**discrimination power of your predictive classification model**. In simple words, it checks how well model is able to distinguish (separates) events and non-events. Suppose you are building a predictive model for bank to identify customers who are likely to buy credit card. In this case case, purchase of credit card is event (or desired outcome) and non-purchase of credit card is non-event.

AUC or ROC curve is a plot of the proportion of true positives (events correctly predicted to be events) versus the proportion of false positives (nonevents wrongly predicted to be events) at different probability cutoffs. True Positive Rate is also called Sensitivity. False Positive Rate is also called (1-Specificity). Sensitivity is on Y-axis and (1-Specificity) is on X-axis. Higher the AUC score, better the model.See below how it works.

Diagonal line represents random classification model. It is equivalent to prediction by tossing a coin. All points along the diagonal line say same true positive and false positive rate.

**Cut-off**represents minimum threshold after that predicted probability would be classified as 'event' (desired outcome). In other words, predictive probability greater than or equal to cut-off would be classified as 1. Let's say cutoff is 0.5. In the case of propensity to buy model, predicted probability >= 0.5 would be classified as 'purchase of product'. To generate ROC curve, we calculate Sensitivity and (1-Specificity) at all possible cutoffs and then we plot them.

Cut-off | Sensitivity | Specificity | 1-specificity |
---|---|---|---|

0 | 1 | 0 | 1 |

0.01 | 0.979 | 0.081 | 0.919 |

0.02 | 0.938 | 0.158 | 0.842 |

…. | |||

…. | |||

…. | |||

0.99 | 0.02 | 0.996 | 0.004 |

1 | 0 | 1 | 0 |

ROC Curve |

## AUC using Concordance and Tied Percent

- Calculate the predicted probability in logistic regression (or any other binary classification model). It is not restricted to logistic regression.
- Divide the data into two datasets. One dataset contains observations having actual value of dependent variable with value 1 (i.e. event) and corresponding predicted probability values. And the other dataset contains observations having actual value of dependent variable 0 (non-event) against their predicted probability scores.
- Compare each predicted value in first dataset with each predicted value in second dataset.
- A pair is concordant if 1 (observation with the desired outcome i.e. event) has a higher predicted probability than 0 (observation without the outcome i.e. non-event).
- A pair is discordant if 0 (observation without the desired outcome i.e. non-event) has a higher predicted probability than 1 (observation with the outcome i.e. event).
- A pair is tied if 1 (observation with the desired outcome i.e. event) has same predicted probability than 0 (observation without the outcome i.e. non-event).
- The final percent values are calculated using the formula below -
- Area under curve (AUC) = (Percent Concordant + 0.5 * Percent Tied)/100

Total Number of pairs to compare =`x`

*`y`

`x`

: Number of observations in first dataset (actual values of 1 in dependent variable)

`y`

: Number of observations in second dataset (actual values of 0 in dependent variable).

In this step, we are performingcartesian product (cross join) of events and non-events. For example, you have 100 events and 1000 non-events. It would create 100k (100*1000) pairs for comparison.

`Percent Concordant = 100*[(Number of concordant pairs)/Total number of pairs]`

`Percent Discordant = 100*[(Number of discordant pairs)/Total number of pairs]`

`Percent Tied = 100*[(Number of tied pairs)/Total number of pairs]`

**Interpretation of Concordant, Discordant and Tied Percent**

**Percent Concordant :**Percentage of pairs where the observation with the desired outcome (event) has a higher predicted probability than the observation without the outcome (non-event).

**Percent Discordant :**Percentage of pairs where the observation with the desired outcome (event) has a lower predicted probability than the observation without the outcome (non-event).

**Percent Tied :**Percentage of pairs where the observation with the desired outcome (event) has same predicted probability than the observation without the outcome (non-event).

**AUC :**Area under curve (AUC) is also known as c-statistics. Some statisticians also call it AUROC which stands for area under the receiver operating characteristics. It is calculated by adding Concordance Percent and 0.5 times of Tied Percent.

