Two-Way Interaction

Let’s start with the contrast of the two-levels of ‘proc’ (i.e., absolute and ambiguous) and code it as p1 = -.5 and p2 = .5. Since we are not contrasting the levels of ‘order’, we will code both levels game1 and survey1 as of equal weight, namely, o1 = .5 and o2 = .5. In this model we are exploring the contrast of absolute and ambiguous proclamation when taking into account the interaction between averaged levels of game1 and survey1. Any time that the higher-order term includes the contrast, for example, the interaction order * proc** includes the contrast for proc, we need to include the contrast coding of the main effect, namely, the difference between game1 and survey1 in the variable proc. Recall that the order of the factors is determined in the mixed command in the SPSS syntax. Since order comes before proc, our contrast coding for the interaction would be the following:

o1p1 = 1/2 * -1/2 = -1/4 o1p2 = 1/2 * 1/2 = 1/4 o2p1 = 1/2 * -1/2 = -1/4 p2o2 = 1/2 * 1/2 = 1/4.

A two-way table for the interaction contrast coding would look like the following:

proc level 1 = -.5 proc level 2 = .5
order level 1 = .5 1/2 * -1/2 = -1/4 1/2 * 1/2 = 1/4
order level 2 = .5 1/2 * -1/2 = -1/4 1/2 * 1/2 = 1/4

Here is the SPSS syntax for the model:

mixed score by order proc time
  /PRINT LMATRIX
 /fixed = time order proc order * proc
 /repeated = time | subject(id) covtype(ar1)
 /test = 'proc 1v2' proc -.5 .5 proc * order -1/4 1/4 -1/4 1/4.

Here are the results for significant effects,

Below we see that the contrast of survey1 - game1 is significant when interacting with averaged levels of order:

Three-Way Model including interaction of proc * time and the three-way interaction

In this example we will examine the same contrast above, namely, a contrast between the levels of proc, but use a model that includes the interaction of proc and time as well as the three-way interaction of order*proc*time. Since proc comes before time, the interaction between proc*time would be p1t1 p1t2 p2t1 p2t2. Since time in not contrast coded, there will be assigned equal weights for the two levels of time, namely, 1/2 and 1/2. The coding scheme for the interaction of proc with time is the following:

p1t1 = -1/2 * 1/2 = -1/4 p1t2 = 1/2 * 1/2 = -1/4 p2t1 = 1/2 * 1/2 = 1/4 p2t2 = 1/2 * 1/2 = 1/4.

A two-way table for the interaction contrast coding would look like the following:

time level 1 = .5 time level 2 = .5
proc level 1 = -.5 -1/2 * 1/2 = -1/4 -1/2 * 1/2 = -1/4
proc level 2 = .5 1/2 * 1/2 = 1/4 1/2 * 1/2 = 1/4

The order*proc*time interaction is coded as o1p1t1 o1p1t2 o1p2t1 o1p2t2 o2p1t1 o2p1t2 o2p2t1 o2p2t2. Since the contrasts is only between levels of proc and not order or time, the levels within order and time are weighted equally. The levels within order and time will be “weighted” equally at 1/2. The contrast coding for the interaction above is therefore o1p1t1 = 1/2 * -1/2 * 1/2 = -1/8 o1p1t2 = 1/2 * -1/2 * 1/2 = -1/8 o1p2t1 = 1/2 * 1/2 * 1/2 = 1/8 o1p2t2 = 1/2 * 1/2 * 1/2 = 1/8 o2p1t1 = 1/2 * -1/2 * 1/2 = -1/8 o2p1t2= 1/2 * -1/2 * 1/2 = -1/8 o2p2t1 = 1/2 * 1/2 * 1/2 = 1/8 o2p2t2 = 1/2 * 1/2 * 1/2 = 1/8. We can visualize this in two-way tables split by levels of order.

