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Multiple Tests, Multivariable Decision Rules, and Studies of Diagnostic Test Accuracy

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### Multiple Tests, Multivariable Decision Rules, and Studies of Diagnostic Test Accuracy

Chapter 8 – Multiple Tests and Multivariable Decision Rules

Chapter 5 – Studies of Diagnostic Test Accuracy

Michael A. Kohn, MD, MPP

10/19/2006

Outline of Topics

- Combining results of multiple tests: importance of test non-independence
- Recursive Partitioning
- Logistic Regression
- Published “rules” for combining test results: importance of validation separate from derivation
- Biases in studies of diagnostic test accuracy

Overfitting bias

Incorporation bias

Referral bias

Double gold standard bias

Spectrum bias

Warning: Different Example

Example of combining two tests in this talk:

Prenatal sonographic Nuchal Translucency (NT) and Nasal Bone Exam (NBE) as dichotomous tests for Trisomy 21*

Example of combining two tests in book**:

Premature birth (GA < 36 weeks) and low birth weight (BW < 2500 grams) as dichotomous tests for neonatal morbidity

*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 Gynecol23(3): 218-23.

**Soon to be replaced

If NT ≥ 3.5 mm Positive for Trisomy 21*

*What’s wrong with this definition?

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

One Dichotomous Test

Trisomy 21

Nuchal D+ D- LR

Translucency

≥ 3.5 mm 212 478 7.0

< 3.5 mm 121 4745 0.4

Total 333 5223

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.)

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%

Clinical Scenario – One TestPre-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

Pre-Test Probability of Tri21 = 6%NT PositivePost-Test Probability of Tri21 = 31%

Using Probabilities

Using Odds

Pre-Test Odds of CAD = 0.064EECG Positive (LR = 7.0)Post-Test Odds of CAD = 0.44

Clinical Scenario – One TestPre-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

Second Dichotomous Test

Nasal Bone Tri21+ Tri21- LR

Absent 229 129 27.8

Present 104 5094 0.32

Total 333 5223

Do you see that this is (229/333)/(129/5223)?

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-Nuclide Probability of Trisomy 21 = ?

Clinical Scenario –Two Tests

Using Probabilities

Using Odds

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

Clinical Scenario – Two TestsPre-Test Probability of Trisomy 21 = 6%NT ≥ 3.5 mm AND Nasal Bone Absent

NT + (LR = 6.96)

|--------------->

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 = 0.064

Prob = 0.06

Odds = 12.4

Prob = 0.925

Odds = 0.44

Prob = 0.31

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 ?

Answer = No

Not 194

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.

Using Odds

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

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

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.

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.

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.

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.

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.

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

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

Ways to Combine Multiple Tests

On a group of patients (derivation set), perform the multiple tests and determine true disease status (apply the gold standard)

- Measure LR for each possible combination of results
- Recursive Partitioning
- Logistic Regression

Determine LR for Each Result Combination

*Assumes pre-test prob = 6%

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.

Determine LR for Each Result Combination

How do you handle continuous tests?

Not practical for most groups of tests.

Recursive PartitioningExamine Nasal Bone FirstCVS if P(Trisomy 21 > 5%)

Recursive PartitioningExamine Nasal Bone FirstCVS if P(Trisomy 21 > 5%)

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”

Tree Pruning: Goldman Rule*

8 “Tests” for Acute MI in ER Chest Pain Patient :

- ST Elevation on ECG;
- CP < 48 hours;
- ST-T changes on ECG;
- Hx of MI;
- Radiation of Pain to Neck/LUE;
- Longest pain > 1 hour;
- Age > 40 years;
- 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.

Recursive Partitioning

- Does not deal well with continuous test results*

*when there is a monotonic relationship between between the rest result and the probability of disease

Logistic Regression

Ln(Odds(D+)) =

a + bNTNT+ bNBENBE + binteract(NT)(NBE)

“+” = 1

“-” = 0

More on this later in ATCR!

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

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

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 8A.)