**Gini coefficient or Somers' D statistic**is closely related to AUC. It is calculated by (2*AUC - 1). It can also be calculated by (Percent Concordant - Percent Discordant)

In general, higher percentages of concordant pairs and lower percentages of discordant and tied pairs indicate a more desirable model.

**SAS and R Code for ROC, Concordant / Discordant :**

Download the CSV data file from

**UCLA website**.

The code below calculates these performance metrics in SAS and R. It executes each step explained above theoretically.

SAS Code

FILENAME PROBLY TEMP; PROC HTTP URL="https://stats.idre.ucla.edu/stat/data/binary.csv" METHOD="GET" OUT=PROBLY; RUN; OPTIONS VALIDVARNAME=ANY; PROC IMPORT FILE=PROBLY OUT=WORK.binary REPLACE DBMS=CSV; RUN; ods graphics on; Proc logistic data= WORK.binary descending plots(only)=roc; class rank / param=ref ; model admit = gre gpa rank; output out = estprob p= pred; run; /*split the data into two datasets- event and non-event*/ Data event nonevent; Set estprob; If admit = 1 then output event; else if admit = 0 then output nonevent; run; /*Cartesian product of event and non-event actual cases*/ Proc SQL noprint; create table pairs as select a.admit as admit1, b.admit as admit0, a.pred as pred1,b.pred as pred0 from event a cross join nonevent b; quit; /*Calculating concordant,discordant and tied percent*/ Data pairs; set pairs; concordant =0; discordant=0; tied=0; If pred1 > pred0 then concordant = 1; else If pred1 < pred0 then discordant = 1; else tied = 1; run; /*Mean values - Final Result*/ proc sql; select mean(Concordant)*100 as Percent_Concordant, mean(Discordant) *100 as Percent_Discordant, mean(Tied)*100 as Percent_Tied, (calculated Percent_Concordant + 0.5* calculated Percent_Tied)/100 as AUC, 2*calculated AUC - 1 as somers_d from pairs; quit;

R Code

# Read Data df = read.csv("https://stats.idre.ucla.edu/stat/data/binary.csv") # Factor Variables df$admit = as.factor(df$admit) df$rank = as.factor(df$rank) # Logistic Model df$rank <- relevel(df$rank, ref='4') mylogistic <- glm(admit ~ ., data = df, family = "binomial") summary(mylogistic)$coefficient # Predict pred = predict(mylogistic, type = "response") finaldata = cbind(df, pred) AUC <- function (actuals, predictedScores){ fitted <- data.frame (Actuals=actuals, PredictedScores=predictedScores) colnames(fitted) <- c('Actuals','PredictedScores') ones <- fitted[fitted$Actuals==1, ] # Subset ones zeros <- fitted[fitted$Actuals==0, ] # Subsetzeros totalPairs <- nrow (ones) * nrow (zeros) # calculate total number of pairs to check conc <- sum (c(vapply(ones$PredictedScores, function(x) {((x > zeros$PredictedScores))}, FUN.VALUE=logical(nrow(zeros)))), na.rm=T) disc <- sum(c(vapply(ones$PredictedScores, function(x) {((x < zeros$PredictedScores))}, FUN.VALUE = logical(nrow(zeros)))), na.rm = T) concordance <- conc/totalPairs discordance <- disc/totalPairs tiesPercent <- (1-concordance-discordance) AUC = concordance + 0.5*tiesPercent Gini = 2*AUC - 1 return(list("Concordance"=concordance, "Discordance"=discordance, "Tied"=tiesPercent, "AUC"=AUC, "Gini or Somers D"=Gini)) } AUC(finaldata$admit, finaldata$pred)

Result |

## Calculate AUC using Integration Method

Trapezoidal Rule Numerical Integration method is used to find area under curve. The area of a trapezoid isIn our case,( x_{i+1}– x_{i}) * ( y_{i}+ y_{i+1}) / 2

**x**refers to values of false positive rate (1-Specificity) at different probability cut-offs,

**y**refers to true positive rate (Sensitivity) at different cut-offs.

**Vector x needs to be sorted**. Any observation with predicted probability that exceeds or equals probability cut-off is predicted to be an event; otherwise, it is predicted to be a nonevent.

