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Wouter Steenbeek writes:
I am a criminologist and mostly do spatial analyses of crime patterns: where does crime occur and why in these neighborhoods / at these locations, and so on. Currently, I am thinking about offender decision-making behavior, specifically his ‘location choice’ of where to offend.
Hey, how about criminologists instead of looking to someone else to solve their problem, do something about maybe taking CPR classes?
But I digress.
Steenbeek continues:
You may be surprised that I contact you as you don’t work on this substantive subject, but the models used to analyze such behaviors are familiar to you as they are used in (voting) choice behavior: multinomial logit. More specifically, most criminological studies use McFadden’s conditional logit model. Just like a voter can choose from 4 political parties, or a traveler can choose between 5 modes of travel behavior (choice between car, bus, taxi, walk, or bike), a criminal can choose where to offend. Characteristics of the offender, characteristics of each choice alternative (e.g. deterrence level at each location), and offender-specific characteristics of each choice alternative (e.g. distance from each offender to each location) are then used to model the choice of crime location.
The main difference is that the offender chooses the *location* where crime is committed, and therefore the choice set is usually much larger, depending on the definition of ‘location’. Often, location refers to “neighborhood” (census tract). When studying offenders within one city, each offender is modeled to choose which neighborhood he commits crime in. This can easily be a choice set of 50-100 neighborhoods. But location can also refer to smaller spatial units of analysis such as street segments (the part of a street between two intersections), leading to a choice set of a few thousand (!) alternatives.
A disadvantage of the conditional logit model is the assumption of the independence of irrelevant alternatives. Especially in spatial analyses where nearby locations are vey similar to each other, this violates the IIA assumption. Exactly *two* studies have used the ‘mixed logit’ model that does not suffer from IIA. The R package RSGHB (https://cran.r-project.org/package=RSGHB) can be used to estimate these models using a Hierarchical Bayesian framework.
(1) I would prefer to work with. But: can such mixed logit models be programmed relatively easily in Stan? Or would you suggest to keep using RSGHB? (RSGHB uses Metropolis-Hastings)
A second question is with regard to the use of an informative prior / knowledge about each offender. One can only commit crimes in locations where one has knowledge of. (Let’s assume for now that offenders only have knowledge of locations that he visits himself, and that the study area is limited to one city and the locations chosen are neighborhoods). In the ideal data situation, we would know exactly how familiar each offender is with each neighborhood. Then one could use multinomial logits with varying choice sets. (An equivalent example from voting behavior would be that a voter in district A can choose from candidates of political parties A, B, C, but a voter in district B can only choose from candidates of political parties A and B: in that case voter[B] should not be modeled as if he can choose C, because he simply *cannot* choose C by definition).
In practice however, we only have (at best) a “likely” familiarity of each offender with each neighborhood, predicted using other sources (such as smartphone travel data of the population). I cannot quite wrap my head around how to incorporate such offender-specific best guesses/proxy of the familiarity with each neighborhood into the model. I suppose I can simply add it as an offender-specific covariate to the model (but there is a lot of uncertainty in the prediction, so this variable will need to incorporate measurement error).
Theory suggests that if the offender is unfamiliar with a neighborhood, then the chance that he commits there is essentially 0. So perhaps I should include an interaction between the offender-specific location-familiarity variable and all other variables to capture this?
But actually, my feeling is that the offender-specific location-familiarity is some kind of “prior”, i.e. our individual-specific prior knowledge of an offender’s choice set. This prior would be on the Y’s, similar to a multinomial model with varying choice sets, but without removing some choice alternative completely (as we cannot be 100% sure that an offender is really totally unfamiliar with a neighborhood). I have no idea if that is a feasible approach, however.
(2) What do you recommend for the situation described above?
My reply:
1. Yes, you should definitely be able to fit those Rsghb models in Stan, this should be no problem at all. If difficulties arise, you can post questions to the Stan users group and you’re likely to get a polite and helpful answer (we’re a Ripley-free zone).
2. If you have information on how “likely familiar” an offender is with each area, I think you should just put this in as a continuous predictor. The model should work fine, and it shouldn’t really matter how many nonzero cells there happen to be.
P.S. I wonder if we could set up this sort of model for junk science: we’d have a grid with research topics in one direction, and journals in the other. The challenge would be to predict where this stuff would be published next.
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