Clearly, the larger the ‘signal-to-noise ratio’, the
easier it is to establish a causal relationship. Thus,
in a clinical drug trial, it is equally important to
measure both noise and signal. How are these
measured? The nature of variability is that the
effect of interest is random. When we measure
the blood pressure of an individual subject repeat-
edly, the measurements will be dispersed around
some central value in a random fashion; some will
be larger and some smaller. The effect, however, is
systematic. If, for example, we measure the blood
pressure of an individual repeatedly before and just
after administering an antihypertensive drug, the
pretreatment and posttreatment measurements will
be dispersed around different central values, the
posttreatment value lower than the pretreatment
value.The magnitude of the effect(signal)isusually
calculated as the mean of the individual effects in a
population of subjects. The variability (noise) is
usually calculated as the standard deviation.
Example: Suppose 10 hypertensive subjects are
treated with a novel antihypertensive drug. The
subjects’ blood pressure is measured at 8:00 a.m.,
just prior to the administration of the drug, and then
again 1 h later. Data are as shown in Table 25.2.
The first and the second rows of the table give the
diastolic blood pressure of subjects before and after
treatment, respectively. The third row gives the
change (D) in diastolic pressure (value in row 1
minus the value in row 2). The mean, given in the
last column, is 12.8 mmHg. On the face of it, 12.8
looks like an impressive effect. However, as we
have discussed earlier, we cannot assess its signifi-
cance without considering the inherent variability,
the noise. Indeed, the values ofDrange from4to
33,a substantial range. To assessD’s variability, we
calculated the deviations of the values ofDabout
their mean 12.8. These values are given in the next
row. Naturally, as the mean is a value somewhere in
the middle, some deviations are positive and others
are negative. One property of the mean is that the
sum of these deviations is always zero. Thus, the
average (mean) of the deviations around the mean
is always zero, and therefore is not useful as a
measure of the variability. Instead, we calculate
the mean of the squares of the deviations about
the mean as a measure of variability. This measure
is called thevariance. Thevariance is an average of
nonnegative numbers and it is, therefore, always a
nonnegative number. It is equal to 0 if and onlyif all
the deviates are equal to zero, meaning that all the
measurements are the same and thus equal to their
mean, that is, there is no variability at all. The
standard deviation(S.D.), the most commonly
used measure of variability, is the square root of
thevariance. In our case S:D:¼p
110 : 16 ¼ 10 :50.
The advantage of using the standard deviation over
the variance is that it is measured with the same
units as the mean. The mean does not represent the
response to treatment of any particular individual.
It does, though, give us an idea of the magnitude of
the response to treatment produced by the drug.
Can we conclude, then, that the drug is efficacious?
If the drug is ineffective, then there should be no
systematic change in blood pressure measurements
taken 1 h after treatment as compared to pretreat-
ment measurements and thus the mean change
should equal approximately to zero. The observed
mean change of 12.8 mmHg is then due entirely to
chance. Statistical theory shows that the likelihood
that a sample of 10 numbers drawn at random from
a set of numbers with mean zero (in our example,Table 25.2 Diastolic blood pressure before and after treatment (mmHg) (hypothetical data)
Subject Mean
1 23456 78910Before treatment 102 78 95 86 109 107 100 86 96 92 95.1
After treatment 75 82 80 81 76 93 92 80 90 74 82.3
Difference (D)27 4 15 5 33 14 8 6 6 18 12.8
(D– meanD) 14.2 16.8 2.2 7.8 20.2 1.2 4.8 6.8 6.8 5.2 0
(D– meanD)^2 201.64 282.24 4.84 60.84 408.04 1.44 23.04 46.24 46.24 27.04 110.16
25.5 VARIABILITY – THE SOURCE OF UNCERTAINTY 317