Agent ranking formula

(Originally published on  Evan Millers’ Blog)

PROBLEM: You are a web programmer. You have users. Your users rate stuff on your site. You want to put the highest-rated stuff at the top and lowest-rated at the bottom. You need some sort of "score" to sort by.

WRONG SOLUTION #1: Score = (Positive ratings) - (Negative ratings)

Why it is wrong: Suppose one item has 600 positive ratings and 400 negative ratings: 60% positive. Suppose item two has 5,500 positive ratings and 4,500 negative ratings: 55% positive. This algorithm puts item two (score = 1000, but only 55% positive) above item one (score = 200, and 60% positive). WRONG.

WRONG SOLUTION #2: Score = Average rating = (Positive ratings) / (Total ratings)

Why it is wrong: Average rating works fine if you always have a ton of ratings, but suppose item 1 has 2 positive ratings and 0 negative ratings. Suppose item 2 has 100 positive ratings and 1 negative rating. This algorithm puts item two (tons of positive ratings) below item one (very few positive ratings). WRONG.

CORRECT SOLUTION: Score = Lower bound of Wilson score confidence interval for a Bernoulli parameter

Say what: We need to balance the proportion of positive ratings with the uncertainty of a small number of observations. Fortunately, the math for this was worked out in 1927 by Edwin B. Wilson. What we want to ask is: Given the ratings I have, there is a 95% chance that the "real" fraction of positive ratings is at least what? Wilson gives the answer. Considering only positive and negative ratings (i.e. not a 5-star scale), the lower bound on the proportion of positive ratings is given by:

(Use minus where it says plus/minus to calculate the lower bound.) Here   is the observed fraction of positive ratings, zα/2 is the (1-α/2) quantile of the standard normal distribution, and n is the total number of ratings. The same formula implemented in Ruby:

require 'statistics2'
def ci_lower_bound(pos, n, confidence)
     if n == 0
         return 0
     end
     z = Statistics2.pnormaldist(1-(1-confidence)/2)
     phat = 1.0*pos/n
     (phat + z*z/(2*n) - z * Math.sqrt((phat*(1-phat)+z*z/(4*n))/n))/(1+z*z/n)
end

pos is the number of positive ratings, n is the total number of ratings, and confidence refers to the statistical confidence level: pick 0.95 to have a 95% chance that your lower bound is correct, 0.975 to have a 97.5% chance, etc. The z-score in this function never changes, so if you don't have a statistics package handy or if performance is an issue you can always hard-code a value here for z. (Use 1.96 for a confidence level of 0.95.)

REFEENCES

Binomial proportion confidence interval (Wikipedia)

Agresti, Alan and Brent A. Coull (1998), "Approximate is Better than 'Exact' for Interval Estimation of Binomial Proportions,"  The American Statistician, 52, 119-126.

Wilson, E. B. (1927), "Probable Inference, the Law of Succession, and Statistical Inference,"  Journal of the American Statistical Association, 22, 209-212.

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