THE FIRST EXAMPLE: SUCCESSES/FAILURES WITH ONE COVARIATE 1 01:09 Thursday, November 3, 2005 The LOGISTIC Procedure Model Information Data Set WORK.LOG1 Response Variable (Events) rr Response Variable (Trials) nn Model binary logit Optimization Technique Fisher's scoring Number of Observations Read 10 Number of Observations Used 10 Sum of Frequencies Read 133 Sum of Frequencies Used 133 Response Profile Ordered Binary Total Value Outcome Frequency 1 Event 66 2 Nonevent 67 Model Convergence Status Convergence criterion (GCONV=1E-8) satisfied. Model Fit Statistics Intercept Intercept and Criterion Only Covariates AIC 186.370 181.087 SC 189.260 186.868 -2 Log L 184.370 177.087 Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 7.2823 1 0.0070 Score 7.1515 1 0.0075 Wald 6.8922 1 0.0087 THE FIRST EXAMPLE: SUCCESSES/FAILURES WITH ONE COVARIATE 2 01:09 Thursday, November 3, 2005 The LOGISTIC Procedure Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -1.0303 0.4264 5.8400 0.0157 xx 1 0.1749 0.0666 6.8922 0.0087 Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits xx 1.191 1.045 1.357 Association of Predicted Probabilities and Observed Responses Percent Concordant 58.6 Somers' D 0.271 Percent Discordant 31.5 Gamma 0.301 Percent Tied 9.9 Tau-a 0.137 Pairs 4422 c 0.636 SECOND EXAMPLE: OBSERVATIONS WITH FOUR COVARIATES 3 Successes/Failures for different covariate values 01:09 Thursday, November 3, 2005 Frequency 14 + YYYYY | YYYYY 13 + YYYYY | YYYYY 12 + YYYYY | YYYYY 11 + YYYYY | YYYYY 10 + YYYYY | YYYYY 9 + NNNNN | NNNNN 8 + NNNNN | NNNNN 7 + YYYYY NNNNN | YYYYY NNNNN 6 + YYYYY NNNNN YYYYY | YYYYY NNNNN YYYYY 5 + NNNNN NNNNN YYYYY | NNNNN NNNNN YYYYY 4 + NNNNN NNNNN NNNNN | NNNNN NNNNN NNNNN 3 + NNNNN NNNNN NNNNN YYYYY | NNNNN NNNNN NNNNN YYYYY 2 + YYYYY NNNNN NNNNN NNNNN YYYYY | YYYYY NNNNN NNNNN NNNNN YYYYY 1 + NNNNN NNNNN NNNNN NNNNN NNNNN | NNNNN NNNNN NNNNN NNNNN NNNNN -------------------------------------------------------------------- 30 40 50 60 70 x1 Midpoint Symbol ysymbol Symbol ysymbol N N Y Y SECOND EXAMPLE: OBSERVATIONS WITH FOUR COVARIATES 4 Successes/Failures for different covariate values 01:09 Thursday, November 3, 2005 Frequency 10 + YYYYY YYYYY | YYYYY YYYYY | YYYYY YYYYY | YYYYY YYYYY 9 + YYYYY YYYYY | YYYYY YYYYY | YYYYY YYYYY | YYYYY YYYYY 8 + YYYYY YYYYY | YYYYY YYYYY | YYYYY YYYYY | YYYYY YYYYY 7 + NNNNN YYYYY | NNNNN YYYYY | NNNNN YYYYY | NNNNN YYYYY 6 + NNNNN NNNNN | NNNNN NNNNN | NNNNN NNNNN | NNNNN NNNNN 5 + NNNNN NNNNN YYYYY | NNNNN NNNNN YYYYY | NNNNN NNNNN YYYYY | NNNNN NNNNN YYYYY 4 + NNNNN NNNNN YYYYY | NNNNN NNNNN YYYYY | NNNNN NNNNN YYYYY | NNNNN NNNNN YYYYY 3 + NNNNN NNNNN NNNNN NNNNN YYYYY | NNNNN NNNNN NNNNN NNNNN YYYYY | NNNNN NNNNN NNNNN NNNNN YYYYY | NNNNN NNNNN NNNNN NNNNN YYYYY 2 + NNNNN NNNNN NNNNN NNNNN YYYYY | NNNNN NNNNN NNNNN NNNNN YYYYY | NNNNN NNNNN NNNNN NNNNN