Summary of Fit for Linear Models

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Fit Analyses

Summary of Fit for Linear Models

The Summary of Fit table for linear models, shown in Figure 39.11, includes the following:

Mean of Response: is the sample mean, ${\overline{y}}$ , of the response variable.
Root MSE: is the estimate of the standard deviation of the error term. It is calculated as the square root of the mean square error.
R-Square: R², with values between 0 and 1, indicates the proportion of the (corrected) total variation attributed to the fit.
Adj R-Sq: An adjusted R² is a version of R² that has been adjusted for degrees of freedom.

Figure 39.11: Summary of Fit, Analysis of Variance Tables for Linear Models

With an intercept term in the model, R² is defined as

R² = 1-(SSE / CSS)

where ${\rm CSS} = \sum_{i=1}^n{( y_{i}-{\overline y})^2}$ is the corrected sum of squares and ${\rm SSE} = \sum_{i=1}^n{( y_{i}-\hat{ y})^2}$ is the sum of squares for error.

The R² statistic is also the square of the multiple correlation, that is, the square of the correlation between the response variable and the predicted values.

The adjusted R² statistic, an alternative to R², is adjusted for the degrees of freedom of the sums of squares associated with R². It is calculated as

Adj R² = 1-[(SSE / (n-p))/(CSS / (n-1))] = 1-[(n-1)/(n-p)] (1- R²)

Without an intercept term in the model, R² is defined as

R² = 1-(SSE / TSS)

where ${\rm TSS} = \sum_{i=1}^n{y_{i}^2}$ is the uncorrected total sum of squares.

The adjusted R² statistic is then calculated as

Adj R² = 1-[(SSE / (n-p))/(TSS / n)] = 1-[n/(n-p)](1- R²)

Note Other definitions of R² exist for models with no intercept term. Care should be taken to ensure that this is the definition desired.

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