Michael A. Kohn, MD, MPP6/9/2011

Combining Tests and Multivariable Decision Rules

Importance of test non-independence

Recursive Partitioning Logistic Regression Variable (Test) Selection Importance of validation separate

from derivation (calibration and discrimination revisited)

Combining Tests/Diagnostic Models

Combining TestsExample

Prenatal sonographic Nuchal Translucency (NT) and Nasal Bone Exam 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.

If NT ≥ 3.5 mm Positive for Trisomy 21*

*What’s wrong with this definition?

>95th Perc.37.9%, 88.6%

> 3.5 mm9.2%, 63.7%

> 4.5 mm3.5%, 43.5%

> 5.5 mm1.9%, 31.2%

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1 - Specificity

Sen

siti

vity

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 undergo 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: 0.06Pre-test odds: 0.06/0.94 = 0.064LR(+) = 7.0Post-Test Odds = Pre-Test Odds x LR(+)

= 0.064 x 7.0 = 0.44Post-Test prob = 0.44/(0.44 + 1) = 0.31

NT Positive

• Pre-test Prob = 0.06

• P(Result|Trisomy 21) = 0.64

• P(Result|No Trisomy 21) = 0.09

• Post-Test Prob = ?

http://www.quesgen.com/PostProbofDisease.php

Slide Rule

Nasal Bone SeenNBA=“No”

Neg for Trisomy 21

Nasal Bone AbsentNBA=“Yes”

Pos for Trisomy 21

Second Dichotomous Test

Nasal Bone Tri21+ Tri21- LR

Absent

Yes 229 129 27.8

No 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%Nasal Bone AbsentPost-NBA Probability of Trisomy 21 = ?

Clinical Scenario –Two Tests

Using Probabilities

Clinical Scenario – Two Tests

Pre-Test Odds of Tri21 = 0.064NT Positive (LR = 7.0)Post-Test Odds of Tri21 = 0.44Nasal Bone Absent (LR = 27.8?)Post-Test Odds of Tri21 = .44 x 27.8?

= 12.4? (P = 12.4/(1+12.4) = 92.5%?)

Using Odds

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

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

NT NBE

Trisomy 21+ %

Trisomy 21- % LR

Pos Pos 158 47% 36 0.7% 69

Pos Neg 54 16% 442 8.5% 1.9

Neg Pos 71 21% 93 1.8% 12

Neg Neg 50 15% 4652 89% 0.2

Total Total 333 100% 5223 100%  

Not 194

158/(158 + 36) = 81%, not 92.5%

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.

Clinical Scenario

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%)

Using Odds

Non-Independence

Non-Independence of NT and NBA

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.

One aspect of Down’s syndrome is slower fetal development; the NT decreases more slowly and the nasal bone ossifies later. Chromosomally NORMAL fetuses that develop slowly will tend to have false positives on BOTH the NT Exam and the Nasal Bone Exam.

Reasons for Non-Independence

Heterogeneity of Disease (e.g. spectrum of severity)*.

Heterogeneity of Non-Disease.

(See EBD page 158.)*In this example, Down’s syndrome is the only chromosomal abnormality considered, so disease is fairly homogeneous

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 (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

Determine LR for Each Result Combination

NT NBA Tri21+ % Tri21- % LRPost Test

Prob*

Pos Pos 158 47% 36 0.7% 69 81%

Pos Neg 54 16% 442 8.5% 1.9 11%

Neg Pos 71 21% 93 1.8% 12 43%

Neg Neg 50 15% 4652 89.1% 0.2 1%

Total Total 333 100% 5223 100%  

*Assumes pre-test prob = 6%

Sort by LR (Descending)

NT NBA Tri21+ % Tri21- % LR

Pos Pos 15847

% 36 0.70% 69

Neg Pos 71

21% 93 1.80% 12

Pos Neg 5416

% 442 8.50% 1.9

Neg Neg 50

15% 4652 89.10% 0.2

Apply Chapter 4 – Multilevel Tests

Now you have a multilevel test (In this case, 4 levels.)

