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Inputting the Crab data, p. 82-83.
data crab; input color spine width satell weight; if satell>0 then y=1; if satell=0 then y=0; n=1; weight = weight/1000; color = color - 1; if color=4 then dark=0; if color < 4 then dark=1; cards; 3 3 28.3 8 3050 4 3 22.5 0 1550 2 1 26.0 9 2300 4 3 24.8 0 2100 4 3 26.0 4 2600 3 3 23.8 0 2100 2 1 26.5 0 2350 4 2 24.7 0 1900 3 1 23.7 0 1950 4 3 25.6 0 2150 4 3 24.3 0 2150 3 3 25.8 0 2650 3 3 28.2 11 3050 5 2 21.0 0 1850 3 1 26.0 14 2300 2 1 27.1 8 2950 3 3 25.2 1 2000 3 3 29.0 1 3000 5 3 24.7 0 2200 3 3 27.4 5 2700 3 2 23.2 4 1950 2 2 25.0 3 2300 3 1 22.5 1 1600 4 3 26.7 2 2600 5 3 25.8 3 2000 5 3 26.2 0 1300 3 3 28.7 3 3150 3 1 26.8 5 2700 5 3 27.5 0 2600 3 3 24.9 0 2100 2 1 29.3 4 3200 2 3 25.8 0 2600 3 2 25.7 0 2000 3 1 25.7 8 2000 3 1 26.7 5 2700 5 3 23.7 0 1850 3 3 26.8 0 2650 3 3 27.5 6 3150 5 3 23.4 0 1900 3 3 27.9 6 2800 4 3 27.5 3 3100 2 1 26.1 5 2800 2 1 27.7 6 2500 3 1 30.0 5 3300 4 1 28.5 9 3250 4 3 28.9 4 2800 3 3 28.2 6 2600 3 3 25.0 4 2100 3 3 28.5 3 3000 3 1 30.3 3 3600 5 3 24.7 5 2100 3 3 27.7 5 2900 2 1 27.4 6 2700 3 3 22.9 4 1600 3 1 25.7 5 2000 3 3 28.3 15 3000 3 3 27.2 3 2700 4 3 26.2 3 2300 3 1 27.8 0 2750 5 3 25.5 0 2250 4 3 27.1 0 2550 4 3 24.5 5 2050 4 1 27.0 3 2450 3 3 26.0 5 2150 3 3 28.0 1 2800 3 3 30.0 8 3050 3 3 29.0 10 3200 3 3 26.2 0 2400 3 1 26.5 0 1300 3 3 26.2 3 2400 4 3 25.6 7 2800 4 3 23.0 1 1650 4 3 23.0 0 1800 3 3 25.4 6 2250 4 3 24.2 0 1900 3 2 22.9 0 1600 4 2 26.0 3 2200 3 3 25.4 4 2250 4 3 25.7 0 1200 3 3 25.1 5 2100 4 2 24.5 0 2250 5 3 27.5 0 2900 4 3 23.1 0 1650 4 1 25.9 4 2550 3 3 25.8 0 2300 5 3 27.0 3 2250 3 3 28.5 0 3050 5 1 25.5 0 2750 5 3 23.5 0 1900 3 2 24.0 0 1700 3 1 29.7 5 3850 3 1 26.8 0 2550 5 3 26.7 0 2450 3 1 28.7 0 3200 4 3 23.1 0 1550 3 1 29.0 1 2800 4 3 25.5 0 2250 4 3 26.5 1 1967 4 3 24.5 1 2200 4 3 28.5 1 3000 3 3 28.2 1 2867 3 3 24.5 1 1600 3 3 27.5 1 2550 3 2 24.7 4 2550 3 1 25.2 1 2000 4 3 27.3 1 2900 3 3 26.3 1 2400 3 3 29.0 1 3100 3 3 25.3 2 1900 3 3 26.5 4 2300 3 3 27.8 3 3250 3 3 27.0 6 2500 4 3 25.7 0 2100 3 3 25.0 2 2100 3 3 31.9 2 3325 5 3 23.7 0 1800 5 3 29.3 12 3225 4 3 22.0 0 1400 3 3 25.0 5 2400 4 3 27.0 6 2500 4 3 23.8 6 1800 2 1 30.2 2 3275 4 3 26.2 0 2225 3 3 24.2 2 1650 3 3 27.4 3 2900 3 2 25.4 0 2300 4 3 28.4 3 3200 5 3 22.5 4 1475 3 3 26.2 2 2025 3 1 24.9 6 2300 2 2 24.5 6 1950 3 3 25.1 0 1800 3 1 28.0 4 2900 5 3 25.8 10 2250 3 3 27.9 7 3050 3 3 24.9 0 2200 3 1 28.4 5 3100 4 3 27.2 5 2400 3 2 25.0 6 2250 3 3 27.5 6 2625 3 1 33.5 7 5200 3 3 30.5 3 3325 4 3 29.0 3 2925 3 1 24.3 0 2000 3 3 25.8 0 2400 5 3 25.0 8 2100 3 1 31.7 4 3725 3 3 29.5 4 3025 4 3 24.0 10 1900 3 3 30.0 9 3000 3 3 27.6 4 2850 3 3 26.2 0 2300 3 1 23.1 0 2000 3 1 22.9 0 1600 5 3 24.5 0 1900 3 3 24.7 4 1950 3 3 28.3 0 3200 3 3 23.9 2 1850 4 3 23.8 0 1800 4 2 29.8 4 3500 3 3 26.5 4 2350 3 3 26.0 3 2275 3 3 28.2 8 3050 5 3 25.7 0 2150 3 3 26.5 7 2750 3 3 25.8 0 2200 4 3 24.1 0 1800 4 3 26.2 2 2175 4 3 26.1 3 2750 4 3 29.0 4 3275 2 1 28.0 0 2625 5 3 27.0 0 2625 3 2 24.5 0 2000 ; run;
Creating the categorical variable for width and plotting the proportion of satellites and whether satellites are present or not (Y = 1, yes; Y=2, no) versus width.
data crab1;
set crab;
wcat=0;
if width<=23.25 then wcat=1;
if 23.25< width<=24.25 then wcat=2;
if 24.25< width<=25.25 then wcat=3;
if 25.25< width<=26.25 then wcat=4;
if 26.25< width<=27.25 then wcat=5;
if 27.25< width<=28.25 then wcat=6;
if 28.25< width<=29.25 then wcat=7;
if 29.25< width then wcat=8;
run;
proc sql;
create table crab2 as
select *, sum(y)/sum(n) as prop, mean(width) as wmidpt, sum(y) as yes, sum(n) as cases
from crab1
group by wcat;
quit;
proc sort data=crab2;
by width;
run;
goption reset=all;
symbol1 v=dot c=blue h=.7;
symbol2 v=dot c=red h=.7;
axis1 order=(0 1) label=(angle = 90 'Presence of Satellites');
axis2 label=('Width');
proc gplot data=crab2;
plot prop*wmidpt y*width/ overlay vaxis=axis1 haxis=axis2;
run;
quit;
Table 5.1, p. 106.
