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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

Coursebook Chapter 8 – Multiple Tests and Multivariable Decision Rules

Coursebook Chapter 5 – Studies of Diagnostic Test Accuracy

Michael A. Kohn, MD, MPP

10/27/2005

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:

Exercise ECG and Nuclide Scan as dichotomous tests for CAD (assumed to be a dichotomous D+/D- disease)*

Example of combining two tests in Coursebook:

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

*Sackett DL, Haynes RB, Guyatt GH, Tugwell P. Clinical epidemiology : a basic science for clinical medicine. 2nd ed. Boston: Little Brown; 1991.

One Dichotomous Test

Exercise ECG CAD+ CAD- LR

Positive 299 44 6.80

Negative 201 456 0.44

Total 500 500

Do you see that this is (299/500)/(44/500)?

Review of Chapter 3: What are the sensitivity, specificity, PPV, and NPV of this test? (Be careful.)

Clinical Scenario – One TestPre-Test Probability of CAD = 33%EECG Positive

Pre-test prob: 0.33

Pre-test odds: 0.33/0.67 = 0.5

LR(+) = 6.80

Post-Test Odds = Pre-Test Odds x LR(+)

= 0.5 x 6.80 = 3.40

Post-Test prob = 3.40/(3.40 + 1) = 0.77

Pre-Test Probability of CAD = 33%EECG PositivePost-Test Probability of CAD = 77%

Using Probabilities

Using Odds

Pre-Test Odds of CAD = 0.50EECG Positive (LR = 6.80)Post-Test Odds of CAD = 3.40

Clinical Scenario – One TestPre-Test Probability of CAD = 33%EECG Positive

EECG + (LR = 6.80)

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

+------------------------------------------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 = 0.50

Prob = 0.33

Odds = 3.40

Prob = 0.77

Second Dichotomous Test

Nuclide Scan CAD+ CAD- LR

Positive 416 190 2.19

Negative 84 310 0.27

Total 500 500

Do you see that this is (416/500)/(190/500)?

Pre-Test Probability of CAD = 33%EECG PositivePost-EECG Probability of CAD = 77%Nuclide Scan PositivePost-Nuclide Probability of CAD = ?

Clinical Scenario –Two Tests

Using Probabilities

Using Odds

Pre-Test Odds of CAD = 0.50EECG Positive (LR = 6.80)Post-Test Odds of CAD = 3.40Nuclide Scan Positive (LR = 2.19?)Post-Test Odds of CAD = 3.40 x 2.19? = 7.44? (P = 7.44/(1+7.44) = 88%?)

Clinical Scenario – Two TestsPre-Test Probability of CAD = 33%EECG Positive

E-ECG + (LR = 6.80)

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

Nuclide + (LR = 2.19)

|------>

E-ECG + Nuclide +

Can we do this? |----------------->|----->

E-ECG + and Nuclide +

+--------------------------------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 = 0.50

Prob = 0.33

Odds = 7.44

Prob = 0.88

Odds = 3.40

Prob = 0.77

Question

Can we use the post-test odds after a positive Exercise ECG as the pre-test odds for the positive nuclide scan?

i.e., can we combine the positive results by multiplying their LRs?

LR(E-ECG +, Nuclide +) = LR(E-ECG +) x LR(Nuclide +) ?

= 6.80 x 2.19 ?

= 14.88 ?

Answer = No

Not 14.88

Non-Independence

A positive nuclide scan does not tell you as much if the patient has already had a positive exercise ECG.

Using Odds

Pre-Test Odds of CAD = 0.50EECG +/Nuclide Scan + (LR = 10.62)Post-Test Odds of CAD = 0.50 x 10.62 = 5.31 (P = 5.31/(1+5.31) = 84%, not 88%)

Non-Independence

E-ECG +

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

Nuclide +

|------>

E-ECG + Nuclide +

if tests were independent |----------------->|----->

E-ECG + and Nuclide +

since tests are dependent |-------------------->

+--------------------------------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 = 0.84

Non-Independence

Instead of the nuclide scan, what if the second test were just a repeat exercise ECG?

A second positive E-ECG would do little to increase your certainty of CAD. 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.

In this example, the gold standard is anatomic narrowing of the arteries, but 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 this 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 = 33%

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 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 ACI;
- 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

Logistic Regression

Ln(Odds(D+)) =

a + bE-ECGE-ECG+ bNuclideNuclide + binteract(E-ECG)(Nuclide)

“+” = 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 TestsReferral 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: Swelling as a test for ankle fracture. Gold standard is a positive X-ray. Patients with swelling are more likely to be referred for x-ray. Only patients who had x-rays are included in the study.

Studies of Diagnostic TestsReferral Bias

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

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

Studies of Diagnostic TestsReferral Bias Example*

Test: A-a O2 gradient

Disease: PE

Gold Standard: VQ scan or pulmonary angiogram

Study Population: Patients who had VQ scan or PA-gram

Results: A-a O2 gradient > 20 mm Hg had very high sensitivity (almost every patient with PE by VQ scan or PA gram had a gradient > 20 mm Hg), but a very low specificity (lots of patients with negative PA grams had gradients > 20 mm Hg).

*McFarlane MJ, Imperiale TF. Use of the alveolar-arterial oxygen gradient in the diagnosis of pulmonary embolism. Am J Med. 1994;96(1):57-62.

Studies of Diagnostic TestsReferral Bias

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

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

*Still concluded test not sensitive enough, so it probably isn’t.

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: A-a O2 gradient

Disease: PE

Gold Standard: VQ scan or pulmonary angiogram in patients who had one, clinical follow-up in patients who didn’t

Study Population: All patients presenting to the ED with dyspnea.

Some patients did not get VQ scan or PA-gram because of normal A-a O2 gradients but would have had positive studies. Instead they had negative clinical follow-up and were counted as true negatives.

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: Pale Conjunctiva as Test for Iron Deficiency Anemia

Assume that conjunctival paleness always occurs at HCT < 25

Pale Conjunctiva as a Test for Iron Deficiency

Sensitivity is HIGHER in the population with more severe disease

Pale Conjunctiva as a Test for Iron Deficiency

Specificity is LOWER in the population with more severe non-disease.

(Patients without the disease in question are more likely to have other diseases that can be confused with the disease in question.)

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)
- 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.

End

Forewarned is forearmed – I will lobby the course faculty to include a question on the New Orleans Criteria (a decision rule) for getting CTs on patients with minor head injuries.

CT Head Rules

New Orleans Criteria -- Haydel MJ, et al. Indications for computed tomography in patients with minor head injury. N Engl J Med 2000;343(2):100-5.

Canadian CT Head Rules -- Stiell IG, et al. The Canadian CT Head Rule for patients with minor head injury. Lancet 2001;357(9266):1391-6.

Association between Clinical Findings and CT Results in 520 Patients with Minor Head Injury (Phase 1)

Haydel, M. J. et al. N Engl J Med 2000;343:100-105

CT Head Rules – New Orleans Criteria*

- Headache
- Vomiting
- Older than 60 years
- Drug or Alcohol Intoxication
- Persistent anterograde amnesia
- Visible trauma above the clavicle
- Seizure

*Selected from 8 (?) candidate variables using recursive paritioning.

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