It can be rather tricky to program the test subcommand when
there are higher order interactions (e.g., three-way interactions, four-way
interactions, etc.) included in the mixed model.  Let’s look at an example where we are using
the mixed command in a repeated measures model.  The data set
exercise was used in our
seminar on repeated measures.  The
data set consists of people who were
randomly assigned to two different diets: low-fat and not low-fat, as well as three different types of exercise: at rest, walking leisurely and running.
Their pulse rate was measured at three different time points during their
assigned exercise: at 1 minute, 15 minutes and 30 minutes. We included all three
variables in our mixed model:  diet which has two levels,
exertype which has three levels
and time which also has three levels.  Even though
time is a repeated factor we can treat it in the same manner as the other variables when we want to test the various
contrasts and interaction contrasts that may be of interest.  Finally, we
will present an example of how to program the test subcommand for
interaction contrasts involving a three-way interaction.

Contrasts for one variable

mixed pulse by diet exertype time 
  /fixed =  time diet exertype
  /repeated = time | subject(id) covtype(arh1)
  /test = 'exertype 12v3' exertype -.5 -.5 1. 

Let’s look at an example where the mixed model includes only the main effects
of diet, exertype and time.  From the graph we see that there might be a difference between exertype
level 3 and the two other levels of exertype.  Therefore, we
will use a reverse Helmert coding for exertype in
the test subcommand in order to test
this particular contrast.  For more information on reverse Helmert coding and other contrast coding
systems please refer to chapter 5 in our
webbook on
regression.

The contrast coding:

mixed pulse by diet exertype time 
  /fixed =  time diet exertype 
  /repeated = time | subject(id) covtype(ar1)
  /test = 'exertype 12v3' exertype -.5 -.5 1. 

The two-way model including the interaction of exertype and diet

Let us look at testing the same contrast, but this time we will use a different model which will
include an interaction between exertype and diet.  In order to implement the test of
the contrast, we have to introduce the idea
of distributing the contrast coding of lower order terms on the higher order terms.  In other
words, when we have higher order terms which include the term being
tested, such as the two-way interaction of exertype and diet, then we need to
include these higher order terms in the test subcommand with the appropriate contrast coding
as well as the contrast coding for the main effect being tested.
The diet*exertype interaction is coded as d1e1 d1e2 d1e3 d2e1 d2e2 d2e3, where d# = coding
for diet level # and e# = coding for exertype level # and d#e# is
the product of the two.  Since we are not contrast coding diet, we just want the
“weight” for each level of diet to be equal; hence, we assign both levels
of diet the same “weight” of 1/2
because there are two levels of diet.  The order of the factors is determined by the order in which
they appear in the mixed command.  In this particular case, diet
appears before exertype in the mixed command, and thus with our coding
system for diet and exertype we would have the following coding
for the interaction:

d1e1 = 1/2*-1/2 = -1/4
d1e2 = 1/2*-1/2 = -1/4
d1e3 = 1/2*1 =
1/2
d2e1 = 1/2*-1/2 = -1/4
d2e2 = 1/2*-1/2 = -1/4
d2e3 = 1/2*1 =
1/2

In order to more easily understand the coding for the interaction, it might help to
visualize it as a matrix which equals the product of the code for diet
as a column matrix
and the contrast coding of exertype as a row matrix.

Now the test subcommand will include both exertype with its
contrast coding and the interaction of exertype*diet with its contrast coding.  Note that the fixed subcommand reflects the change in the model to
include the interaction of diet and exertype.

mixed pulse by diet exertype time
  /fixed =  time diet exertype diet*exertype 
  /repeated = time | subject(id) covtype(ar1)
  /test = 'exertype 12v3' exertype -.5 -.5 1 diet*exertype -1/4 -1/4 1/2 -1/4 -1/4 1/2. 

 

The three-way model including the interaction of exertype*time
and the three-way interaction

Let’s consider testing the same contrast for exertype but using a model
that includes the interaction of exertype and time as well as the
three-way interaction of diet*exertype*time.  In this case we will
need to include three terms in our test subcommand:  the main effect of
exertype, the interaction diet*exertype and the three-way interaction
diet*exertype*time, each with their appropriate contrast coding.  The
coding of exertype will be the same as in the one-way example.

