Base Card Value: Gaussian vs Poisson

All of these mathematical endeavours begin with the presumption that Blizzard has a secret formula it uses to compute the amount of mana a card ought to cost. The value on the card will be different on the card for one of three reasons - the influence of mechanics, the effects of rounding (because a card costs 1 or 2 mana, not 1.5), and tweaking based on how it plays (variance introduced through human evaluation).

What we're ultimatley trying to build (for the base card value) is a model of the basic relationships between the values on the card (Attack, Health, and Mana), and these other effects, to 'reverse-engineer' the basic formula.

A standard lm model uses a family of functions for analyzing the variance of a dataset based on the assumption that there's a gaussian distribution in your data, and is primarily concerned with producing a binary predictor for those values.

As always, model fitting is part art, part science, and part throwing shit at the wall to see what works. Or, it is when I do it, at any rate.

So.

Given our belief that there's a direct, additive relationship between the base card value and the attack/defense values on the card, in theory, a glm on a poisson family should get you a better fit than the gaussian; it better represents the expected behaviour of a count variable. But is it true?

First, the original, gaussian LM fit from last time.

  Call: lm(formula = Mana ~ Attack + Health, data = dataset, subset = CardType == 1 & CardText == "" & Mana > 0)

  Residuals: Min 1Q Median 3Q Max -1.9940 -0.2844 0.1968 0.2218 0.7611 

  Coefficients: Estimate Std. Error t value Pr(>|t|)   
  (Intercept) -0.16376 0.14999 -1.092 0.283   
  Attack 0.50626 0.06172 8.202 2.28e-09 _*_

  ## Health 0.43566 0.06107 7.134 4.27e-08 _*_

  Signif. codes: 0 ‘**_’ 0.001 ‘_**_’ 0.01 ‘_’ 0.05 ‘.’ 0.1 ‘ ’ 1

  Residual standard error: 0.4999 on 32 degrees of freedom Multiple R-squared: 0.9359, Adjusted R-squared: 0.9319 F-statistic: 233.6 on 2 and 32 DF, p-value: < 2.2e-16

The autoplot for lm(formula = Mana ~ Attack + Health), showing a fit using the gaussian family.

Now let's change tack.

If we presume that our outcome value is "count"-like - additive in nature from the base values on the card - we can switch to the generalized linear model, and switch to a poisson distribution - with intriguing results.

glm(formula = Mana ~ Attack + Health, family = family, data = dataset, subset = CardType == 1 & CardText == "" & Mana > 0)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.0093  -0.2435  -0.1382   0.2194   0.9077  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)  
(Intercept) -0.17849    0.23141  -0.771   0.4405  
Attack       0.14891    0.06234   2.389   0.0169 *
Health       0.16466    0.06622   2.487   0.0129 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 44.9140  on 34  degrees of freedom
Residual deviance:  4.4248  on 32  degrees of freedom
AIC: 101.1

Number of Fisher Scoring iterations: 4
The autoplot for lm(formula = Mana ~ Attack + Health, family=poisson), showing a fit using the poisson family.

The autoplot for lm(formula = Mana ~ Attack + Health, family=poisson), showing a fit using the poisson family.

Note the differences in the two sets of graphs:

  • Residuals are healthier. The residuals on the poisson distribution have better deviations; LM range was -1.9940 to 0.7611 (~2.75); our Poisson GLM is -1.0093 to 0.9077 (~1.9).

  • Residual QQ is better. The two graphs are day and night; you now see a nice, clean, quantized Q-Q for the residuals, showing the stair-stepping you'd expect to see when you know that whatever magic formula exists has the effects of rounding (because mana is an integer) applied.

  • Cook's Distance improved. We go from having some pretty strange outliers to being within 0.06 on all modelled values.

Scale-Location and Residuals Vs Leverage also see huge improvements over their normal counterpart.

In short, the poisson family appears to do a much better job of estimating the base value of the card than the normal family.