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 ``` `

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#### Best Answer

The formula for the predicted mean value in your regression is

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

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.

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