( fpr_{i+1}– fpr_{i}) * ( tpr_{i}+ tpr_{i+1}) / 2

`fpr`

represents false positive rate (1- specificity). `tpr`

represents true positive rate (sensitivity). See the image below showing step by step calculation. It includes a very few cut-offs for demonstration purpose.SAS Code

In the SAS program below, we are using`PROC IML`

procedure to perform integration calculations.
FILENAME PROBLY TEMP; PROC HTTP URL="https://stats.idre.ucla.edu/stat/data/binary.csv" METHOD="GET" OUT=PROBLY; RUN; OPTIONS VALIDVARNAME=ANY; PROC IMPORT FILE=PROBLY OUT=WORK.binary REPLACE DBMS=CSV; RUN; ods graphics on; Proc logistic data= WORK.binary descending plots(only)=roc; class rank / param=ref ; model admit = gre gpa rank / outroc=performance; output out = estprob p= pred; run; proc sort data=performance; by _1MSPEC_; run; proc iml; use performance; read all var {_SENSIT_} into sensitivity; read all var {_1MSPEC_} into falseposrate; N = 2 : nrow(falseposrate); fpr = falseposrate[N] - falseposrate[N-1]; tpr = sensitivity[N] + sensitivity[N-1]; ROC = fpr`*tpr/2; Gini= 2*ROC - 1; print ROC Gini;

R Code

# Read Data df = read.csv("https://stats.idre.ucla.edu/stat/data/binary.csv") # Factor Variables df$admit = as.factor(df$admit) df$rank = as.factor(df$rank) # Logistic Model df$rank <- relevel(df$rank, ref='4') mylogistic <- glm(admit ~ ., data = df, family = "binomial") summary(mylogistic)$coefficient # Predict pred = predict(mylogistic, type = "response") finaldata = cbind(df, pred) library(ROCR) predobj <- prediction(finaldata$pred, finaldata$admit) perf <- performance(predobj,"tpr","fpr") plot(perf) # Trapezoidal rule of integration # Computes the integral of Sensitivity (Y) with respect to FalsePosRate (x) x = perf@x.values[[1]] y = perf@y.values[[1]] idx = 2:length(x) testdf=data.frame(FalsePosRate = (x[idx] - x[idx-1]), Sensitivity = (y[idx] + y[idx-1])) (AUROC = sum(testdf$FalsePosRate * testdf$Sensitivity)/2)

## Calculate AUC using Mann–Whitney U Test

Area under curve (AUC) is directly related to Mann Whitney U test. People from analytics community also call it Wilcoxon rank-sum test.This test assumes that the predicted probability of event and non-event are two independent continuous random variables. Area under the curve = Probability that Event produces a higher probability than Non-Event. AUC=P(Event>=Non-Event)

AUC = U_{1}/(n_{1}* n_{2}) Here U_{1}= R_{1}- (n_{1}*(n_{1}+ 1) / 2)where

Uis the Mann Whitney U statistic and_{1}Ris the sum of the ranks of predicted probability of actual event. It is calculated by ranking predicted probabilities and then selecting only those cases where dependent variable is 1 and then take sum of all these cases._{1}nis the number of 1s (event) in dependent variable._{1}nis the number of 0s (non-events) in dependent variable._{2}

nis the total number of pairs (or cross product of number of events and non-events). It is similar to what we have done in concordance method to calculate AUC._{1}*n_{2}

