Regression Diagnostics

The LOGISTIC Procedure

Regression Diagnostics

For binary response data, regression diagnostics developed by Pregibon (1981) can be requested by specifying the INFLUENCE option.

This section uses the following notation:

r_j, n_j: r_j is the number of event responses out of n_j trials for the jth observation. If events/trials syntax is used, r_j is the value of events and n_j is the value of trials. For single-trial syntax, n_j=1, and r_j=1 if the ordered response is 1, and r_j=0 if the ordered response is 2.
w_j: is the total weight (the product of the WEIGHT and FREQ values) of the jth observation.
p_j: is the probability of an event response for the jth observation given by $p_j=F(\alpha+{\beta}'x_j)$ , where F(.) is the inverse link function.
b: is the maximum likelihood estimate (MLE) of $(\alpha,{\beta}^')^'$ .
$\hat{V}_{b}$: is the estimated covariance matrix of b.
$\hat{p}_j,\hat{q}_j$: $\hat{p}_j$ is the estimate of p_j evaluated at b, and $\hat{q}_j= 1-\hat{p}_j$ .

Pregibon suggests using the index plots of several diagnostic statistics to identify influential observations and to quantify the effects on various aspects of the maximum likelihood fit. In an index plot, the diagnostic statistic is plotted against the observation number. In general, the distributions of these diagnostic statistics are not known, so cutoff values cannot be given for determining when the values are large. However, the IPLOTS and INFLUENCE options provide displays of the diagnostic values allowing visual inspection and comparison of the values across observations. In these plots, if the model is correctly specified and fits all observations well, then no extreme points should appear.

The next five sections give formulas for these diagnostic statistics.

Hat Matrix Diagonal

The diagonal elements of the hat matrix are useful in detecting extreme points in the design space where they tend to have larger values. The jth diagonal element is

$h_{jj}= \{ \widetilde{w}_j(1,x'_j) \hat{V}_{b}(1,x'_j)' & {Fisher-Scoring}\\hat{w}_j(1,x'_j) \hat{V}_{b}(1,x'_j)' & {Newton-Raphson} .$

where

$\widetilde{w}_j & = & \frac{w_j n_j}{\hat{p}_j\hat{q}_j[g'(\hat{p}_j)]^2} \ \hat... ...\hat{q}_j-\hat{p}_j)g'(\hat{p}_j)]} { (\hat{p}_j\hat{q}_j)^2 [g'(\hat{p}_j)]^3}$

g'(.) and g''(.) are the first and second derivatives of the link function g(.), respectively.

For a binary response logit model, the hat matrix diagonal elements are

$h_{jj} = w_jn_j\hat{p}_j\hat{q}_j (1, x_j')\hat{V}_{b} ( 1 \ x_j )$

If the estimated probability is extreme (less than 0.1 and greater than 0.9, approximately), then the hat diagonal may be greatly reduced in value. Consequently, when an observation has a very large or very small estimated probability, its hat diagonal value is not a good indicator of the observation's distance from the design space (Hosmer and Lemeshow 1989).

Pearson Residuals and Deviance Residuals

Pearson and Deviance residuals are useful in identifying observations that are not explained well by the model. Pearson residuals are components of the Pearson chi-square statistic and deviance residuals are components of the deviance. The Pearson residual for the jth observation is

$\chi_j=\frac{\sqrt{w_j}(r_j-n_j\hat{p}_j)} {\sqrt{n_j\hat{p}_j\hat{q}_j}}$

The Pearson chi-square statistic is the sum of squares of the Pearson residuals. The deviance residual for the jth observation is

$\hspace*{-0.5in} d_j= & \{ -\sqrt{-2w_jn_j\log(\hat{q}_j)} & {if }r_j=0 \ +-\sqr... ... ]} & {if }0\lt r_j\lt n_j \ \sqrt{-2w_jn_j\log(\hat{p}_j)} & {if }r_j=n_j . \$

where the plus (minus) in is used if r_j/n_j is greater (less) than $\hat{p}_j$ . The deviance is the sum of squares of the deviance residuals.

DFBETAs

For each parameter estimate, the procedure calculates a DFBETA diagnostic for each observation. The DFBETA diagnostic for an observation is the standardized difference in the parameter estimate due to deleting the observation, and it can be used to assess the effect of an individual observation on each estimated parameter of the fitted model. Instead of re-estimating the parameter every time an observation is deleted, PROC LOGISTIC uses the one-step estimate. See the section "Predicted Probability of an Event for Classification". For the jth observation, the DFBETAS are given by

${DFBETA}i_j={\Delta}_i b_j^1 / \hat{\sigma}(b_i)$

where $i=0, 1, ... , s, \hat{\sigma}(b_i)$ is the standard error of the ith component of b, and ${\Delta}_i b_j^1$ is the ith component of the one-step difference

${\Delta}b_j^1 = \frac{w_j(r_j-n_j\hat{p}_j)}{1-h_{jj}}\hat{V}_{b} ( 1 \ x_j )$

${\Delta}b_j^1$ is the approximate change (b- b_j¹) in the vector of parameter estimates due to the omission of the jth observation. The DFBETAs are useful in detecting observations that are causing instability in the selected coefficients.

C and CBAR

C and CBAR are confidence interval displacement diagnostics that provide scalar measures of the influence of individual observations on b. These diagnostics are based on the same idea as the Cook distance in linear regression theory, and by using the one-step estimate, C and CBAR for the jth observation are computed as

$C_j=\chi_j^2 h_{jj} / (1-h_{jj})^2$

and

$\overline{C}_j=\chi_j^2 h_{jj} / (1-h_{jj})$

respectively.

Typically, to use these statistics, you plot them against an index (as the IPLOT option does) and look for outliers.

DIFDEV and DIFCHISQ

DIFDEV and DIFCHISQ are diagnostics for detecting ill-fitted observations; in other words, observations that contribute heavily to the disagreement between the data and the predicted values of the fitted model. DIFDEV is the change in the deviance due to deleting an individual observation while DIFCHISQ is the change in the Pearson chi-square statistic for the same deletion. By using the one-step estimate, DIFDEV and DIFCHISQ for the jth observation are computed as

$\Delta_j D=d_j^2+\overline{C}_j$

and

$\Delta_j \chi^2=\overline{C}_j/h_{jj}$

respectively.

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