assessing information from multilevel (ordinal) tests roc curves and likelihood ratios for results...

Assessing Information from Multilevel (Ordinal) Tests

ROC curves and

Likelihood Ratios for results other than “+” or “-”

Michael A. Kohn, MD, MPP

10/4/2007

Four Main Points

1) Dichotomizing a multi-level test by choosing a fixed cutpoint reduces the value of the test.

2) The ROC curve summarizes the ability of the test to differentiate between D+ and D- individuals.

3) LR(result) = P(result|D+)/P(result|D-) = slope of ROC curve.

4) Pre-Test Odds x LR(result) = Post-Test Odds

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

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

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

Septic ArthritisBacterial infection in a joint.

Clinical ScenarioDoes this Adult Patient Have Septic Arthritis?


A 48-year-old woman with a history of rheumatoid arthritis who has been treated with long-term, low-dose prednisone presents to the emergency department with a 2-day history of a red, swollen right knee that is painful to touch. She reports no prior knee swelling and no recent trauma or knee surgery, illegal drug use, rash, uveitis, or risky sexual behavior. On examination, she is afebrile and has a right knee effusion. Her peripheral white blood cell (WBC) count is 11 000/µL and her erythrocyte sedimentation rate (ESR) is 55 mm/h. An arthrocentesis is performed, and the initial Gram stain is negative.

Margaretten, M. E., J. Kohlwes, et al. (2007). Jama 297(13): 1478-88.

You have the synovial white blood cell (WBC) count.


Assume pre-test probability of septic arthritis is 0.38.

How do you use the synovial WBC result to determine the likelihood of septic arthritis?


Why Not Make It a Dichotomous Test?

Synovial Septic ArthritisWBC Count Yes No

>25,000 77% 27%

≤ 25,000 23% 73%

TOTAL* 100% 100%

*Note that these could have come from a study where the patients with septic arthritis (D+ patients) were sampled separately from those without (D- patients).



Sensitivity = 77%Specificity = 73%

LR(+) = 0.77/(1 - 0.73) = 2.9LR(-) = (1 - 0.77)/0.73 = 0.32

“+” = > 25,000/uL “-” = ≤ 25,000/uL

Clinical ScenarioSynovial WBC = 48,000/mL

(Demonstrate LR Slide Rule?)

(Demonstrate Excel?)

Pre-test prob: 0.38

LR(+) = 2.9

Post-Test prob = ?


Pre-test prob: 0.38

Pre-test odds: 0.38/0.62 = 0.61

LR(+) = 2.9

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

= 0.61 x 2.9 = 1.75

Post-Test prob = 1.75/(1.75+1) = 0.64


Pre-test prob: 0.38

LR(+) = ?

Post-Test prob =?

Clinical Scenario Synovial WBC = 128,000/mL

Pre-test prob: 0.38

Pre-test odds: 0.38/0.62 = 0.61

LR(+) = 2.9 (same as for WBC=48,000!)

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

= 0.61 x 2.9 = 1.75

Post-Test prob = 1.75/(1.75+1) = .64


Because you lose information. The risk associated with a synovial WBC=48,000 is equated with the risk associated with WBC=128,000.

Choosing a fixed cutpoint to dichotomize a multi-level or continuous test throws away information and reduces the value of the test.

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

WBC (/uL) Interval

% of Septic Arthritis

% of No Septic Arthritis

>100,000 29% 1%

>50,000-100,000 33% 7%

>25,000-50,000 15% 19%

0 - 25,000 23% 73%

TOTAL 100% 100%

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80%

0 - 25,000 >25,000-50,000

>50,000-100,000

>100,000

No Septic Arthritis

Septic Arthritis

Synovial Fluid WBC Count

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.

