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 The LOGISTIC Procedure

## Example 39.4: Logistic Regression Diagnostics

In a controlled experiment to study the effect of the rate and volume of air inspired on a transient reflex vaso-constriction in the skin of the digits, 39 tests under various combinations of rate and volume of air inspired were obtained (Finney 1947). The end point of each test is whether or not vaso-constriction occurred. Pregibon (1981) uses this set of data to illustrate the diagnostic measures he proposes for detecting influential observations and to quantify their effects on various aspects of the maximum likelihood fit.

The vaso-constriction data are saved in the data set vaso:

```   data vaso;
length Response \$12;
input Volume Rate Response @@;
LogVolume=log(Volume);
LogRate=log(Rate);
datalines;
3.70  0.825  constrict       3.50  1.09   constrict
1.25  2.50   constrict       0.75  1.50   constrict
0.80  3.20   constrict       0.70  3.50   constrict
0.60  0.75   no_constrict    1.10  1.70   no_constrict
0.90  0.75   no_constrict    0.90  0.45   no_constrict
0.80  0.57   no_constrict    0.55  2.75   no_constrict
0.60  3.00   no_constrict    1.40  2.33   constrict
0.75  3.75   constrict       2.30  1.64   constrict
3.20  1.60   constrict       0.85  1.415  constrict
1.70  1.06   no_constrict    1.80  1.80   constrict
0.40  2.00   no_constrict    0.95  1.36   no_constrict
1.35  1.35   no_constrict    1.50  1.36   no_constrict
1.60  1.78   constrict       0.60  1.50   no_constrict
1.80  1.50   constrict       0.95  1.90   no_constrict
1.90  0.95   constrict       1.60  0.40   no_constrict
2.70  0.75   constrict       2.35  0.03   no_constrict
1.10  1.83   no_constrict    1.10  2.20   constrict
1.20  2.00   constrict       0.80  3.33   constrict
0.95  1.90   no_constrict    0.75  1.90   no_constrict
1.30  1.625  constrict
;
```

In the data set vaso, the variable Response represents the outcome of a test. The variable LogVolume represents the log of the volume of air intake, and the variable LogRate represents the log of the rate of air intake.

The following SAS statements invoke PROC LOGISTIC to fit a logistic regression model to the vaso-constriction data, where Response is the response variable, and LogRate and LogVolume are the explanatory variables. The INFLUENCE option and the IPLOTS option are specified to display the regression diagnostics and the index plots.

```   title 'Occurrence of Vaso-Constriction';
proc logistic data=vaso;
model Response=LogRate LogVolume/influence iplots;
run;
```

Results of the model fit are shown in Output 39.4.1. Both LogRate and LogVolume are statistically significant to the occurrence of vaso-constriction (p=0.0131 and p=0.0055, respectively). Their positive parameter estimates indicate that a higher inspiration rate or a larger volume of air intake is likely to increase the probability of vaso-constriction.

Output 39.4.1: Logistic Regression Analysis for Vaso-Constriction Data

 Occurrence of Vaso-Constriction
 The LOGISTIC Procedure
 Model Information Data Set WORK.VASO Response Variable Response Number of Response Levels 2 Number of Observations 39 Link Function Logit Optimization Technique Fisher's scoring
 Response Profile OrderedValue Response TotalFrequency 1 constrict 20 2 no_constrict 19
 Model Convergence Status Convergence criterion (GCONV=1E-8) satisfied.
 Model Fit Statistics Criterion Intercept Only Intercept and Covariates AIC 56.040 35.227 SC 57.703 40.218 -2 Log L 54.040 29.227
 Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 24.8125 2 <.0001 Score 16.6324 2 0.0002 Wald 7.8876 2 0.0194
 Analysis of Maximum Likelihood Estimates Parameter DF Estimate StandardError Chi-Square Pr > ChiSq Intercept 1 -2.8754 1.3208 4.7395 0.0295 LogRate 1 4.5617 1.8380 6.1597 0.0131 LogVolume 1 5.1793 1.8648 7.7136 0.0055
 Association of Predicted Probabilities andObserved Responses Percent Concordant 93.7 Somers' D 0.874 Percent Discordant 6.3 Gamma 0.874 Percent Tied 0.0 Tau-a 0.448 Pairs 380 c 0.937

The regression diagnostics produced by the INFLUENCE option are shown in Output 39.4.2.

