I an experimental design were I wish to test if the effect of a treatment on several subjects type is different to the effect of the treatment on a control subject type.

(Background: I am a biologist, and have several different cell types. I wish to know for each, if the treatment has a larger effect than the control type).

I have four means, and four sample variances for the 2x2 design. (test subject type vs control subject type vs treatment vs no treatment.

I'm guessing that the variance are not equal because there is a strong correlation between mean and variance for each of the samples.

The two obvious ways to test this would be to test the difference between the two pairs of treatment vs no treatment i.e.

H0: m1 - m2 = m3-m4.

or by using an ANNOVA and looking at the interaction terms (although both of these assume equal variation right?).

However, these methods both compare the absolute difference between treatment and no treatment. I think I want to compare the ratio. I.e:

H0: m1/m2 = m3/m4

You could do this by taking the log of the data and doing the linear comparison, but then the variables in the comparison would no longer be normally distributed.

I read that the ratio of two random variables is a cauchy distribution (although the I gather that the unequal variance might be a problem), but I have no idea how to use this information to construct a test.

Anyone have any ideas on how I might construct a test for this data?

Cheers,

Ian

---