Linearity assumption in Multiple Regression Analysis

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I am confused of the linearity assumption in Multiple Linear Regression. I dont know which of the two is correct?

• Is linearity would mean that EACH independent variable in the model should have linear relationship with the dependent? And, therefore, the linear relationship between each of the independent and dependent should be tested statistically.
• Is linearity would mean that the PREDICTED DEPENDENT and the OBSERVED DEPENDENT should have linear relationship? And, therefore, in testing the linearity one should just use the R-square and check if the regression ANOVA is significant?
  

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关键词：regression assumption regressio Analysis Multiple therefore testing between correct should

Unfortunately, there is not a lot of consistency in how authors talk about "linearity" in linear regression.  Some describe a regression model as linear only if the functional relationship between X and Y is linear.  For these folks, a model including both X and X-squared as predictors (to account for a U-shaped functional relationship) would likely be described as a polynomial regression.

But others emphasize that OLS linear regression models are "linear in the coefficients".  So even if the functional relationship is not  linear--e.g.,X and X-squared as predictors--it is still properly described as a linearregression model (provided it's linear in the coefficients).

The following web-page provides an interesting example: https://onlinecourses.science.psu.edu/stat501/node/235

The page title is "Polynomial Regression".  But notice what the author says in the second paragraph:
"As for a bit of semantics, it was noted at the beginning of the previous course how nonlinear regression (which we discuss later) refers to the nonlinear behavior of the coefficients, which are linear in polynomial regression. Thus, polynomial regression is still considered linear regression!"

Having said all that, I think what you're concerned about is the possibility of non-linear functional relationships.  One good way to check for that is by looking at residual plots.  When you run your model, save the residuals. (Several types of residuals are available, and you may want to look at more than just the raw residuals.  See the Help for details.)  Then make scatter-plots with Y = residual, and X = fitted value of Y, or perhaps X =an explanatory variable of particular interest.  Do a Google search on
<residual plots> or <analysis of residuals> etc to find more info.

Bruce Weaver Professor Lakehead University Canada

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