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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Receiver Operating Characteristic (ROC) Curves Assessing the predictive properties of a test statistic – Decision Theory

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Binary Prediction Problem Conceptual Framework Suppose we have a test statistic for predicting the presence or absence of disease. True Disease Status PosNeg Test Criterion Pos Neg

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Binary Prediction Problem Conceptual Framework Suppose we have a test statistic for predicting the presence or absence of disease. True Disease Status PosNeg Test Criterion Pos Neg

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Binary Prediction Problem Conceptual Framework Suppose we have a test statistic for predicting the presence or absence of disease. True Disease Status PosNeg Test Criterion Pos TP Neg TP = True Positive

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Binary Prediction Problem Conceptual Framework Suppose we have a test statistic for predicting the presence or absence of disease. True Disease Status PosNeg Test Criterion Pos Neg

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Binary Prediction Problem Conceptual Framework Suppose we have a test statistic for predicting the presence or absence of disease. True Disease Status PosNeg Test Criterion Pos FP Neg FP = False Positive

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Binary Prediction Problem Conceptual Framework Suppose we have a test statistic for predicting the presence or absence of disease. True Disease Status PosNeg Test Criterion Pos Neg

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Binary Prediction Problem Conceptual Framework Suppose we have a test statistic for predicting the presence or absence of disease. True Disease Status PosNeg Test Criterion Pos Neg FN FN = False Negative

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Binary Prediction Problem Conceptual Framework Suppose we have a test statistic for predicting the presence or absence of disease. True Disease Status PosNeg Test Criterion Pos Neg

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Binary Prediction Problem Conceptual Framework Suppose we have a test statistic for predicting the presence or absence of disease. True Disease Status PosNeg Test Criterion Pos Neg TN TN = True Negative

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Binary Prediction Problem Conceptual Framework True Disease Status PosNeg Test Criterion Pos TPFP Neg FNTN PNP+ N Suppose we have a test statistic for predicting the presence or absence of disease.

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Binary Prediction Problem Conceptual Framework

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Binary Prediction Problem Test Properties True Disease Status PosNeg Test Criterion Pos TPFP Neg FNTN PNP+ N Accuracy = Probability that the test yields a correct result. = (TP+TN) / (P+N)

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Binary Prediction Problem Test Properties True Disease Status PosNeg Test Criterion Pos TPFP Neg FNTN PNP+ N Sensitivity = Probability that a true case will test positive = TP / P Also referred to as True Positive Rate (TPR) or True Positive Fraction (TPF).

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Binary Prediction Problem Test Properties True Disease Status PosNeg Test Criterion Pos TPFP Neg FNTN PNP+ N Specificity = Probability that a true negative will test negative = TN / N Also referred to as True Negative Rate (TNR) or True Negative Fraction (TNF).

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Binary Prediction Problem Test Properties True Disease Status PosNeg Test Criterion Pos TPFP Neg FNTN PNP+ N 1-Specificity = Prob that a true negative will test positive = FP / N Also referred to as False Positive Rate (FPR) or False Positive Fraction (FPF).

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Binary Prediction Problem Test Properties True Disease Status PosNeg Test Criterion Pos TPFP Neg FNTN PNP+ N Positive Predictive Value (PPV) = Probability that a positive test will truly have disease = TP / (TP+FP)

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Binary Prediction Problem Test Properties True Disease Status PosNeg Test Criterion Pos TPFP Neg FNTN PNP+ N Negative Predictive Value (NPV) = Probability that a negative test will truly be disease free = TN / (TN+FN)

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Binary Prediction Problem Example True Disease Status PosNeg Test Criterion Pos Neg /100 =.27Se = S p =727/900 =.81 FPF = 1- S p =.19 Acc = (27+727)/1000 =.75 PPV = 27/200 =.14 NPV = 727/800 =.91

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Binary Prediction Problem Test Properties Of these properties, only Se and Sp (and hence FPR) are considered invariant test characteristics. Accuracy, PPV, and NPV will vary according to the underlying prevalence of disease. Se and Sp are thus “fundamental” test properties and hence are the most useful measures for comparing different test criteria, even though PPV and NPV are probably the most clinically relevant properties.

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH ROC Curves Now assume that our test statistic is no longer binary, but takes on a series of values (for instance how many of five distinct risk factors a person exhibits). Clinically we make a rule that says the test is positive if the number of risk factors meets or exceeds some threshold (#RF > x ) Suppose our previous table resulted from using x = 4. Let’s see what happens as we vary x.