Note: The variables LCL and UCL are the lower and upper values respectively, of the 95% confidence interval for the predicted probability.
proc logistic data=crab2 desc;
model y = width ;
output out=predict p=pi_hat;
run;
proc sql;
create table pred2 as
select *, sum(pi_hat) as predicted_satell, sum(pi_hat)/sum(n) as predicted_prob
from predict
group by wcat;
quit;
proc sort data=pred2;
by wcat;
run;
data pred3;
set pred2;
by wcat;
if first.wcat;
run;
proc format;
value wcat 1='<=23.25' 2='23.25-24.25' 3='24.25-25.25' 4='25.25-26.25'
5='26.25-27.25' 6='27.25-28.25' 7='28.25-29.25' 8='>29.25';
run;
proc print data = pred3;
format wcat wcat.;
var wcat cases yes prop predicted_prob predicted_satell;
run;
The LOGISTIC Procedure
Model Information
Data Set WORK.CRAB2
Response Variable y
Number of Response Levels 2
Number of Observations 173
Link Function Logit
Optimization Technique Fisher's scoring
Response Profile
Ordered Total
Value y Frequency
1 1 111
2 0 62
Model Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.
Model Fit Statistics
Intercept
Intercept and
Criterion Only Covariates
SC 230.912 204.759
-2 Log L 225.759 194.453
Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 31.3059 1 <.0001
Score 27.8752 1 <.0001
Wald 23.8872 1 <.0001
Analysis of Maximum Likelihood Estimates
Standard
Parameter DF Estimate Error Chi-Square Pr > ChiSq
Intercept 1 -12.3508 2.6287 22.0749 <.0001
width 1 0.4972 0.1017 23.8872 <.0001
The LOGISTIC Procedure
Odds Ratio Estimates
Point 95% Wald
Effect Estimate Confidence Limits
width 1.644 1.347 2.007
Association of Predicted Probabilities and Observed Responses
Percent Concordant 73.5 Somers' D 0.485
Percent Discordant 25.0 Gamma 0.492
Percent Tied 1.5 Tau-a 0.224
Pairs 6882 c 0.742
predicted_ predicted_
Obs wcat cases yes prop prob satell
1 <=23.25 14 5 0.35714 0.25967 3.6354
2 23.25-24.25 14 4 0.28571 0.37900 5.3060
3 24.25-25.25 28 17 0.60714 0.49206 13.7776
4 25.25-26.25 39 21 0.53846 0.62122 24.2277
5 26.25-27.25 22 15 0.68182 0.72445 15.9378
6 27.25-28.25 24 20 0.83333 0.80764 19.3833
7 28.25-29.25 18 15 0.83333 0.86945 15.6502
8 >29.25 14 14 1.00000 0.93443 13.0820
The model can also be estimated using proc genmod as can be seen in the following output. Proc genmod will provide the likelihood ratio confidence interval for all the parameters in the model including chi-squared tests for all the parameters in the model whereas proc logistic will provide the chi-squared tests as well as many other details such as model fit statistics and odds ratios. Another big difference between the two procedures is that proc genmod is a very general procedure that can handle many different distributions and link functions but it does not in general provide a great deal of residuals or built in options that more specific procedures such as proc logistic provides.
proc genmod data=crab; model y = width / dist=bin link=logit waldci lrci; run;
The GENMOD Procedure
Model Information
Data Set WORK.CRAB
Distribution Binomial
Link Function Logit
Dependent Variable y
Observations Used 173
Probability Modeled Pr( y = 0 )
Response Profile
Ordered Ordered
Level Value Count
1 0 62
2 1 111
Criteria For Assessing Goodness Of Fit
Criterion DF Value Value/DF
Deviance 171 194.4527 1.1372
Scaled Deviance 171 194.4527 1.1372
Pearson Chi-Square 171 165.1434 0.9658
Scaled Pearson X2 171 165.1434 0.9658
Log Likelihood -97.2263
Algorithm converged.
Analysis Of Parameter Estimates
Standard Likelihood Ratio 95% Chi-
Parameter DF Estimate Error Confidence Limits Square Pr > ChiSq
Intercept 1 12.3508 2.6287 7.4573 17.8097 22.07 <.0001
width 1 -0.4972 0.1017 -0.7090 -0.3084 23.89 <.0001
Scale 0 1.0000 0.0000 1.0000 1.0000
NOTE: The scale parameter was held fixed.
The predicted probabilities discussed at the top of p. 107.
proc logistic data=crab desc noprint; model y = width ; output out=predict p=pi_hat; run; proc print data=predict; where width=21 or width=26.3 or width=33.5; var width pi_hat; run;
Obs width pi_hat 14 21.0 0.12910 107 26.3 0.67400 141 33.5 0.98670
Variance and covariance can be obtained by using the covout option in the proc statement and the confidence interval for individual predicted values can be obtained by using the upper and lower options in the model statement. Results p. 110.
proc logistic data=crab2 desc covout outest=temp; model y = width ; output out=predict p=pi_hat upper=ucl lower=lcl; run; proc print data=temp; where _type_='COV'; var _name_ intercept width; run; proc sql; select distinct width, pi_hat, lcl, ucl from predict where width= 26.5 ; quit;
The LOGISTIC Procedure
Model Information
Data Set WORK.CRAB2
Response Variable y
Number of Response Levels 2
Number of Observations 173
Link Function Logit
Optimization Technique Fisher's scoring
Response Profile
Ordered Total
Value y Frequency
1 1 111
2 0 62
Model Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.
