Interpretation of quadratic term in Poisson regression

Question

I fit a Poisson regression model where I used $x$ and $x^2$ to predict $y$.

Let's say the coefficients of $x$ and $x^2$ are $\beta_1 = 0.8$ and $\beta_2 = -0.1$.

Exponentiating the coefficient (${\beta_1}$) gives us the multiplicative factor by which the mean count changes when we increase $x$ by one unit: $e^{\beta_1}$ = 2.23. So increasing x by one unit changes the mean of $y$ by a factor of 2.23. This factor is constant over all $x$.

According to this answer, "$e^{\beta_2}$ would be called a ratio of ratio of rates comparing groups differing by 1 unit differing by 1 unit of $X$."

I tried this out with a toy example below with the parameters from above (${\beta_1} = 0.8$ and ${\beta_2} = -0.1$) and indeed the ratio is not constant because of the nonlinear term but the ratio of the ratio is constant. But the ratio of ratios is not equal to $e^{\beta_2}$ but to $e^{(2\beta_2)}$. I think this relates to this part of the referenced answer above "But if you do a difference in differences for $(E[Y|X=x+2] - E[Y|X=x+1]) - (E[Y|X=x+1] - E[Y|X=x]) = 2\beta_2$. So basically the $\beta_1$ is the tangent slope of the quadratic curve at the origin, and $\beta_2$ is a quadratic slope."

With all this, I am not quite sure how to interpret a one-unit change of $x$ on the mean of $y$. Would it be correct to say that for each unit increase in $x$, the linear term ${\beta_1}$ changes the mean of $y$ by a factor of 2.23 while the nonlinear term ${\beta_2}$ leads to a decrease of this factor by $e^{(2\beta_2)}$ = 0.8187 for each increase in $x$. I don't think this is correct because the effect is dependent on the value of $x$.

Is there a way to express the effect on a one-unit increase in $x$ on the mean of $y$ that is independent of $y$ or is this not possible?

And if this is not possible, what is the correct way to specify a change from let's say $x = 1$ to $x = 2$ for this toy example? "Increasing $x$ from 1 to 2 increases the mean of $y$ by a factor of 1.6487"?

x	predicted y	ratio	ratio of ratio
0	1
1	2.01375271	2.01375271
2	3.32011692	1.64872127	0.81873075
3	4.48168907	1.34985881	0.81873075
4	4.95303242	1.10517092	0.81873075
5	4.48168907	0.90483742	0.81873075

Answer 1

You can assemble two useful pieces of information to arrive at the interpretation of the linear and quadratic term in a Poisson GLM:

1. As you already stated, $\exp(\beta)$ for any first-order coefficient $\beta$ in a Poisson model can be interpreted as a relative rate of the Poisson process comparing groups differing by 1 in the corresponding variable being modeled.

2. In a linear model with identity link, like OLS, the interpretation of the quadratic term $\beta_2$ is readily seen as the rate of change of the effect, that is for every unit difference in $X$ the slope relating $X$ and $Y$ changes by a factor of $\beta_2$. Similarly, the linear term $\beta_1$ can be seen as the instantaneous rate of change relating $X$ and $Y$ when $X=0$. That is to say, since the shape of the predicted trend between $X$ and $Y$ is quadratic, $\beta_1$ is the slope of the tangent curve at $X=0$.

So, in your model with $\beta_1 = 0.8$ and $\beta_2 = -0.1$, comparing groups differing by 1 unit "near 0" (ignoring the issue of prediction at the means), has a relative rate ratio of 2.22, and groups differing by 1 unit have a ratio of relative rate ratios of 0.91