ANOVA-like analysis for geometric distribution

I have a dataset (>10,000 samples total) that strongly appears to be geometrically distributed. For this dataset I have a way of partitioning that makes sense theoretically and I would like to know if my theory is backed by said dataset, i.e., the same question usually answered using ANOVA analysis in the continuous, normally distributed case. My question is: how is this done with geometrically distributed data?

My approach would be to fit a geometric distribution (using least squares) to the whole dataset, and then to each partition of the partitioned dataset. I would then compute the ratio of the sum of errors in the partitioned case to the squared error for the total distribution, i.e., $$\frac{\sum_{part}\textrm{RMSE}_{part}^2}{\textrm{RMSE}_{total}^2}$$, to get a test score, similar to an ANOVA.

This is where I am stuck, as I don't know which distribution to compare it to. Can somebody give me an ELI5 explanation of why it is stupid and wrong to solve my problem in this way, and then tell me how to actually do it?