Model Fit Statistics
Intercept
Intercept and
Criterion Only Covariates
AIC 227.759 198.453
SC 230.912 204.759
-2 Log L 225.759 194.453
Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 31.3059 1 <.0001
Score 27.8752 1 <.0001
Wald 23.8872 1 <.0001
Analysis of Maximum Likelihood Estimates
Standard
Parameter DF Estimate Error Chi-Square Pr > ChiSq
Intercept 1 -12.3508 2.6287 22.0749 <.0001
width 1 0.4972 0.1017 23.8872 <.0001
The LOGISTIC Procedure
Odds Ratio Estimates
Point 95% Wald
Effect Estimate Confidence Limits
width 1.644 1.347 2.007
Association of Predicted Probabilities and Observed Responses
Percent Concordant 73.5 Somers' D 0.485
Percent Discordant 25.0 Gamma 0.492
Percent Tied 1.5 Tau-a 0.224
Pairs 6882 c 0.742
Obs _NAME_ Intercept width
2 Intercept 6.91023 -0.26685
3 width -0.26685 0.01035
Lower 95% Upper 95%
Estimated Confidence Confidence
width Probability Limit Limit
---------------------------------------------
26.5 0.695465 0.612054 0.767747
Table 5.2, p. 112.
proc sql;
create table pred2 as
select pi_hat, wcat, sum(y) as Num_yes, sum(1-y) as Num_no, sum(pi_hat) as Fitted_yes,
sum(1-pi_hat) as Fitted_no
from predict
group by wcat;
quit;
proc sort data=pred2;
by wcat;
run;
data pred3;
set pred2;
by wcat;
if first.wcat;
run;
proc print data = pred3;
format wcat wcat.;
var wcat Num_yes Num_no Fitted_yes Fitted_no;
run;
Fitted_ Fitted_ Obs wcat Num_yes Num_no yes no 1 <=23.25 5 9 3.6354 10.3646 2 23.25-24.25 4 10 5.3060 8.6940 3 24.25-25.25 17 11 13.7776 14.2224 4 25.25-26.25 21 18 24.2277 14.7723 5 26.25-27.25 15 7 15.9378 6.0622 6 27.25-28.25 20 4 19.3833 4.6167 7 28.25-29.25 15 3 15.6502 2.3498 8 >29.25 14 0 13.0820 0.9180
Inputting the grouped Crab data, p. 271.
data grouped; input width cases satell; cards; 22.69 14 5 23.84 14 4 24.77 28 17 25.84 39 21 26.79 22 15 27.74 24 20 28.67 18 15 30.41 14 14 run;
Formatting the variable width to make the output look nice.
proc format;
value width 22.69='<=23.25' 23.84='23.25-24.25' 24.77='24.25-25.25' 25.84='25.25-26.25'
26.79='26.25-27.25' 27.74='27.25-28.25' 28.67='28.25-29.25' 30.41='>29.25';
run;
Likelihood-ratio model comparisons test from the deviance of the model with width as the predictor and the deviance of the model without any predictors, p. 114.Note: In proc logistic SAS includes the -2log likelihood for the full model and for the model without any predictors. Moreover, the output includes various goodness of fit test in the table labeled Testing Global Null Hypothesis: BETA=0.
proc logistic data=grouped desc; model satell/cases = width ; run;
The LOGISTIC Procedure
Model Information
Data Set WORK.GROUPED
Response Variable (Events) satell
Response Variable (Trials) cases
Number of Observations 8
Link Function Logit
Optimization Technique Fisher's scoring
Response Profile
Ordered Binary Total
Value Outcome Frequency
1 Event 111
2 Nonevent 62
Model Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.
Model Fit Statistics
Intercept
Intercept and
Criterion Only Covariates
AIC 227.759 201.694
SC 230.912 208.001
-2 Log L 225.759 197.694
Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 28.0644 1 <.0001
Score 25.6828 1 <.0001
Wald 22.2312 1 <.0001
Analysis of Maximum Likelihood Estimates
Standard
Parameter DF Estimate Error Chi-Square Pr > ChiSq
Intercept 1 -11.5128 2.5488 20.4031 <.0001
width 1 0.4646 0.0985 22.2312 <.0001
The LOGISTIC Procedure
Odds Ratio Estimates
Point 95% Wald
Effect Estimate Confidence Limits
width 1.591 1.312 1.930
Association of Predicted Probabilities and Observed Responses
Percent Concordant 66.3 Somers' D 0.454
Percent Discordant 20.9 Gamma 0.520
Percent Tied 12.8 Tau-a 0.210
Pairs 6882 c 0.727
Table 5.3, p. 116.Note: The influence option in the model statement provides the deviance residual, the diagonal element of the hat matrix, two confidence interval displacement diagnostics (C and CBAR), the change in the Pearson chi-square statistic (DIFCHSQ), and the change in the deviance (DIFDEV). This option was shown for the model using width as a predictor.
data grouped; set grouped; id = _n_; run; proc logistic data = grouped desc noprint; model satell/cases = ; output out=temp1 reschi=pearsona p=pi_hata; run; data temp1; set temp1; keep id Fitted_yesa pearsona; Fitted_yesa= pi_hata*cases; run; proc logistic data = grouped desc; model satell/cases= width /influence; output out=temp2 reschi=pearson p=pi_hat h=h; run; data temp2; set temp2; Fitted_yes=pi_hat*cases; adjres = pearson/sqrt(1-h); keep pearson Fitted_yes adjres cases satell width id pi_hat; run; data combo; merge temp1 temp2; by id; run; proc print data = combo; format width width.; var width cases satell fitted_yesa pearsona Fitted_yes pearson adjres; run;
The LOGISTIC Procedure
Model Information
Data Set WORK.GROUPED
Response Variable (Events) satell
Response Variable (Trials) cases
Number of Observations 8
Link Function Logit
Optimization Technique Fisher's scoring
Response Profile
Ordered Binary Total
Value Outcome Frequency
1 Event 111
2 Nonevent 62
Model Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.
