The Essentials of Biostatistics for Physicians, Nurses, and Clinicians

6.11 Special Tests in Clinical Research 83

The estimate of the unconditional probability of a false positive is
estimated to be b / n based on this sample. The estimate of the uncondi-
tional false negative probability is c / n. Perhaps of greater interest are
the conditional probabilities of error. These rates are estimates as c / m
for false negative probability, given the patient has the disease (by
the gold standard), and the conditional false positive probability
b /( b + d ) = b /( n − m ).
Now we shall defi ne the specialized terms called sensitivity and
specifi city. Sensitivity is defi ned as the probability that a screening test
declares the patient diseased given that the patient has the disease.
Mathematically, the estimate of sensitivity for the above table is 1 − c /
( a + c ) = a /( a + c ) = 1 − c / m. So sensitivity is 1 - probability of a false
positive.
Specifi city is the probability that the screening test declares
the patient well given that the patient the patient does not have the
disease (based on the gold standard). Mathematically, the specifi city
estimate is 1 − b /( b + d ) = d /( b + d ) = 1 − b /( n − m ). So specifi city is
1 - probability of a false negative.
Ideally, a test should have high sensitivity and specifi city. However,
measurement error and imperfect discrimination rules prevent perfec-
tion (i.e., specifi city = 1 and sensitivity = 1). But just as there is a
tradeoff of type I and type II error when n is fi xed, but the threshold is
allowed to change sensitivity, and specifi city can be changed to increase
one at the cost of the other. So it is usually important to decide which
type error is the most serious for the application and make the tradeoff
accordingly. Friis and Sellers ( 1999 ) provide more detail regarding
screening tests.
The curve that shows the tradeoff between specifi city and sensitiv-
ity is called the receiver operating characteristic (ROC) curve. Useful
references on diagnostic testing that include discussion of ROC curves
are Pepe ( 2004 ), Zhou et al. ( 2002 ), Krzanowski and Hand ( 2009 ),
G ö nen ( 2007 ) and Broemeling ( 2007 ).

6.11 SPECIAL TESTS IN CLINICAL RESEARCH

Superiority testing is the standard testing approach in clinical trials and

involves testing a null hypothesis that the treatment is no different from

the control or worse than the control versus a one - sided alternative that

the treatment is better or superior to the control. This is simply a