1. Definitions:
Null Hypothesis: Observed data are the result of chance
Type 1 error: False rejection of null hypothesis
Type 2 error: False acceptance of null hypothesis
TP: true positive = diseased patients who test positive (1-a)
TN: true negative = well patients who test negative (1-þ)
FN: false negative = diseased patients who test negative (a)
FP: false positive = well patients who test negative (þ)
POP: all patients tested

2. Fractions, rates or ratios (F or R):
TPF or TPR: true (+) = SENS FNF or FNR: false (-) = 1-SENS
TNF or TNR: true (-) = SPEC FPF or FPR: false (+) = 1-SPEC
PREV: = [ D+ / POP ] people with disease in POP
ACCUR: accuracy
PPV: positive predictive value
NPV: negative predictive value

3. Bayes' Rules for post-test probabilities: Predicts a post-test probability. Requires knowledge of pretest probability and test accuracy (SENS & SPEC).

where:
p(D+|T+) = post test probability that disease is present given a positive test
p(D+|T-) = post test probability that disease is present given a negative test
p(D-|T+) = post test probability that disease is not present given a positive test
p(D-|T-) = post test probability that disease is not present given a negative test
p(T+|D+) = SENS or TPF p(T-|D+) = 1-SENS or FNF
p(T+|D-) = 1-SPEC or FPF
p(T-|D-) = SPEC or TNF
p(D+) = pre-test probability that disease is present (PreTP)
p(D-) = pre-test probability that disease is not present (1-PreTP)

4. Linking serial tests using Bayes' Rules:

a. Step 1: The pretest probability (prevalence) of disease and the first test sensitivity and specificity are entered into the appropriate equation. This yields the post-test probability for the first test.
b. Step 2: The post-test probability from Step 1 and the second test sensitivity and specificity are entered into the appropriate equation. This yields the post-test probability for the second test.

II. Odds ratios, Likelihood and Relative Risk analyses: Although Bayes Theorem provides a reliable method for modifying disease probabilities based on diagnostic test results, the method seems unintuitive by many and therefore difficult to remember. Use of odds analyses on the other hand seem far more familiar.

1. Odds: odds in favor of the presence of disease (D+)
where p(D+) = odds of disease / odds of not having the disease.

e.g. when p(D+) = (2/3) / (1/3) the odds in favor of having disease (D+) have the ratio of 2:1

2. Posterior odds: When B represents a test result, the posterior odds favoring the presence of disease (POFD) are

where p([D+][+T]) = p( D+ and T+) recall that

and

posterior odds in favor of disease being present can be calculated from the ratio

(EQ 14)

which yields the fundamental relationship where the likelihood ratio is the ratio of the conditional probabilities of p(D+|T+) / p(D-|T+).

Definitions:
Types of studies: Cohort or prospective
Case control retrospective
LRP: Likelihood ratio for a positive test
LRN: Likelihood ratio for a negative test
RRP: Relative risk for disease given the risk
RRN: Relative risk for no disease given the risk

Estimated odds or likelihood for a (+) test (LRP)

Estimated odds or likelihood for a (-) test (LRN)

Odds ratio for LRP/LRN = (TPF/FNF) / (FPF/TNF).

Linking pretest probabilities with likelihood ratios:

The estimate of probability of having disease varies with the pattern of the test result. Certain scan patterns are more specific than others. The particular formula taken from ACR 20 by Dr. McNeill is given below.

1. Marginal probabilities (probability of a single event occurring (+T)):
Population a+b+c+d 700 + 200 + 300 + 800 = 2000
Positive test: a+b / a+b+c+d p(+D) = 700 + 200 / 2000 = .45
Negative test: c+d / a+b+c+d p(-D) = 300 + 800 / 2000 = .55
Diseased: a+c / a+b+c+d p(+T) = 700 + 300 / 2000 = .50
Not diseased: b+d / a+b+c+d p(-T) = 200 + 800 / 2000 = .50

2. Conditional probabilities:
TPF (SENS) a / a+c p(T+|D+) = 700 / 1000 = .70
FPF (1-SPEC) b / b+d p(T+|D-) = 200 / 1000 = .20
TNF (SPEC) d / b+d p(T-|D-) = 800 / 1000 = .80
FNF (1-SENS) c / a+c p(T-|D+) = 300 / 1000 = .30