Model Fit Statistics
Intercept
Intercept and
Criterion Only Covariates
AIC 227.759 201.694
SC 230.912 208.001
-2 Log L 225.759 197.694
Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 28.0644 1 <.0001
Score 25.6828 1 <.0001
Wald 22.2312 1 <.0001
Analysis of Maximum Likelihood Estimates
Standard
Parameter DF Estimate Error Chi-Square Pr > ChiSq
Intercept 1 -11.5128 2.5488 20.4031 <.0001
width 1 0.4646 0.0985 22.2312 <.0001
The LOGISTIC Procedure
Odds Ratio Estimates
Point 95% Wald
Effect Estimate Confidence Limits
width 1.591 1.312 1.930
Association of Predicted Probabilities and Observed Responses
Percent Concordant 66.3 Somers' D 0.454
Percent Discordant 20.9 Gamma 0.520
Percent Tied 12.8 Tau-a 0.210
Pairs 6882 c 0.727
The LOGISTIC Procedure
Regression Diagnostics
Pearson Residual Deviance Residual
Covariates
Case (1 unit = 0.14) (1 unit = 0.18)
Number width Value -8 -4 0 2 4 6 8 Value -8 -4 0 2 4 6 8
1 22.6900 0.6901 | | * | 0.6719 | | * |
2 23.8400 -0.8196 | * | | -0.8370 | * | |
3 24.7700 1.1443 | | *| 1.1487 | | * |
4 25.8400 -1.0606 | * | | -1.0485 | * | |
5 26.7900 -0.3772 | * | | -0.3727 | * | |
6 27.7400 0.4272 | | * | 0.4372 | | * |
7 28.6700 -0.3146 | * | | -0.3072 | * | |
8 30.4100 1.0113 | | * | 1.4051 | | *|
Regression Diagnostics
Hat Matrix Diagonal
Intercept
Case (1 unit = 0.02) DfBeta (1 unit = 0.07)
Number Value 0 2 4 6 8 12 16 Value -8 -4 0 2 4 6 8
1 0.3458 | *| 0.5603 | | *|
2 0.2244 | * | -0.3956 | * | |
3 0.2807 | * | 0.4775 | | * |
4 0.2726 | * | -0.0365 | *| |
5 0.1741 | * | 0.0821 | |* |
6 0.2551 | * | -0.2012 | * | |
7 0.2382 | * | 0.1646 | | * |
8 0.2092 | * | -0.5316 |* | |
Regression Diagnostics
Confidence Interval Displacement C
width
Case DfBeta (1 unit = 0.07) (1 unit = 0.04)
Number Value -8 -4 0 2 4 6 8 Value 0 2 4 6 8 12 16
1 -0.5410 |* | | 0.3848 | * |
2 0.3740 | | * | 0.2505 | * |
3 -0.4295 | * | | 0.7102 | *|
4 -0.0150 | * | 0.5794 | * |
5 -0.0935 | *| | 0.0363 | * |
6 0.2149 | | * | 0.0839 | * |
7 -0.1721 | * | | 0.0406 | * |
8 0.5469 | | *| 0.3422 | * |
The LOGISTIC Procedure
Regression Diagnostics
Confidence Interval Displacement CBar Delta Deviance
Case (1 unit = 0.03) (1 unit = 0.14)
Number Value 0 2 4 6 8 12 16 Value 0 2 4 6 8 12 16
1 0.2517 | * | 0.7032 | * |
2 0.1943 | * | 0.8948 | * |
3 0.5109 | *| 1.8304 | * |
4 0.4215 | * | 1.5209 | * |
5 0.0300 | * | 0.1689 | * |
6 0.0625 | * | 0.2537 | * |
7 0.0310 | * | 0.1253 | * |
8 0.2706 | * | 2.2448 | *|
Regression Diagnostics
Delta Chi-Square
Case (1 unit = 0.11)
Number Value 0 2 4 6 8 12 16
1 0.7280 | * |
2 0.8660 | * |
3 1.8204 | *|
4 1.5464 | * |
5 0.1723 | * |
6 0.2449 | * |
7 0.1300 | * |
8 1.2933 | * |
Fitted_ Fitted_
Obs width cases satell yesa pearsona yes pearson adjres
1 <=23.25 14 5 8.9827 -2.21972 3.8473 0.69012 0.85323
2 23.25-24.25 14 4 8.9827 -2.77706 5.4975 -0.81957 -0.93058
3 24.25-25.25 28 17 17.9653 -0.38043 13.9724 1.14434 1.34923
4 25.25-26.25 39 21 25.0231 -1.34344 24.2136 -1.06063 -1.24356
5 26.25-27.25 22 15 14.1156 0.39321 15.7962 -0.37724 -0.41511
6 27.25-28.25 24 20 15.3988 1.95862 19.1604 0.42716 0.49492
7 28.25-29.25 18 15 11.5491 1.69621 15.4644 -0.31464 -0.36050
8 >29.25 14 14 8.9827 2.79639 13.0469 1.01131 1.13725
Fig. 5.3, p. 116.
data temp2;
set temp2;
prop = satell/cases;
run;
goption reset = all;
symbol1 v=diamond c=blue h=1 i=spline;
symbol2 v=dot c=red h=.8 i=none;
axis1 order=(0 to 1 by .2) label=(angle=90 'Proportion with Satellites');
axis2 order=(22 to 32 by 2);
legend1 label=none value=(height=1 font=swiss 'Fitted' 'Observed' )
position=(bottom right inside) mode=share cborder=black;
proc gplot data=temp2;
plot (pi_hat prop)*width/ overlay legend=legend1 vaxis=axis1 haxis=axis2;
run;
quit;
goptions reset=all;
Table 5.4, p. 118.
