How to interpret parameters of GLM output with Gamma log link

I am having tough time interpreting the output of my GLM model with Gamma family and log link function. My dependent variable if "Total Out-of-pocket cost" and my independent variables are "Private health insurance(yes/no)", "year of diagnosis" and "interaction with private health insurance and year". I am trying to find the trend over a period of 4 years in Out-of-pocket costs depending on the insurance status. Also please tell me what does +ve and -ve coeffiecients mean.

My code and the output is as mentioned below.

Thank you very much for the help.

Call:
glm(formula = total_oop ~ private_insur2 + year + private_insur2 *
year, family = Gamma(link = "log"), data = dfq5.1)

Deviance Residuals:
Min       1Q   Median       3Q      Max
-3.2932  -1.2051  -0.5681   0.2311   4.8237

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)            -278.75702  128.19627  -2.174   0.0298 *
private_insur2Yes       166.72653  150.45167   1.108   0.2680
year                      0.14184    0.06370   2.227   0.0261 *
private_insur2Yes:year   -0.08207    0.07475  -1.098   0.2725
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for Gamma family taken to be 1.911381)

Null deviance: 2631.2  on 1399  degrees of freedom
Residual deviance: 2098.6  on 1396  degrees of freedom
AIC: 24676

Number of Fisher Scoring iterations: 7
$$$$


The formula for the predicted mean value in your regression is

$$\textrm{total_oop} = \exp \left(\beta_0 + \beta_1 \cdot \textrm{PI} + \beta_2 \cdot\textrm{year} + \beta_3 \cdot \textrm{PI} \cdot\textrm{year} \right)$$

where PI is a dummy variable equal to 0 if someone doesn't have private insurance and 1 if they do.

• The intercept ($$\beta_0$$) is the expected log of OOP costs for someone without insurance in year 0 (!!). The very small value (-278) is pretty much nonsense, it means that if you extrapolated back to the year 0 you'd expect a non-privately-insured person to be paying about $$\exp(-278) \approx 10^{-121}$$ dollars (?or whatever your unit of cost is).
• The private insurance differential $$\beta_1$$ is a huge positive number, but it also applies in year 0, so it's also somewhat nonsensical. The value of 166 means you'd expect someone with private insurance to be paying about $$\exp(166) \approx 10^{72}$$ times as much for insurance as someone without, in year zero. (Put another way, the expected cost for someone privately insured in year zero is about $$\exp(-278+166) \approx 10^{-49}$$ dollars.)

These coefficients will be much easier to interpret if you center your year variable, by subtracting the minimum value or the mean (e.g. let your year variable run from 0 to 9 instead of 2010 to 2019).

The other two parameters are a little easier since they don't depend on the zero-point of the year variable.

• $$\beta_2$$ is the expected difference in log-costs per year for a non-privately-insured person: 0.142 means a multiplicative increase of $$\exp(0.142) \approx 1.153$$ per year (small values of $$\beta$$ can be read approximately as proportional differencs).
• $$\beta_3$$ is the difference in slope between privately insured and non-privately-insured people: privately insured people's OOP costs increase slower than non-privately-insured people. They increase at a multiplicative rate of $$\exp(0.142-0.082) \approx 1.06$$ per year.
• Hi @Ben Bolker, Thank you very much for your reply and the explanation. Could you please elaborate on your suggestion of "center" the year variable and how to do it? Thank you, Pavan Oct 15, 2019 at 23:34
• change year to year-min(year) or year-mean(year) (or derive a new variable called c_year` with that value. See e.g. Schielzeth 2010 Methods in Ecology and Evolution "Simple means to improve the interpretability of regression coefficients" Oct 15, 2019 at 23:53
• Thank you very much. It was useful and the intercepts make sense now. Can you suggest me some links to understand how to interpret residual plots in GLM. Oct 22, 2019 at 1:17
• Hello Ben, thank you for your answer. I would like to know what is the role of "Gamma" here? If we put "gaussian", what would be the difference? Thank you Jun 15, 2020 at 9:16
• The response (conditional) distribution determines the variance-mean relationship, in this case that the variance is proportional to the mean ... you might need to read up on generalized linear models ... Jun 15, 2020 at 15:36