The values of the explanatory variables (LogRate and LogVolume) are listed for each observation (Output 39.4.2). For each diagnostic, the case number, representing the sequence number of the observation, is displayed along with the diagnostic value. Also displayed is a plot where the vertical axis represents the case number and the horizontal axis represents the value of the diagnostic statistic.

Output 39.4.2: Regression Diagnostics from the INFLUENCE Option

 Occurrence of Vaso-Constriction
 The LOGISTIC Procedure
 Regression Diagnostics CaseNumber Covariates Pearson Residual Deviance Residual Hat Matrix Diagonal Intercept DfBeta Value ` (1 unit = 0.13) -8 -4 0 2 4 6 8 ` LogRate DfBeta Value ` (1 unit = 0.12) -8 -4 0 2 4 6 8 ` LogVolume DfBeta Value ` (1 unit = 0.13) -8 -4 0 2 4 6 8 ` Confidence Interval Displacement C Confidence Interval DisplacementCBar Delta Deviance Delta Chi-Square LogRate LogVolume Value ` (1 unit = 0.44) -8 -4 0 2 4 6 8 ` Value ` (1 unit = 0.28) -8 -4 0 2 4 6 8 ` Value ` (1 unit = 0.02) 0 2 4 6 8 12 16 ` Value ` (1 unit = 0.08) 0 2 4 6 8 12 16 ` Value ` (1 unit = 0.07) 0 2 4 6 8 12 16 ` Value ` (1 unit = 0.4) 0 2 4 6 8 12 16 ` Value ` (1 unit = 0.85) 0 2 4 6 8 12 16 ` 1 -0.1924 1.3083 0.2205 `| |* |` 0.3082 `| |* |` 0.0927 `| * |` -0.0165 `| * |` 0.0193 `| * |` 0.0556 `| * |` 0.00548 `|* |` 0.00497 `|* |` 0.1000 `|* |` 0.0536 `|* |` 2 0.0862 1.2528 0.1349 `| * |` 0.1899 `| |* |` 0.0429 `| * |` -0.0134 `| * |` 0.0151 `| * |` 0.0261 `| * |` 0.000853 `|* |` 0.000816 `|* |` 0.0369 `|* |` 0.0190 `|* |` 3 0.9163 0.2231 0.2923 `| |* |` 0.4049 `| |* |` 0.0612 `| * |` -0.0492 `| * |` 0.0660 `| |* |` 0.0589 `| * |` 0.00593 `|* |` 0.00557 `|* |` 0.1695 `|* |` 0.0910 `|* |` 4 0.4055 -0.2877 3.5181 `| | *|` 2.2775 `| | *|` 0.0867 `| * |` 1.0734 `| | *|` -0.9302 `|* | |` -1.0180 `|* | |` 1.2873 `| *|` 1.1756 `| *|` 6.3626 `| *|` 13.5523 `| *|` 5 1.1632 -0.2231 0.5287 `| |* |` 0.7021 `| | * |` 0.1158 `| * |` -0.0832 `| *| |` 0.1411 `| |* |` 0.0583 `| * |` 0.0414 `| * |` 0.0366 `|* |` 0.5296 `| * |` 0.3161 `|* |` 6 1.2528 -0.3567 0.6090 `| |* |` 0.7943 `| | * |` 0.1524 `| * |` -0.0922 `| *| |` 0.1710 `| |* |` 0.0381 `| * |` 0.0787 `| * |` 0.0667 `| * |` 0.6976 `| * |` 0.4376 `| * |` 7 -0.2877 -0.5108 -0.0328 `| * |` -0.0464 `| * |` 0.00761 `|* |` -0.00280 `| * |` 0.00274 `| * |` 0.00265 `| * |` 8.321E-6 `|* |` 8.258E-6 `|* |` 0.00216 `|* |` 0.00109 `|* |` 8 0.5306 0.0953 -1.0196 `| * | |` -1.1939 `| * | |` 0.0559 `| * |` -0.1444 `| *| |` 0.0613 `| |* |` 0.0570 `| * |` 0.0652 `| * |` 0.0616 `| * |` 1.4870 `| * |` 1.1011 `| * |` 9 -0.2877 -0.1054 -0.0938 `| * |` -0.1323 `| * |` 0.0342 `| * |` -0.0178 `| * |` 0.0173 `| * |` 0.0153 `| * |` 0.000322 `|* |` 0.000311 `|* |` 0.0178 `|* |` 0.00911 `|* |` 10 -0.7985 -0.1054 -0.0293 `| * |` -0.0414 `| * |` 0.00721 `|* |` -0.00245 `| * |` 0.00246 `| * |` 0.00211 `| * |` 6.256E-6 `|* |` 6.211E-6 `|* |` 0.00172 `|* |` 0.000862 `|* |` 11 -0.5621 -0.2231 -0.0370 `| * |` -0.0523 `| * |` 0.00969 `| * |` -0.00361 `| * |` 0.00358 `| * |` 0.00319 `| * |` 0.000014 `|* |` 0.000013 `|* |` 0.00274 `|* |` 0.00138 `|* |` 12 1.0116 -0.5978 -0.5073 `| *| |` -0.6768 `| * | |` 0.1481 `| * |` -0.1173 `| *| |` 0.0647 `| |* |` 0.1651 `| |* |` 0.0525 `| * |` 0.0447 `| * |` 0.5028 `| * |` 0.3021 `|* |` 13 1.0986 -0.5108 -0.7751 `| * | |` -0.9700 `| * | |` 0.1628 `| * |` -0.0931 `| *| |` -0.00946 `| * |` 0.1775 `| |* |` 0.1395 `| * |` 0.1168 `| * |` 1.0577 `| * |` 0.7175 `| * |` 14 0.8459 0.3365 0.2559 `| |* |` 0.3562 `| |* |` 0.0551 `| * |` -0.0414 `| * |` 0.0538 `| * |` 0.0527 `| * |` 0.00404 `|* |` 0.00382 `|* |` 0.1307 `|* |` 0.0693 `|* |` 15 1.3218 -0.2877 0.4352 `| |* |` 0.5890 `| | * |` 0.1336 `| * |` -0.0940 `| *| |` 0.1408 `| |* |` 0.0643 `| |* |` 0.0337 `|* |` 0.0292 `|* |` 0.3761 `| * |` 0.2186 `|* |` 16 0.4947 0.8329 0.1576 `| * |` 0.2215 `| |* |` 0.0402 `| * |` -0.0198 `| * |` 0.0234 `| * |` 0.0307 `| * |` 0.00108 `|* |` 0.00104 `|* |` 0.0501 `|* |` 0.0259 `|* |` 17 0.4700 1.1632 0.0709 `| * |` 0.1001 `| * |` 0.0172 `| * |` -0.00630 `| * |` 0.00701 `| * |` 0.00914 `| * |` 0.000089 `|* |` 0.000088 `|* |` 0.0101 `|* |` 0.00511 `|* |` 18 0.3471 -0.1625 2.9062 `| | * |` 2.1192 `| | * |` 0.0954 `| * |` 0.9595 `| | * |` -0.8279 `| * | |` -0.8477 `| * | |` 0.9845 `| * |` 0.8906 `| * |` 5.3817 `| * |` 9.3363 `| * |` 19 0.0583 0.5306 -1.0718 `| * | |` -1.2368 `| * | |` 0.1315 `| * |` -0.2591 `| * | |` 0.2024 `| | * |` -0.00488 `| * |` 0.2003 `| * |` 0.1740 `| * |` 1.7037 `| * |` 1.3227 `| * |` 20 0.5878 0.5878 0.2405 `| |* |` 0.3353 `| |* |` 0.0525 `| * |` -0.0331 `| * |` 0.0421 `| * |` 0.0518 `| * |` 0.00338 `|* |` 0.00320 `|* |` 0.1156 `|* |` 0.0610 `|* |` 21 0.6931 -0.9163 -0.1076 `| * |` -0.1517 `| *| |` 0.0373 `| * |` -0.0180 `| * |` 0.0158 `| * |` 0.0208 `| * |` 0.000465 `|* |` 0.000448 `|* |` 0.0235 `|* |` 0.0120 `|* |` 22 0.3075 -0.0513 -0.4193 `| *| |` -0.5691 `| * | |` 0.1015 `| * |` -0.1449 `| *| |` 0.1237 `| |* |` 0.1179 `| |* |` 0.0221 `|* |` 0.0199 `|* |` 0.3437 `| * |` 0.1956 `|* |` 23 0.3001 0.3001 -1.0242 `| * | |` -1.1978 `| * | |` 0.0761 `| * |` -0.1961 `| *| |` 0.1275 `| |* |` 0.0357 `| * |` 0.0935 `| * |` 0.0864 `| * |` 1.5212 `| * |` 1.1355 `| * |` 24 0.3075 0.4055 -1.3684 `| * | |` -1.4527 `| * | |` 0.0717 `| * |` -0.1281 `| *| |` 0.0410 `| * |` -0.1004 `| *| |` 0.1558 `| * |` 0.1447 `| * |` 2.2550 `| * |` 2.0171 `| * |` 25 0.5766 0.4700 0.3347 `| |* |` 0.4608 `| | * |` 0.0587 `| * |` -0.0403 `| * |` 0.0570 `| * |` 0.0708 `| |* |` 0.00741 `|* |` 0.00698 `|* |` 0.2193 `| * |` 0.1190 `|* |` 26 0.4055 -0.5108 -0.1595 `| * |` -0.2241 `| *| |` 0.0548 `| * |` -0.0366 `| * |` 0.0329 `| * |` 0.0373 `| * |` 0.00156 `|* |` 0.00147 `|* |` 0.0517 `|* |` 0.0269 `|* |` 27 0.4055 0.5878 0.3645 `| |* |` 0.4995 `| | * |` 0.0661 `| * |` -0.0327 `| * |` 0.0496 `| * |` 0.0788 `| |* |` 0.0101 `|* |` 0.00941 `|* |` 0.2589 `| * |` 0.1423 `|* |` 28 0.6419 -0.0513 -0.8989 `| * | |` -1.0883 `| * | |` 0.0647 `| * |` -0.1423 `| *| |` 0.0617 `| |* |` 0.1025 `| |* |` 0.0597 `| * |` 0.0559 `| * |` 1.2404 `| * |` 0.8639 `| * |` 29 -0.0513 0.6419 0.8981 `| | * |` 1.0876 `| | * |` 0.1682 `| * |` 0.2367 `| | * |` -0.1950 `| * | |` 0.0286 `| * |` 0.1961 `| * |` 0.1631 `| * |` 1.3460 `| * |` 0.9697 `| * |` 30 -0.9163 0.4700 -0.0992 `| * |` -0.1400 `| * |` 0.0507 `| * |` -0.0224 `| * |` 0.0227 `| * |` 0.0159 `| * |` 0.000554 `|* |` 0.000526 `|* |` 0.0201 `|* |` 0.0104 `|* |` 31 -0.2877 0.9933 0.6198 `| |* |` 0.8064 `| | * |` 0.2459 `| *|` 0.1165 `| |* |` -0.0996 `| *| |` 0.1322 `| |* |` 0.1661 `| * |` 0.1253 `| * |` 0.7755 `| * |` 0.5095 `| * |` 32 -3.5066 0.8544 -0.00073 `| * |` -0.00103 `| * |` 0.000022 `|* |` -3.22E-6 `| * |` 3.405E-6 `| * |` 2.48E-6 `| * |` 1.18E-11 `|* |` 1.18E-11 `|* |` 1.065E-6 `|* |` 5.324E-7 `|* |` 33 0.6043 0.0953 -1.2062 `| * | |` -1.3402 `| * | |` 0.0510 `| * |` -0.0882 `| *| |` -0.0137 `| * |` -0.00216 `| * |` 0.0824 `| * |` 0.0782 `| * |` 1.8744 `| * |` 1.5331 `| * |` 34 0.7885 0.0953 0.5447 `| |* |` 0.7209 `| | * |` 0.0601 `| * |` -0.0425 `| * |` 0.0877 `| |* |` 0.0671 `| |* |` 0.0202 `|* |` 0.0190 `|* |` 0.5387 `| * |` 0.3157 `|* |` 35 0.6931 0.1823 0.5404 `| |* |` 0.7159 `| | * |` 0.0552 `| * |` -0.0340 `| * |` 0.0755 `| |* |` 0.0711 `| |* |` 0.0180 `|* |` 0.0170 `|* |` 0.5295 `| * |` 0.3091 `|* |` 36 1.2030 -0.2231 0.4828 `| |* |` 0.6473 `| | * |` 0.1177 `| * |` -0.0867 `| *| |` 0.1381 `| |* |` 0.0631 `| * |` 0.0352 `|* |` 0.0311 `|* |` 0.4501 `| * |` 0.2641 `|* |` 37 0.6419 -0.0513 -0.8989 `| * | |` -1.0883 `| * | |` 0.0647 `| * |` -0.1423 `| *| |` 0.0617 `| |* |` 0.1025 `| |* |` 0.0597 `| * |` 0.0559 `| * |` 1.2404 `| * |` 0.8639 `| * |` 38 0.6419 -0.2877 -0.4874 `| *| |` -0.6529 `| * | |` 0.1000 `| * |` -0.1395 `| *| |` 0.1032 `| |* |` 0.1397 `| |* |` 0.0293 `|* |` 0.0264 `|* |` 0.4526 `| * |` 0.2639 `|* |` 39 0.4855 0.2624 0.7053 `| | * |` 0.8987 `| | * |` 0.0531 `| * |` 0.0326 `| * |` 0.0190 `| * |` 0.0489 `| * |` 0.0295 `|* |` 0.0279 `|* |` 0.8355 `| * |` 0.5254 `| * |`