proc logistic data = grouped desc noprint; model satell/cases= ; output out=temp1 difchisq=Pearson_diffa difdev=Likelihood_ratio_diffa; run; data temp1; set temp1; keep id Pearson_diffa Likelihood_ratio_diffa; run; proc logistic data = grouped desc noprint; model satell/cases = width ; output out=temp2 difchisq=Pearson_diff difdev=Likelihood_ratio_diff c=c dfbetas=dfbeta_int dfbeta_width; run; data temp2; set temp2; keep id width Pearson_diff Likelihood_ratio_diff c dfbeta_width; run; data combo; merge temp2 temp1; by id; run; proc print data = combo; format width width.; var width dfbeta_width c Pearson_diff Likelihood_ratio_diff Pearson_diffa Likelihood_ratio_diffa; run;
dfbeta_ Pearson_ Likelihood_ Pearson_ Likelihood_ Obs width width c diff ratio_diff diffa ratio_diffa 1 <=23.25 -0.54098 0.38481 0.72800 0.70315 5.36099 5.0931 2 23.25-24.25 0.37397 0.25049 0.86599 0.89483 8.39114 8.0007 3 24.25-25.25 -0.42952 0.71024 1.82042 1.83044 0.17268 0.1708 4 25.25-26.25 -0.01498 0.57945 1.54644 1.52094 2.33013 2.2704 5 26.25-27.25 -0.09352 0.03633 0.17232 0.16890 0.17714 0.1799 6 27.25-28.25 0.21486 0.08387 0.24494 0.25366 4.45410 4.9507 7 28.25-29.25 -0.17208 0.04064 0.12996 0.12534 3.21126 3.5837 8 >29.25 0.54688 0.34220 1.29335 2.24483 8.50836 13.1139
Inputting the aids data, table 5.5, p. 119.
data aids1;
input race1 azt1 symptoms freq race2 azt2 race3 azt3;
cards;
1 1 1 14 0 0 1 1
1 1 0 93 0 0 1 1
1 0 1 32 0 1 1 -1
1 0 0 81 0 1 1 -1
0 1 1 11 1 0 -1 1
0 1 0 52 1 0 -1 1
0 0 1 12 1 1 -1 -1
0 0 0 43 1 1 -1 -1
;
run;
Fitting a Logit model using different dummy and effect coding. In table 5.6, p. 121, the Last=zero column corresponds to the parameter estimates obtained by using dummy variables race1 and azt1. The First=zero column corresponds to the parameter estimates obtained using dummy variables race2 and azt2. The Sum=zero column corresponds to the parameter estimates obtained using the effect coded variables race3 and azt3.Note: The option descending (desc) in the proc statement so that the lower value, in this case symptoms = zero, is defined as the nonevent.
proc genmod data=aids1 descending; model symptoms = race1 azt1/ dist=bin link=logit; weight freq; run; proc genmod data=aids1 desc; model symptoms = race2 azt2/ dist=bin link=logit; weight freq; run; proc genmod data=aids1 desc; model symptoms = race3 azt3/ dist=bin link=logit; weight freq; run;
The GENMOD Procedure
Model Information
Data Set WORK.AIDS1
Distribution Binomial
Link Function Logit
Dependent Variable symptoms
Scale Weight Variable freq
Observations Used 8
Probability Modeled Pr( symptoms = 1 )
Response Profile
Ordered Ordered
Level Value Count
1 1 69
2 0 269
Criteria For Assessing Goodness Of Fit
Criterion DF Value Value/DF
Deviance 5 335.1512 67.0302
Scaled Deviance 5 335.1512 67.0302
Pearson Chi-Square 5 338.3142 67.6628
Scaled Pearson X2 5 338.3142 67.6628
Log Likelihood -167.5756
Algorithm converged.
Analysis Of Parameter Estimates
Standard Wald 95% Confidence Chi-
Parameter DF Estimate Error Limits Square Pr > ChiSq
Intercept 1 -1.0736 0.2629 -1.5889 -0.5582 16.67 <.0001
race1 1 0.0555 0.2886 -0.5102 0.6212 0.04 0.8476
azt1 1 -0.7195 0.2790 -1.2662 -0.1727 6.65 0.0099
Scale 0 1.0000 0.0000 1.0000 1.0000
NOTE: The scale parameter was held fixed.
The GENMOD Procedure
Model Information
Data Set WORK.AIDS1
Distribution Binomial
Link Function Logit
Dependent Variable symptoms
Scale Weight Variable freq
Observations Used 8
Probability Modeled Pr( symptoms = 1 )
Response Profile
Ordered Ordered
Level Value Count
1 1 69
2 0 269
Criteria For Assessing Goodness Of Fit
Criterion DF Value Value/DF
Deviance 5 335.1512 67.0302
Scaled Deviance 5 335.1512 67.0302
Pearson Chi-Square 5 338.3142 67.6628
Scaled Pearson X2 5 338.3142 67.6628
Log Likelihood -167.5756
Algorithm converged.
Analysis Of Parameter Estimates
Standard Wald 95% Confidence Chi-
Parameter DF Estimate Error Limits Square Pr > ChiSq
Intercept 1 -1.7375 0.2404 -2.2087 -1.2664 52.25 <.0001
race2 1 -0.0555 0.2886 -0.6212 0.5102 0.04 0.8476
azt2 1 0.7195 0.2790 0.1727 1.2662 6.65 0.0099
Scale 0 1.0000 0.0000 1.0000 1.0000
NOTE: The scale parameter was held fixed.
The GENMOD Procedure
Model Information
Data Set WORK.AIDS1
Distribution Binomial
Link Function Logit
Dependent Variable symptoms
Scale Weight Variable freq
Observations Used 8
Probability Modeled Pr( symptoms = 1 )
Response Profile
Ordered Ordered
Level Value Count
1 1 69
2 0 269
Criteria For Assessing Goodness Of Fit
Criterion DF Value Value/DF
Deviance 5 335.1512 67.0302
Scaled Deviance 5 335.1512 67.0302
Pearson Chi-Square 5 338.3142 67.6628
Scaled Pearson X2 5 338.3142 67.6628
Log Likelihood -167.5756
Algorithm converged.
Analysis Of Parameter Estimates
Standard Wald 95% Confidence Chi-
Parameter DF Estimate Error Limits Square Pr > ChiSq
Intercept 1 -1.4056 0.1467 -1.6931 -1.1181 91.82 <.0001
race3 1 0.0277 0.1443 -0.2551 0.3106 0.04 0.8476
azt3 1 -0.3597 0.1395 -0.6331 -0.0863 6.65 0.0099
Scale 0 1.0000 0.0000 1.0000 1.0000
NOTE: The scale parameter was held fixed.