The exertype*time interaction is coded as e1t1 e1t2 e1t3 e2t1 e2t2 e2t3
e3t1 e3t2 e3t3, where e# = coding
for exertype level # and t# = coding for time level # and e#t# is the product of
the two.  Since we are not contrast coding time, we just want the
“weight” for each level of time to be equal; hence, we assign
all three levels of time the same “weight” of 1/3.  The order of the factors
is determined by the order in which
they appear in the mixed command.  In this particular case
exertype
appears before time in the mixed command, and thus with our coding
system for exertype and time we would have the following coding
for the interaction:

e1t1 = 1/3*-1/2 = -1/6
e1t2 = 1/3*-1/2 = -1/6
e1t3 = 1/3*-1/2 =
-1/6
e2t1 = 1/3*-1/2 = -1/6
e2t2 = 1/3*-1/2 = -1/6
e2t3 = 1/3*-1/2 =
-1/6
e3t1 = 1/3*1 = 1/3
e3t2 = 1/3*1 = 1/3
e3t3 = 1/3*1 =
1/3

In order to more easily understand the coding for the interaction, it might help to
visualize it as a matrix which equals the product of the contrast code for exertype
as a column matrix and the coding of time as a row matrix.

The contrast coding for the three-way interaction is slightly more
complicated.  The diet*exertype*time interaction is coded as d1e1t1
d1e1t2 d1e1t3 d1e2t1 d1e2t2 d1e2t3 d1e3t1 d1e3t2 d1e3t3 d2e1t1 d2e1t2 d2e1t3
d2e2t1 d2e2t2 d2e2t3 d2e3t1 d2e3t2 d2e3t3.  For this code d# = coding for
diet level #, e# = coding for exertype
level #, t# = coding for time level # and d#e#t# is the product of the
three. Since we are only contrasting levels of exertype and not levels of
either diet or time, we want to code each level of diet
to be “weighted” equally, so they are both coded as 1/2; likewise, we want to code each level of
time to be “weighted” equally, so all three levels of time
are coded as 1/3.  In this case the coding for the interaction is:

d1e1t1 = 1/2*-1/2*1/3 = -1/12
d1e1t2 = 1/2*-1/2*1/3 = -1/12

d1e1t3 = 1/2*-1/2*1/3 = -1/12
d1e2t1 = 1/2*-1/2*1/3 = -1/12

d1e2t2 = 1/2*-1/2*1/3 = -1/12
d1e2t3 = 1/2*-1/2*1/3 = -1/12

d1e3t1 = 1/2*1*1/3 = 1/6
d1e3t2 = 1/2*1*1/3 = 1/6
d1e3t3 = 1/2*1*1/3 =
1/6

d2e1t1 = 1/2*-1/2*1/3 = -1/12
d2e1t2 = 1/2*-1/2*1/3 =
-1/12
d2e1t3 = 1/2*-1/2*1/3 = -1/12
d2e2t1 = 1/2*-1/2*1/3 =
-1/12
d2e2t2 = 1/2*-1/2*1/3 = -1/12
d2e2t3 = 1/2*-1/2*1/3 =
-1/12
d2e3t1 = 1/2*1*1/3 = 1/6
d2e3t2 = 1/2*1*1/3 = 1/6

d2e3t3 = 1/2*1*1/3 = 1/6

This can be more conveniently visualized as matrices.  The coding
for time, which is the last factor in the mixed command, can be thought of as
the row matrix that is multiplied by the coding for exertype which is the second
to last factor in the mixed command and which can be thought of as the column
matrix.  The matrix which is the product is then multiplied by the coding
for each level of diet which appears before exertype and time in the
mixed command.

For diet level 1 = 1/2:

For diet level 2 = 1/2:

mixed pulse by diet exertype time
  /fixed =  time diet exertype diet*exertype diet*exertype*time 
  /repeated = time | subject(id) covtype(ar1)
  /test = 'exertype 12v3' exertype -.5 -.5 1 
   diet*exertype -1/4 -1/4 1/2 -1/4 -1/4 1/2 
   diet*exertype*time -1/12 -1/12 -1/12 -1/12 -1/12 -1/12 1/6 1/6 1/6
                      -1/12 -1/12 -1/12 -1/12 -1/12 -1/12 1/6 1/6 1/6.  

Interaction contrasts

The model including only two-way interactions

Let’s consider an interaction contrast involving diet and exertype
in a mixed model which includes diet, exertype, time and
all the two-way interactions.  From the graphs we see that exertype
level 3 is higher than both levels 1 and 2 for both diets but that level 3
increases at different rates depending on which diet is considered.  We
would like to contrast exertype level 3 in diet 2 with the other
groups in order to see if they are really different from the other groups.  In
order to accomplish, this we will need to use an interaction contrast where
exertype is coded with a reverse Helmert coding contrasting level three
versus levels 1 and 2 and diet has a contrasts coding
contrasting level1 versus level 2.  For more information on reverse Helmert coding and other contrast coding
systems please refer to chapter 5 in our
webbook on
regression.

