If we use the model
we have to estimate IJ means and 
(a total of IJ+1 parameters) using only IJ observations!
Since we can't estimate all of our parameters, we will change models
(slightly),
where 
is the effect of factor A and 
is the effect of factor B. Now we only have to estimate I+J+1
parameters, which is now possible. (Actually, we also assume 
which leaves us with only I+J-1 parameters to estimate.)
A slightly more general additive model is
where 
are the number of replications at each combination of factor A
and factor B levels.
NOTE:\ When k is small,
especially when k=1, we are forced to use the additive model.
There will be more about this in Section 10.2.2.
The ANOVA table for the additive model is given by
The relevant null hypotheses are
and are tested by 
and 
, respectively. In words, these hypotheses are
EXAMPLE:\ In a study of
automobile traffic and air pollution, air samples taken at four
different times and at five different locations were analyzed to obtain
the amount of particulate matter present in the air. Is there any
difference in true average amount of particulate matter present in the
air due either to different sampling times or to different locations?
Notice that in this case, both
and
are significantly greater than one. Thus, there is an effect due both to
time and location.
When the additive model holds, there is no interaction between
factors A and B. In other words, the effect of factor A
is the same no matter what the level of factor B is. When the
additive model doesn't hold, we have to go to a model which allows A
and B to interact.
 | Two-way
ANOVA with Interaction
|
We will use the model
where 
,  ,
and
, but represent it in the form
where 
is the interaction of factors A and B.
The relevant null hypotheses are
and are tested by their respective F values in the following
ANOVA table.
|
|
and 
. The test is very simple and consists of combining the two samples into
one sample of size 
, sorting the result, assigning ranks to the sorted values (giving the
average rank to any `tied' observations), and then letting T be
the sum of the ranks for the observations in the first sample. If the
two populations have the same distribution then the sum of the ranks of
the first sample and those in the second sample should be close to the
same value. Stataquest returns a p value for the null hypothesis
that the two distributions are the same.
, we combine all the samples into one large sample, sort the result from
smallest to largest and assign ranks (again assigning the average rank
to any observation in a group of tied observations), and then find 
, the average of the ranks of the observations in the ith sample.
The test statistic is then
. Again, Stataquest will return the p value for the test.
ANOVA 
