Question

Adjusted Coefficient of Determination For Exercise 2, why is it better to use values of adjusted \({R^2}\)instead of simply using values of \({R^2}\)?

Short answer

When more variables are added, the unadjusted\({R^2}\)increases (or stays the same). But the adjusted\({R^2}\)adjusts according to the number of variables and the sample size. It increases/decreases depending on whether the added variable explains the variation in the model better than the previous one.

The unadjusted \({R^2}\)incorrectly suggests that the best multiple regression equation is generated by adding all available variables. On the other hand, the adjusted \({R^2}\) is more helpful in eliminating factors that should not be included as they add nothing useful to the original model.

Step-by-step solution

Given information

A regression equation is computed to predict the weight of a bear (in lb) by using the variables “weight”, “length,” and “chest size” and the adjusted\({R^2}\)value is noted.

Further, another predictor variable “neck size” is added to the equation and the adjusted \({R^2}\)is noted again.

Step 2: \({R^2}\) vs. adjusted \({R^2}\)

It is always better to consider the value of adjusted \({R^2}\) as compared to the value of simple \({R^2}\) whenever another predictor variable is added to a given regression equation because of the following factors:

• The unadjusted value of \({R^2}\) always increases if a new predictor variable is added to an equation, irrespective of whether the new variable significantly adds to the explanation of the response variable.
• On the other hand, the value of adjusted \({R^2}\) increases for the new equation only when the new variable significantly adds to the explanation of variation in the response variable. It can decrease if the variable added is useless.

Since the value of adjusted \({R^2}\)considers the number of variables as well as the sample size, it helps in eliminating the variables that do not add anything to the model. Therefore, it helps to correctly identify the best available regression model.

Here, a new variable, “neck size,” is added to the given regression equation.

If the value of unadjusted \({R^2}\) is considered, it will either increase or remain the same. Thus, it cannot be determined whether the new variable significantly enhances the original model.

When the value of unadjusted \({R^2}\) is considered, it increases. If the new variable “neck size” is added, it explains the variation in the model better. Thus, it should be used for future purposes.

Therefore, the value of adjusted\({R^2}\)is better than that ofsimple\({R^2}\).