Relativity and Discount
In the Interacting 2 Categorical Variables section, I get slightly confused on the example regarding occupancy class 2 with a sprinkler. It states:
"The key point now is to understand the relativities for the remaining occupancy classes with sprinklers are relative to the class 1 and sprinkler status.
For a risk in occupancy class 2 which has sprinklers, the relativity is e^(−0.2895+−0.2847) =0.563, or a 43.7% discount."
We don't include the occupancy:2 coefficient (0.2303) in calculating that 43.7% discount. To me, this would mean the 43.7% discount is relative to a class 2 risk with no sprinkler, not to class 1 and sprinkler status. Is that correct?
Comments
Hi,
In short, yes! We'll update the wiki soon but in the meantime hopefully what follows will help clear things up.
Reading these tables is tricky so let's start at the beginning. If you're given a risk's occupancy class and sprinkler status along with Table 2 then, assuming a log-link function, you can produce the claim frequency estimate by summing the estimate values the intercept and each line item that your risk matches before exponentiating. (Remember, Table 2 refers to a frequency model with Poisson error distribution.)
For example, a risk in class 2 that has sprinklers has an overall claim frequency of exp(-10.8690 + 0.2303 -0.2895-0.2847) = 0.0000135. A risk that has no sprinklers and is in class 1 has an overall claim frequency of exp(-10.8690+0+0) = 0.0000190. The relativity between these risks is then 0.0000135/0.0000190 = 0.709 for the class 2 with sprinklers relative to class 1 without sprinklers.
Notice the e^{intercept estimate} cancels out when you perform the relativity calculation.
It's instructive to write out the linear predictor as intercept + b_class + b_sprinkler + b_interaction. Then remembering exp(A+B) = exp(A)exp(B) we can see the following
The relativity exp(-0.2895) comes from exp(-0.2895) = exp(intercept + b_class + -0.2895 + b_interaction) / exp(intercept + c_class + c_sprinkler + c_interaction). Now since exp(intercept) always cancels out and we implicitly have with b_class =0 and b_interaction =0, this means c_class = 0 and c_interaction =0 i.e. this is relative to class 1 with no sprinklers which is what the text/wiki says. A 25.1% discount for class 1 risks with sprinklers compared to class 1 risks without sprinklers.
Next, the relativity exp(-0.2895+-0.2847) comes from exp(-0.2895+-0.2847) = exp(b_intercept + b_class + -0.2895 + -0.2847) / exp(intercept + c_class + c_sprinkler + c_interaction). Again, the intercept cancels out and we can't have b_class = 0 as this would mean b_interaction = 0. From b_interaction = -0.2847 we know b_class must be 0.2303 and for this to also cancel out, c_class must be 0.2303. Lastly, it follows that c_interaction must be 0 for the equation to hold, so exp(-0.2895+-0.2847) is relative to class 2, not class 1 as the text and wiki currently say. So the 43.7% discount is for a class 2 risk with sprinklers relative to a class 2 risk without sprinklers.
Calculating from first principles:
a.) A risk in class 2 with sprinklers has overall claim frequency of 0.0000135
b.) A risk in class 2 without sprinklers has overall claim frequency of 0.0000240
c.) A risk in class 1 with sprinklers has overall claim frequency of 0.0000143
d.) A risk in class 1 without sprinklers has overall claim frequency of 0.0000190
a.) relative to b.) = 0.563 = 43.7% discount
c.) relative to d.) = 0.753 = 24.7% discount (if you don't round you get the 25.1% discount shown in the text)
a.) relative to b.) is a much deeper discount than c.) relative to d.) which confirms what the text is trying to say. Namely, the magnitude of the sprinkler discount should depend on the occupancy class.