#### Transcript Added Predictive Ability

A SAS Macro to Compute Added Predictive Ability of New Markers in Logistic Regression Kevin Kennedy, MS Saint Luke’s Hospital, Kansas City, MO Kansas City Area SAS User Group Meeting September 3, 2009 Acknowledgment • A special thanks to Michael Pencina (PhD, Biostatistics Dept, Boston University, Harvard Clinical Research Institute) for his valuable input, ideas, and additional formulas to strengthen output. Motivation • Predicting dichotomous outcomes are important – Will a patient develop a disease? – Will an applicant default on a loan? – Will KSU win a game in the NCAA tournament? • Improving an already usable models should be a continued goal Motivation • Many published models exist – Risk of Bleeding after PCI • Predicts patient risk of bleeding after a PCI (Percutaneous Coronary Intervention) based on 10 patient characteristics: Age, gender, shock, prior intervention, kidney function, etc.. • Example: a 78 year old female with Prior Shock and Congestive Heart Failure has a ~8% chance of bleeding after procedure (national avg=2-3%) – Why important? We can treat those at high risk appropriately Mehta et al. Circ Intervention, June 2009 However….. • These models aren’t etched in stone • New markers (variables) should be investigated to improve model performance • Important Question: How do we determine if we should add these new markers to the model? Project Goal • Compare: model1: logit( y) 1 var1 ..... n varn With model2: logit( y) 1 var1 ..... n varn n1 varn1 • How Much Does the new variable add to model performance? • Output of Interest: Predicted Probability of Event (one for each model) computed with the logistic regression model Outline • Traditional Comparisions – Receiver Operating Characteristic Curve • New Measures – IDI, NRI, Vickers Decision Curve • SAS macro to obtain output Traditional Approach-AUCs • Common to plot the Receiver Operating Characteristic Curve (ROC) and report the area underneath (AUC) or c-statistic • Measures model discrimination • Equivalent to the probability that the predicted risk is higher for an event than a non-even[4] AUC/ROC Background • ROC/AUC plot depicts trade-off between benefit (true positives) and costs (false positives) by defining a “cut off” to determine positive and negative individuals Cut Off Events Non-Events False Neg Predicted Probabilities True Positive Rate (Sensitivity)=.9 False Positive Rate (1-Specificity)=.4 ROC curve 100% AUC .9 True Positive Rate (sensitivity) AUC 1 P erfect Discrimination ~ .9 ~ .7 .65 .5 Yippy! Ehh... Ugh... No Discrimination 0% 0% 100% False Positive Rate (1-specificity) AUC Computation • All possible pairs are made between events and nonevents – A dataset with 100 events and 1000 non-events would have 100*1000=100000 pairs – If the predicted probability for the event is higher for the subject actually experiencing the event give a ‘1’ (concordant) otherwise ‘0’ (discordant) – C-statistic is the average of the 1’s and 0’s (.5 for ties) • Now use methods by DeLong to compare AUCs of Model1 and Model2 Advantages • Used all the time • Recommended guidelines exist for Excellent/Good/Poor discrimination • Default output in proc logistic, with new extensions in version 9.2 (roc and roccontrast statements) along with other computing packages Disadvantages • Rank based – A comparison of .51 to .50 is treated the same as .7 to .1 • Doesn’t imply a useful model – Example: all events have probability=.51 and non-event have probability=.5---Perfect discrimination (c=1) but not useful • Extremely hard to find markers that result in high AUC – Pepe [5] claims an odds ratio of 3 doesn’t yield good discrimination Alternatives • Pencina and D’Agostino in 2008 (Statistics in Medicine) suggest 2 additional statistics: – IDI (Integrated Discrimination Improvement) – NRI (Net Reclassification Improvement) • Vickers [2,3] developed graphical techniques of comparing models IDI • Measure of improved sensitivity without sacrificing specificity • Formula measures how much increase in ‘p’ for events, and how much decrease for the non-event Absolute_ IDI ( p _ event _ 2 p _ event _ 1) ( p _ nonevent_ 1 p _ nonevent_ 2) Re lative_ IDI p _ event _ 2 p _ nonevent_ 2 1 p _ event _ 1 p _ nonevent_ 1 where: p _ event _ 2 is the mean predictedprobability of the eventsin the2nd model IDI Absolute_IDI=(.15-.12)+(.25-.2)=.08 Model1 Model2 Mean Predicted Probability in % 50 Relative_IDI Mean ‘p’ decreased by 3% for non-events .25 .12 1 1.6 .2 .15 160% Relative improvement Mean ‘p’ increased by 5% for events 40 30 25 20 20 15 12 10 0 Non-Events Events NRI • Measures how well the new model reclassifies event and non-event • Dependant on how we decide to classify observations – Example: 0-10% Low, 10-20% Moderate, >20% High • Questions: do the patients experiencing an event go up in risk? Mod to High between model1 and 2 • Do Patients not experiencing an event go down in risk? Moderate to Low? NRI • Formula: # Eventsmoving up # Eventsmovingdown NRI # of Events # of Events # Non - eventsmovingdown # Non - eventsmoving up # of Non - events # of Non - events NRI Computation Example • 3 groups defined – <10% (low) – 10-20% (moderate) – >20% (high) • Each individual will have a group for model 1 and 2 • 2 cross-tabulation tables (events and non-events) NRI Computation Example • Crosstab1: Events (100 Events) Model 2 Low Mod High 20 Events moving up Model 1 10 Events moving down Low 10 8 2 Mod 3 30 10 High 2 5 30 70 Events not moving Net of 10/100 (10%) of events getting reclassified correctly NRI Computation Example • Crosstab2: Non-Events (200 Non-Events) Model 2 Model 1 Low Mod High Low 50 5 0 Mod 20 40 10 High 5 10 60 15 Non-events moving up 35 Non-events moving down 150 Non-events not moving Net of 20/200 (10%) of nonevents getting reclassified correctly 10 35 15 20 NRI .2 100 100 200 200 NRI Caveats • Dependant on Groups – Would we reach similar conclusions with groups: <15, 15-30, >30????? • Alternative ways to define groups – Any up/down movement. A change in p from .15 to .151 would be an ‘up’ movement – A threshold: ex. a change of 3% would constitute an up/down. i.e. .33 to .37 would be an ‘up’ but .33 to .34 would be no movement • Good News: Macro handles all these cases and you can request all at once Vickers Decision Curve • A graphical comparison of model 1 and 2 based off of ‘net’ benefit (first attributed to Peirce-1884) • Useful if a threshold is important. • Example: If a persons predicted probability of an outcome is greater than 10% we treat with strategy A – Here we’d want to compare the models at this threshold True P ositives FalseP ositives pt NetBenefit 1 pt n n Vickers Decision Curve • Example: – N=1000, dichotomous event, 10% as threshold Predicted Probability Outcome ≥10% <10% True Positive count Yes No 80 200 20 False Positive count 700 80 200 .1 .0578 NetBenefit 1000 1000 1 .1 5.78 net true positive results per 100 compared with treating all as negative Vickers Decision Curve No difference between 1 2 and treat all 0.06 0.05 Net Benefit 0.04 Model 2 seems to outperform 1 0.03 0.02 0.01 0.00 -0.01 0 10 20 30 40 Threshold Probability in % PLOT Treat All Model1 Model2 50 SAS Macro to Obtain Output • %added_pred(data= ,id= , y= , model1cov= ,model2cov= , nripoints=ALL, nriallcut=%str(), vickersplot=FALSE, vickerpoints=%str()); • Model1cov=initial model • Model2cov=new model • Nripoints=nri levels (eg: <10, 10-20, >20) insert nripoints=.1 .2 • Nriallcut=if you want to test amount of increase or decrease instead of levels, i.e. if you want to know if a person increase/decreases by 5% insert nriallcut=.05 • Vickerpoints=thresholds to test (eg, 10%) AUC section of Macro • AUC Comparisions are easy with SAS 9.2 PROC LOGISTIC DATA=&DATA; MODEL &y=&model1cov &model2cov; ROC ‘FIRST’ &model1cov; ROC ‘SECOND’ &model2cov; ROCCONTRAST REFERENCE=‘FIRST’; ODS OUTPUT ROCASSOCIATION=ROCASS ROCCONTRASTESTIMATE=ROCDIFF; RUN; • If working from an earlier version the %roc macro will be called from the SAS website[6] AUC Section of Macro • Sample output • %added_pred(data=data, id=id, y=event, model1cov=x1 x2 x3 x4, model2cov=x1 x2 x3 x4 x5, …….); Model1 AUC Model2 AUC Difference in AUC 0.77907 0.79375 0.0147 Std Error for Difference P-value for difference 95% CI for Difference .0042 0.0005 (0.00646,0.0229) IDI Section of Macro • Use proc logistic output dataset for model1 and model2 PROC LOGISTIC DATA=&DATA; MODEL &y=&model1cov; OUTPUT OUT=OLD PRED=P_OLD; RUN; PROC LOGISTIC DATA=&DATA; MODEL &y=&model2cov; OUTPUT OUT=NEW PRED=P_NEW; RUN; proc sql noprint; create table probs as select *, (p_new-p_old) as pdiff from old(keep=&id &y &model1cov &model2cov p_old ) as a join new(keep=&id &y &model1cov &model2cov p_new ) as b on a.