Screening Test for Occult Cancer

100 patients with occult cancer: 95 have "x" in their blood

100 patients without occult cancer: 95 do not have "x" in their blood

5 out of every 1000 randomly selected individuals will have occult cancer

SENSITIVITY

SPECIFICITY

PREVALENCE

2 X 2 Table

Occult CancerPresent

Occult CancerAbsent

"x" present "x" absent

100,000

99,500

50025475

4,975 94,525

5,450 94,550

If a patient has “x” in his blood, chance of occult canceris 475 / 5450 = 8.7%

Standard Terminology

DiseasePresent

DiseaseAbsent

Test positive

Testnegative

EntirePopulation

FP + TN

TP + FNTrue

Positives(TP’s)

FalsePositives

(FP’s)

TrueNegatives

(TN’s)

FalseNegatives

(FN’s)

TP + FP FN + TN

Definitions

SENSITIVITY = TPTP + FN

= P(T+|D+)

SPECIFICITY = TNFP + TN

= P(T -|D-)

PV + = PREDICTIVE VALUE = TPTP + FP

= P(D+|T+)

Positive Predictive Value Formula

(Sens)(Prev) + (1-Spec)(1-Prev)PV+ =

(Sens)(Prev)

Detection of Prostatic Cancer by Solid-Phase Radioimmunoassay of Serum Prostatic Acid Phosphatase

Editorial

“The clear implication of the accompanying report is that mass screening on the basis of a blood test alone can reverse this gloomy experience [of fatal delays in diagnosis of prostate cancer].”

New England Journal of MedicineDecember 22, 1977

Medical Journal

Advertisement to Physicians

Medical Journal

Advertisement to Physicians

Posed by a Professional Model

Medical Journal

Advertisement to Physicians

Advertisement

“(You should be aware of) a new blood test called the Male-P.A.P. test ... a new, more sensitive method that your physician can use to detect chemical signals of a cancerous growth in the prostate. ... And even though all lab tests must be ordered by a physician, we believe that you should know the facts.”

New York Times, January 21, 1979

Sensitivity

Patients with prostate 113 79 70%cancer

Stage I 24 8 33%

Stage II 33 26 79%

Stage III 31 22 71%

Stage IV 25 23 92%

# ofpatients

# of positivetests

sensitivity

Specificity

Patients without prostate 217 13 94%cancer

Normal controls 50 0

BPH 36 2

After total prosta- 28 1tectomy

Other cancers 83 9

Misc. GI disorders 20 1

# ofpatients

# of positivetests

specificity

Use As Screening Test

• Without rectal examination:–Sensitivity = 70% Specificity = 94%

–Prevalence 33/100,000

–PV+ = 0.41% (i.e., 1 in 244 subjects)

• With rectal examination:–Sensitivity = 33% Specificity = 94%

–Prevalence 33/100,000

–PV+ = 0.19% (i.e., 1 in 526 subjects)

When is the test useful for screening?

Suppose patient has a nodule on rectal examination:

–Sensitivity (Stage 2 of disease) = 79%

–Specificity = 94%

–Prevalence = 50% !!

PV+ = 93% (chance of cancer if acid phosphatase is positive)

PV- = 82% (chance that there is no cancer if acid phosphatase is negative)

Predictive Values in Patients with a Nodule

PAP+

PAP-

50%

93%

18%

Combining Tests For Screening:

If a prostate biopsy is now performed, it needs to be considered as another test.

Specificity = 100%

Sensitivity depends on talent and statistics of surgeon doing the procedure

Prevalence is 50% if acid phosphatase has not been measured, but is 93% if acid phosphatase is positive and 18% if acid phosphatase is negative.

Sequential Testing

PAP+

PAP-

50%

93%

18%

BX+

BX+

100%

100%

Chance of Cancer after Negative Biopsy

Sensitivity of Biopsy

Acid Phosphatase positive (93% chance before biopsy)

Acid Phosphatase negative (18% chance before biopsy)

50%

87%

10%

70%

80%

6%

90%

56%

2%

Sequential Testing

PAP+

PAP-

50%

93%

18%

BX+

BX+

BX-

BX-

100%

100%

56%

< 2%

BAYES’ THEOREM

OR

P(D+|T+) = P(T+|D+) P(D+)P(T+|D+) P(D+) + [1-P(T -|D-)] [1- P(D+)]

PV + = (Sens )(Prev )(Sens )(Prev ) + (1-Spec )(1-Prev )

Typical Assumptions with theUse of Bayes' Theorem

• Completeness (for example, all men either have or do not have prostate cancer; there are no other possibilities)

• Mutual exclusivity (for example, if a man has prostate cancer, he cannot simultaneously NOT have prostate cancer)

• Conditional independence (for example, acid phosphatase and a biopsy result ARE conditionally independent tests; rectal exams and acid phosphatase may NOT be conditionally independent)

References

Foti et al. “Detection of prostate cancer by solid-phase radioimmunoassay of serum prostatic acid phosphatase.” New England Journal of Medicine 297:1357-1361 (1977)

Watson, R.A. and Tang, D.B. “The predictive value of prostatic acid phosphatase as a screening test for prostatic cancer.” New England Journal of Medicine 303:497-499 (1980)

Berwick, D.M., Fineberg, H.V., and Weinstein, M.C. “When doctors meet numbers.” American Journal of Medicine 71:991 (1981)

What is a “Positive Test”?

• All the analysis has assumed that it is clear whether a test is positive or negative

• In reality, many tests involve continuous values so that one result may be “more positive” than another

• How should one define the cut-off at which a test is judged to be abnormal?

Continuously Valued Variables

Normal

Diseased

Result

“Normal” cutoff False

Positives

False Negatives

True Negatives

True Positives

Continuously Valued Variables

Normal

Diseased

Result

“Normal” cutoff

• Fewer false positives (more “conservative”)

• More false negatives• Higher specificity• Lower sensitivity

Continuously Valued Variables

Normal

Diseased

Result

“Normal” cutoff

• Fewer false negatives (more “aggressive”)

• More false positives• Higher sensitivity• Lower specificity

Receiver Operating Characteristic(ROC) Curves

Tru

e P

osi

tive

Rat

e =

Sen

siti

vity

False Positive Rate = 1 - Specificity

Test ATest B

ROC curve shifts to left, indicating the new test (B) is “better” – or a better indicator to disease presence (more discriminatory).

The Importance of the Gold Standard

• Evaluating the value of a new test requires having some other method for determining “truth”

• Methods for determining truth are called gold standards

• Gold standards are often expensive, time consuming, uncomfortable, or risky– Biopsies

– Major invasive procedures or surgery

– Autopsies

– Integrated opinions of “super experts”

• We often seek simple, inexpensive, rapid, and safe tests that can perform almost as well as the gold standard