Inputting the aids data and fitting a logit model using the code in Table A.9, p. 272.Note: The parameter estimates from the first model corresponds to the column labeled Last=zero and the estimated from the second model corresponds to the column labeled First=zero.
data aids; input race $ azt $ yes no @@; cases = yes + no; cards; white y 14 93 white n 32 81 black y 11 52 black n 12 43 ; run; proc genmod data=aids order=data; class race azt; model yes/cases = race azt / dist=bin link=logit obstats type3; run; proc genmod data = aids desc; class race azt; model yes/cases = race azt/ dist=bin link=logit; run;
The GENMOD Procedure
Model Information
Data Set WORK.AIDS
Distribution Binomial
Link Function Logit
Response Variable (Events) yes
Response Variable (Trials) cases
Observations Used 4
Number Of Events 69
Number Of Trials 338
Class Level Information
Class Levels Values
race 2 white black
azt 2 y n
Criteria For Assessing Goodness Of Fit
Criterion DF Value Value/DF
Deviance 1 1.3835 1.3835
Scaled Deviance 1 1.3835 1.3835
Pearson Chi-Square 1 1.3910 1.3910
Scaled Pearson X2 1 1.3910 1.3910
Log Likelihood -167.5756
Algorithm converged.
Analysis Of Parameter Estimates
Standard Wald 95% Confidence Chi-
Parameter DF Estimate Error Limits Square Pr > ChiSq
Intercept 1 -1.0736 0.2629 -1.5889 -0.5582 16.67 <.0001
race white 1 0.0555 0.2886 -0.5102 0.6212 0.04 0.8476
race black 0 0.0000 0.0000 0.0000 0.0000 . .
azt y 1 -0.7195 0.2790 -1.2662 -0.1727 6.65 0.0099
azt n 0 0.0000 0.0000 0.0000 0.0000 . .
Scale 0 1.0000 0.0000 1.0000 1.0000
NOTE: The scale parameter was held fixed.
The GENMOD Procedure
LR Statistics For Type 3 Analysis
Chi-
Source DF Square Pr > ChiSq
race 1 0.04 0.8473
azt 1 6.87 0.0088
Observation Statistics
Observation yes cases race azt Pred Xbeta Std HessWgt
Lower Upper Resraw Reschi Resdev
StResdev StReschi Reslik
1 14 107 white y 0.1496245 -1.737549 0.2403848 13.614362
0.0989724 0.2198735 -2.009824 -0.544703 -0.554665
-1.200988 -1.179418 -1.184051
2 32 113 white n 0.2653998 -1.018089 0.1985145 22.03079
0.1966808 0.3477355 2.009824 0.4281964 0.4252503
1.171303 1.1794176 1.1783512
3 11 63 black y 0.1427012 -1.793034 0.2843628 7.707267
0.087036 0.2251866 2.009824 0.7239488 0.7034699
1.1460546 1.1794176 1.1669593
4 12 55 black n 0.2547241 -1.073574 0.2629407 10.441185
0.1695348 0.3639596 -2.009824 -0.62199 -0.63259
-1.199517 -1.179418 -1.185042
The GENMOD Procedure
Model Information
Data Set WORK.AIDS
Distribution Binomial
Link Function Logit
Response Variable (Events) yes
Response Variable (Trials) cases
Observations Used 4
Number Of Events 69
Number Of Trials 338
Class Level Information
Class Levels Values
race 2 black white
azt 2 n y
Criteria For Assessing Goodness Of Fit
Criterion DF Value Value/DF
Deviance 1 1.3835 1.3835
Scaled Deviance 1 1.3835 1.3835
Pearson Chi-Square 1 1.3910 1.3910
Scaled Pearson X2 1 1.3910 1.3910
Log Likelihood -167.5756
Algorithm converged.
Analysis Of Parameter Estimates
Standard Wald 95% Confidence Chi-
Parameter DF Estimate Error Limits Square Pr > ChiSq
Intercept 1 -1.7375 0.2404 -2.2087 -1.2664 52.25 <.0001
race black 1 -0.0555 0.2886 -0.6212 0.5102 0.04 0.8476
race white 0 0.0000 0.0000 0.0000 0.0000 . .
azt n 1 0.7195 0.2790 0.1727 1.2662 6.65 0.0099
azt y 0 0.0000 0.0000 0.0000 0.0000 . .
Scale 0 1.0000 0.0000 1.0000 1.0000
NOTE: The scale parameter was held fixed.
Results from the logistic model with width and dummy variables for the categorical variable color as predictors. In order generate the predicted probabilities at the bottom of p. 123 two new observations were added to the dataset.
data dummy; if _n_=1 then do; width=26.3; c1=1; c2=0; c3=0; output; width=26.3; c1=0; c2=0; c3=0; output; end; set crab; c1 = 0; if color=1 then c1=1; c2 = 0; if color=2 then c2=1; c3 = 0; if color=3 then c3=1; output; run; proc logistic data = dummy desc; model y = c1 c2 c3 width; output out=temp p=pi_hat; run; proc print data = temp; where y=.; var width c1 c2 c3 pi_hat; run;
The LOGISTIC Procedure
Model Information
Data Set WORK.DUMMY
Response Variable y
Number of Response Levels 2
Number of Observations 173
Link Function Logit
Optimization Technique Fisher's scoring
Response Profile
Ordered Total
Value y Frequency
1 1 111
2 0 62
NOTE: 2 observations were deleted due to missing values for the response or explanatory
variables.
Model Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.
Model Fit Statistics
Intercept
Intercept and
Criterion Only Covariates
AIC 227.759 197.457
SC 230.912 213.223
-2 Log L 225.759 187.457
Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 38.3015 4 <.0001
Score 34.3384 4 <.0001
Wald 27.6788 4 <.0001
The LOGISTIC Procedure
Analysis of Maximum Likelihood Estimates
Standard
Parameter DF Estimate Error Chi-Square Pr > ChiSq
Intercept 1 -12.7151 2.7618 21.1965 <.0001
c1 1 1.3299 0.8525 2.4335 0.1188
c2 1 1.4023 0.5484 6.5380 0.0106
c3 1 1.1061 0.5921 3.4901 0.0617
width 1 0.4680 0.1055 19.6573 <.0001
Odds Ratio Estimates
Point 95% Wald
Effect Estimate Confidence Limits
c1 3.781 0.711 20.102
c2 4.065 1.387 11.909
c3 3.023 0.947 9.646
width 1.597 1.298 1.964
Association of Predicted Probabilities and Observed Responses
Percent Concordant 76.9 Somers' D 0.543
Percent Discordant 22.6 Gamma 0.546
Percent Tied 0.5 Tau-a 0.251
Pairs 6882 c 0.771
Obs width c1 c2 c3 pi_hat
1 26.3 1 0 0 0.71546
2 26.3 0 0 0 0.39942
Fig. 5.4, p. 124.
