ROC

ROC curve

• Binary case:
  • Y = true positive rate = sensitivity,
  • X = false positive rate = 1-specificity
• Area under the curve AUC from the wikipedia: the probability that a classifier will rank a randomly chosen positive instance higher than a randomly chosen negative one (assuming 'positive' ranks higher than 'negative').

[math]\displaystyle{ A = \int_{\infty}^{-\infty} \mbox{TPR}(T) \mbox{FPR}'(T) \, dT = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} I(T'\gt T)f_1(T') f_0(T) \, dT' \, dT = P(X_1 \gt X_0) }[/math]

where [math]\displaystyle{ X_1 }[/math] is the score for a positive instance and [math]\displaystyle{ X_0 }[/math] is the score for a negative instance, and [math]\displaystyle{ f_0 }[/math] and [math]\displaystyle{ f_1 }[/math] are probability densities as defined in previous section.

• Interpretation of the AUC. A small toy example (n=12=4+8) was used to calculate the exact probability [math]\displaystyle{ P(X_1 \gt X_0) }[/math] (4*8=32 all combinations).
  • It is a discrimination measure which tells us how well we can classify patients in two groups: those with and those without the outcome of interest.
  • Since the measure is based on ranks, it is not sensitive to systematic errors in the calibration of the quantitative tests.
  • The AUC can be defined as The probability that a randomly selected case will have a higher test result than a randomly selected control.
  • Plot of sensitivity/specificity (y-axis) vs cutoff points of the biomarker
  • The Mann-Whitney U test statistic (or Wilcoxon or Kruskall-Wallis test statistic) is equivalent to the AUC (Mason, 2002)
  • The p-value of the Mann-Whitney U test can thus safely be used to test whether the AUC differs significantly from 0.5 (AUC of an uninformative test).
• Calculate AUC by hand. AUC is equal to the probability that a true positive is scored greater than a true negative.
• How to calculate Area Under the Curve (AUC), or the c-statistic, by hand or by R
• Introduction to the ROCR package. Add threshold labels
• http://freakonometrics.hypotheses.org/9066, http://freakonometrics.hypotheses.org/20002
• Illustrated Guide to ROC and AUC
• ROC Curves in Two Lines of R Code
• Gini and AUC. Gini = 2*AUC-1.
• Generally, an AUC value over 0.7 is indicative of a model that can distinguish between the two outcomes well. An AUC of 0.5 tells us that the model is a random classifier, and it cannot distinguish between the two outcomes.

Survival data

'Survival Model Predictive Accuracy and ROC Curves' by Heagerty & Zheng 2005

• Recall Sensitivity= [math]\displaystyle{ P(\hat{p_i} \gt c | Y_i=1) }[/math], Specificity= [math]\displaystyle{ P(\hat{p}_i \le c | Y_i=0 }[/math]), [math]\displaystyle{ Y_i }[/math] is binary outcomes, [math]\displaystyle{ \hat{p}_i }[/math] is a prediction, [math]\displaystyle{ c }[/math] is a criterion for classifying the prediction as positive ([math]\displaystyle{ \hat{p}_i \gt c }[/math]) or negative ([math]\displaystyle{ \hat{p}_i \le c }[/math]).
• For survival data, we need to use a fixed time/horizon (t) to classify the data as either a case or a control. Following Heagerty and Zheng's definition (Incident/dynamic), Sensitivity(c, t)= [math]\displaystyle{ P(M_i \gt c | T_i = t) }[/math], Specificity= [math]\displaystyle{ P(M_i \le c | T_i \gt 0 }[/math]) where M is a marker value or [math]\displaystyle{ Z^T \beta }[/math]. Here sensitivity measures the expected fraction of subjects with a marker greater than c among the subpopulation of individuals who die at time t, while specificity measures the fraction of subjects with a marker less than or equal to c among those who survive beyond time t.
• The AUC measures the probability that the marker value for a randomly selected case exceeds the marker value for a randomly selected control
• ROC curves are useful for comparing the discriminatory capacity of different potential biomarkers.

Confusion matrix, Sensitivity/Specificity/Accuracy

		Predict
		1	0
True	1	TP	FN	Sens=TP/(TP+FN)=Recall FNR=FN/(TP+FN)
	0	FP	TN	Spec=TN/(FP+TN)
		PPV=TP/(TP+FP) FDR=FP/(TP+FP)	NPV=TN/(FN+TN)	N = TP + FP + FN + TN

• Sensitivity = TP / (TP + FN) = Recall
• Specificity = TN / (TN + FP)
• Accuracy = (TP + TN) / N
• False discovery rate FDR = FP / (TP + FP)
• False negative rate FNR = FN / (TP + FN)
• Positive predictive value (PPV) = TP / # positive calls = TP / (TP + FP) = 1 - FDR
• Negative predictive value (NPV) = TN / # negative calls = TN / (FN + TN)
• Prevalence = (TP + FN) / N.
• Note that PPV & NPV can also be computed from sensitivity, specificity, and prevalence:
  • PPV is directly proportional to the prevalence of the disease or condition..
  • For example, in the extreme case if the prevalence =1, then PPV is always 1.

[math]\displaystyle{ \text{PPV} = \frac{\text{sensitivity} \times \text{prevalence}}{\text{sensitivity} \times \text{prevalence}+(1-\text{specificity}) \times (1-\text{prevalence})} }[/math]

[math]\displaystyle{ \text{NPV} = \frac{\text{specificity} \times (1-\text{prevalence})}{(1-\text{sensitivity}) \times \text{prevalence}+\text{specificity} \times (1-\text{prevalence})} }[/math]

Precision recall curve

• Precision and recall
  • Y-axis: Precision = tp/(tp + fp) = PPV, large is better
  • X-axis: Recall = tp/(tp + fn) = Sensitivity, large is better
• The Relationship Between Precision-Recall and ROC Curves. Remember ROC is defined as
  • Y-axis: Sensitivity = tp/(tp + fn) = Recall
  • X-axis: 1-Specificity = fp/(fp + tn)

Incidence, Prevalence

https://www.health.ny.gov/diseases/chronic/basicstat.htm

Calculate area under curve by hand (using trapezoid), relation to concordance measure and the Wilcoxon–Mann–Whitney test

• https://stats.stackexchange.com/a/146174
• The meaning and use of the area under a receiver operating characteristic (ROC) curve J A Hanley, B J McNeil 1982

genefilter package and rowpAUCs function

• rowpAUCs function in genefilter package. The aim is to find potential biomarkers whose expression level is able to distinguish between two groups.

# source("http://www.bioconductor.org/biocLite.R")
# biocLite("genefilter")
library(Biobase) # sample.ExpressionSet data
data(sample.ExpressionSet)

library(genefilter)
r2 = rowpAUCs(sample.ExpressionSet, "sex", p=0.1)
plot(r2[1]) # first gene, asking specificity = .9

r2 = rowpAUCs(sample.ExpressionSet, "sex", p=1.0)
plot(r2[1]) # it won't show pAUC

r2 = rowpAUCs(sample.ExpressionSet, "sex", p=.999)
plot(r2[1]) # pAUC is very close to AUC now

Use and Misuse of the Receiver Operating Characteristic Curve in Risk Prediction

http://circ.ahajournals.org/content/115/7/928

Performance evaluation

• Testing for improvement in prediction model performance by Pepe et al 2013.

Some R packages

• Some R Packages for ROC Curves
• ROC animation

Comparison of two AUCs

• Statistical Assessments of AUC. This is using the pROC::roc.test function.

Confidence interval of AUC

How to get an AUC confidence interval. pROC package was used.