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I have output data from an aggregator website comparing insurance premiums. Each unique input (incl variables such as age, medical screening score, territorial limits, duration, etc) produces an insurer, a premium quote, and their respective rank in the output. I have 500+ inputs.

While some of the input variables are categorical, others like territorial limits aren't, I want to produce a couple of multiple linear regressions to explain the variation in premium and rank. How should I approach this?

(NB. I'm a maths/stats student, so if there is something blatantly obvious that I'm not seeing or forgetting in my description, please let me know.)

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It sounds like you have the potential for two different models here, one that predicts rank and one that predicts premium.

For the rank model, something like an ordinal logistic regression may be appropriate. For the premium model, a linear regression may work. Both models can accommodate continuous and categorical predictors and can be implemented in a number of software packages.

Something to consider in your dataset is the concept of nesting. By that I mean that some observations may violate the assumption of independence required by both of these models. Specifically, you can certainly imagine a scenario in which, for a variety of reasons, individual cases from the same state or an even smaller geographic region share certain characteristics that increase their similarity to each each other and their dissimilarity with the remaining cases in the set. Technically, this becomes an issue of correlated error terms in your model (i.e., the model tends to incorrectly predict values for sets of cases in the same way), which is a violation of one of the key assumptions of models I mentioned above.

Should you find that your data are nested or there is reason to believe that something like the scenario I describe above applies, you can use a multilevel extension of the models I mentioned. Again numerous software packages exist that offer implementation of this slightly more complex family of models.

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