# What do you call the coefficients other than the intercept in a linear regression

I'm trying to refer to the coefficients other than the intercept. Is there a word/jargon that refers to coefficients other than the intercept? (I'm currently calling them 'other coefficients', which is mildly descriptive in the context, but not ideal.)

• Some people refer to the other coefficients as "slopes". Others refer to them as "effects" or "associations", depending on the type of study. – Isabella Ghement Mar 24 at 2:26
• I call them coefficients. I call the intercept the intercept. I thought this was standard. – The Laconic Mar 24 at 2:41
• "beta coefficients" might work. I'm not sure. – Sal Mangiafico Mar 24 at 3:34

## 1 Answer

Consider the multiple linear regression model

$$y=X\beta+\varepsilon$$

Here $$y$$ is the response vector, $$X$$ is the design matrix with (say) $$p+1$$ columns (the first column in this matrix is a vector of all ones corresponding to the intercept), $$\beta=(\beta_0,\beta_1,\ldots,\beta_p)^T$$ is the vector of regression coefficients and $$\varepsilon$$ is the random error.

Without resorting to vectors, we can write the model as

$$y=\beta_0+\beta_1 x_1+\beta_2 x_2+\cdots+\beta_p x_p+\varepsilon$$

In this model with $$p$$ regressors or predictor variables, the parameters $$\beta_j,\,j=0,1,\ldots,p$$ are simply called the regression coefficients. In fact, $$\beta_j$$ represents the expected change in the response $$y$$ per unit change in $$x_j$$ when all of the remaining regressor variables $$x_i\,(i\ne j)$$ are held constant. For this reason, the parameters $$\beta_j,\,j=1,2,\ldots,p$$ are often called partial regression coefficients. The parameter $$\beta_0$$ is of course separately called the intercept.

In simple linear regression we have $$p=1$$ and the regression coefficient $$\beta_1$$ is simply called the slope.

• I am happy with anyone referring to all of them -- all the $\beta$ elements -- as coefficients. Another name for intercept is just the constant. – Nick Cox Mar 24 at 7:47
• Gradient for slope is common also. – Nick Cox Mar 24 at 8:11
• – StubbornAtom Mar 24 at 8:22