Interpreting Poisson output in R [duplicate]

I am having some issues with interpreting the results from a Poisson log linear model done in R. I will give my thoughts and it would be great if somebody would be kind enough to expand on it. I just need help with interpreting the coefficients.

I saw some interpretations online but almost all of use use the main effects or just one effect to explain. Also, the answers on stack exchange are not so simple that a layman could understand. Thank you in advance.

The data is from a paper titled "A Microeconometric Model of the Demand for Health Care and Health Insurance in Australia"

This is the back transformed data with intervals.

I drew some preliminary inferences,

We can infer from this that the expected number of visits by a doctor to a female at age zero is 0.23 (the intercept) with CI’s 0.195 and 0.271.

For every one extra male, the expected number of visits by a doctor increases by 0.45 with CI’s 0.349 and 0.576.

As age increases by one unit, the number of visits by a doctor increases by 1.009 for a female with CI’s 1.006 and 1.012.

Similarly, as age increases by one unit, the number of visits by a doctor if the patient is a male increases by 1.012 with CI’s 1.007 and 1.017.

Is this correct?

• The coefficients are given on the log scale. You must convert them back. – SmallChess Apr 6 '17 at 2:49
• @SmallChess Done. Modified the question – Charles Apr 6 '17 at 16:11
• I think you will find the information you need in the linked thread. Please read it. If it isn't what you want / you still have a question afterwards, come back here & edit your question to state what you learned & what you still need to know. Then we can provide the information you need without just duplicating material elsewhere that already didn't help you. – gung - Reinstate Monica Apr 6 '17 at 18:34
• If you are wondering about how to interpret interaction terms, that is actually orthogonal to whether the model is a Poisson GLM or not. You can find information on that in many places on Cross Validated, eg, here: Interpretation of interaction term. – gung - Reinstate Monica Apr 6 '17 at 18:38

Since it's a Poisson model, the expected value of the dependent variable is related to the independent variables by the log link, which is to say

$E(y) = \exp(\beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_1 x_2)$

where here, x1 = 0 if female and 1 if male, x2 = age, and the $\beta_0$ to $\beta_3$ are the estimated coefficients in the order shown in the R output.

The three independent variables here are all equal to zero when you have a female with age zero. So the expected number of visits for a female with age zero is $\exp(-1.466168) = 0.23$ That's the meaning of the intercept. If you take its exponential, you get the baseline number of visits, where the baseline means that all the independent variables are set to zero.

The expected number of visits for a male with age zero is $\exp(-1.466168 - 0.801987) = 0.10$ or $\exp(-.801987) = 0.45$ times the expected number of visits for a female with age zero.

As you increase the age by one, the expected number of visits for a female increases by a factor of $\exp(0.009322) = 1.009$ or about 1%.

As you increase the age by one, the expected number of visits for a male increases by a factor of $\exp(0.009322 + 0.012186) = 1.022$ or about 2%.

So, overall, you expect about half the number of visits for newborn males compared to females, but the expected number of visits increases with age at about twice the rate it does for females.

The AIC isn't helpful in isolation. You'd compare it to the AIC of some alternative model. Roughly speaking, whichever model has a lower AIC has a better fit after adjusting for the number of parameters.

You can use the deviance to do a goodness-of-fit test; basically, whether whatever unexplained variation is due to the kind of random variation you'd expect from a Poisson distribution.

There isn't a closed-form solution for the parameters of the Poisson model in general; they have to be computed using numerical methods. The Fisher scoring iterations tell how many iterations the optimizer had to go through before the deviance (I think) was minimized to within some acceptable tolerance. You would probably only worry about this if the number of iterations were really high, which might point to a poorly-specified model (which you would probably spot from abnormally large parameter values and/or standard errors anyway).

• Have you answered about interactions? – SmallChess Apr 6 '17 at 3:35
• I don't think 1.022 is equal to 2%. – Charles Apr 6 '17 at 16:11
• @Charles, $1.022$ means a $2.2\%$ increase. – gung - Reinstate Monica Apr 6 '17 at 18:33
• @SmallChess, I did answer about interactions; in particular, I showed how the sex/age interaction enters the model and how to interpret the coefficient. Maybe I could have done a better job of it, but I didn't ignore it. – The Laconic Apr 7 '17 at 2:58
• @TheLaconic Thanks. I gave you +1 for your efforts. – SmallChess Apr 7 '17 at 3:07