How do I interpret generalized linear regression with continues independent variable with Gaussian family and log link

Question

I am running generalized linear regression Gaussian family and log link.

Independent variable is Time (continues variable).

Dependent variables:

1. years of practice (continues variable). Interpretation something like: each unit increase of years of practice gives increase of what kind of unit? of time?

2. impact (binary variable, with categories "type one" or "type two"). Interpretation something like: "impact type 2" in comparison to "impact type 2" gives increase of what kind of unit? of time?

How do I interpret results?

Answer 1

Let $Y$ be Time (your independent variable), $P$ be years of practice, and $I$ be impact (0 for type one, 1 for type 2).

Then, a glm with Gaussian family and log link fits

$$\log(\mathbb{E}(Y))=\beta_0 + \beta_pP + \beta_II$$

Exponentiating, we get

$$\mathbb{E}(Y) = e^{\beta_0 + \beta_1P + \beta_2I}$$

With a Gaussian family, our model is $$ Y \sim N(e^{\beta_0 + \beta_1P + \beta_2I}, \sigma^2)$$

Note that we can break up the exponential term:

$$e^{\beta_0 + \beta_1P + \beta_2I}=e^{\beta_0}e^{\beta_1P}e^{\beta_2I}$$

The relationship between Y and the predictors is no longer additive as it would be with identity link. Instead, it is multiplicative. For example, a unit increase in $P$ mulitplies the value of the response by $\beta_1$, rather than increasing the response by $\beta_1$. Likewise, a person with impact of type 2 ($I=1$) will have $\beta_2$ times as many years $Y$ as a person with impact of type 1 who has the same number of years in practice $P$.

When performing inference on the coefficients, our estimates for the $\beta$'s do not have an exact Normal distribution, as they do with Gaussian family and identity link. However, there are several methods for testing coefficients that can be run on any glm. @gung's answer gives an excellent overview of the three types.