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Currently I have insurance claim loss data (say roughly N=10,000), with independent variables state (California, Washington, etc.), policy number (uniquely identifies a policyholder), type of car (sedan, SUV, etc.), and product type (each product has different claim loss structure), so something like the following:

Car type           State         Product type    Policy Number              Loss
Sedan               CA                 A          918319                    1000  
SUV                 CA                 B          918319                    400
Sedan               WA                 C          931239                    600 
Truck               CO                 A          138234                    300

Also, the amount of data that exists for each car type is uneven; for instance, there exists N=3000 entries for sedan but only N=30 entries for utility vehicles.

I'd like to test which car type has higher risk, i.e. which car type, generally speaking, has a higher average loss per policy number.

I'm wondering if there are any issues with the following approach, or important points I need to consider:

Right now my plan is to conduct t-tests by comparing the average loss per policy for each car type vs. the overall average, e.g. null hypothesis might be: average loss per policy for sedans is equal to the average loss per policy for all car types, and alternative is average loss per policy for sedans is higher than average loss per policy for all car types.

For the t-test specifically for sedans, since claim losses are typically not normally distributed, to help satisfy the normality assumption, I plan to randomly assign each data entry with sedan as the car type to a random sample, so that each sedan sample has roughly 30 entries (to get each resultant sample closer to a normal distribution), and then apply the t-test on the resulting sampling distribution.

Are there other approaches? For car types without a lot of entries, my guess is a non-parametric approach might be better, but I'm not sure what would be the best test.

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I think the most straightforward approach would be to fit a linear model adjusting for car type, state, and product type (since policy number is unique, we can exclude it unless it appears several times within the data in which case we would need to do something about that).

The loss claims are likely right skewed conditional on the other variables, but I don't think that matters all that much. If you think it does, a gamma GLM might be best, or maybe a Tweedie GLM. Its been a while since I thought about insurance modelling so hopefully someone else can recommend something better than a gaussian GLM, which is what I would try first. After fitting the model, you can examine coefficients for the different car types to determine which has the largest impact on average loss.

The reason why t-testing is not a great idea is because the affect of some car types might be impacted by state. Maybe people in Oregon love their Sedans and also happen to drive drunk after having too many craft beers. In that case, the effect of sedan might not be what you think it is (it would actually be the state, but the use of sedans in the Oregon is what is causing the observed loss).

With N>10, 000 you're going to find significant effects. I would encourage you not to concentrate too hard on them since they could potentially be small. Take an estimation first approach here unless you have a priori hypotheses on which cars might lead to which kind of losses.

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  • $\begingroup$ Good points, thank you. I completely separated hypothesis testing and modeling in my head so fitting a linear model was something that I didn't even consider. $\endgroup$
    – platypus17
    Commented Sep 28, 2020 at 23:11
  • $\begingroup$ Besides linear modeling, I was wondering what if you fixed multiple categories? For instance instead of just looking at one specific car type, you would look at one car type, one state, and one product (that one might self-select with a high risk among these different categories) vs. the overall average? Would t-testing still not be a good idea here? $\endgroup$
    – platypus17
    Commented Sep 29, 2020 at 1:47
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    $\begingroup$ @platypus17 You're doing far too many comparisons at this point. You won't find any significant differences (even if you just looked at states, you're looking at something on the order of 1000 comparisons so any post hoc correction will require p values to be extremely small to be "significant" at this point. Not that significance matters IMO, like I said, you should do an estimation first approach). $\endgroup$ Commented Sep 29, 2020 at 2:01
  • $\begingroup$ @platypus17 If all you're looking for is the type of car which leads to higher loss, adjust for state and policy and look at the coefficients for vehicle. $\endgroup$ Commented Sep 29, 2020 at 2:02

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