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0 - 25,000 >25,000-50,000

>50,000-100,000

>100,000

Negative Positive

Cutoff = ∞Sensitivity = 0%1 - Specificity = 0%

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0 - 25,000 >25,000-50,000

>50,000-100,000

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Negative Positive

Cutoff = 100,000Sensitivity = 29%1 - Specificity = 1%

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0 - 25,000 >25,000-50,000

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Negative Positive


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0 - 25,000 >25,000-50,000

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Negative Positive

Cutoff = 0Sensitivity = 100%1 - Specificity = 100%

WBC Count (x1000/uL)

Sensitivity 1 - Specificity

> ∞ 0% 0%

> 100 29% 1%

> 50 62% 8%

> 25 77% 27%

≥ 0 100% 100%


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1 - Specificity

Sen

sitiv

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Cutoff > ∞

Cutoff > 100k

Cutoff > 50k

Cutoff > 25k

Cutoff ≥ 0

Area Under Curve = 0.8114

0 5 10 15 20 25 30

WBC Count (1000/uL)

D-: No BacteremiaD+: Bacteremia

0

0.2

0.4

0.6

0.8

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0 0.2 0.4 0.6 0.8 1

Test Discriminates Well Between D+ and D-

-40 -20 0 20 40 60Test Result

D-D+

Test Discriminates Well Between D+ and D-

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0 0.2 0.4 0.6 0.8 1

1 - Specificity

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Test Discriminates Poorly Between D+ and D-

-40 -20 0 20 40 60Test Result

D-D+

Test Discriminates Poorly Between D+ and D-

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1 - Specificity

Sen

sitiv

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Cutoff > ∞

Cutoff > 100k

Cutoff > 50k

Cutoff > 25k

Cutoff ≥ 0

Area Under Curve = 0.8114

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


• 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

Area under ROC curve = 0.8405S

en

sitiv

ity

1 - Specificity0.00 0.25 0.50 0.75 1.00

0.00

0.25

0.50

0.75

1.00

“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)…

“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.

Synovial WBC Count in 5 Patients with Septic Arthritis

PatientWBC Count(x 1000/uL)

D1 128

D2 92

D3 64

D4 37

D5 24

Synovial WBC Count in 10 Patient Without Septic ArthritisPatient WBC Count (x 1000)

N1 71

N2 48

N3 37

N4 23

N5 12

N6 12

N7 8

N8 7

N9 6

N10 0

Septic Arthritis No Septic Arthritis

128

92

71

64

48

37 37

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23

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0

DDNDN(DN)DN(NN)NNNN

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D-

D+

Area under ROC curve = 0.8405S

en

sitiv

ity

1 - Specificity0.00 0.25 0.50 0.75 1.00

0.00

0.25

0.50

0.75

1.00

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

ROC Curve Describes the Test, Not the Patient

Clinical Scenario

Synovial WBC count = 48,000


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0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

1 - Specificity

Sen

sitiv

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Cutoff > ∞

Cutoff > 100k

Cutoff > 50k

Cutoff > 25k

Cutoff ≥ 0

Sensitivity, Specificity, LR(+), and LR(-) of the Synovial Fluid WBC Count for Septic Arthritis at 3

Different Cutoffs

WBC (/uL) Sensitivity Specificity LR+ LR-

>100,000 29% 99% 29.0 0.7

>50,000 62% 92% 7.8 0.4

>25,000 77% 73% 2.9 0.3

Synovial WBC Count = 48,000/uL

Which LR should we use?

Likelihood Ratios

LR(+) = Sensitivity/(1 – Specificity) = P(+|D+)/(1-P(-|D-)) = P(+|D+)/P(+|D-)

LR(-) = (1 – Sensitivity)/Specificity = (1-P(+|D+))/P(-|D-) = P(-|D+)/P(-|D-)

Likelihood Ratios

LR(result) = P(result|D+)/P(result|D-)

P(Result) in patient WITH disease

------------------------------------------------------

P(Result) in patients WITHOUT disease

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0 - 25,000 >25,000-50,000

>50,000-100,000

>100,000

No Septic Arthritis

Septic Arthritis

Likelihood RatiosThe ratio of the height of the D+ distribution to the height of the D- distribution

15%19%

LR = 15%/19% = 0.8

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1 - Specificity

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> 50k

> 25k

15%

19%Slope = 15%/19% =0.8

Likelihood Ratio

WBC (/uL) Interval % of D+ % of D-

Interval LR

>100,000 29% 1% 29.0

>50,000-100,000 33% 7% 4.7

>25,000-50,000 15% 19% 0.8

0 - 25,000 23% 73% 0.3

Common Mistake

When given an “ROC Table,” it is tempting to calculate an LR(+) or LR(-) as if the test were “dichotomized” at a particular cutoff.