Optimal Cutoff for a Single Continuous Test

Depends on

- Pre-test Probability of Disease
- ROC Curve (Likelihood Ratios)
- Relative Misclassification Costs

Cannot choose an optimal cutoff with just the ROC curve.

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.

Options include:

- Recursive partitioning
- Automated stepwise logistic regression*

Choice of variables in derivation data set requires confirmation in a separate validation data set.

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.

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.”

VALIDATION

No matter what technique (CART or logistic regression) is used, the “rule” for combining multiple test results must be tested on a data set different from the one used to derive it.

Beware of “validation sets” that are just re-hashes of the “derivation set”.

(This begins our discussion of potential problems with studies of diagnostic tests.)

Studies of Diagnostic Test AccuracySackett, EBM, pg 68

- Was there an independent, blind comparison with a reference (“gold”) standard of diagnosis?
- Was the diagnostic test evaluated in an appropriate spectrum of patients (like those in whom we would use it in practice)?
- Was the reference standard applied regardless of the diagnostic test result?
- Was the test (or cluster of tests) validated in a second, independent group of patients?

Bias in Studies of Diagnostic Test Accuracy

Index Test = Test Being Evaluated

Gold Standard = Test Used to Determine True Disease Status

Studies of Diagnostic TestsSackett, EBM, pg 68

- Was there an independent, blind comparison with a reference (“gold”) standard of diagnosis?
- Was the diagnostic test evaluated in an appropriate spectrum of patients (like those in whom we would use it in practice)?
- Was the reference standard applied regardless of the diagnostic test result?
- Was the test (or cluster of tests) validated in a second, independent group of patients?

Studies of Diagnostic TestsIncorporation Bias

Index Test is “incorporated” into gold standard.

Consider a study of the usefulness of various findings for diagnosing pancreatitis. If the "Gold Standard" is a discharge diagnosis of pancreatitis, which in many cases will be based upon the serum amylase, then the study can't quantify the accuracy of the amylase for this diagnosis.

Studies of Diagnostic TestsIncorporation Bias

A study* of BNP in dyspnea patients as a diagnostic test for CHF also showed that the CXR performed extremely well in predicting CHF.

The two cardiologists who determined the final diagnosis of CHF were blinded to the BNP level but not to the CXR report, so the assessment of BNP should be unbiased, but not the assessment CXR.

*Maisel AS, Krishnaswamy P, Nowak RM, McCord J, Hollander JE, Duc P, et al. Rapid measurement of B-type natriuretic peptide in the emergency diagnosis of heart failure. N Engl J Med 2002;347(3):161-7.

Studies of Diagnostic TestsSackett, EBM, pg 68

- Was there an independent, blind comparison with a reference (“gold”) standard of diagnosis?
- Was the diagnostic test evaluated in an appropriate spectrum of patients (like those in whom we would use it in practice)?
- Was the reference standard applied regardless of the diagnostic test result?
- Was the test (or cluster of tests) validated in a second, independent group of patients?

Studies of Diagnostic TestsVerification Bias*

The study population only includes those to whom the gold standard was applied, but patients with positive index tests are more likely to be referred for the gold standard.

Example: V/Q Scan as a test for PE. Gold standard is a PA-gram. Patients with negative V/Q scans are less frequently referred for PA-gram than those with positive V/Q scans. Only patients who had PA-grams are included in the study.

*AKA Work-up, Referral Bias, or Ascertainment Bias

Studies of Diagnostic TestsVerification Bias

Sensitivity (a/(a+c)) is biased UP.

Specificity (d/(b+d)) is biased DOWN.

Studies of Diagnostic TestsDouble Gold Standard Bias

One gold standard (e.g. biopsy) is applied in patients with positive index test, another gold standard (e.g., clinical follow-up) is applied in patients with a negative index test.

Studies of Diagnostic TestsDouble Gold Standard

Test: V/Q Scan

Disease: PE

Gold Standard: PA-gram in patients who had one, clinical follow-up in patients who didn’t

Study Population: All patients presenting to the ED who received a V/Q scan.

