RSquares

The model with the highest \(R^2\) and adjusted \(R^2\) is the preferable of all candidate models The quadratic model is the preferable model in that case.

• This coefficient is a measure of how much of the variance in the observed values of the dependent variable (Y) can be explained by it's relationship to the independent variable (X).

• In other words, the coefficient of determination is the percent of the variation explained by the .

• The coefficient of determination is also called or .
• It can be computed by simply squaring the (Pearson) correlation coefficient value ().

• \(R^2\) is a measure of variation explained by regression.

• The following coefficient has a natural interpretation as amount of variability in the data that is explained by the regression fit: \(R^2 = SSLR/SST = 1 - SSR/SST\).

• A similar interpretation is given to the adjusted coefficient \(R^2_{adj}\) which is given by $R^2_{adj} = 1 - MSR/MST $; where MSR is the mean squared error due to residuals, and MST is the total mean squared error.

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• Model selection is the task of selecting a statistical model from a set of potential models, given data.
• In a multiple linear regression model, adjusted R square measures the proportion of the variation in the dependent variable accounted for by the independent variables.
  
• Adjusted R square is generally considered to be a more accurate goodness-of-fit measure than R square.
• The adjusted coefficient is accounting for the degrees of freedom used for each source of variation and is often a more reliable indicator of variability than \(R^2\). \(R^2_{adj}\) is always smaller than \(R^2\).
• \(R^2\) is a measure of variation explained by regression.

• The coefficient of determination \(R^2\) is the proportion of variability in a data set that is accounted for by the linear model.

• Equivalently \(R^2\) provides a measure of how well future outcomes are likely to be predicted by the model.

• For simple linear regression, it can be computed by squaring the correlation coefficient. It is not specifically defined that way.

• This relationship is co-incidental when there are just two variables.

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The adjusted R-square value is found on the summary output for a fitted model. It is called because it takes into account the number of predictor variables being used. The law of parsimony states the simplest model that adequately explains the outcomes is the best. The candidate model with the higher adjusted R squared is considered preferable.