CODEWITHSUNDEEP
statisticscoefficient-of-determination
Meth Bandaranaike
Meth Bandaranaike

Reputation: 11

R-Squared (Coefficient of determination) range beyond 0 and 1?

I was under the impression that the R-squared value is bound between 0 and 1. However after reading some online literature/forums and also some personal experience, I now feel that the R-squared value (coefficient of determination) is:

a) bound between negative infinity and 1

b) adjusted R-squared value can go beyond 1 (I'm assuming positive infinity?)

Reason for a is that I have fitted some regression models that have returned a negative R-squared value. This is when the best fit line performs worse than the average value of your dependent variable. i.e. the RSS > TSS.

Reason for b is that when I used a PLS regression model on my data I got an R-squared value of about 0.94 (I can't remember the actual figure), but plugging it into the adjusted R-squared equation gave me a value above 1. I attributed this to the fact that there were fewer observations than there were input variables (140 odd observations and 228 input variables.) I have attached the formula of the adjusted R-squared value below.

enter image description here

After a few combinations of p and N I observed that if N > p, the adjusted R-squared does not go beyond 1. However in my case, N < p, the adjusted R-squared does exceed 1.

TL;DR So I would like to know:

  1. Am I correct in thinking that the R-squared value can range from negative infinity to 1 and the adjusted R-squared value from 0 to positive infinity instead of being bound between 0 and 1?
  2. If not, is there a different range that R-squared and adjusted R-squared is bound by?

Upvotes: 0

Views: 311

Answers (1)

Sachin kaushik
Sachin kaushik

Reputation: 11

I had similar question regarding range of adjusted r2_score. I have implemented and came to conclusion that both negative inf and positive infinity values are possible.

def adj_r2(r2,n,k):
    return 1 - (((1-r2)*(n-1))/(n-k-1))

# case 1: r2=-inf
adj_r2(r2=-4000,n=100,k=10) # returns -4449.550561797752

# case 2: n<k
adj_r2(r2=-4000,n=5,k=10) # returns 2668.3333333333335

adj_r2(r2=0.8,n=5,k=10)# returns 1.1333333333333333

Upvotes: 0

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