Assume some patients did not get PA-gram because of normal/low probability V/Q scans but would have had positive PA-grams. Instead they had negative clinical follow-up and were counted as true negatives. If they had had PA-grams, they would have been counted as false negatives.

*PIOPED. JAMA 1990;263(20):2753-9.

Studies of Diagnostic TestsDouble Gold Standard

Sensitivity (a/(a+c)) biased UP

Specificity (d/(b+d)) biased UP

Studies of Diagnostic TestsSackett, EBM, pg 68

- Was there an independent, blind comparison with a reference (“gold”) standard of diagnosis?
- Was the reference standard applied regardless of the diagnostic test result?
- Was the test (or cluster of tests) validated in a second, independent group of patients?

Studies of Diagnostic TestsSpectrum Bias

So far, we have said that PPV and NPV of a test depend on the population being tested, specifically on the prevalence of D+ in the population.

We said that sensitivity and specificity are properties of the test and independent of the prevalence and, by implication at least, the population being tested.

In fact, …

Studies of Diagnostic TestsSpectrum Bias

Sensitivity depends on the spectrum of disease in the population being tested.

Specificity depends on the spectrum of non-disease in the population being tested.

Studies of Diagnostic TestsSpectrum Bias

D+ and D- groups are not homogeneous.

D-/D+ really is D-,D+, D++, or D+++

D-/D+ really is (D1-, D2-, or D3-)/D+

Studies of Diagnostic TestsSpectrum Bias

Example: Absence of Nasal Bone (on 13-week ultrasound) as a Test for Chromosomal Abnormality

Spectrum BiasAbsence of Nasal Bone as a Test for Chromosomal Abnormality

Nasal D+ D- LR

Bone

Absent 229 129 7.0

Present 104 5094 0.4

Total 333 5223

Sensitivity = 229/333 = 69%

BUT

the D+ group only included fetuses with Trisomy 21

Spectrum BiasAbsence of Nasal Bone as a Test for Chromosomal Abnormality

D+ group excluded 295 fetuses with other chromosomal abnormalities (esp. Trisomy 18)

If the purpose of the nasal bone exam is to determine on whom to get CVS, these 295 fetuses with chromosomal abnormalities other than trisomy 21 should be included in the D+ group.

95/295 (32%, not 69%) had absent nasal bone.

Spectrum BiasAbsence of Nasal Bone as a Test for Chromosomal Abnormality

Nasal D+ D- LR

Bone

Absent 229+95 =324 478 7.0

Present 104+200=304 4745 0.4

Total 333+295=628 5223

Sensitivity = 324/628 = 52%

NOT 69% obtained

when the D+ group only included fetuses with Trisomy 21

Spectrum BiasAbsence of Nasal Bone as a Test for Chromosomal Abnormality

By excluding chromosomal abnormalities other than Trisomy 21 from the D+ group, the study exaggerates the sensitivity of the Nasal Bone Exam (NBE) for chromosomal abnormalities.

“True” Sensitivity of NBE for chromosomal abnormalities = 52%

Biased estimate due to spectrum bias (excluding other chromosomal problems) = 69%

Biases in Studies of Tests

- Overfitting Bias – “Data snooped” cutoffs take advantage of chance variations in derivations set making test look falsely good.
- Incorporation Bias – index test part of gold standard (Sensitivity Up, Specificity Up)
- Verification/Referral Bias – positive index test increases referral to gold standard (Sensitivity Up, Specificity Down)
- Double Gold Standard – positive index test causes application of definitive gold standard, negative index test results in clinical follow-up (Sensitivity Up, Specificity Up)
- Spectrum Bias
- D+ sickest of the sick (Sensitivity Up)
- D- wellest of the well (Specificity Up)

Biases in Studies of Tests

Don’t just identify potential biases, figure out how the biases could affect the conclusions.

Studies concluding a test is worthless are not invalid if biases in the design would have led to the test looking BETTER than it really is.

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