# Adapting two-sample $t$-test of a ratio for log transformation

I have some data, belonging to paired groups $$A,B$$. From each group I get a non-negative statistic $$d^A,d^B$$ which is averaged on all samples in group. My interest is in the ratio $$\frac{d^A}{d^B}$$, i.e. how far are these statistics from each other (straight subtraction makes no sense). My equivalence hypotheses (inspired by the 80/125 rule of bioequivalence) are: $$H_0:\frac{E[d^A]}{E[d^B]}>\epsilon\text{ or }\frac{E[d^A]}{E[d^B]}<\epsilon^{-1},\quad H_1:\epsilon^{-1}\le \frac{E[d^A]}{E[d^B]}\le\epsilon$$

The null hypothesis can be rewritten as $$H_0:E[d^A]-\epsilon E[d^B]>0 \text{ or }E[d^A]-\epsilon^{-1} E[d^B]<0$$ which makes things a bit easier: While we don't know the actual (or theoretical) distribution of the ratio, we can relate to the term $$\bar{d}_{U}=\frac{1}{N}(\sum_i d^A_i-\epsilon\sum_i d^B_i)$$ as the sample mean, calculate the sample variance $$s^2_{U}=Var(d^A_i-\epsilon d^B_i)$$ and then the statistic $$t_U=\frac{\bar{d}_{U}-0}{\sqrt{s^2_{U}}}$$ has $$t$$ distribution and we can test whether or not $$t_U \le t_{\alpha,N-1}$$.

In the same manner we define $$\bar{d}_{L}=\frac{1}{N}(\sum_i d^A_i-\epsilon^{-1}\sum_i d^B_i),s^2_{L}=Var(d^A_i-\epsilon^{-1} d^B_i)$$ and $$t_L=\frac{\bar{d}_{L}-0}{\sqrt{s^2_{L}}}$$, in order to test whether or not $$t_L\ge t_{1-\alpha,N-1}$$. Finally, we reject $$H_0$$ if $$t_U \le t_{\alpha,N-1}~\text{ and }~t_L\ge t_{1-\alpha,N-1}$$.

All of the above is nice, but now we want to make it a one-sided test. That is, our updated hypotheses are

$$H_0:\left|\log\left(\frac{E[d^A]}{E[d^B]}\right)\right|>\epsilon,\quad H_1:\left|\log\left(\frac{E[d^A]}{E[d^B]}\right)\right|\le\epsilon$$

We can obviously write $$|\log(E[d^A])-\log(E[d^B])|-\log(\epsilon)>0$$ but I'm not really sure how does this help me in formulating a test, what are the sample statistics I get here or anything else. As the sum and log operations are not commutative, I'm not really sure how to progress here of is it a smart move at all. It is not at all like the case of $$t$$-testing a log-transformed data (unlike here and here), because I only transform my statistic (and not the whole data).

Would love to hear ideas and thoughts.