# Interpretation regression intercept when only numerical predictors are standardized

I have a mixed-effects model in which the dependent variable is inverted (1/y) and the numerical predictors are standardized, but the binary variables are not.  (Intercept) 3.48 binary variable 1 0.39 binary variable 2 0.31 ... standardized variable 1 -2.06 standardized variable 2 -0.07 ...  I'm trying to figure out how to interpret the regression intercept, because in my analysis I have to multiply standardized variable 1 with its upper bound (about 1.77 standard deviations) and add it to the constant term giving me a negative value (-0.168). My response cannot be negative, so this does not make sense.

Does anyone know how I should interpret the 3.48 value?

• How are the binary variables coded (0 and 1, -1 and 1, etc.)? – Ian_Fin Aug 8 '16 at 14:42
• They are coded as (0,1) – Jeroen Aug 8 '16 at 14:45
• Although your response is never negative, there is nothing to preclude a negative fit for the response in the model, nor should that be considered an error--assuming the model is appropriate or reasonably accurate overall. If a concern about negative fits is your basic problem, then look at some of our threads on non-negative responses, GLMs, logistic regression, and other procedures to handle such situations. – whuber Aug 8 '16 at 14:59

In a regression equation, the coefficient for the intercept is the estimated value of the outcome when all of the predictors are equal to zero.

In a case like this where continuous predictors have been standardized but binary predictors have been coded as 0 and 1, the intercept represents the estimated value of the outcome for the mean value of the continuous predictor(s) and whatever level of the binary predictor(s) is coded as 0.

To make it a little more concrete, imagine I have two predictors, gender (male: 0, female:1) and age (ranging from 20-40, mean of 30). If I standardise age, and fit the regression model and get the following coefficients

predictor beta
Intercept 5
Gender    10
Age       1


then the intercept tells us that for a male of average age (i.e. 30), the estimated outcome would be 5 ($5 + 0\times10 + 0\times1$). For a female of average age, the estimated outcome would be 15 ($5 + 1\times10 + 0\times1$).

As the number of predictors increase, interpretation may become a little more difficult, but the basic point remains. The intercept is the estimated value of $y$ when all $x$s are zero.

• Thanks for the answer. What I forgot to mention in my question is that the dependent variable yis inverted (1/y). What I read is that this transformation has the effect of reversing the order of your scores. Does it also change the way the intercept should be interpreted? – Jeroen Aug 10 '16 at 8:06
• @Jeroen No, the intercept is still just the value of the transformed outcome when all predictors are 0. – Ian_Fin Aug 10 '16 at 8:27