Sampling refers to - ‘the process of selecting a portion or part to represent the whole’.
SAMPLING :
PROBABILITY PROFILE
Sampling refers to - ‘the process of selecting a portion or part to represent the whole’.
In usual
practice, a ‘sterility test’ attempts
to infer and ascertain the state (sterile
or non-sterile) of a particular batch
; and, therefore, it designates predominantly a ‘statistical operation’.
Let us
consider that ‘p’ duly refers to the
proportion of infected containers in
a batch, and ‘q’ the proportion of
corresponding non-infected containers. Then, we may have :
p + q = 1
or q = 1
– p
Further,
we may assume that a specific ‘sample’
comprising of two items is duly
withdrawn from a relatively large batch containing 10% infected containers. Thus, the probability of a single item taken at random contracting infection is usually given by the following
expression :
p = 0.1 [i.e., 10% = 0.1]
whereas,
the probability of such an item being
non-infected is invariably represented by the following expression :
q = 1 – p = 1 – 0.1 = 0.9
Probability Status—The probability status of the said two items may be obtained virtually in three
different forms, such as :
(a) When both items get infected : p2 = 0.01
(b) When both items being non-infected : q2 = (1 – p)2 = (0.9)2 = 0.81, and
(c) When one item gets infected and the other one non-infected : 1 – (p2 + q2)
or = 1 –
(0.01 + 0.81) = 1 – (0.82)
or = 0.18
i.e., = 2pq
Assumption : In a particular ‘sterility test’ having a ‘sample’ size of ‘n’ containers, the ensuing
probability p of duly accomplishing ‘n’
consecutive ‘steriles’ is
represented by the following expres-sion :
qn = (1 – p)n
Consequently,
the ensuing values for various levels of ‘p’*
having essentially a constant sample
size are as provided in the
following. Table 8 : 4A, that evidently illustrates that the ‘sterility test’ fails to detect rather low levels of contamination contracted/present in the ‘sample’.
Likewise,
in a situation whereby different sample
sizes were actually used**, it may be em-phatically demonstrated that as
the sample size enhances, the probability component of the batch being passed as sterile also gets decreased accordingly.
Table 8.4 : Sampling in Sterility Testing
1. A : First Sterility Test : Calculated
from P = (1 – p)20 = q20
2. B : First Re-Test : Calculated
from P = (1 – p)20 [2 – (1
– p)20]
[Adapted
From : Hugo and Russell : Pharmaceutical
Microbiology, PG Publishing Pvt. Ltd., New Delhi, 3rd edn., 1984]
In actual
practice, however, the additional tests, recommended by BP (1980), enhances substan-tially the very chances of passing a specific batch essentially comprising of a
proportion or part of the infected items
(see Table : 8.4B). Nevertheless, it may be safely deduced by making use of
the following mathematical formula :
(1 – p)n
[2 – (1 – p)n]
that
provides adequate chance in the ‘First
Re-Test’ of passing a batch comprising of a proportion or part ‘p’ of the infected containers.
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