The four conditions ("LINE") that comprise the multiple linear regression model generalize the simple linear regression model conditions to take account of the fact that we now have multiple predictors:

• Linear Function: The mean of the response, \(\mbox{E}(Y_i)\), at each set of values of the predictors, \((x_{1i}, x_{2i},...)\), is a Linear function of the predictors.
• Independent: The errors, \( \epsilon_{i}\), are Independent.
• Normally Distributed: The errors, \( \epsilon_{i}\), at each set of values of the predictors, \((x_{1i}, x_{2i},...)\), are Normally distributed.
• Equal variances (denoted \(\alpha^{2}\)): The errors, \( \epsilon_{i}\), at each set of values of the predictors, \((x_{1i}, x_{2i},...)\), have Equal variances (denoted \(\alpha^{2}\)).

An equivalent way to think of the first (linearity) condition is that the mean of the error, \(\epsilon_i\), at each set of values of the predictors, \((x_{1i},x_{2i},\dots)\), is zero. An alternative way to describe all four assumptions is that the errors, \(\epsilon_i\), are independent normal random variables with mean zero and constant variance, \(\sigma^2\).

As in simple linear regression, we can assess whether these conditions seem to hold for a multiple linear regression model applied to a particular sample dataset by looking at the estimated errors, i.e., the residuals, \(e_i = y_i-\hat{y}_i\).