Incident rate ratios with log-transformed variables in Poisson regression

Question

I'm in a bit of a mess with interpreting the output of a Poisson regression model with log-transformed predictors. The predictors are counts, and the log transforms them for the linear part of model.

glm(y ~ log(x1) + log(x2), data=dt, family=poisson(link="log"))

I've been asked to present them as incidence rate ratios (IRR), which are the exponentiated coefficients: exp(coef), rather than coefficeints themselves. These are more intuitive to my audience.

I've found a number of other posts on interpreting coefficients/poisson regression:How to interpret coefficients in a Poisson regression?, How to interpret parameter estimates in Poisson GLM results and Interpreting coefficients for Poisson regression . None of these relate to transformed variables.

Does the log transformation pose any problems with the usual IRR interpretation. Can it be exponentiation out in some manner? What would this do to the IRRs?

As it stands my IRRs would be for Intercept, log(x1) and log(x2).

Answer 1

It seems this didn't generate any interest, so I'll answer with what I've found after a few weeks.

IRRs are multiplicative, so: 'holding all other parameters constant, the change in y is y * IRR(x) for each increase of 1 in x.'

In the case of transformed variables like above, the transformation remains in the IRR:

for log(x1): y = y * IRR(log(x1)) for each increase in 1 in x

i.e. the IRR represents log(x1), and exponentiation to x1 would not be helpful, as it's not the relationship fitted by the model.

Answer 2

Let's write down the regression equation for your model: $$ \log(E(Y\mid X_1, X_2)) = \beta_0 + \beta_1\log(X_1) + \beta_2\log(X_2).$$

As usual, you can express $\beta_1$ as the difference of $\log(E(Y\mid X_1 + c, X_2))$ and $\log(E(Y\mid X_1, X_2))$ for some constant $c$; if you hadn't transformed $X_1$ using a logarithm, you would simply take $c=1$, but now we have to find $c$ that satisfies the following:

\begin{align*} \beta_1 &= \log(E(Y\mid X_1 + c, X_2)) - \log(E(Y\mid X_1, X_2))\\ &= \beta_0 + \beta_1\log(X_1 + c) + \beta_2\log(X_2) - \beta_0 - \beta_1\log(X_1) - \beta_2\log(X_2)\\ &= \beta_1(\log(X_1 + c) - \log(X_1))\\ &= \beta_1\log\left(\frac{X_1 + c}{X_1}\right). \end{align*}

Therefore, we want $c$ such that $$ \frac{X_1+c}{X_1} = e.$$ By rearranging the last equation, you get $$c = X_1(e-1).$$

In other words, $\beta_1$ is the ratio of the expected outcome at $X_1+c=eX_1$ and $X_1$, i.e. it describes the change in rate for an $e$-fold increase in the value of the predictor $X_1$.

Of course, $e$-fold changes are maybe not as intuitive as 10-fold changes, so you may want to use a different base for the logarithm.