YYYYY | NNNNN NNNNN NNNNN NNNNN YYYYY 1 + NNNNN NNNNN NNNNN NNNNN NNNNN YYYYY | NNNNN NNNNN NNNNN NNNNN NNNNN YYYYY | NNNNN NNNNN NNNNN NNNNN NNNNN YYYYY | NNNNN NNNNN NNNNN NNNNN NNNNN YYYYY ------------------------------------------------------------------ 45 60 75 90 105 120 x2 Midpoint Symbol ysymbol Symbol ysymbol N N Y Y SECOND EXAMPLE: OBSERVATIONS WITH FOUR COVARIATES 5 Successes/Failures for different covariate values 01:09 Thursday, November 3, 2005 Frequency 14 + YYYYY | YYYYY 13 + YYYYY | YYYYY 12 + YYYYY | YYYYY 11 + YYYYY | YYYYY 10 + YYYYY | YYYYY 9 + YYYYY YYYYY | YYYYY YYYYY 8 + NNNNN YYYYY | NNNNN YYYYY 7 + NNNNN YYYYY | NNNNN YYYYY 6 + NNNNN YYYYY YYYYY | NNNNN YYYYY YYYYY 5 + NNNNN NNNNN YYYYY | NNNNN NNNNN YYYYY 4 + NNNNN NNNNN NNNNN | NNNNN NNNNN NNNNN 3 + NNNNN NNNNN NNNNN | NNNNN NNNNN NNNNN 2 + NNNNN NNNNN NNNNN NNNNN | NNNNN NNNNN NNNNN NNNNN 1 + NNNNN NNNNN NNNNN NNNNN NNNNN | NNNNN NNNNN NNNNN NNNNN NNNNN -------------------------------------------------------------------- 20 30 40 50 60 x3 Midpoint Symbol ysymbol Symbol ysymbol N N Y Y SECOND EXAMPLE: OBSERVATIONS WITH FOUR COVARIATES 6 Successes/Failures for different covariate values 01:09 Thursday, November 3, 2005 Frequency 13 + YYYYY | YYYYY 12 + YYYYY | YYYYY 11 + YYYYY | YYYYY 10 + YYYYY | YYYYY 9 + YYYYY | YYYYY 8 + NNNNN | NNNNN 7 + NNNNN YYYYY | NNNNN YYYYY 6 + NNNNN YYYYY YYYYY | NNNNN YYYYY YYYYY 5 + YYYYY NNNNN NNNNN NNNNN | YYYYY NNNNN NNNNN NNNNN 4 + YYYYY NNNNN NNNNN NNNNN | YYYYY NNNNN NNNNN NNNNN 3 + YYYYY NNNNN NNNNN NNNNN | YYYYY NNNNN NNNNN NNNNN 2 + YYYYY NNNNN NNNNN NNNNN | YYYYY NNNNN NNNNN NNNNN 1 + NNNNN NNNNN NNNNN NNNNN NNNNN | NNNNN NNNNN NNNNN NNNNN NNNNN -------------------------------------------------------------------- 52.5 67.5 82.5 97.5 112.5 x4 Midpoint Symbol ysymbol Symbol ysymbol N N Y Y SECOND EXAMPLE: OBSERVATIONS WITH FOUR COVARIATES 7 LOGISTIC REGRESSION FOR FULL MODEL 01:09 Thursday, November 3, 2005 The LOGISTIC Procedure Model Information Data Set WORK.LOG2 Response Variable yy Number of Response Levels 2 Model binary logit Optimization Technique Fisher's scoring Number of Observations Read 32 Number of Observations Used 32 Response Profile Ordered Total Value yy Frequency 1 1 12 2 0 20 Probability modeled is yy=1. Model Convergence Status Convergence criterion (GCONV=1E-8) satisfied. Model Fit Statistics Intercept Intercept and Criterion Only Covariates AIC 44.340 29.063 SC 45.806 36.391 -2 Log L 42.340 19.063 Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 23.2774 4 0.0001 Score 15.4648 4 0.0038 Wald 7.0455 4 0.1335 SECOND EXAMPLE: OBSERVATIONS WITH FOUR COVARIATES 8 LOGISTIC REGRESSION FOR FULL MODEL 01:09 Thursday, November 3, 2005 The LOGISTIC Procedure Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 1.3752 4.9996 0.0757 0.7833 x1 1 -0.1490 0.0881 2.8635 0.0906 x2 1 