Model Fit and Diagnostic Statistics

The REG Procedure

Model Fit and Diagnostic Statistics

This section gathers the formulas for the statistics available in the MODEL, PLOT, and OUTPUT statements. The model to be fit is $Y=X{\beta}+ {\epsilon}$ , and the parameter estimate is denoted by b = (X'X)^-X'Y. The subscript i denotes values for the ith observation, the parenthetical subscript (i) means that the statistic is computed using all observations except the ith observation, and the subscript jj indicates the jth diagonal matrix entry. The ALPHA= option in the PROC REG or MODEL statement is used to set the $\alpha$ value for the t statistics.

Table 55.5 contains the summary statistics for assessing the fit of the model.

Table 55.5: Formulas and Definitions for Model Fit Summary Statistics

${MODEL Option} \ {or Statistic}$	Definition or Formula
n	the number of observations
p	the number of parameters including the intercept
i	1 if there is an intercept, 0 otherwise
$\hat{\sigma}^2$	the estimate of pure error variance from the SIGMA= option or from fitting the full model
SST₀	the uncorrected total sum of squares for the dependent variable
SST₁	the total sum of squares corrected for the mean for the dependent variable
SSE	the error sum of squares
MSE	$\rule{0in}{0cm}\displaystyle \frac{SSE}{n-p}$
R²	$\rule{0in}{0cm}1 - \displaystyle \frac{SSE}{{SST}_i}$
ADJRSQ	$\rule{0in}{0cm}1 - \displaystyle\frac{(n-i)(1-R^2)}{n-p}$
AIC	$\rule{0in}{0cm}\displaystyle n \ln ( \frac{SSE}n ) + 2p$
BIC	$\rule{0in}{0cm}\displaystyle n \ln ( \frac{SSE}n ) + 2(p+2)q - 2q^2 { where } q = \frac{n \hat{\sigma}^2}{SSE}$
CP (C_p)	$\rule{0in}{0cm}\displaystyle \frac{SSE}{\hat{\sigma}^2} + 2p - n$
GMSEP	[( MSE(n+1)(n-2))/(n(n-p-1))] = [1/n] S_p(n+1)(n-2)
JP (J_p)	[(n+p)/n] MSE
PC	[(n+p)/(n-p)] (1 - R²) = J_p ( [n/( SST_i)] )
PRESS	the sum of squares of predr_i (see Table 55.6)
RMSE	$\sqrt{MSE}$
SBC	n ln( [ SSE/n] ) + p ln(n)
SP (S_p)	[ MSE/(n-p-1)]

Table 55.6 contains the diagnostic statistics and their formulas; these formulas and further information can be found in Chapter 3, "Introduction to Regression Procedures," and in the "Influence Diagnostics" section. Each statistic is computed for each observation.

Table 55.6: Formulas and Definitions for Diagnostic Statistics

${MODEL Option} \ {or Statistic}$	Formula
PRED ( $\displaystyle \hat{Y}_i$ )	X_ib
RES (r_i)	$Y_i - \hat{Y}_i$
H (h_i)	x_i(X'X)^-x_i'
STDP	$\sqrt{h_i\hat{\sigma}^2}$
STDI	$\sqrt{(1+h_i)\hat{\sigma}^2}$
STDR	$\sqrt{(1-h_i)\hat{\sigma}^2}$
LCL	$\displaystyle \hat{Y}_i-t_{\frac{\alpha}2}$ STDP
LCLM	$\displaystyle \hat{Y}_i-t_{\frac{\alpha}2}$ STDI
UCL	$\displaystyle \hat{Y}_i+t_{\frac{\alpha}2}$ STDP
UCLM	$\displaystyle \hat{Y}_i+t_{\frac{\alpha}2}$ STDI
STUDENT	[(r_i)/( STDR_i)]
RSTUDENT	$\displaystyle \frac{r_i}{{\hat{\sigma}}_{(i)}\sqrt{1-h_i}}$
COOKD	[1/p] STUDENT²([ STDP/( STDR²)])
COVRATIO	$\displaystyle \frac{{det}({\hat{\sigma}^2}_{(i)}(X_{(i)}'x_{(i)})^{-1}}{{det}({\hat{\sigma}^2}(X'X)^{-1})}$
DFFITS	$\displaystyle \frac{(\hat{Y}_i-\hat{Y}_{(i)})}{({\hat{\sigma}}_{(i)}\sqrt{h_i})}$
DFBETAS_j	$\displaystyle \frac{b_j-b_{(i)j}}{{\hat{\sigma}}_{(i)}\sqrt{(X'X)_{jj}}}$
PRESS(predr_i)	[(r_i)/(1-h_i)]

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