Example: LR(+,25,000) = 77%/27% = 2.9

This is NOT the LR of a particular result (e.g. WBC >25,000 and ≤ 50,000); it is the LR(+) if you divide “+” from “-” at 25,000.

WBC (/uL) Sensitivity Specificity LR+ LR-

>100,000 29% 99% 29.0 0.7

>50,000 62% 92% 7.8 0.4

>25,000 77% 73% 2.9 0.3

Common Mistake

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1 - Specificity

Sen

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Slope = 77%/27% = 2.9

Slope = 0.8

27%

77%

> 25,000

Common Mistake

Main Point 3 Likelihood Ratio

P(Result) in patient WITH disease

------------------------------------------------------

P(Result) in patients WITHOUT disease

Slope of ROC Curve

Do not calculate an LR(+) or LR(-) for a multilevel test.

Clinical ScenarioSynovial WBC = 48,000/uL

Pre-test prob: 0.38

Pre-test odds: 0.38/0.62 = 0.61

LR(WBC btw 25,000 and 50,000) = 0.8

Post-Test Odds = Pre-Test Odds x LR(48)

= 0.61 x 0.8 = 0.49

Post-Test prob = 0.49/(0.49+1) = 0.33

Clinical ScenarioSynovial WBC = 128,000/uL

Pre-test prob: 0.38

Pre-test odds: 0.38/0.62 = 0.61

LR(128,000/uL) = 29

Post-Test Odds = Pre-Test Odds x LR(128)

= 0.61 x 29 = 17.8

Post-Test prob = 17.8/(17.8+1) = 0.95

Clinical Scenario

WBC = 48,000/uL Post-Test Prob = 0.33

WBC = 128,000/uL Post-Test Prob = 0.95

(Recall that dichotomizing the WBC with a fixed cutpoint of 25,000/uL meant that WBC = 48,000/uL would be treated the same as WBC = 128,000/uL and post-test prob = 0.64)

Main Point 4Bayes’s Rule

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

What you knew before + What you learned = What you know now

Summary

1) Dichotomizing a multi-level 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

(NOTE: Do not calculate an LR(+) or LR(-) for a multilevel test.)

4) Pre-Test Odds x LR(result) = Post-Test Odds

Calculating the c StatisticIn the “walking man” approach to tracing out the ROC curve, the actual values of the test are not important for the shape of the ROC curve or the area under it--only the ranking of the values.

The c statistic for the area under an ROC curve is calculated using the same information as the Wilcoxon Rank Sum statistic (or Mann-Whitney U, which is equivalent) and gives identical P values.

Non-parametric equivalent of the t test statistic comparing two means.

Septic Arthritis No Septic Arthritis

128

92

71

64

48

37 37

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0 1 2 3 4 5 6 7 8 9 10

D-

D+

Boxes under Curve = 43.5

Total Boxes = 50

Area Under Curve = 43.5/50 = 0.87

BACTEREMIA NO BACTEREMIA

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6.5 6.5

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10.5

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15

S = 21.5

Replace Test Results with Ranks

S = 21.5Smin = d(d+1)/2 = 5(6)/2 = 15Smax = dn + Smin = 5(10) + 15 = 65

C = (Smax – S) / (Smax – Smin)* = (65 – 21.5) / (65 – 15) = 43.5/50 = 0.87* Smax – Smin = dn

Calculating the C Statistic

assessing information from multilevel (ordinal) tests roc curves and likelihood ratios for results...

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