How to fairly compare a model vs. averages at discrete points

Forgive me for my ignorance, but I've been asked a question that I need some stats help on (I'm not a statistician, and have only a rudimentary knowledge!).

I've been supplied a data set which compares a daily average rank against a success KPI. The daily average rank varies from 1-10 (continuous) while the KPI varies from 0 to 1 (also continuous).

From this data, I have fitted a decay curve, of the form y ~ rank^-x , and simplistically run a grid search over values of x between 0.01 to 9.99 to maximise the adjusted r2. This may or may not be a good method (!) but the shape looks sensible and it's a better fit than a simple linear model.

The question I have been asked is: how much better is this at fitting the data than if I just take an average of the KPI at discrete ranks (i.e. rank=1, rank=2, ... rank=10 - given that the real world data [non-averaged] exists only at these positions) ?

I'm a bit stumped on how to actually quantify this! Can I calculate an "adjusted r2" for the combination of 10 discrete data points?

Thanks for any help!

• The adjusted R^2 is a property of the model, not individual data points – mkt Mar 16 '18 at 10:34
• What you probably want to do is compare a model with an intercept only (i.e. the prediction of y is just the average of all x-values) to a model with both an intercept and a slope (or power, I guess). – mkt Mar 16 '18 at 10:37
• Take a look at the model comparison tag for methods to do this (information criteria, etc): stats.stackexchange.com/questions/tagged/model-comparison – mkt Mar 16 '18 at 10:38
• Thanks for your help mkt. I think on reflection that there is no comparison between the two because one extrapolates while the other simply gives the best answer for a small set of discrete points. The model takes in to consideration other data, which helps smooth erraticity that can occur for specific points. I get it better now. Ta – Jon Mar 19 '18 at 10:07