Management Guide

In the regression models, what is the significance of adjusted r square? Why is it needed once you calculate the coefficient of determination (r square)? Furthermore how are r and r square different?

( Bishal Shrestha says : )

I would like to proceed from the last part of the question.

The objective of any regression model is to predict the unknown variable(response or dependent variable) based on the known variable (explanatory or independent variable) by trying to determine a relationship between the two. In a regression model, r (which is the coefficient of correlation) is used to determine the strength of the relationship between the response and the explanatory variables e.g. the annual sales volume (response) and the annual spending in advertising (explanatory). The value of r also tells us the nature of the linear relationship w.r.t. positive or negative linear relationship i.e. whether the values of the dependent variable increases with an increase in the value of the independent variable  (positive linear relationship e.g.  increase in  the number of trade executions with increase in the number of inquiries per day) or whether the value of the dependent variable decreases with an increase in the value of the independent variable (negative linear relationship e.g. decrease in annual sales of a certain product for every increase in price of that product). An r value with a positive sign denotes a positive linear relationship and an r value with a negative sign denotes a negative linear relationship between the variables. However the r value does not denote whether the relationship is causal or not. It only tells us how strong a relationship is between the two variables of interest.

On the other hand, the coefficient of determination (r square) tells us what proportion or percentage of the variation of the dependent variable can be explained by the independent variable which is also to say, how strong a relationship exists between the two variables in question. This however does not say anything about the nature of the relationship.

Having said this, however, when the number of independent variables increases in the regression model (multiple regression) e.g. annual sales vs advertising spending, promotional expenses, price, spending on training of sales team, etc., things become a little complicated with respect to comparison of models with different number of independent variables. As the number of independent variables increase in a model, the coefficient of multiple determination, by its very nature, increases so if we are to compare a model (trying to predict annual sales) which has three independent variables and another model (which also tries to predict annual sales) but has five variables, and if we take the coefficient of multiple determination as the only comparison factor, then there is always a bias towards the model with the more number of independent variables and so taking the absolute value of the coefficient of multiple determination may give erroneous interpretations. So in order to to avoid this bias, the coefficient of multiple determination of both the models need to be adjusted to reflect the number of independent variables in each model which will then give a more true indicator for comparison. So Adjusted r square (adjusted coefficient of multiple determination) of both these models can then be used to judge which is the better model for predicting the response variable of interest.


(Bishal Shrestha currently teaches Data Analysis and Modeling II to the MBA students at Ace Institute of Management. He can be contacted at bishal@ace.edu.np)