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I am currently working on analyzing a case and I need some help with the statistics.

When a certain woman tried to find an insurance for her car, she was told that the premium she had to pay would be more than average, based on the fact that she was a woman. According to the insurance company, on average, women were a higher risk. She won the court case: the court decided that her individual risk could not be based on averages of other women. Now my questions are:

(a) What kind of probability is used by the insurance company? (b) Can you comment on the kind of probability that is suggested by the verdict of the court?

I am really interested in the difference of the type of probability used by the insurance company and the court.

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What do you mean by "type of probability"? I don't think the court is using probability at all...it sounds to me like they are making some sort of legal, discrimination issue. – Michael McGowan May 13 '12 at 22:00
"Type of probability" doesn't make sense - it seems like you may mean something more along the lines of "How do insurance companies estimate risk?" – Macro May 13 '12 at 22:01
Well the court decided that the insurance company was basing the individual risk on the averages of all other women (uthe insurance company claimed was higher overall than men). I feel like the probability the insurance company uses is P(car damage|she is a woman) so that damage took place given that she is a woman. And I feel like the court didn't see it this way? – DinaH May 13 '12 at 22:10
I don't think that there is an argument about how the two sides estimate the probability of a car accident given the fact that the individual is a woman. I think the court's point is that the predictive models that the insurance company has developed actually are not all that predictive. So although the insurance company can accurately compute population-level summary statistics, and thus infer that women as a group are more prone to car accidents, they can't actually use their data to build models that reliably predict the risk of having car accidents in new individuals. – Alexander May 13 '12 at 22:41
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(+1) I'm not sure I understand the reason for the downvote on this question; it is well-formed and the question of interest is pretty clearly spelled out even if it's apparent that the OP is struggling to find the right terminology. – cardinal May 13 '12 at 22:53

2 Answers

The insurance company may have simply noted that, according to their data, women, on average, get into more motor vehicle collisions than men. In the simplest case, they could have just calculated the average number of motor vehicle collisions per insured person, stratified by gender. However, just because women, on a population-wide level, get into more accidents than men, does not tell us much about an individual woman's chances of getting into an accident. This is one of the central points of the court. As Michael notes below, this method is not designed to predict the outcome of an individual (unlike, for instance, the use of logistic regression or other predictive statistical models).

You might be interested in a paper by Kennaway that looks at the use of statistical trends to make predictions in a more mathematical manner:

A frequent use of statistical trends is to make predictions about individuals. Aptitude tests and credit rating are two major applications, especially in the latter case if ratings are derived from rules generated from statistical analysis by data mining applications. An individual to whom such tests are applied is, in effect, participating in a lottery. If the test is valid, the lottery is biased to a greater or lesser extent in his favour, but it is a lottery nonetheless. Such tests say little about any individual being tested.

He later states that:

The population relationship is a property only of the population, and not of any individual in it.

I think this final point is important to keep in mind. It reminds us that in any population in which some population-wide relationship is measured, there will invariably be individuals who don't "fit the pattern", so to speak.

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Very interesting! Thank you very much! – DinaH May 13 '12 at 22:11
This is a method that is not designed to predict the outcome for the individual. It just makes it easier for the insurance company to apply different rates and be safer from losses. – Michael Chernick May 14 '12 at 1:01
Thanks for your comment, Michael. I've edited my answer to reflect that fact. – Alexander May 16 '12 at 15:34

This is not a matter of different types of probability. It is really a matter of what you can use to estimate it. In the case of the woman they want to predict the probability of an accident say over a 3 year period given personal characteristics. The insurance company can look at people that have been insured and can compare those that had accidents over a 3 year period with those that don't. Then with this data they can create a logistic regression model using characteristics such as age and gender. The model is then applied with the characteristics for the woman in question to estimate the probability that she will have an accident in the next 3 years. The cost of the premium will then depend on how high this estimated probability is. The difference between the court and the insurance company is that the insurance company found that gender was useful to include in their model. The court is saying that inspite of that they see using gender as being discriminatory against women. So they are telling the insurance company that they cannot use gender in their model. They would probably also rule against race as a variable to use in the model but would allow age. Once gender is taken out the new model will give a different estimate of the probability for this woman and probably it would be lower which may mean a lower premium.

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Very good points! – Alexander May 13 '12 at 22:53
Many states and provinces in North America already explicitly prohibit the use of the usual protected classes as risk factors for pricing car insurance, and this includes both gender and age. An interesting additional one that is prohibited is geographic location of garaging, though this is likely highly predictive. I am unsure how strong one could really argue that removing gender would actually lower a particular woman's rate, since it is not as simple as treating the positive coefficient as if it were zero. – cardinal May 13 '12 at 23:08
I wonder how much information would be lost by posing this as a logistic regression problem rather than predicting the expected loss, which would seem to be closer to the quantity of interest. Though I suspect the median number of accidents is zero, knowing for example whether I'm driving a Camry or a Bugatti would be important for predicting the probability of an accident certainly, but much more important for the expected loss. – cardinal May 13 '12 at 23:16
I think age must be allowed in most states. Everywhere I have lived the teenage drivers are charged much higher rates and I think it must have to do with age.I don't see using individual characteristics as ever discriminatory whether the covariates are age, gender, race or anything else. It is just a judgment that if you belong to a specific class you are more likely ot have accidents. If the insurance companies are allowed to use some of these characteristics then they should be allowed to use all or any that they find helpful. – Michael Chernick May 13 '12 at 23:37
I do not have a ready compendium, but I do know of several in which it is illegal. Your state appears to be more friendly to the insurance companies in this respect. I would think that using number of years of driving experience would be one way to "penalize" teenagers without explicitly using age. – cardinal May 14 '12 at 0:02
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