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the test (with respect to the confidence
E
). Equivalently, instead of considering the
p-value, the null hypothesis can be rejected when
V
≥
V
0
.
7.6.3
Sample Significance Indexes
Analyzing a population can be aimed at selecting individuals of the population hav-
ing some property of interest. In the following we express some sampling situations
in terms of clinical tests [223]. A population
D
is partitioned in two sub-populations
D
+
and
D
−
of people having a disease, and people without the disease, respectively.
The same population is selected in other two sub-populations
T
+
and
T
−
of people
resulting positive to a clinical test
T
and people resulting negative to the test, respec-
tively. Some indexes can be defined for the test
T
, which characterize its capacity in
discriminating correctly, a priori and a posteriori, with respect to disease
D
.
Table 7.2 defines four important indexes of a clinical test
T
for a disease
D
.
Ta b l e 7 . 2
Indexes of adequacy for clinical tests
Sensitivity
|
T
+
∩
D
+
|/|
T
+
|
Specificity
|
T
−
∩
D
−
|/|
T
−
|
Positive predictability
|
T
+
∩
D
+
|/|
D
+
|
Negative predictability
|
T
−
∩
D
−
|/|
D
−
|
We can interpret a test as a measure of reliability of statistical inference with
respect to a property of having a pathology. Table 7.3 explains the meaning of sta-
tistical inferences.
Ta b l e 7 . 3
Statistical inferential value of test indexes
High value
Reliability
Sensitivity
Positivity
⇒
Disease
Specificity
Negativity
⇒
Health
Positive predictability
Disease
⇒
Positivity
Negative predictability
Health
⇒
Negativity
Given a population
X
, the population of the samples of
X
is a
second level
pop-
ulation which is very important in the statistical analysis of
X
. In fact, it provides
some statistical distributions, called
sample distributions
of the samples taken from
X
. The power of statistics is based on the evidence that sample distributions follow
precise mathematical laws. These are the basis for important procedures inferring
properties of populations from their samples.