Thread by @IsabellaGhement

I found a tweet on Poisson regression this morning on my feed and thought it would be a good time to write a tweetorial on Poisson regression. To keep things simple, I will assume we have a count response variable Y and a single predictor variable X. Here we go! #rstats

The response variable in Poisson regression is a count variable (e.g., the number of fish in a lake, measured every year). The marginal distribution of this variable can be visualized with an R command like:

plot(table(Data$Y), xlab = "Y", ylab = "Frequency")

The relationship between the response variable Y (e.g., yearly count of fish in a lake) and the predictor variable X (e.g., year of study, labelled as 0, 1, 2, etc.) can be plotted with the commands:

library(lattice)

xyplot(Y ~ X, data = Data, cex = 1.5, pch=19)

Notice the apparent non-linear relationship between year of study X and number of fish in the lake Y apparent from the plot of Y versus X in our example. Poisson regression will allow us to capture this relationship.

A Poisson regression model is easy to formulate and fit in R:

Model <- glm(Y ~ X,
family = poisson(link = "log"),
data = Data)

summary(Model)

What can be tricky though is interpreting the model summary.

To interpret the model summary of a Poisson regression model, we need to first understand what we are modelling:

We are modelling the log-transformed (true) 'mean' value of Y as a linear function of X.

Other people may use the words 'expected' or average' instead of 'mean'.

So how do we interpret the estimated intercept in our Poisson regression model? It is the estimated log-transformed 'mean' value of Y when X = 0. In other words, the log 'mean' count of fish in the lake in the first year of study is 0.25964.

The intercept becomes more interpretable if we exponentiate its value to get rid of the pesky log-transformation:

exp(0.25964) = 1.296463 is the 'mean' count of fish in the lake in the first year of study (which can be rounded for reporting purposes).

What about interpreting the estimated slope of X in our Poisson regression model?

Each additional year (i.e., each 1-unit increase in X) is associated with an (additive) increase of 0.30677 units in the log-transformed 'mean' count of fish in the lake.

The log-transformation strikes again! The estimated slope of X becomes more interpretable after exponentiation:

exp(0.30677) = 1.36.

Each additional year is associated with a multiplicative increase by a factor of 1.36 in the 'mean' count of fish in the lake (a 36% increase).

The effects package of R makes it easy to visualize the (additive) effect of X on the log-transformed 'mean' value of Y given X or the (multiplicative) effect of X on the 'mean' value of Y given X.