proc sort data=temp;
by width;
run;
data temp1;
set temp;
if c1=1 then pi1 = pi_hat ;
else pi1= . ;
if c2=1 then pi2=pi_hat;
else pi2=.;
if c3=1 then pi3=pi_hat;
else pi3=.;
if c2=0 and c1=0 and c3=0 then pi4=pi_hat;
if c1 = 1 or c2=1 or c3=1 then pi4=.;
run;
goptions reset=all;
symbol1 c=blue i=spline width=2 ;
symbol2 c=red i=spline w=2 ;
symbol3 c=green i=spline w=2 ;
symbol4 c=cyan i=spline w=2 ;
axis1 order=(0 to 1 by .1) label=(angle=90 'Est. prob.');
legend1 label=none value=(height=1 font=swiss 'Color 1' 'Color 2' 'Color 3' 'Color 4' )
position=(bottom right inside) mode=share cborder=black;
proc gplot data=temp1;
plot (pi1 pi2 pi3 pi4)*width/vaxis=axis1 overlay legend=legend1;
run;
quit;
Testing the main effect of color by testing the three dummy variables simultaneously, p. 124.Note: There are several ways to accomplish this. One way is to add a test statement which uses a Wald Chi-squared test and SAS will provide the test statistic and a p-value. Another way is to run the model with and without the variables to be tested and then take the difference of the -2logL in the Model Fit Statistics table and compare this difference to the Chi-squared distribution with n degrees of freedom (where n=number of variables being tested).
proc logistic data = dummy desc; model y = c1 c2 c3 width; test c1=c2=c3=0; run; proc logistic data = dummy desc; model y=width; run;
The LOGISTIC Procedure
Model Information
Data Set WORK.DUMMY
Response Variable y
Number of Response Levels 2
Number of Observations 173
Link Function Logit
Optimization Technique Fisher's scoring
Response Profile
Ordered Total
Value y Frequency
1 1 111
2 0 62
NOTE: 2 observations were deleted due to missing values for the response or explanatory
variables.
Model Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.
Model Fit Statistics
Intercept
Intercept and
Criterion Only Covariates
AIC 227.759 197.457
SC 230.912 213.223
-2 Log L 225.759 187.457
Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 38.3015 4 <.0001
Score 34.3384 4 <.0001
Wald 27.6788 4 <.0001
The LOGISTIC Procedure
Analysis of Maximum Likelihood Estimates
Standard
Parameter DF Estimate Error Chi-Square Pr > ChiSq
Intercept 1 -12.7151 2.7618 21.1965 <.0001
c1 1 1.3299 0.8525 2.4335 0.1188
c2 1 1.4023 0.5484 6.5380 0.0106
c3 1 1.1061 0.5921 3.4901 0.0617
width 1 0.4680 0.1055 19.6573 <.0001
Odds Ratio Estimates
Point 95% Wald
Effect Estimate Confidence Limits
c1 3.781 0.711 20.102
c2 4.065 1.387 11.909
c3 3.023 0.947 9.646
width 1.597 1.298 1.964
Association of Predicted Probabilities and Observed Responses
Percent Concordant 76.9 Somers' D 0.543
Percent Discordant 22.6 Gamma 0.546
Percent Tied 0.5 Tau-a 0.251
Pairs 6882 c 0.771
Linear Hypotheses Testing Results
Wald
Label Chi-Square DF Pr > ChiSq
Test 1 6.6246 3 0.0849
The LOGISTIC Procedure
Model Information
Data Set WORK.DUMMY
Response Variable y
Number of Response Levels 2
Number of Observations 173
Link Function Logit
Optimization Technique Fisher's scoring
Response Profile
Ordered Total
Value y Frequency
1 1 111
2 0 62
NOTE: 2 observations were deleted due to missing values for the response or explanatory
variables.
Model Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.
Model Fit Statistics
Intercept
Intercept and
Criterion Only Covariates
AIC 227.759 198.453
SC 230.912 204.759
-2 Log L 225.759 194.453
Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 31.3059 1 <.0001
Score 27.8752 1 <.0001
Wald 23.8872 1 <.0001
The LOGISTIC Procedure
Analysis of Maximum Likelihood Estimates
Standard
Parameter DF Estimate Error Chi-Square Pr > ChiSq
Intercept 1 -12.3508 2.6287 22.0749 <.0001
width 1 0.4972 0.1017 23.8872 <.0001
Odds Ratio Estimates
Point 95% Wald
Effect Estimate Confidence Limits
width 1.644 1.347 2.007
Association of Predicted Probabilities and Observed Responses
Percent Concordant 73.5 Somers' D 0.485
Percent Discordant 25.0 Gamma 0.492
Percent Tied 1.5 Tau-a 0.224
Pairs 6882 c 0.742
Estimating the main effects with Crab data, table 5.7, p. 127.Note: The type3 option tells SAS to test the main effects as well as the dummy variables for the categorical variables.
proc genmod data = crab desc; class color spine ; model y = color spine width weight/ dist=bin link=logit type3; run;
The GENMOD Procedure
Model Information
Data Set WORK.CRAB
Distribution Binomial
Link Function Logit
Dependent Variable y
Observations Used 173
Probability Modeled Pr( y = 1 )
Class Level Information
Class Levels Values
color 4 1 2 3 4
spine 3 1 2 3
Response Profile
Ordered Ordered
Level Value Count
1 1 111
2 0 62
Criteria For Assessing Goodness Of Fit
Criterion DF Value Value/DF
Deviance 165 185.2020 1.1224
Scaled Deviance 165 185.2020 1.1224
Pearson Chi-Square 165 169.7557 1.0288
Scaled Pearson X2 165 169.7557 1.0288
Log Likelihood -92.6010
Algorithm converged.