0.2026 0.0878 5.3250 0.0210 x3 1 0.00193 0.0676 0.0008 0.9773 x4 1 -0.1681 0.0679 6.1254 0.0133 Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits x1 0.862 0.725 1.024 x2 1.225 1.031 1.455 x3 1.002 0.878 1.144 x4 0.845 0.740 0.966 Association of Predicted Probabilities and Observed Responses Percent Concordant 93.8 Somers' D 0.879 Percent Discordant 5.8 Gamma 0.883 Percent Tied 0.4 Tau-a 0.425 Pairs 240 c 0.940 SECOND EXAMPLE: OBSERVATIONS WITH FOUR COVARIATES 9 STEPWISE LOGISTIC REGRESSION 01:09 Thursday, November 3, 2005 The LOGISTIC Procedure Model Information Data Set WORK.LOG2 Response Variable yy Number of Response Levels 2 Model binary logit Optimization Technique Fisher's scoring Number of Observations Read 32 Number of Observations Used 32 Response Profile Ordered Total Value yy Frequency 1 1 12 2 0 20 Probability modeled is yy=1. Stepwise Selection Procedure Step 0. Intercept entered: Model Convergence Status Convergence criterion (GCONV=1E-8) satisfied. -2 Log L = 42.340 Residual Chi-Square Test Chi-Square DF Pr > ChiSq 15.4648 4 0.0038 Step 1. Effect x2 entered: Model Convergence Status Convergence criterion (GCONV=1E-8) satisfied. SECOND EXAMPLE: OBSERVATIONS WITH FOUR COVARIATES 10 STEPWISE LOGISTIC REGRESSION 01:09 Thursday, November 3, 2005 The LOGISTIC Procedure Model Fit Statistics Intercept Intercept and Criterion Only Covariates AIC 44.340 36.317 SC 45.806 39.249 -2 Log L 42.340 32.317 Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 10.0228 1 0.0015 Score 8.8315 1 0.0030 Wald 6.7022 1 0.0096 Residual Chi-Square Test Chi-Square DF Pr > ChiSq 9.6376 3 0.0219 NOTE: No effects for the model in Step 1 are removed. Step 2. Effect x4 entered: Model Convergence Status Convergence criterion (GCONV=1E-8) satisfied. Model Fit Statistics Intercept Intercept and Criterion Only Covariates AIC 44.340 29.115 SC 45.806 33.512 -2 Log L 42.340 23.115 SECOND EXAMPLE: OBSERVATIONS WITH FOUR COVARIATES 11 STEPWISE LOGISTIC REGRESSION 01:09 Thursday, November 3, 2005 The LOGISTIC Procedure Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 19.2253 2 <.0001 Score 14.2935 2 0.0008 Wald 7.1590 2 0.0279 Residual Chi-Square Test Chi-Square DF Pr > ChiSq 3.6904 2 0.1580 NOTE: No effects for the model in Step 2 are removed. NOTE: No (additional) effects met the 0.05 significance level for entry into the model. Summary of Stepwise Selection Effect Number Score Wald Step Entered Removed DF In Chi-Square Chi-Square Pr > ChiSq 1 x2 1 1 8.8315 0.0030 2 x4 1 2 7.3427 0.0067 Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -1.2708 3.4850 0.1330 0.7154 x2 1 0.1147 0.0455 6.3664 0.0116 x4 1 -0.1253 0.0550 5.1955 0.0226 Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits x2 1.122 1.026 1.226 x4 0.882 0.792 0.983 SECOND EXAMPLE: OBSERVATIONS WITH FOUR COVARIATES 12 STEPWISE LOGISTIC REGRESSION 01:09 