We know by now: multiple regression means more than one predictor.
We will move briskly through the basics of multiple regression,
emphasizing what is different from simple regression.
Background: In 1975, Congress reacted to the 1973 oil embargo imposed by the Organization of Petroleum Exporting Countries (OPEC) by establishing the Corporate Average Fuel Economy (CAFE) Program as part of the Energy Policy and Conservation Act. For the current political debate about CAFE, search the internet for "CAFE fuel efficieny".
The automative industry as well as car buyers have an interest in understanding what factors in cars affect fuel efficiency. The industry can approach this problem with engineering knowledge, but the car-buying public and policy makers may want to collect publicly available data and perform an empirical analysis. Even car manufacturers may be interested in an empirical analysis that includes competitors' products.
"Fit Model": Analyze > Fit Model > "GP1000M City":Y, "Weight(lb)":Add, "Horsepower":Add; Run Model
In the output focus on the sections "Summary of Fit" and "Parameter Estimates".
the average increase in y per unit increase in the predictor,
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Estimated mean fuel consumption in GP1000M
Compare the above regression onto Weight and Horsepower with the simple regression on Weight:
Estimated mean fuel consumption in GP1000M
These coefficients are different from those of the multiple regression:
Regression coefficients depend on what other predictors are present in the regression!!! |
Reason: Each coefficient describes an average change in the response per unit change in the predictor, holding the other predictors fixed.
Example: If Horsepower is dropped from the regression, it is no longer held fixed, and the coefficient of Weight changes.
Jargon: The coefficients of a regression with some predictors dropped are called "marginal coefficients". The coefficients of a regression with the larger set of predictors are called "partial coefficients". Simple regression coefficients are always "marginal".
For a deeper discussion of partial vs. marginal coefficients: see [slide 4-10]
In general: