1. Multiple Tests, Multivariable Decision Rules, and Prognostic Tests Michael A. Kohn, MD, MPP 10/25/2007 Chapter 8 – Multiple Tests and Multivariable Decision Rules Chapter 7 – Prognostic and Genetic Tests

2. Digression: Screening Merenstein, “Winners and Losers,” about his experience being sued in 2002 by a man with incurable prostate cancer for not obtaining a prostate-specific antigen (PSA) test in 1999 when the man was 53 years old. Hard copy handed out last week and posted on course website. Discuss in section today?

3. Outline of Topics Combining Tests –Importance of test non-independence –Recursive Partitioning –Logistic Regression –Variable (Test) Selection –Importance of validation separate from derivation Prognostic tests –Differences from diagnostic tests and risk factors –Quantifying prediction: calibration and discrimination –Value of prognostic information –Common problems with studies of prognostic tests

4. Combining Tests Example Prenatal sonographic Nuchal Translucency (NT) and Nasal Bone Exam (NBE) as dichotomous tests for Trisomy 21* *Cicero, S., G. Rembouskos, et al. (2004). "Likelihood ratio for trisomy 21 in fetuses with absent nasal bone at the 11- 14-week scan." Ultrasound Obstet Gynecol 23(3): 218-23.

5. If NT ≥ 3.5 mm Positive for Trisomy 21* *What’s wrong with this definition?

7. In general, don’t make multi-level tests like NT into dichotomous tests by choosing a fixed cutoff I did it here to make the discussion of multiple tests easier I arbitrarily chose to call ≥ 3.5 mm positive

8. One Dichotomous Test Trisomy 21 Nuchal D+ D- LR Translucency ≥ 3.5 mm212 4787.0 < 3.5 mm12147450.4 Total3335223 Do you see that this is (212/333)/(478/5223)? Review of Chapter 3: What are the sensitivity, specificity, PPV, and NPV of this test? (Be careful.)

9. Nuchal Translucency Sensitivity = 212/333 = 64% Specificity = 4745/5223 = 91% Prevalence = 333/(333+5223) = 6% (Study population: pregnant women about to under go CVS, so high prevalence of Trisomy 21) PPV = 212/(212 + 478) = 31% NPV = 4745/(121 + 4745) = 97.5%* * Not that great; prior to test P(D-) = 94%

10. Clinical Scenario – One Test Pre-Test Probability of Down’s = 6% NT Positive Pre-test prob: 0.06 Pre-test odds: 0.06/0.94 = 0.064 LR(+) = 7.0 Post-Test Odds = Pre-Test Odds x LR(+) = 0.064 x 7.0 = 0.44 Post-Test prob = 0.44/(0.44 + 1) = 0.31

11. Pre-Test Probability of Tri21 = 6% NT Positive Post-Test Probability of Tri21 = 31% Clinical Scenario – One Test Pre-Test Odds of CAD = 0.064 EECG Positive (LR = 7.0) Post-Test Odds of CAD = 0.44 Using Probabilities Using Odds

12. Clinical Scenario – One Test Pre-Test Probability of Tri21 = 6% NT Positive NT + (LR = 7.0) |---------------> +-------------------------X---------------X------------------------------+ | | | | | | | Log(Odds) 2 -1.5 -1 -0.5 0 0.5 1 Odds 1:100 1:33 1:10 1:3 1:1 3:1 10:1 Prob 0.01 0.03 0.09 0.25 0.5 0.75 0.91 Odds = 0.064 Prob = 0.06 Odds = 0.44 Prob = 0.31

13. Nasal Bone Seen NBE Negative for Trisomy 21 Nasal Bone Absent NBE Positive for Trisomy 21

14. Second Dichotomous Test Nasal Bone Tri21+ Tri21-LR Absent229 12927.8 Present10450940.32 Total3335223 Do you see that this is (229/333)/(129/5223)?