R Code

# AUC using Mann–Whitney test auc_mannWhitney <- function(y, pred){ y <- as.logical(y) n1 <- sum(y) n2 <- sum(!y) R1 <- sum(rank(pred)[y]) U1 <- R1 - n1 * (n1 + 1)/2 U1/(n1 * n2) } auc_mannWhitney(as.numeric(as.character(finaldata$admit)), finaldata$pred)

SAS Code

ods select none; ods output WilcoxonScores=WilcoxonScore; proc npar1way wilcoxon data= estprob ; where admit^=.; class admit; var pred; run; ods select all; data AUC; set WilcoxonScore end=eof; retain v1 v2 1; if _n_=1 then v1=abs(ExpectedSum - SumOfScores); v2=N*v2; if eof then do; d=v1/v2; Gini=d * 2; AUC=d+0.5; put AUC= GINI=; keep AUC Gini; output; end; proc print noobs; run;

## Calculate AUC using Cumulative Events and Non-Events

In this method, we will see how we can calculate area under curve using decile (binned) data.- Sort predicted probabilities in descending order. It means customer having high likelihood to buy a product should appear at top (in case of propensity model)
- Split or rank into 10 parts. It is similar to concept of calculating decile.
- Calculate number of cases in each decile level. It would be same in each level as we divided the data in 10 equal parts.
- Calculate number of 1s (event) in each decile level. Maximum 1s should be captured in first decile (if your model is performing fine!)
- Calculate cumulative percent of 1s in each decile level. Last decile should have 100% as it is cumulative in nature.
- Similar to the above step, we will calculate cumulative percent of 0s in each decile level.
- AUC would be calculated using trapezoidal rule numeric integration formula. In this case,
`x`

is cumulative % of 0s and`y`

is cumulative % of 1s

Very precise and clear explanation of concordance and discordance. Also the code helps in better understanding of the phenomenon. Thanks.

ReplyDeleteThank you for your appreciation. Cheers!

DeleteNeat explanations, really helpful to understood these definitions. Thanks!

ReplyDeleteVery clear explanation, thank you :)

ReplyDeleteThanks for the post! Shouldn't it be proc logistic with descending option? as we are treating 1s as events and 0 as nonevents

ReplyDeleteCorrected! Thanks for pointing it out.

DeleteFirst time I understood concordance and discordance. Thanks

ReplyDeleteFor a good model what should be the concordance?

ReplyDeleteConcordance Percent should be 80 or above.

DeleteVery good explanation

ReplyDeleteVery informative, clear, and to the point

ReplyDeleteVery good explanation and informative. Thanks Buddy keep sharing

ReplyDeleteCan you please give the calculation of concordance and disconcordance in excel format with example which will be easy to understand the calculation.

ReplyDeleteThe above codes are very useful. Any suggestions for weighted data?

ReplyDeleteHello, I want to know, what to do in cases where tied percentage is high, say 20%. How to reduce tied percentage?

ReplyDeleteExcellent Work. Thanks for such detailed description.

ReplyDeleteNot as clear as needed

ReplyDeleteSorry

ReplyDeleteThorough and very useful. However can you let me know how to derive the equation: AUC = (Percent Concordant + 0.5 * Percent Tied)/100. Basically I want to know the steps to get the above equation

ReplyDeleteThanks for the article, but cross join is quite heavy and won't be possible on large datasets.

ReplyDeleteReally great explanation. I had one more doubt You said Gini is a special case of SomerS' D. Can you explain how are gini and Somers' D related and what is the difference ?

ReplyDeleteYour website is a god sent for students like me. Thank you for providing such easy and clear explainations.

ReplyDeleteVery few know about Percent Tied and its role in AUC. I am very thankful that you introduce it here. Could you please provide reference to the 4 methods you introduced and the proves that they are equivalent?

ReplyDeleteVery few notice the existence of Percent Tied and its role in AUC. It is so nice that you introduced it here. Could you provide proves that the four methods you introduced here are equivalent? Any references would be highly appreciated!

ReplyDelete