Analysis Of Parameter Estimates
Standard Wald 95% Confidence Chi-
Parameter DF Estimate Error Limits Square Pr > ChiSq
Intercept 1 -9.2734 3.8378 -16.7954 -1.7514 5.84 0.0157
color 1 1 1.6087 0.9355 -0.2250 3.4423 2.96 0.0855
color 2 1 1.5058 0.5667 0.3951 2.6164 7.06 0.0079
color 3 1 1.1198 0.5933 -0.0430 2.2826 3.56 0.0591
color 4 0 0.0000 0.0000 0.0000 0.0000 . .
spine 1 1 -0.4003 0.5027 -1.3856 0.5850 0.63 0.4259
The GENMOD Procedure
Analysis Of Parameter Estimates
Standard Wald 95% Confidence Chi-
Parameter DF Estimate Error Limits Square Pr > ChiSq
spine 2 1 -0.4963 0.6292 -1.7294 0.7369 0.62 0.4302
spine 3 0 0.0000 0.0000 0.0000 0.0000 . .
width 1 0.2631 0.1953 -0.1197 0.6459 1.82 0.1779
weight 1 0.8258 0.7038 -0.5537 2.2053 1.38 0.2407
Scale 0 1.0000 0.0000 1.0000 1.0000
NOTE: The scale parameter was held fixed.
LR Statistics For Type 3 Analysis
Chi-
Source DF Square Pr > ChiSq
color 3 7.60 0.0551
spine 2 1.01 0.6038
width 1 1.80 0.1801
weight 1 1.41 0.2351
Obtaining the same results using dummy variables and proc logistic.Note: Only proc logistic will provide the likelihood-ratio test comparing the full model to the null model.
data main; set crab; c1 = 0; if color=1 then c1=1; c2 = 0; if color=2 then c2=1; c3 = 0; if color=3 then c3=1; spine1=0; if spine=1 then spine1=1; spine2=0; if spine=2 then spine2=1; run; proc logistic data = main desc; model y = c1 c2 c3 spine1 spine2 width weight; test c1=c2=c3=0; test spine1=spine2=0; run;
The LOGISTIC Procedure
Model Information
Data Set WORK.MAIN
Response Variable y
Number of Response Levels 2
Number of Observations 173
Link Function Logit
Optimization Technique Fisher's scoring
Response Profile
Ordered Total
Value y Frequency
1 1 111
2 0 62
Model Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.
Model Fit Statistics
Intercept
Intercept and
Criterion Only Covariates
AIC 227.759 201.202
SC 230.912 226.428
-2 Log L 225.759 185.202
Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 40.5565 7 <.0001
Score 36.3068 7 <.0001
Wald 29.4763 7 0.0001
The LOGISTIC Procedure
Analysis of Maximum Likelihood Estimates
Standard
Parameter DF Estimate Error Chi-Square Pr > ChiSq
Intercept 1 -9.2734 3.8378 5.8386 0.0157
c1 1 1.6087 0.9355 2.9567 0.0855
c2 1 1.5058 0.5667 7.0607 0.0079
c3 1 1.1198 0.5933 3.5624 0.0591
spine1 1 -0.4003 0.5027 0.6340 0.4259
spine2 1 -0.4963 0.6292 0.6222 0.4302
width 1 0.2631 0.1953 1.8152 0.1779
weight 1 0.8258 0.7038 1.3765 0.2407
Odds Ratio Estimates
Point 95% Wald
Effect Estimate Confidence Limits
c1 4.996 0.799 31.259
c2 4.508 1.485 13.687
c3 3.064 0.958 9.803
spine1 0.670 0.250 1.795
spine2 0.609 0.177 2.089
width 1.301 0.887 1.908
weight 2.284 0.575 9.073
Association of Predicted Probabilities and Observed Responses
Percent Concordant 77.6 Somers' D 0.555
Percent Discordant 22.1 Gamma 0.557
Percent Tied 0.3 Tau-a 0.257
Pairs 6882 c 0.778
Linear Hypotheses Testing Results
Wald
Label Chi-Square DF Pr > ChiSq
Test 1 7.1610 3 0.0669
Test 2 1.0105 2 0.6034
The first four rows of the deviance and df columns of table 5.8, p. 128.
ods listing close; proc genmod data=crab desc ; class color spine; model y = color|spine|width / dist=bin link=logit type3; ods output modelfit=temp; run; proc genmod data=crab desc; class color spine; model y = color|spine|width@2 / dist=bin link=logit type3; ods output modelfit=temp1; run; proc genmod data=crab desc; class color spine; model y = color spine width color*spine spine*width / dist=bin link=logit type3; ods output modelfit=temp2; run; proc genmod data=crab desc; class color spine; model y = color spine width color*width spine*width / dist=bin link=logit type3; ods output modelfit=temp3; run; ods output close; ods listing; data combo; set temp temp1 temp2 temp3; run; proc print data = combo; where Criterion='Deviance'; var criterion df value; run;
>Obs Criterion DF Value 1 Deviance 152 170.4404 6 Deviance 155 173.6738 11 Deviance 158 177.3357 16 Deviance 161 181.5588
The Diarrhea Example:
data diarrhea;
input cep age stay case count;
datalines;
0 0 0 0 385
0 0 1 5 233
0 1 0 3 789
0 1 1 47 1081
1 1 1 5 5
;
run;
proc logistic data = diarrhea descending exactonly;
model case/count = age stay cep;
exact 'parm' age stay cep /estimate = parm;
run;
The LOGISTIC Procedure
Model Information
Data Set WORK.DIARRHEA
Response Variable (Events) case
Response Variable (Trials) count
Number of Observations 5
Model binary logit
Optimization Technique Fisher's scoring
Response Profile
Ordered Binary Total
Value Outcome Frequency
1 Event 60
2 Nonevent 2433
Exact Conditional Analysis
Conditional Exact Tests for 'parm'
--- p-Value ---
Effect Test Statistic Exact Mid
age Score 3.4067 0.0750 0.0624
Probability 0.0252 0.0750 0.0624
stay Score 34.4965 <.0001 <.0001
Probability 9.03E-11 <.0001 <.0001
cep Score 98.8190 <.0001 <.0001
Probability 2.19E-7 <.0001 <.0001
Exact Parameter Estimates for 'parm'
95% Confidence
Parameter Estimate Limits p-Value
age 0.8514 -0.0782 2.0300 0.0800
stay 2.6775 1.5411 4.2937 <.0001
cep 4.9592* 2.9497 Infinity <.0001
NOTE: * indicates a median unbiased estimate.
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