15. Pre-Test Probability of Trisomy 21 = 6% NT Positive for Trisomy 21 (≥ 3.5 mm) Post-NT Probability of Trisomy 21 = 31% NBE Positive for Trisomy 21 (no bone seen) Post-NBE Probability of Trisomy 21 = ? Clinical Scenario –Two Tests Using Probabilities

16. Clinical Scenario – Two Tests Pre-Test Odds of Tri21 = 0.064 NT Positive (LR = 7.0) Post-Test Odds of Tri21 = 0.44 NBE Positive (LR = 27.8?) Post-Test Odds of Tri21 =.44 x 27.8? = 12.4? (P = 12.4/(1+12.4) = 92.5%?) Using Odds

17. Clinical Scenario – Two Tests Pre-Test Probability of Trisomy 21 = 6% NT ≥ 3.5 mm AND Nasal Bone Absent Odds = 0.064 Prob = 0.06 Odds = 0.44 Prob = 0.31 NT + (LR = 7.0) |---------------> NBE + (LR = 27.8) |---------------------------> NT + NBE + Can we do this? |--------------->|---------------------------> NT + and NBE + +---------------X----------------X----------------------------X-+ | | | | | | | Log(Odds) 2 -1.5 -1 -0.5 0 0.5 1 Odds 1:100 1:33 1:10 1:3 1:1 3:1 10:1 Prob 0.01 0.03 0.09 0.25 0.5 0.75 0.91 Odds = 12.4 Prob = 0.925

18. Question Can we use the post-test odds after a positive Nuchal Translucency as the pre- test odds for the positive Nasal Bone Examination? i.e., can we combine the positive results by multiplying their LRs? LR(NT+, NBE +) = LR(NT +) x LR(NBE +) ? = 7.0 x 27.8 ? = 194 ?

19. Answer = No NTNBE Trisomy 21 +% Trisomy 21 -%LR Pos 15847%360.7% 69 PosNeg5416%4428.5% 1.9 NegPos7121%931.8% 12 Neg 5015%465289% 0.2 Total 333100%5223100% Not 194 158/(158 + 36) = 81%, not 92.5%

20. Non-Independence Absence of the nasal bone does not tell you as much if you already know that the nuchal translucency is ≥ 3.5 mm.

21. Clinical Scenario Pre-Test Odds of Tri21 = 0.064 NT+/NBE + (LR =68.8) Post-Test Odds = 0.064 x 68.8 = 4.40 (P = 4.40/(1+4.40) = 81%, not 92.5%) Using Odds

22. Non-Independence NT + |---------------> NBE + |---------------------------> NT + NBE + if tests were independent|--------------->|----------------------------> NT + and NBE + since tests are dependent|-----------------------------------> +---------------X----------------X------------------X----------+ | | | | | | | Log(Odds) 2 -1.5 -1 -0.5 0 0.5 1 Odds 1:100 1:33 1:10 1:3 1:1 3:1 10:1 Prob 0.01 0.03 0.09 0.25 0.5 0.75 0.91 Prob = 0.81

23. Non-Independence of NT and NBE Apparently, even in chromosomally normal fetuses, enlarged NT and absence of the nasal bone are associated. A false positive on the NT makes a false positive on the NBE more likely. Of normal (D-) fetuses with NT < 3.5 mm only 2.0% had nasal bone absent. Of normal (D-) fetuses with NT ≥ 3.5 mm, 7.5% had nasal bone absent. Some (but not all) of this may have to do with ethnicity. In this London study, chromosomally normal fetuses of “Afro-Caribbean” ethnicity had both larger NTs and more frequent absence of the nasal bone. In Trisomy 21 (D+) fetuses, normal NT was associated with the presence of the nasal bone, so a false negative on the NT was associated with a false negative on the NBE.

24. Non-Independence Instead of looking for the nasal bone, what if the second test were just a repeat measurement of the nuchal translucency? A second positive NT would do little to increase your certainty of Trisomy 21. If it was false positive the first time around, it is likely to be false positive the second time.

25. Reasons for Non-Independence Tests measure the same aspect of disease. Consider exercise ECG (EECG) and radionuclide scan as tests for coronary artery disease (CAD) with the gold standard being anatomic narrowing of the arteries on angiogram. Both EECG and nuclide scan measure functional narrowing. In a patient without anatomic narrowing (a D- patient), coronary artery spasm could cause false positives on both tests.

26. Reasons for Non-Independence Spectrum of disease severity. In the EECG/nuclide scan example, CAD is defined as ≥70% stenosis on angiogram. A D+ patient with 71% stenosis is much more likely to have a false negative on both the EECG and the nuclide scan than a D+ patient with 99% stenosis.

27. Reasons for Non-Independence Spectrum of non-disease severity. In this example, CAD is defined as ≥70% stenosis on angiogram. A D- patient with 69% stenosis is much more likely to have a false positive on both the EECG and the nuclide scan than a D- patient with 33% stenosis.

28. Counterexamples: Possibly Independent Tests For Venous Thromboembolism: CT Angiogram of Lungs and Doppler Ultrasound of Leg Veins Alveolar Dead Space and D-Dimer MRA of Lungs and MRV of leg veins

29. Unless tests are independent, we can’t combine results by multiplying LRs

30. Ways to Combine Multiple Tests On a group of patients (derivation set), perform the multiple tests and (independently*) determine true disease status (apply the gold standard) Measure LR for each possible combination of results Recursive Partitioning Logistic Regression *Beware of incorporation bias

31. Determine LR for Each Result Combination NTNBETri21+%Tri21-%LR Post Test Prob* Pos 15847%360.7%6981% PosNeg5416%4428.5%1.911% NegPos7121%931.8%1243% Neg 5015%465289.1% 0.21% Total 333100%5223100% *Assumes pre-test prob = 6%

32. Determine LR for Each Result Combination 2 dichotomous tests: 4 combinations 3 dichotomous tests: 8 combinations 4 dichotomous tests: 16 combinations Etc. 2 3-level tests: 9 combinations 3 3-level tests: 27 combinations Etc.

33. Determine LR for Each Result Combination How do you handle continuous tests? Not practical for most groups of tests.

34. Recursive Partitioning Measure NT First

35. Recursive Partitioning Examine Nasal Bone First

36. Recursive Partitioning Examine Nasal Bone First CVS if P(Trisomy 21 > 5%)

38. Recursive Partioning Same as Classification and Regression Trees (CART) Don’t have to work out probabilities (or LRs) for all possible combinations of tests, because of “tree pruning”

39. Tree Pruning: Goldman Rule* 8 “Tests” for Acute MI in ER Chest Pain Patient : 1.ST Elevation on ECG; 2.CP < 48 hours; 3.ST-T changes on ECG; 4.Hx of MI; 5.Radiation of Pain to Neck/LUE; 6.Longest pain > 1 hour; 7.Age > 40 years; 8.CP not reproduced by palpation. *Goldman L, Cook EF, Brand DA, et al. A computer protocol to predict myocardial infarction in emergency department patients with chest pain. N Engl J Med. 1988;318(13):797-803.

40. 8 tests  2 8 = 256 Combinations

42. Recursive Partitioning Does not deal well with continuous test results* *when there is a monotonic relationship between the test result and the probability of disease

43. Logistic Regression Ln(Odds(D+)) = a + b NT NT+ b NBE NBE + b interact (NT)(NBE) “+” = 1 “-” = 0 More on this later in ATCR!

44. Why does logistic regression model log-odds instead of probability? Related to why the LR Slide Rule’s log-odds scale helps us visualize combining test results.

47. Logistic Regression Approach to the “R/O ACI patient” *Selker HP, Griffith JL, D'Agostino RB. A tool for judging coronary care unit admission appropriateness, valid for both real- time and retrospective use. A time-insensitive predictive instrument (TIPI) for acute cardiac ischemia: a multicenter study. Med Care. Jul 1991;29(7):610-627. For corrected coefficients, see http://medg.lcs.mit.edu/cardiac/cpain.htm CoefficientMV Odds Ratio Constant-3.93 Presence of chest pain1.233.42 Pain major symptom0.882.41 Male Sex0.712.03 Age 40 or less-1.440.24 Age > 500.671.95 Male over 50 years**-0.430.65 ST elevation1.3143.72 New Q waves0.621.86 ST depression0.992.69 T waves elevated1.0952.99 T waves inverted1.133.10 T wave + ST changes**-0.3140.73

48. Clinical Scenario* 71 y/o man with 2.5 hours of CP, substernal, non-radiating, described as “bloating.” Cannot say if same as prior MI or worse than prior angina. Hx of CAD, s/p CABG 10 yrs prior, stenting 3 years and 1 year ago. DM on Avandia. ECG: RBBB, Qs inferiorly. No ischemic ST- T changes. *Real patient seen by MAK 1 am 10/12/04

50. CoefficientClinical Scenario Constant-3.93Result-3.93 Presence of chest pain1.231 Pain major symptom0.881 Sex0.711 Age 40 or less-1.4400 Age > 500.671 Male over 50 years-0.431 ST elevation1.31400 New Q waves0.6200 ST depression0.9900 T waves elevated1.09500 T waves inverted1.1300 T wave + ST changes-0.31400 -0.87 Odds of ACI0.418952 Probability of ACI30%

51. What Happened to Pre-test Probability? Typically clinical decision rules report probabilities rather than likelihood ratios for combinations of results. Can “back out” LRs if we know prevalence, p[D+], in the study dataset. With logistic regression models, this “backing out” is known as a “prevalence offset.” (See Chapter 8.)

