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Darlene Goldstein 29 January 2003 Receiver Operating Characteristic Methodology

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Outline Introduction Hypothesis testing ROC curve Area under the ROC curve (AUC) Examples using ROC Concluding remarks

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Introduction to ROC curves ROC = Receiver Operating Characteristic Started in electronic signal detection theory (1940s - 1950s) Has become very popular in biomedical applications, particularly radiology and imaging Also used in machine learning applications to assess classifiers Can be used to compare tests/procedures

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ROC curves: simplest case Consider diagnostic test for a disease Test has 2 possible outcomes: –‘postive’ = suggesting presence of disease – ‘negative’ An individual can test either positive or negative for the disease Prof. Mean...

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Hypothesis testing refresher 2 ‘competing theories’ regarding a population parameter: –NULL hypothesis H (‘straw man’) –ALTERNATIVE hypothesis A (‘claim’, or theory you wish to test) H: NO DIFFERENCE –any observed deviation from what we expect to see is due to chance variability A: THE DIFFERENCE IS REAL

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Test statistic Measure how far the observed data are from what is expected assuming the NULL H by computing the value of a test statistic (TS) from the data The particular TS computed depends on the parameter For example, to test the population mean , the TS is the sample mean (or standardized sample mean) The NULL is rejected fi the TS falls in a user-specified ‘rejection region’

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True disease state vs. Test result not rejected rejected No disease (D = 0) specificity X Type I error (False +) Disease (D = 1) X Type II error (False -) Power 1 - ; sensitivity Disease Test

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Specific Example Test Result Pts with disease Pts without the disease

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Test Result Call these patients “negative”Call these patients “positive” Threshold

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Test Result Call these patients “negative”Call these patients “positive” without the disease with the disease True Positives Some definitions...

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Test Result Call these patients “negative”Call these patients “positive” without the disease with the disease False Positives

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Test Result Call these patients “negative”Call these patients “positive ” without the disease with the disease True negatives

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Test Result Call these patients “negative”Call these patients “positive” without the disease with the disease False negatives

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Test Result without the disease with the disease ‘‘-’’‘‘+’’ Moving the Threshold: right

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Test Result without the disease with the disease ‘‘-’’‘‘+’’ Moving the Threshold: left

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True Positive Rate (sensitivity) 0% 100% False Positive Rate (1-specificity) 0% 100% ROC curve

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True Positive Rate 0%0% 100% False Positive Rate 0%0% 100% True Positive Rate 0%0% 100% False Positive Rate 0%0% 100% A good test: A poor test: ROC curve comparison

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Best Test: Worst test: True Positive Rate 0%0% 100% False Positive Rate 0%0% 100 % True Positive Rate 0%0% 100% False Positive Rate 0%0% 100 % The distributions don’t overlap at all The distributions overlap completely ROC curve extremes

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‘Classical’ estimation Binormal model: –X ~ N(0,1) in nondiseased population –X ~ N(a, 1/b) in diseased population Then ROC(t) = (a + b -1 (t)) for 0 < t < 1 Estimate a, b by ML using readings from sets of diseased and nondiseased patients

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ROC curve estimation with continuous data Many biochemical measurements are in fact continuous, e.g. blood glucose vs. diabetes Can also do ROC analysis for continuous (rather than binary or ordinal) data Estimate ROC curve (and smooth) based on empirical ‘survivor’ function (1 – cdf) in diseased and nondiseased groups Can also do regression modeling of the test result Another approach is to model the ROC curve directlyas a function of covariates

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Area under ROC curve (AUC) Overall measure of test performance Comparisons between two tests based on differences between (estimated) AUC For continuous data, AUC equivalent to Mann- Whitney U-statistic (nonparametric test of difference in location between two populations)

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True Positive Rate 0%0% 100% False Positive Rate 0%0% 100 % True Positive Rate 0%0% 100% False Positive Rate 0%0% 100 % True Positive Rate 0%0% 100% False Positive Rate 0%0% 100 % AUC = 50% AUC = 90% AUC = 65% AUC = 100% True Positive Rate 0%0% 100% False Positive Rate 0%0% 100 % AUC for ROC curves

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Interpretation of AUC AUC can be interpreted as the probability that the test result from a randomly chosen diseased individual is more indicative of disease than that from a randomly chosen nondiseased individual: P(X i X j | D i = 1, D j = 0) So can think of this as a nonparametric distance between disease/nondisease test results

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Problems with AUC No clinically relevant meaning A lot of the area is coming from the range of large false positive values, no one cares what’s going on in that region (need to examine restricted regions) The curves might cross, so that there might be a meaningful difference in performance that is not picked up by AUC

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Examples using ROC analysis Threshold selection for ‘tuning’ an already trained classifier (e.g. neural nets) Defining signal thresholds in DNA microarrays (Bilban et al.) Comparing test statistics for identifying differentially expressed genes in replicated microarray data (Lönnstedt and Speed) Assessing performance of different protein prediction algorithms (Tang et al.) Inferring protein homology (Karwath and King)

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Homology Induction ROC

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Concluding remarks – remaining challenges in ROC methodology Inference for ROC curve when no ‘gold standard’ Role of ROC in combining information? Incorporating time into ROC analysis Alternatives to ROC for describing test accuracy? Generalization of positive/negative predictive value to continuous test? (+/-) predictive value = proportion of patients with (+/-) result who are correctly diagnosed = True/(True + False)

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