3. Predictive values:
Accuracy: a+d / a+b+c+d 700 + 800 / 2000 = .75
Positive test: a / a+b p(D+|T+) = 700 / 900 = .78
Negative test: d / c+d p(D+|T-) = 800 / 1100 = .73

4. Joint probabilities (probability of 2 events occurring simultaneously)
this involves building a 2 x 2 probability table

where: p(A|B) = p(A and B) /p(B)
p(D+) = .60 (information external to the table)

p(T+ and D+): a / a+c p(T+|D+)p(D+) 700/1000 * .6 = .42
p(T- and D-): d / b+d p(T-|D-)p(D-) 800/1000 * .4 = .32
p(T- and D+): c / a+c p(T-|D+)p(D+) 300/1000 * .6 = .18
p(T+ and D-): b / b+d p(T+|D-)p(D-) 200/1000 * .4 = .08

alternately p(T- and D+) and p(T+ and D-) can be calculated using
the Addition rule which states:

p(D+) = p(T+ and D+) + p(T- and D+)
p(D-) = p(T+ and D-) + p(T- and D-)

p(T- and D+) = p(D+) - p(T+ and D+) = .60 - .42 = .18
p(T+ and D-) = p(D-) - p(T- and T-) = .40 - .32 = .08

p(D+|T+) = p(D+ and T+) / p(T+) = .42 / .50 = 84%
p(D-|T+) = p(D- and T+) / p(T+) = .08 / .50 = 16%
p(D-|T-) = p(D- and T-) / p(T-) = .32 / .50 = 64%
p(D+|T-) = p(D+ and T-) / p(T-) = .08 / .50 = 36%

A positive test increases the P(D+) from 60% to 84%.
A negative test increases the p(D-) from 40% to 64%.

5. Bayes Theorem:

6. Linking serial tests using Bayes' Rules: Approximate pretest probability of coronary artery disease as a function of age, sex and type of chest pain
(JACC 1:444-455, 1983 and NEJM 300:1350-58, 1979).

given: p(D+) = .40 for a 50 year old _ with atypical chest pain
TM SENS = .62 SPEC = .83
TL SENS = .84 SPEC = .94

1. Serial analysis given abnormal test results:
Step 1: Calculate the post-test probability for p(D+|T+) using p(D+) = (.40).

Step 2: Calculate the post-test probability for p(D+|T+) using p(D+) =(.71).

2. Serial analysis given normal test results:
Step 1: Calculate the post-test probability for p(D-|T-) using p(D+) = (.40).

Step 2: Calculate the post-test probability for p(D-|T-) using p(D+) =(1-.77).

6. Odds ratio, likelihood ratio and relative risk: Factor VIII ratio test results for 34 normals and 34 obligatory hemophilia carriers.

given: test subject has mother who is a carrier: p(D+) = .50
p(T+|D+) = 32 / 34 = .94
p(T-|D-) = 28 / 34 = .82

Posterior probability:
p(D+|T+) = (.94)(.50) / (.94)(.50) + (.18)(.50) = .84

Posterior Odds:
LRP = SENS / 1-SPEC
LRN = 1-SENS / SPEC

LRP = (32/34) / (6/34) = 5.3
LRN = (2/34) / (28/34) = .07

Posterior Odds: Determines odds that test subject is a carrier. The posterior odds favoring D+ = (prior odds) * (likelihood ratio)

T+ POFD = (.50/.50) * 5.30 = 5.30
T- POFD = (.50/.50) * 0.07 = 0.07

Conversion of odds to probability p(D+) = odds / (1+odds)

p(D+|T+) = 5.30 / (1+5.30) = (5.03/6.30) = .84
p(D+|T-) = 0.07 / (1+0.07) = (0.07/1.07) = .065

Posterior Odds ratio: Used in case control or retrospective studies.

POFD = LRP/LRN = (30/33)/(970/1967) / (3/33)/(997/1967)
= 1.84/.179 = 3.03

Relative risk: Used in cohort or prospective studies. Analysis by row permitted here since outcome is not predetermined.

RR = RRP/RRN = (30/1000) / (3/1000) = (0.03/0.003) = 10