Say I want to investigate the rate of crime by the density of police inside a city. The sampling unit is per city.(adsbygoogle = window.adsbygoogle || []).push({});

So according to a problem I saw, it claims that we can model it by

##log(\mu_{i})\sim PoliceDensity_{i}+log(PopulationSize_{i})##

Where ##\mu_{i}## is the mean count of crime in the ith city.

The last term is the offset term.

First question,

does this assume that there isn't collinearity when this model is put together? I mean, Density presumably is (Num of police officers/Population size)

So that means that the offset term and the density term are a function of each other. Is that ok? If I want to avoid collinearity, then would I just have to be sure that the density isn't related to the population size in a linear way? For example, if I have ##x ##and ##x^2## in as predictors in a regression, then that should be ok right? As long as its not ##x and x##.

Second question,

Ok now I want to update the model in include the proportion of improversed citizens in a city. So the new model, I supposed to figure out.

##log(\mu_{i})\sim PoliceDensity_{i}+PovertyRate_{i}+log(PopulationSize_{i})##

where ##PovertyRate_{i}## is the proportion of poverty in a city.

Now I'm guessing that that is the wrong way to go about seeing if poverty rate is associated with because intuitively, the rate would depend on large cities. Large urban cities have more crime in general. So assuming that, by having the the poverty rate and the offset, log(Population Size) would be giving redundant information? But then again, it might not as its not redundant in a linear sense.

On the other hand, ##PovertyRate_{i}## and ##PoliceDensity_{i}## might be redundant linearly. As high crime areas would have more crime police per area.

Or it could just be confounding such that it isn't so much as to be redundant. Poverty rate could be associated with the the rate of crimes and associated with police density, making it a potential confounder.

The problem is, it could be that ##PovertyRate->CrimeRate->PoliceDensity## making the response inside the causal pathway to density. Would this be something to worry about?

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# A Collinearity Confusion

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