11.3 Interpretation of parameters
Finally, we come to the discussion of parameters of a model. As mentioned earlier, each one of them represents the slope of the model. But there is more to the meaning of parameters of the model. Consider the coefficients of the previously estimated model:
## (Intercept) cyl disp hp drat wt ## 17.88963741 -0.41459575 0.01293240 -0.02084886 1.10109551 -3.92064847 ## qsec gear carb ## 0.54145693 1.23321026 -0.25509911
Each of the parameters of this model shows an average effect of each variable on the mileage. They have a simple interpretation and show how the response variable will change on average with the increase of a variable by 1 unit, keeping all the other variables constant. For example, the parameter for
wt (weight) shows that with the increase of weight of a car by 1000 pounds, the mileage would decrease on average by 3.921 miles per gallon, if all the other variables do not change. I have made the word “average” boldface three times in this paragraph for a reason. This is a very important point to keep in mind - the parameters will not tell you how variable will change for any specific observation. They do not show how it will change for each point. The regression model capture average tendencies and thus the word “average” is very important in the interpretation. In each specific case, the increase of weight by 1 will lead to different decreases (and even increases in some cases). But if we take the arithmetic mean of those individual effects, it will be close to the value of the parameter in the model. This however is only possible if all the assumptions of regression hold (see Section 15).