ROC
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ROC curve
- Binary case:
- Y = true positive rate = sensitivity,
- X = false positive rate = 1-specificity = 假陽性率
- 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
- Learning Data Science: Understanding ROC Curves
- 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), 1-Spec=FPR | |
| PPV=TP/(TP+FP) FDR=FP/(TP+FP) |
NPV=TN/(FN+TN) | N = TP + FP + FN + TN | ||
- Sensitivity 敏感度 = TP / (TP + FN) = Recall
- Specificity 特異度 = TN / (TN + FP)
- Accuracy = (TP + TN) / N
- False discovery rate FDR = FP / (TP + FP)
- False negative rate FNR = FN / (TP + FN)
- False positive rate FPR = FP / (FP + TN)
- True positive rate = TP / (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 盛行率 = (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]
- Prediction of heart disease and classifiers’ sensitivity analysis Almustafa, 2020
- Positive percent agreement (PPA) and negative percent agreement (NPA)
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
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.