Predict satisfaction score given a shift of Service Level Agreement I would like to make predictions for Overall Satisfaction based on a shift of Service Level Agreement (SLA).
I have number of days taken to complete a single service, and a satisfaction score (n=4000)
num_days   score
       1       5
       1       4
       2       5
       2       3
      10       1

The observed data is given SLA = 10 days. For those that are unaware of Service Level Agreements: this means that contractually a supplier promises to deliver a service/product within 10 days.
Please see the plot for illustrative purpose. The orange line is the average satsifaction per day and it has confidence bounds.

How can I predict the Overall Satisfaction for the whole data given that we would move the SLA to 9 days?
Some assumptions:


*

*The observed frequencies will "move" towards the left (but how to model it?)

*I expect the average satisfaction will get lower (or equal) for x=1 given SLA=9 compared to x=1 given SLA=10, since we are relatively slower


How can we estimate the shift of frequencies? I thought maybe I can use a poisson distribution but it is not a good fit. I also thought perhaps to fit a log/exp curve, but I am looking for someone who has knowledge how to theoretically do this.
In the end: what would we expect the total average satisfaction to be when we shift the SLA from 10 to 9?
EDIT: Maybe another useful image is to show the frequencies (x=num_days, y=frequency density)

 A: Do you want a proportional odds logistic regression model? https://data.library.virginia.edu/fitting-and-interpreting-a-proportional-odds-model/   There are variations on this basic idea which may/may not be a better fit for your data.
The response variables are ordinal (1 to 5) and the predictor variables are quantitative (?).
I think Agresti wrote the definite book on this https://www.wiley.com/en-gb/Categorical+Data+Analysis%2C+3rd+Edition-p-9780470463635, I don't know if you can find a copy of the relevant chapter if that helps?
I made a quick sketch as to how I understand it.   Your linear predictor for any value of time gives you a random latent variable which is sketched in red (sorry they're not very smooth, very rough sketch).  As you can see, the top "latent variable distribution" is to the right of the bottom "latent variable distribution".   The location of this latent variable distribution is specified by your linear predictor.
The cutpoints are sketched in grey.   Each zone delimeted by grey dotted lines corresponds to a different satisfaction score, from 1, 2, 3, 4, 5, upwards.  From this, you can work out the probability that you get whichever score.   So you can assess quality of fit by whether this matches the number of responses in each box.   But you can also tell whether overall satisfaction has gone up or down

