This is a follow-up to the question https://stats.stackexchange.com/questions/533160/how-to-approach-time-data-that-arent-time-series. 

I realized that my prior question was missing an important nuance, so I have to change the hypothetical scenario entirely. 

Imagine the following scenario: suppose I, as a researcher, **hypothesize that teachers who have more experience teaching tend to give their students more As. As with the prior question, significance tests and effect sizes are desired.** Because teachers have to receive certification to teach through obtaining an academic degree, I suspect that there's an association between how long ago a teacher graduated from their certification program, and the rate at which teachers issue As to their students out of all students taught so far. 

You may assume that I have available the following in a data set: points $\{(u_i, t_i, \mathbf{y}_i)\}_{i=1}^{N}$ where

* $u_i$ is the unique identifier for a teacher
* $t_i$ is the year at which the teacher graduated from their certification program
* $\mathbf{y}_i = (y_{1i}, y_{2i})$ is a vector consisting of two components for each teacher: the count of As ($y_{1i}$), and the count of grades that are not As ($y_{2i}$) since the teacher graduated from their certification program.

Like with the prior question, I don't think this is a time series problem, but upon re-reading the prior answer, I don't think this is a panel data or longitudinal data problem either. We run, however, into similar problems to the ones I pointed out in the prior question:

* We don't have an equal number of teachers in each year for which teachers graduate. 
* If my sample is insufficient so that some calendar years were skipped, I will have data for some calendar years but not for others, breaking the usual "equal-spacing" assumption with time series.
* The data are probably not stationary if we were to consider using the graduation year as an index for time.
* We could attempt to bin graduation years together, but such a mechanism has no precedent (i.e., assume no similar study has been done), and would be quite arbitrary to implement. 
* Given $t_i$, we also suspect that $\sum_{\{i : u_i \text{ graduated in year }t_i\}}\dfrac{y_{1i}}{y_{1i} + y_{2i}}$ (the rate at which As were given) would be similar for values of $t_i$ that are close to each other, so any binning of graduation years we do would ignore this to an extent that I would not be comfortable with without adequate justification.

Additionally, there's an aspect of this that should be controlled for: teachers who have graduated at an earlier year naturally have higher counts in $\mathbf{y}_i$ than those who graduated at a later year.

I suspect this is more complex than viewing this as a time series or longitudinal data problem. What technique(s) would you suggest for approaching this problem?