How to decide which glm family to use?

Question

I have fish density data that I am trying to compare between several different collection techniques, the data has lots of zeros, and the histogram looks vaugley appropriate for a poisson distribution except that, as densities, it is not integer data. I am relatively new to GLMs and have spent the last several days looking online for how to tell which distribution to use but have failed utterly in finding any resources that help make this decision. A sample histogram of the data looks like the following:

I have no idea how to go about deciding on the appropriate family to use for the GLM. If anyone has any advice or could give me a resource I should check out, that would be fantastic.

Answer 1

GLM families comprise a link function as well as a mean-variance relationship. For Poisson GLMs, the link function is a log, and the mean-variance relationship is the identity. Despite the warnings that most statistical software gives you, it's completely reasonable to model a relationship in continuous data in which the relationship between two variables is linear on the log scale, and the variance increases in accordance with the mean.

This, essentially, is the rationale for choosing the link and variance function in a GLM. Of course, there are several assumptions behind this process. You can make a more robust model by using quasilikelihood (see ?quasipoisson) or robust standard errors (see package sandwich or gee).

You have correctly noted that many densities are 0 in your data. Under Poisson probability models, it is appropriate to occasionally sample 0s in the data, so it's not necessarily the case that these observations are leading to bias in your estimates of rates.

To inspect the assumptions behind GLMs, it is usually helpful to look at the Pearson residuals. These account for the mean variance relationship and show the statistician whether particular observations, such as these 0s, are egregiously affecting estimation and results.

Answer 2

Generalized linear model is defined in terms of linear predictor

$$ \eta = \boldsymbol{X} \beta $$

that is passed through the link function $g$:

$$ g(E(Y\,|\,\boldsymbol{X})) = \eta $$

It models the relation between the dependent variable $Y$ and independent variables $\boldsymbol{X} = X_1,X_2,\dots,X_k$. More precisely, it models a conditional expectation of $Y$ given $\boldsymbol{X}$,

$$ E(Y\,|\,\boldsymbol{X} ) = \mu = g^{-1}(\eta) $$

so the model can be defined in probabilistic terms as

$$ Y\,|\,\boldsymbol{X} \sim f(\mu, \sigma^2) $$

where $f$ is a probability distribution of the exponential family. So first thing to notice is that $f$ is not the distribution of $Y$, but $Y$ follows it conditionally on $\boldsymbol{X}$. The choice of this distribution depends on your knowledge (what you can assume) about the relation between $Y$ and $\boldsymbol{X}$. So anywhere you read about the distribution, what is meant is the conditional distribution.

• If your outcome is continuous and unbounded, then the most "default" choice is the Gaussian distribution (a.k.a. normal distribution), i.e. the standard linear regression (unless you use other link function then the default identity link).

• If you are dealing with continuous non-negative outcome, then you could consider the Gamma distribution, or Inverse Gaussian distribution.

• If your outcome is discrete, or more precisely, you are dealing with counts (how many times something happen in given time interval), then the most common choice of the distribution to start with is Poisson distribution. The problem with Poisson distribution is that it is rather inflexible in the fact that it assumes that mean is equal to variance, if this assumption is not met, you may consider using quasi-Poisson family, or negative binomial distribution (see also Definition of dispersion parameter for quasipoisson family).

• If your outcome is binary (zeros and ones), proportions of "successes" and "failures" (values between 0 and 1), or their counts, you can use Binomial distribution, i.e. the logistic regression model. If there is more then two categories, you would use multinomial distribution in multinomial regression.

On another hand, in practice, if you are interested in building a predictive model, you may be interested in testing few different distributions, and in the end learn that one of them gives you more accurate results then the others even if it is not the most "appropriate" in terms of theoretical considerations (e.g. in theory you should use Poisson, but in practice standard linear regression works best for your data).

Answer 3

This is a somewhat broad question, you are asking for how to do modelling, and there are entire books dedicated to that. For example, when dealing with count data, consider the following:

In addition to choosing a distribution, you have to choose a link function. With count data you could try poisson or negative binomial distribution, and log link function. A reason for the log link is given here: Goodness of fit and which model to choose linear regression or Poisson If your patches have very different areas, maybe you should include logarithm of area as an offset, to model counts per unit area and not absolute counts. For an explication of offset in count data regression, see When to use an offset in a Poisson regression?

EDIT 

This answer was originally posted to another question, which was merged with this one. While the answer is general, it commented specifics of a data set and problem which no more are in the question. The original question can be found in the following link: Family in GLM - how to choose the right one?