The linear function Y = b0 + b1*X1 + b2*X2 + ... + bk*Xk + e[i] is the correct model, where Y[i] is the ith observed value of Y, Xj[i] is the ith observed value of the jthX variable, and e[i] is the error term. Equivalently, the expected value of Y for a given value of X is Y = b0 + b1*X1 + b2*X2 + ... + bk*Xk. The intercept is b0, the expected value of Y when the value for each X variable is 0.
The Xj variable (predictor variable) values are fixed (i.e., none of the Xj is a random variable).
The Y variable (response variable) observations are independent.
The variable Y is normally distributed with the same variance as the e[i]. For a given set of X variable values, the variable Y has constant mean.
The normality assumption is required for
hypothesis tests, but not for estimation.
The X variables are also known as the independent variables.
The Y variable is also known as the dependent variable.
The coefficients are bj, the amount by which the expected value of Y increases when Xj increases by a unit amount, when all the other X variables are held constant. This interpretation of the coefficients does not hold if some of the X variables are functions of the others, such as an interaction term Xj*Xk.
Note that it is not assumed that the X variables are independent of each other.
Ways to detect before performing the multiple linear regression whether your data violate any assumptions.
Ways to examine multiple linear regression results to detect assumption violations.
Possible alternatives if your data or multiple linear regression results indicate assumption violations.
To properly analyze and interpret the results of multiple linear regression, you should be familiar with the following terms and concepts:
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