st: Modeling Interactions and Interpretation using ONLY factorial interactions (and having imputed data)

[Andrea Bennett <mac.stata@gmail.com>]

Hi!

I'd like to know whether the following way of implementing and interpreting an interaction between say -female- (0 "Male" 1 "Female) 
and -treatment- ( 1 "Control" 2 "Treatment A" 3 "Treatment B") is correct.

I've learned to model an interaction as: reg y i.female##i.treatment. However, is the following also correct?
reg y i.female#i.treatment (using ONLY factorial interactions). The way I understand it, this generates 6 dummies for each
combination of paired interactions whereas the dummy for male & control group is the base and hence omitted from the output. I therefore
would prefer this approach over the more frequently applied.

Output:
female#treatment
0 2 | 3.85 (p=0.000)
0 3 | 2.87 (p=0.002)
1 1 | 1.97 (p=0.001)
1 2 | 4.39 (p=0.000)
1 3 | 5.622783 (p=0.000)

If I want to test whether females perform significantly better in treatment B compared to treatment A, 
I would have to run: mi estimate (diff: _b[1.female#2.treatment]-_b[1.female#3.treatment]): reg  y i.female#i.treatment + controls

If I wanted to test whether females perform significantly better compared to males in treatment A,
I would have to run: mi estimate (diff: _b[0.female#2.treatment]-_b[1.female#2.treatment]): reg  y i.female#i.treatment + controls


When I apply the standard procedure reg y i.female##i.treatment + controls the output is as follows:

treatment
2 | 3.85 (p=0.000)
3 | 2.87 (p=0.002)

female
1 | 1.97 (p=0.001)

female#treatment
1 2 | -1.43 (p=0.127)
1 3 |.77 (p=0.403)

The BIG question:
What (always) confuses me is that the pure interaction term of 1 3 is highly insignificant. But we cannot conclude from this that the interaction between female and treatment B is indeed insignificant because we have to take into account 1) the treatment effect of 3, 2) the gender effect, 
and 3) the interaction term. Only if these three are jointly insignificant I cannot reject the hypothesis that females do not change behavior in
treatment B.

Doing the first test of the above procedure would then be:
mi estimate (diff: (_b[1.female]+_b[2.treatment]+_b[1.female#2.treatment])-(_b[1.female]+_b[3.treatment]+_b[1.female#3.treatment]))

Is this correct, and if not, how can I make sure that I do it right. Any good article or book reference is highly welcomed, too.

Best regards and many thanks in advance,

Andrea




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