# lm in R: What's better? “ugly” model with high $R^2$ or a “beautiful” one with worse $R$? How much VIF is too much?

I have been trying to model with lm in R. I have several variables: Initial cell concentration $(X1)$, macroscopic appearance $(X2, X3, X4, X5)$, microscopic appearance $(X6,X7,X8)$, % of cells moving fast, slow or not moving $(X9, X10, X11)$, % of cells with microscopic features $(X12, X13,...X18)$, etc. The final alive cell concentration obtained $(Y)$ is a function of these variables; I can'd decide which one to choose.

The theoretical model that would fit best is something like

1:   Y ~ X2 + (...) + X8 + X9:X1 + X10:X1 + X11:X1 + (...) + X18:X1


(that means, the initial cell concentration is a fundamental factor in this model). Here $R^2$ approaches $0.55$ at best (which is sort of understandable, given the high variability and observer-dependent and VIFs are very high, X1:X9 reach 30000.

However the model with the highest $R^2=0.77$ is this (VIF reaching 25 for X1:X9):

2:   log(Y) ~ X1*X9 + X2 + X3 + (...) + X18


or this:

3:   log(Y) ~ X2 + (...) + X8 + X9:X1 + X10:X1 + X11:X1 + (...) + X18:X1


(in which I use absolute concentrations instead or numbers, getting $R^2= 0.68$ and VIF reaching 30000 for X9:X1, as expected for a combination but still high). Changing all percentages to absolute values has unexpectedly reduced $R^2$, but the coefficients make more sense in this model than in the second one (any link between Y and X9 or other percentaes was probably was a spurious association).

I also have this:

4:    Y ~ X2 + (...) + X8 + X9*X1 + X10*X1 + X11*X1 + (...) + X18*X1


in which I include both the percentage and the actual value (I think this is an overfitting model, and it has $R^2$ around 0.78 and very high VIFs).

How would you deal with this situation? I think I should choose a good model a priori (probably 3, or maybe 1). Residual graphics look better in the models with better R as well (and they look particularly worse in the first one)...

• Why are you fitting the model? What's your goal? How will you be using the result? – Aaron Mar 11 '18 at 1:45
• I want to predict the output of the process. I have all these variables early in the technical process. Later phases of the process can be optimized for poor-quality cells. This predictive score could be validated prospectively I think... – A-B Mar 11 '18 at 2:10
• Why are you using OLS? Why not something like a random forest? – generic_user Mar 11 '18 at 2:25
• What's your sample size? VIF of 30,000 is unacceptable. Also, what's in (...)? – Peter Flom Mar 11 '18 at 13:59
• When you transform the response in any nonlinear fashion, the model changes so completely that it's meaningless to compare $R^2.$ – whuber Mar 11 '18 at 16:16