The contrast coding:

The diet*exertype interaction is coded as d1e1 d1e2 d1e3 d2e1 d2e2 d2e3, where d# = coding
for diet level # and e# = coding for exertype level # and d#e# is the product of
the two.  The contrast coding for diet is level 1 = 1
and level 2 = -1 and the contrast coding for exertype is the same as in
the previous example, namely, level 1 = -1/2, level 2 = -1/2 and level 3 = 1.  The order of the factors
is determined by the order in which they appear in the mixed command.  In this particular case diet
appears before exertype in the mixed command, and thus with our coding
system for diet and exertype, we would have the following coding for the interaction:

d1e1 = 1*-1/2 = -1/2
d1e2 = 1*-1/2 = -1/2
d1e3 = 1*1 =
1
d2e1 = -1*-1/2 = 1/2
d2e2 = -1*-1/2 = 1/2
d2e3 = -1*1 =
-1

In order to more easily understand the coding for the interaction, it might help to
visualize it as a matrix which equals the product of the code for diet
as a column matrix and the contrast coding of exertype as a row matrix.

We don’t need to include any other terms besides the interaction diet*exertype since the model does not
include any higher order terms that contain diet*exertype.  If a term, such as the interaction
exertype*time,
contains only part of the interaction being tested, then it is not included in the
test subcommand.  Only those
higher order terms, which contain the complete term diet*exertype being tested
are included.

mixed pulse by diet exertype time 
  /fixed =   diet exertype time diet*exertype 
  /repeated = time | subject(id) covtype(ar1)
  /test = 'exertype 12v3 & diet 1v2' diet*exertype  -.5 -.5 1 .5 .5 -1. 

 

Let’s test the same interaction contrast but now we will consider a model that includes all
two-way interactions and the three-way interaction.  Just as in the
previous example we now have to “distribute” the contrast coding of the lower
order term across the higher order terms.  In other words, the test subcommand
will include the contrast coding for the interaction of diet*exertype
that we developed in the above example.  We also have to include the
three-way interaction with the appropriate contrast coding because the three-way
interaction includes the term, namely diet*exertype, being
tested.

The contrast coding for the three-way interaction is slightly more
complicated.  The diet*exertype*time interaction is coded as d1e1t1 d1e1t2
d1e1t3 d1e2t1 d1e2t2 d1e2t3 d1e3t1 d1e3t2 d1e3t3 d2e1t1 d2e1t2 d2e1t3 d2e2t1
d2e2t2 d2e2t3 d2e3t1 d2e3t2 d2e3t3.  In this code, d# = coding for diet level #, e# = coding for exertype
level #, t# = coding for time level # and d#e#t# is the product of the
three.  The contrast coding for exertype is level 1 = -1/2, level 2 = -1/2 and level 3 = 1; the
contrast coding for diet is level 1 = 1 and level 2 = -1.  Since we are not implementing a contrast
for time we want to code each level of time to be “weighted” equally, so all
three levels are coded as 1/3.  Note that the coding is very similar to the
second example in the contrast section of this FAQ, but that instead of coding
diet as 1/2 for each level, we are now giving diet a contrast coding
because we are testing in interaction contrast.  In this case the coding for the interaction is:

d1e1t1 = 1*-1/2*1/3 = -1/6
d1e1t2 = 1*-1/2*1/3 = -1/6
d1e1t3 = 1*-1/2*1/3 =
-1/6
d1e2t1 = 1*-1/2*1/3 = -1/6
d1e2t2 = 1*-1/2*1/3 = -1/6
d1e2t3 = 1*-1/2*1/3 =
-1/6
d1e3t1 = 1*1*1/3 = 1/3
d1e3t2 = 1*1*1/3 = 1/3
d1e3t3 = 1*1*1/3 =
1/3

d2e1t1 = -1*-1/2*1/3 = 1/6
d2e1t2 = -1*-1/2*1/3 =
1/6
d2e1t3 = -1*-1/2*1/3 = 1/6
d2e2t1 = -1*-1/2*1/3 = 1/6
d2e2t2 = -1*-1/2*1/3 =
1/6
d2e2t3 = -1*-1/2*1/3 = 1/6
d2e3t1 = -1*1*1/3 = –1/3
d2e3t2 = -1*1*1/3 = –1/3
d2e3t3 = -1*1*1/3 = –1/3

This can be more conveniently visualized as matrices.  The coding
for time which is the last factor in the mixed command can be thought of as
the row matrix that is multiplied by the coding for exertype which is the second
to last factor in the mixed command and which can be thought of as the column
matrix.  The matrix which is the product is then multiplied by the coding
for each level of diet which appears before exertype and time in the
mixed command.

For diet level 1 = 1:

For diet level 2 = -1:

mixed pulse by diet exertype time
  /fixed = time diet exertype time*exertype diet*exertype time*diet diet*exertype*time
  /repeated = time | subject(id) covtype(cs)
  /test = 'exertype 12v3, diet 1v2' diet*exertype -.5 -.5 1 .5 .5 -1
    diet*exertype*time -1/6 -1/6 -1/6 -1/6 -1/6 -1/6  1/3  1/3  1/3 
                        1/6  1/6  1/6  1/6  1/6  1/6 -1/3 -1/3 -1/3.