17.1: Linear or Poisson Regression?

To illustrate some of these observations, let us create a count dataset, fit it with a simple linear model, fit it with a Poisson model, and then compare the results. The data that we will use for this example, fakepoisson, was fabricated so that we know the parameters. As such, we can compare the estimates we get from the three modeling techniques to the true parameters.

Here is the code I used to create the fakepoisson data set:

set.seed(370)

n=75
x = sort( runif(n, min=0, max=2) )
beta0 = 0
beta1 = 2
lambda = exp( beta0 + beta1*x )

y = rpois(n, lambda)

By this point, you should be able to determine what each line of code does. You should also take note of how the parameter lambda is defined and keep this in mind as you read forward.

Figure \(\PageIndex{1}\): Plot of the pseudo data with three regression equations overlaying. The linear regression is in red, the linear regression on the log-transformed data is in green, and the Poisson regression is in blue. The black curve is the "correct" curve.

For this example, the true parameters are \(\tilde{\beta}_0 = 0\) and \(\tilde{\beta}_1 = 2\). Both of these are in log units (the tildes to serve as reminders of this). Except for those provided for the linear model, it is difficult to compare the estimates of the true value. It is much easier to compare the prediction curves.

OLS Model (Untransformed)

The OLS (untransformed) model can easily be performed. However, it does not fit the data well at all (see the figure above). If you decide to perform the three usual numeric tests, you will find all three violated. Yikes!

OLS Model (Log-Transformed)

The transformed OLS model has its own problem. Logically, a log transform would be appropriate (only bounded below). However, the dependent variable takes on a \(0\) value. This means you should either perform an additional transformation (add 1 to each dependent value) or drop the records with \(y=0\).

If we add 1 to each dependent variable before performing the log transform (that is, we perform the transformation \(y^* = \log[y+1]\).) We see that there is a lingering issue with heteroskedasticity. Is it ignorable? Perhaps, but let us try Poisson regression.

Poisson Model

Here, we easily perform Poisson regression and check the requirements. What was the code to run the analysis?

model.c <- glm(y~x, family=poisson)
summary(model.c)

That's all.

At this point, you should be able to guess what the lines do and what information in the first line is missing because it is the default link… as we see shortly.

Which is the best model of the three?

It may not be clear that these three cannot be directly compared. The linear model makes no adjustments, the log-transformed model does, as does the Poisson regression model. None of the transformations are the same. This means that we cannot use information criteria to compare the models.

So, how can we determine which of these models is best? A first step is to determine which is appropriate by checking the assumptions. The untransformed OLS model violates all three requirements. The transformed OLS model is heteroskedastic. The Poisson model, however, violates none of the requirements. Thus, on the basis of meeting assumptions, the third model is the best of these three.

If all we care about is estimates (and not confidence intervals), we could look at the graphic comparing the data and the estimations from the model (see the figure above). Numerically, we could also check how much the uncertainty in \(y\) has changed. The uncertainty using the null model (predicting \(y = \bar{y}\)) is \(184.96\). The uncertainty with the linear model is \(43.6\) — a reduction of 76%. The log-transformed linear model has an uncertainty of \(8.1\), which is a reduction of 96%. This is quite much different from the pure linear model. The Poisson model also has an uncertainty of \(8.1\) and a total reduction of 96%.

Thus, if all you care about is the estimate (which scientists should not), finding some adjustment so that the curve fits the data works. If you are a true scientist, then the confidence interval (and p-values) are important. This means assumptions about homoskedasticity are important — if they exist. Some modeling requires homoskedasticity others do not. Poisson regression does not (at least, not really).

Remember

Remember that we can use AIC, BIC, etc. only when the y-values are the same. This is not true here, as the y-values are all transformed differently.