For order level 1 = 1/2
time level 1 = .5 time level 2 = .5
proc level 1 = -.5 1/2 * -1/2 * 1/2 = -1/8 1/2 * -1/2 * 1/2 = -1/8
proc level 2 = .5 1/2 * 1/2 * 1/2 = 1/8 1/2 * 1/2 * 1/2 = 1/8
For order level 2 = 1/2
time level 1 = .5 time level 2 = .5
proc level 1 = -.5 1/2 * -1/2 * 1/2 = -1/8 1/2 * -1/2 * 1/2 = -1/8
proc level 2 = .5 1/2 * 1/2 * 1/2 = 1/8 1/2 * 1/2 * 1/2 = 1/8 
mixed score by order proc time
  /fixed =  time order proc order*proc order*proc*time 
  /repeated = time | subject(id) covtype(ar1)
  /test = 'prov 1v2' proc -.5 .5 
   order*proc -1/4 1/4 -1/4 1/4 
   order*proc*time -1/8 -1/8 1/8 1/8 -1/8 -1/8 1/8 1/8.

Here is the main and interaction effects:

The contrast for the three-way model is the same with the contrast model with the two-way interaction.

##Interaction Contrasts Perhaps you want to consider the contrast between absolute and ambiguous proclamation when survey1 is presented first. This would require the two-way contrast. In the previous example, we considered the contrast between the two levels of proc averaged over the two levels of order. Here we will need to contrast code the levels for both proc and order. Below are the graphs of absolute versus ambiguous within each level of order. As you can see, the graphs are very different:

The order * proc interaction is coded as o1p1 o1p2 o2p1 o2p2. The contrast coding for order at level 1 =1 and at level 2 = -1 (recall, we are not averaging but contrasting). The contrast coding for proc is level 1 =1 and level 2 = -1. Coding for the interaction would look like this: o1p1 = 1 * 1 = 1 o1p2 = 1 * -1 = -1 o2p1 = -1 * 1 = -1 o2p2 = -1 * -1 = 1

Here is the SPSS syntax for the interaction contrast for order * proc:

DATASET ACTIVATE DataSet1.
  mixed score by order proc time 
  /fixed =   order proc time order*proc 
  /repeated = time | subject(id) covtype(ar1)
  /test = 'prov 1v2 & order1v2' order*proc  1 -1 -1 1.

Here are the fixed effects specified in the syntax file (and in the model):

Here is the custom interaction contrast:

This could be interpreted as the difference between absolute and ambiguous proclamation is significantly different when game1 is presented first versus when survey1 is presented first. The next model will perform the same contrast as above, but while including all the two-way and three-way interactions. We have already coded the order * proc interaction. Now we have to code the three-way interaction as we had done previously: o1p1t1 o1p1t2 o1p2t1 o1p2t2 o2p1t1 o2p1t2 o2p2t1 o2p2t2. The contrast coding for order is the same as above, namely, order at level 1 =1 and at level 2 = -1, as well as the contrast coding for proc, that is, proc at level 1 =1 and at level 2 = -1.Like previously, we will weight all the levels of time as the same (i.e., 1/2): o1p1t1 = 1 * 1 * 1/2 = 1/2; o1p1t2 = 1 * 1 * 1/2 = 1/2; o1p2t1 = 1 * -1 * 1/2 = -1/2; o1p2t2 = 1 * -1 * 1/2 = -1/2; o2p1t1 = -1 * 1 * 1/2 = -1/2; o2p1t2= -1 * 1 * 1/2 = -1/2; o2p2t1 = -1 * -1 * 1/2 = 1/2; o2p2t2 = -1 * -1 * 1/2 = 1/2.

mixed score by order proc time
  /fixed = time order proc time*proc order*proc time*order order*proc*time
  /repeated = time | subject(id) covtype(cs)
  /test = 'proc 1v2, order 1v2' order*proc 1 -1 1 -1 1
    order*proc*time 1/2 1/2 -1/2 -1/2 -1/2 -1/2 1/2 1/2.

Note how we include the ‘lower-order’ interaction contrast of interest, namely ‘prov 1v2, order 1v2’, with the three-way interaction, since the interaction contrast is contained within the three-way contrast. In this model we will consider all the two-way and three-way interaction effects:

When the interaction contrast is taken over the average effect of time, the interaction contrast is identical to the results when we did not include the three-way interaction.