52. Combining 2 Continuous Tests (Mainly in case we assign Problem 8-3)

53. Optimal Cutoff Line for Two Continuous Tests

54. Optimal Cutoff for a Single Continuous Test Depends on 1)Pre-test Probability of Disease 2)ROC Curve (Likelihood Ratios) 3)Relative Misclassification Costs Cannot choose an optimal cutoff with just the ROC curve.

55. Effect of Prevalence

56. Choosing Which Tests to Include in the Decision Rule Have focused on how to combine results of two or more tests, not on which of several tests to include in a decision rule. Variable Selection Options include: Recursive partitioning Automated stepwise logistic regression Choice of variables in derivation data set requires confirmation in a separate validation data set.

57. Variable Selection Especially susceptible to overfitting

58. Need for Validation: Example* Study of clinical predictors of bacterial diarrhea. Evaluated 34 historical items and 16 physical examination questions. 3 questions (abrupt onset, > 4 stools/day, and absence of vomiting) best predicted a positive stool culture (sensitivity 86%; specificity 60% for all 3). Would these 3 be the best predictors in a new dataset? Would they have the same sensitivity and specificity? *DeWitt TG, Humphrey KF, McCarthy P. Clinical predictors of acute bacterial diarrhea in young children. Pediatrics. Oct 1985;76(4):551- 556.

59. Need for Validation Develop prediction rule by choosing a few tests and findings from a large number of possibilities. Takes advantage of chance variations* in the data. Predictive ability of rule will probably disappear when you try to validate on a new dataset. Can be referred to as “overfitting.” e.g., low serum calcium in 12 children with hemolytic uremic syndrome and bad outcomes

60. VALIDATION No matter what technique (CART or logistic regression) is used, the tests included in a “rule” and the way in which their results are combined must be tested on a data set different from the one used to derive the rule. Beware of studies that use a “validation set” to “tweak” the rule. This is really just a second derivation step.

61. Prognostic Tests Chapter 7

62. Assessment of Prognostic Tests Difference from diagnostic tests and risk factors Quantifying accuracy (calibration and discrimination) Value of prognostic information Common problems with studies of prognostic tests

63. Potential confusion: “cross- sectional” means 2 things Cross-sectional sampling means sampling does not depend on either the predictor variable or the outcome variable. (E.g., as opposed to case-control sampling) Cross-sectional time dimension means that predictor and outcome are measured at the same time -- opposite of longitudinal

64. Cohort Studies Start with a Cross-Sectional Study (Substitute “Outcome” for “Disease”) Eliminate subjects who already have disease

65. Difference from Diagnostic Tests Longitudinal rather than cross-sectional time dimension Incidence rather than prevalence Sensitivity, specificity, prior probability confusing Time to an event may be important Harder to quantify accuracy in individuals –Exceptions: short time course, continuous outcomes

66. Difference from Risk Factors Causality not important Absolute risk very important –Sampling scheme makes a much bigger difference because absolute risks are less generalizable than relative risks –Can be informative even if no bad outcomes!

67. How accurate are the predicted probabilities? –Break the population into groups –Compare actual and predicted probabilities for each group Calibration is important for decision making and giving information to patients Quantifying Prediction 1: Calibration

68. How well can the test separate subjects in the population from the mean probability to values closer to zero or 1? May be more generalizable Often measured with C-statistic Quantifying Prediction 2: Discrimination

69. Illustrations Perfect calibration, no discrimination: –Predicted = actual 5-year mortality = 45% (for everyone) Perfect discrimination, poor calibration –Overall predicted mortality* is 15%, but every patient who dies has a predicted mortality > 20% and every patient who survives has a predicted mortality of < 20% * ∑P i (mortality) / N

70. Calibration Mackillop, W. J. and C. F. Quirt (1997). "Measuring the accuracy of prognostic judgments in oncology." J Clin Epidemiol 50(1): 21-9.

71. Calibration Glare, P., K. Virik, et al. (2003). "A systematic review of physicians' survival predictions in terminally ill cancer patients." Bmj 327(7408): 195-8.

72. Quantifying Discrimination: Dichotomize outcome at time t Then can calculate –Sensitivity and specificity –Likelihood ratios –ROC curves, c-statistic –Can provide these for multiple time points. In each case, probabilities are for an event on or before time t.

73. Discrimination Mackillop, W. J. and C. F. Quirt (1997). "Measuring the accuracy of prognostic judgments in oncology." J Clin Epidemiol 50(1): 21-9.

74. Discrimination Mackillop, W. J. and C. F. Quirt (1997). "Measuring the accuracy of prognostic judgments in oncology." J Clin Epidemiol 50(1): 21-9.

