How to incorporate costs (into logit model) of false positive, false negative, true positive, true negative responses, if they are different costs ? Is it possible to do that on the level of likelihood function ?
Edition :
I see know that likeligooe function could be quite easily modified to incorporate costs, but then likelihood function become discontinuous :
y_i == 1 oraz f(x_i*B) > 0.5 cost = cost11
y_i == 1 oraz f(x_i*B) < 0.5 cost = cost10
y_i == 0 oraz f(x_i*B) > 0.5 cost = cost01
y_i == 0 oraz f(x_i*B) < 0.5 cost = cost00
non-modified likelihood function :
f(x_i*B)^y_i * ( 1 – f(x_i*B) ) ^ (1 – y_i)
modified likelihood function :
(cost11*positive(f(x_i*B)-0.5) + cost10*negative(f(x_i*B)-0.5) ) ^ y_i
*
(cost00*negative(f(x_i*B)-0.5) + cost01*positive(f(x_i*B)-0.5) ) ^ (1 – y_i)
where :
positive(x) = 1 if x > 0
positive(x) = 0 if x < 0
negative(x) = 1 if x > 0
negative(x) = 0 if x < 0
Best Answer
This is best thought of, in my opinion, not of a change to the likelihood function, but as a way to translate estimated risk into an optimum decision. We maximize the (standard or standard penalized) likelihood for a reason – to get optimal models. Then optimal decisions are made one individual at a time based on that individual's loss function. As typically loss functions vary from subject to subject, the final decision has to be deferred and cannot be made by an analyst. Most commonly, the loss function is not articulated but is used implicitly by the subject to make her own decision. It will depend on deeply held beliefs as well as prevailing conditions (e.g., resource availability).
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