The index plots produced by the IPLOTS option are essentially the same plots as those produced by the INFLUENCE option with a 90-degree rotation and perhaps on a more refined scale. The vertical axis of an index plot represents the value of the diagnostic and the horizontal axis represents the sequence (case number) of the observation. The index plots are useful for identification of extreme values.

 The LOGISTIC Procedure
 ``` ---------------+----+----+----+----+----+----+----+----+--------------- RESCHI | | P 4 + + e | * | a | | r | * | s | | o 2 + + n | | | | R | * * * * | e | * * * ** * * * *** | s 0 + * * *** ** * * * * + i | * * * | d | * * * | u | * * ** * | a | | l -2 + + | | ---------------+----+----+----+----+----+----+----+----+--------------- 0 5 10 15 20 25 30 35 40 Case Number INDEX ```
 ``` ---------------+----+----+----+----+----+----+----+----+--------------- D RESDEV | | e 4 + + v | | i | | a | | n | * | c 2 + * + e | | | * | R | ** * *** * | e | * * *** * * * | s 0 + * * *** * * * * + i | * * | d | ** * | u | * * * * * * | a | * | l -2 + + | | ---------------+----+----+----+----+----+----+----+----+--------------- 0 5 10 15 20 25 30 35 40 Case Number INDEX ```