Thursday, November 3, 2005 The LOGISTIC Procedure Association of Predicted Probabilities and Observed Responses Percent Concordant 91.3 Somers' D 0.825 Percent Discordant 8.8 Gamma 0.825 Percent Tied 0.0 Tau-a 0.399 Pairs 240 c 0.913 SECOND EXAMPLE: OBSERVATIONS WITH FOUR COVARIATES 13 BACKWARDS STEPWISE LOGISTIC REGRESSION 01:09 Thursday, November 3, 2005 The LOGISTIC Procedure Model Information Data Set WORK.LOG2 Response Variable yy Number of Response Levels 2 Model binary logit Optimization Technique Fisher's scoring Number of Observations Read 32 Number of Observations Used 32 Response Profile Ordered Total Value yy Frequency 1 1 12 2 0 20 Probability modeled is yy=1. Backward Elimination Procedure Step 0. The following effects were entered: Intercept x1 x2 x3 x4 Model Convergence Status Convergence criterion (GCONV=1E-8) satisfied. Model Fit Statistics Intercept Intercept and Criterion Only Covariates AIC 44.340 29.063 SC 45.806 36.391 -2 Log L 42.340 19.063 Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 23.2774 4 0.0001 Score 15.4648 4 0.0038 Wald 7.0455 4 0.1335 SECOND EXAMPLE: OBSERVATIONS WITH FOUR COVARIATES 14 BACKWARDS STEPWISE LOGISTIC REGRESSION 01:09 Thursday, November 3, 2005 The LOGISTIC Procedure Step 1. Effect x3 is removed: Model Convergence Status Convergence criterion (GCONV=1E-8) satisfied. Model Fit Statistics Intercept Intercept and Criterion Only Covariates AIC 44.340 27.063 SC 45.806 32.926 -2 Log L 42.340 19.063 Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 23.2766 3 <.0001 Score 15.2522 3 0.0016 Wald 7.0368 3 0.0707 Residual Chi-Square Test Chi-Square DF Pr > ChiSq 0.0008 1 0.9773 Step 2. Effect x1 is removed: Model Convergence Status Convergence criterion (GCONV=1E-8) satisfied. Model Fit Statistics Intercept Intercept and Criterion Only Covariates AIC 44.340 29.115 SC 45.806 33.512 -2 Log L 42.340 23.115 SECOND EXAMPLE: OBSERVATIONS WITH FOUR COVARIATES 15 BACKWARDS STEPWISE LOGISTIC REGRESSION 01:09 Thursday, November 3, 2005 The LOGISTIC Procedure Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 19.2253 2 <.0001 Score 14.2935 2 0.0008 Wald 7.1590 2 0.0279 Residual Chi-Square Test Chi-Square DF Pr > ChiSq 3.6904 2 0.1580 NOTE: No (additional) effects met the 0.05 significance level for removal from the model. Summary of Backward Elimination Effect Number Wald Step Removed DF In Chi-Square Pr > ChiSq 1 x3 1 3 0.0008 0.9773 2 x1 1 2 2.9023 0.0885 Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -1.2708 3.4850 0.1330 0.7154 x2 1 0.1147 0.0455 6.3664 0.0116 x4 1 -0.1253 0.0550 5.1955 0.0226 Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits x2 1.122 1.026 1.226 x4 0.882 0.792 0.983 Association of Predicted Probabilities and Observed Responses Percent Concordant 91.3 Somers' D 0.825 Percent Discordant 8.8 Gamma 0.825 Percent Tied 0.0 Tau-a 0.399 Pairs 240 c 0.913