Assessing Information from Multilevel (Ordinal) and Continuous Tests

1. Assessing Information from Multilevel (Ordinal) and Continuous Tests ROC curves and Likelihood Ratios for results other than “+” or “-” Michael A. Kohn, MD, MPP 10/6/2005

2. Four Main Points 1) Dichotomizing a multi-level or continuous test by choosing a fixed cutpoint reduces the value of the test. 2) The ROC curve summarizes the discriminatory ability of the test. 3) LR(result) = P(result|D+)/P(result|D-) = slope of ROC curve. 4) Pre-Test Odds x LR(result) = Post-Test Odds

3. Many Tests Are Not Dichotomous Ordinal • “-”, “+”, “++”, “+++” for leukocyte esterase on urine dip stick • “Normal”, “Low Prob”, “Intermediate Prob”, “High Prob” on VQ scan Continuous • Systolic Blood Pressure • WBC Count

4. Evaluating the Test--Test Characteristics • For dichotomous tests, we discussed sensitivity P(+|D+) and specificity P(-|D-) • For multi-level and continuous tests, we will discuss the Receiver Operating Characteristic (ROC) curve

5. Using the Test Result to Make Decisions about a Patient • For dichotomous tests, we use the LR(+) if the test is positive and the LR(-) if the test is negative • For multilevel and continuous tests, we use the LR(r), where r is the result of the test

6. Clinical Scenario 5-month old boy with fever 39.7. You have the results of a WBC count. How do you use this WBC result to determine whether to treat empirically for possible bacteremia?

7. Why Not Make It a Dichotomous Test? WBC Count (x1000/uL) Bacteremia No Bacteremia >15 109 2028 0 -14.99 18 6601 Total 127 8629 Lee GM, Harper MB. Risk of bacteremia for febrile young children in the post-Haemophilus influenzae type b era. Arch Pediatr Adolesc Med. 1998;152(7):624-628.

8. Why Not Make It a Dichotomous Test? Sensitivity = 109/127 = 0.86 Specificity = 6601/8629 = 0.76 LR(+) = 0.86/(1 - 0.76) = 3.65 LR(-) = (1 - 0.86)/0.76 = 0.19 Equivalently LR(+) = P(+|D+)/P(+|D-) = (109/127)/(2028/8629) = 3.65 LR(-) = P(-|D+)/P(-|D-) = (18/127)/(6601/8629) = 0.19

9. Clinical ScenarioWBC = 16,000/mL (Demonstrate LR Slide Rule?) Pre-test prob: 0.03 LR(+) = 3.65 Post-Test prob = ?

10. Clinical ScenarioWBC = 16,000/mL Pre-test prob: 0.03 Pre-test odds: 0.03/0.97 = 0.031 LR(+) = 3.65 Post-Test Odds = Pre-Test Odds x LR(+) = 0.031 x 3.65 = .113 Post-Test prob = .113/(.113+1) = .10

11. Clinical ScenarioWBC = 28,000/mL Pre-test prob: 0.03 LR(+) = ? Post-Test prob =?

12. Clinical ScenarioWBC = 28,000/mL Pre-test prob: 0.03 Pre-test odds: 0.03/0.97 = 0.031 LR(+) = 3.65 (same as for WBC=16,000!) Post-Test Odds = Pre-Test Odds x LR(+) = 0.031 x 3.65 = .113 Post-Test prob = .113/(.113+1) = .10

13. Why Not Make It a Dichotomous Test? Because you lose information. The risk associated with WBC=16,000 is equated with the risk associated with WBC=28,000. Choosing a fixed cutpoint to dichotomize a multi-level or continuous test throws away information and reduces the value of the test.

14. Main Point 1: Avoid Making Multilevel Tests Dichotomous Dichotomizing a multi-level or continuous test by choosing a fixed cutpoint reduces the value of the test

15. Lee GM, Harper MB. Risk of bacteremia for febrile young children in the post-Haemophilus influenzae type b era. Arch Pediatr Adolesc Med. 1998;152(7):624-628.

18. Histogram • Does not reflect prevalence of D+ (Dark D+ columns add to 100%, Open D- columns add to 100%) • Sensitivity and specificity depend on the cutpoint chosen to separate “positives” from “negatives” • The ROC curve is drawn by serially lowering the cutpoint from highest (most abnormal) to lowest (least abnormal).* * Just said that choosing a fixed cutpoint reduces the value of the test. The key issues are 1) the ROC curve is for evaluating the test, not the patient, and 2) drawing the ROC curve requires varying the cutpoint, not choosing a fixed cutpoint.

28. 10,000/uL 5,000/uL 15,000/uL 20,000/uL Area Under Curve (AUC) = 0.86 25,000/uL 30,000/uL

33. WBC Cutoff = 12,000/μL

34. Test Discriminates Well Between D+ and D- D+ D- Test Result

35. Test Discriminates Well Between D+ and D-

36. Test Discriminates Poorly Between D+ and D- D+ D- Test Result

37. Test Discriminates Poorly Between D+ and D-

38. Area Under ROC Curve 10,000/uL 5,000/uL 15,000/uL 20,000/uL Area Under Curve (AUC) = 0.86 25,000/uL 30,000/uL

39. Area Under ROC Curve Summary measure of test’s discriminatory ability Probability that a randomly chosen D+ individual will have a more positive test result than a randomly chosen D- individual e.g. randomly choose 1 of the 127 bacteremic children and 1 of the 8629 non-bacteremic children. The probability that the bacteremic child’s WBC will fall in a higher WBC interval than the non-bacteremic child is 0.86

40. Area Under ROC Curve • Corresponds to the Mann-Whitney (Wilcoxan Rank Sum) Test Statistic, which is the non-parametric equivalent of Student’s t test. • Also corresponds to the “c statistic” reported in logistic regression models

42. “Walking Man” Approach to ROC Curves • Divide vertical axis into d steps, where d is the number of D+ individuals • Divide horizontal axis into n steps, where n is the number of D- individuals • Sort individuals from most to least abnormal test result • Moving from the first individual (with the most abnormal test result) to the last (with the least abnormal test result)…

43. “Walking Man” (continued) • …call out “D” if the individual is D+ and “N” if the individual is D- • Let the walking man know when you reach a new value of the test • The walking man takes a step up every time he hears “D” and a step to the right every time he hears “N” • When you reach a new value of the test, he drops a stone.

44. WBC Count in 5 Bacteremic Children

45. WBC Count in 10 Non-Bacteremic Children

47. DDNDN(DN)DN(NN)NNNN

49. Main Point 2ROC Curve Describes the Test, Not the Patient • Describes the test’s ability to discriminate between D+ and D- individuals • Not particularly useful in interpreting a test result for a given patient

50. ROC Curve Describes the Test, Not the Patient Clinical Scenario WBC count = 16,000 WBC count = 28,000