Measuring the treatment effect of a binary variable on a binary outcome in R I have a data set with 10000 entries of projects that take part in an auction for financial support. In that auction all of the bids below a certain cutpoint receive the support.
The data includes the bids, the distance of the bid to the cutpoint, the received support (0,1) and if the project was realized (0,1) in the end.
This is what my data looks like
The links have been automatically created by StackExchange since i have not yet earned the privilege of directly showing pictures in the post.
                     | quantities      | 
| ------------------ | --------------- |
| support = 0        | 5036            |
| ------------------ | --------------- |
| Support = 1        | 4964            |
| ------------------ | --------------- |
| realization = 0    | 5513            |
| ------------------ | --------------- |
| realization = 1    | 4487            |
| ------------------ | --------------- |

                     | support = 0     | support = 1   | 
| ------------------ | --------------- | ------------- |
| realization = 0    | 4035            | 1478          |
| ------------------ | --------------- | ------------- |
| realization = 1    | 1001            | 3486          |
| ------------------ | --------------- | ------------- |

This is the visualization of the discontinuity of the outcome at the cutpoint
I now want to measure the treatment-effect at the cutpoint with a regression discontinuity approach. I tried to do that with a logistic regression.
model_bandwith1 <- glm(realization ~ support + bid_centered ,family = binomial(link= "logit"),
                  data = filter(auction,
                             bid_centered <= 1,
                             bid_centered >= -1))

From there on i'm not sure what to do. Do i measure the odds ratio or the marginal effects to measure the size of the discontinuity at the cutpoint?
Thank you all in advance!
My approach to measuring the discontinuity at the cutpoint:
library(margins)
#gives the AME as default for probit and logit
model_bandwith1 <- glm(realization ~ support + bid_centered ,family = binomial(link= "logit"),
               data = filter(auction,
                             bid_centered <= 1,
                             bid_centered >= -1))
logitmargins <- margins(model_bandwith1, type = "response")
tidy(logitmargins)


# Marginal effects
library(mfx) # marginal effect at the mean (MEM)
# base model no weights
model_logit <- logitmfx(formula = realization ~ support + bid_centered, data = filter(auction,
                             bid_centered <= 1,
                             bid_centered >= -1))

model_logitor <- logitor(formula = realization ~ support + 
bid_centered , data = filter(auction,
                             bid_centered <= 1,
                             bid_centered >= -1))

The estimate of the AME for support is 0.404 while the estimate of the MEM for support is 0.4977.
The Odds Ratio is 9.227.
How can i interpret the difference between the AME and the MEM? the MEM seems to measure the gap shown in the visualization of the discontinuity pretty accurately.

Before you move on to the best way to describe the discontinuity, I'd suggest that you take advantage of the size of your data set to model the continuous predictor more closely near the discontinuity.
So far, you've only assumed a linear association (around the discontinuity attributed to the support predictor) between log-odds of project realization and the continuous bid_centered predictor. You restricted analysis to values close to the cutoff to emphasize local behavior near there, but you have enough data to try more flexible modeling with regression splines to get a more refined model.
Once you've done that, you can express the magnitude of the discontinuity in any terms that make sense to you and to your audience. The coefficient for support describes the discontinuity in log-odds scale. It might be easier to explain if you use a prediction from the model to translate that into a difference in probability of project realization at the bid_centered cutoff of 0.
