Introduction
Adjusted R2 is a statistical measure used to evaluate the performance of a linear regression model. This measure is used to compare the predictive power of two or more regression models with different numbers of predictor variables. It is a commonly used metric to measure the goodness of fit of a regression model. The adjusted R2 is calculated by subtracting the ratio of the residual sum of squares to the total sum of squares from one. The adjusted R2 allows for a more accurate comparison of models with different numbers of predictors.

Background
The coefficient of determination, R2, is a measure of how well a linear regression model fits a given data set. It is the proportion of the variation in the dependent variable that can be explained by the predictor variables in the model. However, when comparing two or more models with different numbers of predictor variables, it is necessary to use an adjusted coefficient of determination. The adjusted R2 is a more accurate measure of the predictive power of the model because it adjusts for the number of predictor variables that are included in the model.

Calculation
The adjusted R2 is calculated by subtracting the ratio of the residual sum of squares to the total sum of squares from one. The residual sum of squares is the sum of the squared differences between the observed values of the dependent variable and the predicted values of the dependent variable. The total sum of squares is the sum of the squared differences between the observed values of the dependent variable and the mean of the dependent variable. The adjusted R2 is then calculated as follows:

Adjusted R2 = 1 – (Residual Sum of Squares / Total Sum of Squares)

Interpretation
The adjusted R2 is interpreted in the same way as the R2. A higher adjusted R2 indicates a better fit of the model to the data. The adjusted R2 should be used when comparing two or more models with different numbers of predictor variables. This is because the adjusted R2 takes into account the number of predictors in the model and is therefore a more accurate measure of the predictive power of the model.

Conclusion
The adjusted R2 is a measure of the goodness of fit of a regression model. It is used to compare the predictive power of two or more models with different numbers of predictor variables. The adjusted R2 is calculated by subtracting the ratio of the residual sum of squares to the total sum of squares from one. The adjusted R2 allows for a more accurate comparison of models with different numbers of predictors.

References
Hair, J.F., Black, W.C., Babin, B.J., Anderson, R.E., & Tatham, R.L. (2010). Multivariate Data Analysis (7th ed.). Upper Saddle River, NJ: Prentice Hall.

Kutner, M.H., Nachtsheim, C.J., Neter, J., & Li, W. (2004). Applied Linear Statistical Models (5th ed.). Boston, MA: McGraw-Hill.

Maxwell, S.E., & Delaney, H.D. (2004). Designing Experiments and Analyzing Data (2nd ed.). Belmont, CA: Thomson/Wadsworth.

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