I am running a discontinuous regression to see the effect of a cash transfer on an outcome using a poverty index as the running variable. The problem is that the score is not a very good predictor of the treatment, with a compliance of about 40%. Moreover, the literature suggests that I should look for a minimum detectable effect size of about 0.1 standard deviations. Therefore, when doing the power calculations, I get that, although my sample is not small (~ 16 000 obs.), the power is still insufficient.
Reading the paper of Cattaneo et al. (2024; A practical introduction to regression discontinuity designs: Extensions), I notice that, in cases where there are not many observations, one can assume a random assignment just above and below the cutoff, and use Fisherian inference (which assumes a non stochastic potential outcome and uses a sharp null hypothesis) to still get robust results (although you have to give up point estimation).
Can the same logic be applied when you the results are underpowered not because of a small sample size, but due to imperfect compliance?
That is, if I get an non significant result with the common regression discontinuity approach and then I also get a non significant result when running the same regression but with Fisherian inference, is this last result reassuring? Can I be fairly confident that the non significance is not due to chance, but due to the absence of an effect?