Have LR for each test result Can create ROC curve and calculate

AUROC Given pre-test probability and

treatment threshold probability (C/(B+C)), can find optimal cutoff.

Create ROC Table

NTNBE Tri21+

Sens

Tri21- 1 - Spec LR

      0%   0%  

Pos Pos 158 47% 36 0.70% 69

Neg Pos 71 68% 93 3% 12

PosNeg 54 84% 442 11% 1.9

NegNeg 50

100% 4652 100% 0.2

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Sensitivity

1 - S

pecific

ity

AUROC = 0.896

Optimal Cutoff

NT NBE LRPost-Test

Prob

Pos Pos 69 0.81

Neg Pos 12 0.43

Pos Neg 1.9 0.11

Neg Neg 0.2 0.01

Assume

• Pre-test probability = 6%

• Threshold for CVS is 2%

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 always practical for groups of tests.

Recursive PartitioningMeasure NT First

Nuchal Translucency

Nasal Bone

< 3.5 mm ≥ 3.5 mm

31%2.5%

Present

1 %

Suspected Trisomy 21 (P = 6%)

43 %

Nasal Bone

Absent Present

11 %

Absent

81 %

Recursive PartitioningExamine Nasal Bone First

Nasal Bone

Nuchal Translucency

< 3.5 mm≥ 3.5 mm

64%2 %

Present

1 %

Suspected Trisomy 21 (P = 6%)

11 % 43 %

Absent

81 %

< 3.5 mm≥ 3.5 mm

Nuchal Translucency

Do Nasal Bone Exam First

Better separates Trisomy 21 from chromosomally normal fetuses

If your threshold for CVS is between 11% and 43%, you can stop after the nasal bone exam

If your threshold is between 1% and 11%, you should do the NT exam only if the NBE is normal.

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

Nasal Bone

Nuchal Translucency

< 3.5 mm≥ 3.5 mm

64%2%

Present

1 %

Suspected Trisomy 21 (P = 6%)

11 % 43 %

Absent

81 %

< 3.5 mm≥ 3.5 mm

Nuchal Translucency

No NT, CVS

CVSNo CVS

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

Nasal Bone

Nuchal Translucency

< 3.5 mm

64%2%

Present

1 %

Suspected Trisomy 21 (P = 6% )

11 %

Absent

≥ 3.5 mmCVS

CVSNo CVS

Recursive Partitioning

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”

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

Logistic Regression

Ln(Odds(D+)) = a + bNTNT+ bNBANBA + binteract(NT)(NBA)

“+” = 1“-” = 0

Needs a course of its own!

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.

Probability of Trisomy 21 vs. Maternal Age

Ln(Odds) of Trisomy 21 vs. Maternal Age

Combining 2 Continuous Tests

> 1% Probability of Trisomy 21

< 1% Probability of Trisomy 21

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.

Variable Selection

Especially susceptible to overfitting

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 ValidationDevelop prediction rule by choosing a few

tests and findings from a large number of candidates.

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

VALIDATION

No matter what technique (CART or logistic regression) is used, the tests included in a model 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 model. This is really just a second derivation step.

Prognostic Tests and Multivariable Diagnostic Models

Commonly express results in terms of a probability

-- risk of the outcome by a fixed time point (prognostic test)

-- posterior probability of disease (diagnostic model)

Need to assess both calibration and discrimination.

Validation Dataset

Measure all the variables needed for the model.

Determine disease status (D+ or D-) on all subjects.

VALIDATIONCalibration

-- Divide dataset into probability groups (deciles, quintiles, …) based on the model (no tweaking allowed).-- In each group, compare actual D+ proportion to model-predicted probability in each group.

VALIDATIONDiscrimination

Discrimination-- Test result is model-predicted probability of disease.-- Use “Walking Man” to draw ROC curve and calculate AUROC.

Importance of test non-independence

Recursive Partitioning Logistic Regression Variable (Test) Selection Importance of validation separate

from derivation (calibration and discrimination revisited)

Combining Tests/Diagnostic Models