# In a linear regression model with both categorical and continuous predictors, what is the interpretation of a categorical predictor coefficient?

Suppose we have a linear regression model where the response $Y$ is continuous and have $X_1, X_2$ as continuous predictor variables and $I_1, I_2, I_3, I_4$ as categorical predictors that refer to the seasons, i.e., $I_1$ is fall, $I_2$ is winter, etc.

Suppose we have:

$$Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + \beta_3I_1 + \beta_4I_2 + \beta_5I_3 + \epsilon$$

where $I_4$ was dropped to prevent multicollinearity.

Normally, the interpretation of $\beta_3$ in a model where there are ONLY categorical predictors is that it is the difference in effects between $I_4$ and $I_1$, and that the intercept is the case where $\beta_0$ is the effect of $I_4$.

However, in this model, how do we interpret $\beta_0$ and $\beta_3$? Do we set the continuous predictors to zero? Thanks.

Effect of Categorical Variable
In the absence of an interaction* with a continuous predictor, the effect of the categorical predictor is the difference with the intercept ($\beta_0$) at any given level of the continuous predictor. After all, if there is no interaction, they are presumed to affect the outcome independently of each other.

Do we set the continuous predictors to zero?

No, at least not necessarily.

*If there would be an interaction with a continuous predictor, you can only consider the effect of the interaction (i.e. the slope per category), due to the principle of marginality.

Edit
To put it more simple, $\beta_3$ is already the difference between $I_4$ and $I_1$. There is no need to take the continuous predictor into consideration. To whom downvoted this, care to elaborate?

• I am not sure that a variable season would be any different from four categories. It would still be treated the exact same way in the regression (i.e. a 4x4 dummy variable matrix with a column dropped due to multicolinearity) – JWH2006 Oct 10 '17 at 12:41
• So why then opt to include separate dummy variables? A single factor seems simpler to me. – Frans Rodenburg Oct 10 '17 at 14:25
• Depends on the audience? Some reviewers want the model spelled out, some are happy with just having a single variable represent levels. All in all, its mathematically equivalent. – JWH2006 Oct 10 '17 at 15:50
• Very well, I'll remove that part – Frans Rodenburg Oct 10 '17 at 21:24