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH ROC Curves Impact of using a threshold of 3 or more RFs True Disease Status PosNeg Test Criterion Pos Neg /100 =.45Se = S p =727/900 =.78 FPF = 1- S p =.22 Acc = (27+727)/1000 =.75 PPV = 27/200 =.18 NPV = 727/800 =.93 Se , Sp , and interestingly both PPV and NPV

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH ROC Curves Summary of all possible options ThresholdTPRFPR As we relax our threshold for defining “disease,” our true positive rate (sensitivity) increases, but so does the false positive rate (FPR). The ROC curve is a way to visually display this information.

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH ROC Curves Summary of all possible options ThresholdTPRFPR x=5 x=4 x=2 The diagonal line shows what we would expect from simple guessing (i.e., pure chance). What might an even better ROC curve look like?

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH ROC Curves Summary of a more optimal curve ThresholdTPRFPR Note the immediate sharp rise in sensitivity. Perfect accuracy is represented by upper left corner.

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH ROC Curves Use and interpretation The ROC curve allows us to see, in a simple visual display, how sensitivity and specificity vary as our threshold varies. The shape of the curve also gives us some visual clues about the overall strength of association between the underlying test statistic (in this case #RFs that are present) and disease status.

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH ROC Curves Use and interpretation The ROC methodology easily generalizes to test statistics that are continuous (such as lung function or a blood gas). We simply fit a smoothed ROC curve through all observed data points.

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH ROC Curves Use and interpretation See demo from

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH ROC Curves Area under the curve (AUC) The total area of the grid represented by an ROC curve is 1, since both TPR and FPR range from 0 to 1. The portion of this total area that falls below the ROC curve is known as the area under the curve, or AUC.

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Area Under the Curve (AUC) Interpretation The AUC serves as a quantitative summary of the strength of association between the underlying test statistic and disease status. An AUC of 1.0 would mean that the test statistic could be used to perfectly discriminate between cases and controls. An AUC of 0.5 (reflected by the diagonal 45° line) is equivalent to simply guessing.

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Area Under the Curve (AUC) Interpretation The AUC can be shown to equal the Mann- Whitney U statistic, or equivalently the Wilcoxon rank statistic, for testing whether the test measure differs for individuals with and without disease. It also equals the probability that the value of our test measure would be higher for a randomly chosen case than for a randomly chosen control.

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Area Under the Curve (AUC) Interpretation FPR TPR 1 01 ROC Curve AUC ~ cases controls

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH AUC ~.95 TPR 1 01 FPR ROC Curve Area Under the Curve (AUC) Interpretation cases controls

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Area Under the Curve (AUC) Interpretation What defines a “good” AUC? Opinions vary Probably context specific What may be a good AUC for predicting COPD may be very different than what is a good AUC for predicting prostate cancer

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Area Under the Curve (AUC) Interpretation = excellent = good = fair = poor = fail Remember that <.50 is worse than guessing!

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH Area Under the Curve (AUC) Interpretation = excellent = very good = good = fair

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH ROC Curves Comparing multiple ROC curves Suppose we have two candidate test statistics to use to create a binary decision rule. Can we use ROC curves to choose an optimal one?

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH ROC Curves Comparing multiple ROC curves Adapted from curves at:

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH ROC Curves Comparing multiple ROC curves iki/Receiver_operating_ characteristic

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH ROC Curves Comparing multiple ROC curves We can formally compare AUCs for two competing test statistics, but does this answer our question? AUC speaks to which measure, as a continuous variable, best discriminates between cases and controls? It does not tell us which specific cutpoint to use, or even which test statistic will ultimately provide the “best” cutpoint.

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH ROC Curves Choosing an optimal cutpoint The choice of a particular Se and Sp should reflect the relative costs of FP and FN results. What if a positive test triggers an invasive procedure? What if the disease is life threatening and I have an inexpensive and effective treatment? How do you balance these and other competing factors? See excellent discussion of these issues at

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH ROC Curves Generalizations These techniques can be applied to any binary outcome. It doesn’t have to be disease status. In fact, the use of ROC curves was first introduced during WWII in response to the challenge of how to accurately identify enemy planes on radar screens.

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© 2009, KAISER PERMANENTE CENTER FOR HEALTH RESEARCH ROC Curves Final cautionary notes We assume throughout the existence of a gold standard for measuring “disease,” when in practice no such gold standard exists. COPD, asthma, even cancer (can we truly rule out the absence of cancer in a given patient?) As a result, even Se and Sp may not be inherently stable test characteristics, but may vary depending on how we define disease and the clinical context in which it is measured. Are we evaluating the test in the general population or only among patients referred to a specialty clinic? Incorrect specification of P and N will vary in these two settings.

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