 The LOGISTIC Procedure
 ``` ----------------+----+----+----+----+----+----+----+----+----------------- H | | 0.3 + + | | H | | a | * | t | | 0.2 + + D | | i | * * * | a | * * * | g | * * | o 0.1 + * * * * + n | * ** | a | * * * * **** * *** * * | l | * * * * | | * | 0.0 + * ** * + | | ----------------+----+----+----+----+----+----+----+----+----------------- 0 5 10 15 20 25 30 35 40 Case Number INDEX ```
 ``` --------------+----+----+----+----+----+----+----+----+--------------- DFBETA0 | | I 1.5 + + n | | t | | e | * | r 1.0 + * + c | | e | | p | | t 0.5 + + | | D | * | f | * | B 0.0 + *** * *** * ** ** *** * * ** * + e | ** * ** * * * * * *** | t | * * | a | | -0.5 + + --------------+----+----+----+----+----+----+----+----+--------------- 0 5 10 15 20 25 30 35 40 Case Number INDEX ```

 The LOGISTIC Procedure
 ``` --------------+----+----+----+----+----+----+----+----+--------------- DFBETA1 | | 0.5 + + L | | o | | g | * * | R | * * * * ** ** * * ***** | a 0.0 + ** * *** * ** ** * ** * ** * + t | * | e | * | | | D | | f -0.5 + + B | | e | | t | * | a | * | -1.0 + + | | --------------+----+----+----+----+----+----+----+----+--------------- 0 5 10 15 20 25 30 35 40 Case Number INDEX ```
 ``` --------------+----+----+----+----+----+----+----+----+--------------- DFBETA2 | | L 0.5 + + o | | g | | V | ** | o | * * * * ** * * * ** * ***** | l 0.0 + * ** *** ** * * * * ** ** * + u | * | m | | e | | | | D -0.5 + + f | | B | | e | * | t | | a -1.0 + * + | | --------------+----+----+----+----+----+----+----+----+--------------- 0 5 10 15 20 25 30 35 40 Case Number INDEX ```

 The LOGISTIC Procedure
 ``` ----------------+----+----+----+----+----+----+----+----+----------------- C C | | o 1.5 + + n | | f | * | i | | d | | e 1.0 + * + n | | c | | e | | | | I 0.5 + + n | | t | | e | * * * * | r | * * ** * * * * | v 0.0 + *** * * *** **** *** *** * * *** ** + a | | l ----------------+----+----+----+----+----+----+----+----+----------------- 0 5 10 15 20 25 30 35 40 D Case Number INDEX ```
 ``` ----------------+----+----+----+----+----+----+----+----+---------------- C CBAR | | o 1.5 + + n | | f | | i | * | d | | e 1.0 + + n | * | c | | e | | | | I 0.5 + + n | | t | | e | * * | r | * * * ** * * * * | v 0.0 + *** * * **** **** *** *** * * *** ** + a | | l ----------------+----+----+----+----+----+----+----+----+---------------- 0 5 10 15 20 25 30 35 40 D Case Number INDEX ```

 The LOGISTIC Procedure
 ``` ---------------+----+----+----+----+----+----+----+----+--------------- DIFDEV | | 8 + + D | | e | | l | * | t 6 + + a | * | | | D | | e 4 + + v | | i | | a | * | n 2 + * + c | * * * * | e | * * * * * | | ** * * * * *** * | 0 + *** * *** * ** ** ** * * + ---------------+----+----+----+----+----+----+----+----+--------------- 0 5 10 15 20 25 30 35 40 Case Number INDEX ```
 ``` --------------+----+----+----+----+----+----+----+----+-------------- DIFCHISQ | | D 15 + + e | * | l | | t | | a | | 10 + + C | * | h | | i | | S | | q 5 + + u | | a | | r | * * | e | * * * * ** * * * | 0 + *** *** **** **** *** *** * * *** * + | | --------------+----+----+----+----+----+----+----+----+-------------- 0 5 10 15 20 25 30 35 40 Case Number INDEX ```

The index plots of the Pearson residuals and the deviance residuals indicate that case 4 and case 18 are poorly accounted for by the model. The index plot of the diagonal elements of the hat matrix suggests that case 31 is an extreme point in the design space. The index plots of DFBETAS indicate that case 4 and case 18 are causing instability in all three parameter estimates. The other four index plots also point to these two cases as having a large impact on the coefficients and goodness of fit.

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