Derive coefficient of determination

To derive the coefficient of determination, let's start by understanding some key concepts.

The coefficient of determination, often denoted as R^2, is a statistical measure that represents the proportion of the variance in the dependent variable that can be predicted from the independent variables in a regression model. In simpler terms, it assesses how well the independent variables explain the variations in the dependent variable.

The coefficient of determination can be obtained by squaring the Pearson correlation coefficient (r) between the actual values of the dependent variable and the predicted values from the regression model. The correlation coefficient measures the linear relationship between two variables and ranges between -1 and 1, where -1 represents a perfectly negative linear relationship, 1 represents a perfectly positive linear relationship, and 0 represents no linear relationship.

To derive the coefficient of determination:

1. Calculate the Pearson correlation coefficient (r) between the actual values of the dependent variable and the predicted values from the regression model. This can be done using the following formula:

r = (Σ((xᵢ - ȳ)(yᵢ - ȳ))) / √(Σ(xᵢ - ȳ)²) * √(Σ(yᵢ - ȳ)²),

where xᵢ represents the predicted values and yᵢ represents the actual values, and ȳ represents the mean of the actual values.

2. Square the correlation coefficient:

R^2 = r².

This will give you the coefficient of determination, which represents the proportion of the variance in the dependent variable that can be predicted from the independent variables in the regression model.

It is important to note that the coefficient of determination ranges between 0 and 1, where 0 indicates that the independent variables do not explain any variance in the dependent variable, and 1 indicates that the independent variables explain all the variance in the dependent variable.