# Comparing effects of different methods when each method has multiple levels

I'm working in retail and we are trying to determine the effect (on units sold) of reducing an expense for a set of products within a product group.

We have two methods of reaching this goal. One is to reduce the number of products bearing this expense, the other is to reduce the amount of expense on each product. We do not expect to use both methods at the same time.

We are also looking to do each method at different levels, an overall control group, a 50% reduction, and a 100% reduction.

So we have:

Level        Method1 Method2
Control        0%      0%
1             50%     50%
2            100%    100%


We also have a number of covariates we are trying to control for, but thankfully few have interactons with the treatment. (The overall test would be conducted all at once, with one set of control stores used to compare against both treatments).

Importantly we want to know if M1,50% is different than M2,50% and M1,100% is different than M2,100% as well as M1,50% vs M2,100%.

I've thought about using some dummy variables and creating one model:

 y = Intercept + Method + Treatment + Method * Treatment + Covariates


However, I'm unsure how to interpret the results at the mid-level area between treatments. Especially, whether the difference in coefficients are statistically different from each other.

At this point I'm wanting to create two identical models for each method

 y(m1)= Intercept + Treatment(m1) + Covariates
y(m2)= Intercept + Treatment(m2) + Covariates


The test will contain the same number of stores at each level, and the covariates would not change. Comparing the coefficients using the formula:

(b1-b1)/sqrt(seb1^2 + seb2^2)


I'm not positive which is the better route to go, and which are the pitfalls with each one.

Any help or guidance would be greatly appreciated.

$Q=\beta_0+\beta_1 M1_{50} + \beta_2 M1_{100} + \beta_3 M2_{50} + \beta_4 M2_{100} + \Sigma \gamma X$