest centers on the specific
factor levels chosen.
The two-factor interaction effects in a three-factor study are defined
in the same way as for a two-factor study, except all means are
averaged over the third factor.
Thus, following our previous definition, we define the two-factor
Sometimes there is interest in estimating the overall mean . We
know that Ecfw_Yij = . thus an unbiased estimator when all factor
sample sizes are equal is
. = Y .
The variance of the estimator is
2 n + 2
2 cfw_Y . =
To demonstrate how the ANOVA tests are done for unequal sample
sizes we will use an example about growth hormones.
Growth hormones were injected into children. The researcher was
interested in the effects of a child's gender (factor A
/* There are 3 windows in SAS: the editor window, the log window, and the output window.
The editor window has been saved as SAS1. It is where we type our commands. It is color
coded to let us know when a word is a command (blue), when we are entering dat
one case per treatment
When there is only one case per treatment, we cant use our standard
two-factor ANOVA model with interaction.
SSE = (Yijk Yij .)
The SSE is a sum of squares made up of components measuring the
We will partition the total deviation of an observation Yijk from the
overall mean Y(bar) in two stages. First, we obtain a
decomposition of the total deviation Yijk - Y(bar) by viewing the
study as consisting of ab treatments.
Yijk Y .
For a single factor ANOVA, we will denote the number of levels of
the factor under study with an r, and will denote any of these
levels by the index i (i = 1,r).
For example, for the production example, r= 4 types of incentive
An experimental unit is the smallest unit of experimental material
to which a treatment can be assigned; the experimental unit is thus
determined by the method of randomization.
Consider an experiment to study the effects of 3 differe
Corresponding to the decomposition of the total sum of squares,
we can also obtain a breakdown of the associated degrees of
SSTO has nT -1 degrees of freedom associated with it. There
are nT deviations but one degree of fr
The Scheffe multiple comparison procedure applies when the family of
interest is the set of all possible contrasts among the factor level
L = ci i where
The family confidence level for the Scheffe procedure
Factor Level Means
We previously discussed the F test for determining whether or not
the factor level means differ. This is a preliminary test to establish
whether detailed analysis of the factor level means is warranted.
If we do not reject t
It is not necessary, nor is it usually possible, that an ANOVA model
fit the data perfectly.
ANOVA models are reasonably robust against certain types of
departures from the model, such as the error term not being exactly
It is possible for several hypothesis tests to be performed. For
instance, if a factor has several levels and we compare each of
these levels to each other.
When many tests are performed multiple comparison procedures
are required. Th
Another definition of . is as a weighted average of all factor level
. = wi i
Note that the wi are weights defined so that their sum is 1. The
restriction on the i implied by this definition of . is
In a two-factor study, when the two factors do not interact, the
analysis of the factor level effects usually only involves the factor
level means i. and .j
Unbiased estimators are
We have that
i . = Yi .
. j = Y . j.
(Yi . ) =
Rank F test
When transformations are not successful in bringing the
distributions of the error terms close to normality, a nonparametric
procedure may be used.
Nonparametric procedures do not depend on the distribution of the
error terms; of
The table above contains data for the learning example when there is
interaction between age and gender. Note that gender has no effect
on learning for young people but has a substantial effect for old
Thus the effect of gender
Analysis of covariance (ANCOVA) is a technique that combines
features of ANOVA and regression. The basic idea is to augment
the ANOVA model with one or more quantitative variables.
This is done in order to reduce the variance of the error terms.
A junior college system studied the effects of teaching method (factor
A) and students quantitative ability (factor B) on learning of college
mathematics. There were two teaching methods (standard and
abstract) and three ability levels
/* using cell means model so dummy coding is 1 or 0 for all factor levels not just the
first r-1. We will then run the model with no intercept.
The weights have already been calculated and are
a part of the data set*/
data abt_regression; input i j force