&id=b.&id order by &y; quit; • Now use proc means or sql to obtain: p_event_new, p_event_old, p_nonevent_new, p_nonevent_old IDI Section of Macro • Sample Output IDI IDI Std Err Z-value Pvalue 95% CI Probability change for events Probability change for non-events Relative IDI .0207 .0064 6.3 <.0001 (0.0143, 0.0272) .0186 -.002125 .167 NRI Section of Macro • In 3 parts: – All groups (any up/down movement) – User defined (eg. <10,10-20,>20) – Threshold (eg. a change of 3%) • Coding is more involved here containing – Counts for number of groups – Do-loops for various # of thresholds and user groups NRI Section of Macro User Defined Groups(eg. <10, 10-20,>20 nripoints=.1 .2) %if &nripoints^=ALL %then %do; %let numgroups=%eval(%words(&nripoints)+1); /*figure out how many ordinal groups*/ data nriprobs; set probs; /*define first ordinal group for pre and post*/ if 0<=p_old<=%scan(&nripoints,1,' ') then group_pre=1; if 0<=p_new<=%scan(&nripoints,1,' ') then group_post=1; %let i=1; %do %until(&i>%eval(&numgroups-1)); if %scan(&nripoints,&i,' ')<p_old then do;group_pre=&i+1;end; if %scan(&nripoints,&i,' ')<p_new then do;group_post=&i+1;end; %let i=%eval(&i+1); %end; if &y=0 then do; if group_post>group_pre then up_nonevent=1; if group_post<group_pre then down_nonevent=1; end; if &y=1 then do; if group_post>group_pre then up_event=1; if group_post<group_pre then down_event=1 end; if up_nonevent=. then up_nonevent=0; if down_nonevent=. then down_nonevent=0; if up_event=. then up_event=0; if down_event=. then down_event=0; run; NRI Section of Macro • Sample Output • %added_pred(data=,…..,nripoints=.1 .2, nriallcut=.03,…..); Group NRI STD ERR Zvalue PVALUE 95% CI % of events correctly reclassified % of non-event correctly reclassified ALL .454 .09 9.7 <.0001 (0.3649,0.5447) (10%) 56% CUT_.03 .101 .08 2.46 .014 (0.0205,0.1817) 4% 6% USER .127 .05 4.95 <.0001 (0.0769,0.177) 5% 8% Vickers’ Section of Macro • Default is no analysis • Can test multiple thresholds • Uses bootstrapping techniques to create 95% CI’s • If testing thresholds run time will increase due to Bootstrapping Vickers Section of Macro %added_pred(data=,…..,vickersplot=TRUE,…..) Vickers Decision Curve 0.11 0.10 0.09 Net Benefit 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 -0.01 0 10 20 30 40 50 60 70 Threshold Probability in % PLOT Treat All Model1 Model2 80 90 100 Conclusions • Don’t rely only on AUC and statistical significance to access added marker predictive ability, but a combination of methods • Future: extend to time-to-event analysis Q’s or Comments • If you want to use the macro or obtain literature contact me at: • Email: [email protected] or [email protected] References 1) Pencina MJ, D'Agostino RB Sr, D’Agostino RB Jr. Evaluating the added predictive ability of a new marker: From area under the ROC curve to reclassification and beyond. Stat Med 2008; 27:157-72. 2) Vickers AJ, Elkin EB. Decision curve analysis: a novel method for evaluating prediction models. Medical Decision Making. 2006 NovDec;26(6):565-74 3) Vickers AJ, Cronin AM, Elkin EB, Gonen M. Extensions to decision curve analysis, A novel method for evaluating diagnostic tests, prediction models and molecular markers. BMC Medical Informatics and Decision Making. 2008 Nov 26;8(1):53. 4) Cook NR. Use and Misuse of the receiver operating characteristics curve in risk prediction. Circulation 2007; 115:928-935. 5) Pepe MS, Janes H, Longton G, Leisenring W, Newcomb P. Limitations of the odds ratio in gauging the performance of a diagnostic, prognostic, or screening marker. Am J Epidemiol. 2004;159:882-890. 6) http://support.sas.com/kb/25/017.html