Suresh Joshi's Blog

Covariance: It is a measure of the degree to which two random variables (X, Y) change together.

Cov_xy = Sum of products of errors / (n-1)

r² (Coefficient of determination):

Indicates how well data points fit a line or curve. It is mainly used in models to predict future outcomes or test hypotheses on the basis of other related information.

The squared correlation coefficient (r²) is the proportion of variance in Y that can be accounted for by knowing X. Conversely, it is the proportion of variance in X that can be accounted for by knowing Y. Further, I can say that it is a statistic which indicates the percentage change in the amount of the outcome variable (dependent) that is ‘explained by’ the changes in the predictor variable (independent). In other words, shared / common variance, means how much percentage of the relationship can be explained by the regression (or predicted) and rest is unexplained. It is the indicator of accuracy of a prediction. We can also call it a proportion of the variance in outcome variable (Y) that is predictable from the predictor variable (X) or it is the fraction of the variation in Y that is explained by least-squares regression of Y on X.

1.     The coefficient of determination ranges from 0 to 1 (proportion or percentage, cannot be more than 1 or 100% respectively).

2.     It is important to note that a high coefficient of determination does not guarantee that a cause-and-effect relationship exists. However, a cause-and-effect relationship between the independent variable and the dependent variable will result in a high coefficient of determination.

3.     An R² of 0 means that the dependent variable cannot be predicted from the independent variable.

4. An R² of 1 means the dependent variable can be predicted without error from the independent variable.

These notes are written by S C Joshi during EPSY 635 Course, Fall 2015, Texas A&M University. Acknowledgements to Dr. Bob Hall, Professor, EPSY, Texas A&M University for his assistance in understanding these terms during the course