Contrasts with interactions

Question

I have a couple of questions about contrasts but would also appreciate any suggestions for relevant literature.

I am carrying out a GLM (binomial). There are 3 Factors (A,B,C) - A has 3 levels, B & C have 2. As main effects all factors are significant, and for factor A I can test the significance between the levels with orthogonal contrasts (-1, 0.5, 0.5)(0, -1, 1).

There is also a significant interaction between A & B and this is where I am getting stuck/confused.The interaction between A & B produces 6 treatments (A1*B1, A1*B2, A2*B1, A2*B2, A3*B1, A3*B2) Do I need to construct a matrix with 6 levels? For example...

(-1, 0.2, 0.2, 0.2, 0.2, 0.2) ( 0, -1, 0.25, 0.25, 0.25 0.25) ( 0, 0, -1, 0.33, 0.33, 0.33) ( 0, 0, 0, -1, 0.5, 0.5) ( 0, 0, 0, 0, -1, 1)

Second question - Say I am only really interested in one contrast, for example between A1*B1 and A1*B2. Is there anything stopping me doing this? I get the impression you need to create the full matrix but I have not read anywhere that it is a must? I'm using jmp and it allows me to run a single contrast I'm just not really sure if it is ok statistically?

Answer 1

Here is what you are describing:

Y = sample(0:1,400,replace=T)
A = sample(factor(c("A1","A2","A3")),400,replace=T)
B = sample(factor(c("B1","B2")),400,replace=T)
C = sample(factor(c("C1","C2")),400,replace=T)
mod = glm(Y ~ A + B + C + A:B, family=binomial)
summary(mod)

which gives:

glm(formula = Y ~ A + B + C + A:B, family = binomial)

Deviance Residuals: 
   Min      1Q  Median      3Q     Max  
-1.362  -1.152   1.003   1.193   1.280  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)
(Intercept)  0.01955    0.27106   0.072    0.943
AA2         -0.05528    0.33304  -0.166    0.868
AA3         -0.16445    0.33920  -0.485    0.628
BB2         -0.15186    0.34719  -0.437    0.662
CC2          0.07330    0.20193   0.363    0.717
AA2:BB2     -0.04942    0.49982  -0.099    0.921
AA3:BB2      0.64755    0.49179   1.317    0.188

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 554.51  on 399  degrees of freedom
Residual deviance: 551.15  on 393  degrees of freedom
AIC: 565.15

With all of the factors appropriately irrelevant (random), if some were relevant you would see them flagged ('.','*','**','***'). You can see all the factors of A, B and C. The base factors are not there because if A1=1 then A2=0 and A3=0. This link will show you how to change the contrasts in R and give you an interpretation of the summary with factors and contrasts.