75. Value of Prognostic Information Why do you want to know prognosis? -- ALS, slow vs rapid progression -- GBM, expected survival -- Ambulatory ECG monitoring after AMI to predict recurrent MI (Problem 7-2) -- Na-MELD Score It’s not like deciding whether to carry an umbrella.

76. Value of Prognostic Information To inform treatment or other clinical decisions To inform (prepare) patients and their families To stratify by disease severity in clinical trials Altman, D. G. and P. Royston (2000). "What do we mean by validating a prognostic model?" Stat Med 19(4): 453-73.

77. Doctors and patients like prognostic information But hard to assess its value Most objective approach is decision- analytic. Consider: –What decision is to be made –Costs of errors –Cost of test Value of Prognostic Information

78. DECISION: Treat with more aggressive regimen BEFORE test: 5-year mortality = 25% AFTER test: 5-year mortality either 10% or 50% BUT: do we know how bad it is: –To treat patient with 10% mortality with more aggressive regimen? –To treat patient with 50% mortality with less aggressive regimen? Example

79. Common Problems with Studies of Prognostic Tests- 1 Referral/selection bias – e.g. too many studies from tertiary centers Poorly defined cohort; heterogeneous inclusion criteria – (See Problem 7-1 about predicting neurologic and audiologic sequelae in congenital CMV infection.) Effects of prognosis on treatment and effects of treatment on prognosis –Effective treatments blunt relationships –End-of-life decisions may accentuate relationships

80. Common Problems with Studies of Prognostic Tests- 2 Multiple and composite outcomes –If multiple outcomes collected, is the one best predicted highlighted? –If a study combines multiple outcomes, will the composite outcome be dominated by the most frequent? More on composite outcomes in Chapter 9 (next week)

81. Common Problems with Studies of Prognostic Tests- 3 Loss to follow-up Blinding Next week

82. Common Problems with Studies of Prognostic Tests- 4 Overfitting – Already discussed –Which variables are included –How they are combined Inadequate sample size –Small sample size results in imprecise absolute risk estimates. Quantifying effect size –Watch for comparison of 1 st and 5 th quintiles, units for hazard or odds ratios –How much NEW information? Frequently assessed with multivariable techniques Publication bias

83. Example: A multigene assay to predict recurrence of tamoxifen-treated, node- negative breast cancer* 10-year distant recurrence risk in low, medium, high risk: 6.8%, 14.3%, and 30.5% Hazard ratio 3.21 per 50 point change –51% had scores <12 –Age, tumor size dichotomized –Tumor grade reproducibility only fair (κ=0.34-0.37) Authors of paper patented the test ($3500 charge) *Paik et al. N Engl J Med 2004;351(27):2817-26.

84. Additional Slides Prognostic Test Accuracy Calibration and Discrimination Should I carry an umbrella tomorrow? Watch the weather man on TV tonight

85. Good Weather Man Predicted Chance of Rain Days Expected Rainy Days Actual Rainy Days Actual Percent Actual - Expected 0%120000% 10%919.111%-9% 20%499.848%-12% 30%206735%5% 40%228.81255%15% 50%2110.51467%17% 60%1911.41474%14% 70%149.81286%16% 80%545100%20% 90%32.73100%10% 100%111 0% 36573.173

86. Calibration

88. Bad Weather Man Predicted Chance of Rain Days Expected Rainy Days Actual Rainy Days Actual Percent Actual - Expected 0%000 10%10010 10%0% 20%25050 20%0% 30%000 40%000 50%000 60%000 70%000 80%000 90%1513.51387%-3% 100%000 36573.573

89. Calibration

91. Good Weather Man

92. Same Discrimination, Poor Calibration

93. Bad Weather Man

94. Compare Weathermen

95. Compare Weather Men Failing to carry an umbrella on rainy day (“Wet”) just as bad as carrying an umbrella on a sunny day (“Tired”). Cutoff = 50% chance of rain Good Weatherman: Wet = 24, Tired = 14, Total = 38 Bad Weatherman: Wet = 60, Tired = 2, Total = 62

96. Compare Weather Men Failing to carry an umbrella on rainy day (“Wet”) 4X as bad as carrying an umbrella on a sunny day (“Tired”). Cutoff = 20% chance of rain. Good Weatherman: Wet = 1, Tired = 82, Total = 1 × 4 + 82 = 86 Bad Weatherman: Wet = 10, Tired = 202, Total = 10 × 4 + 202 = 242