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From figure 2.1-4B it is also obvious that a standart deviation calculated from three
values (df=2) (unfortunately not an exception in validation literature) is rather
meaningless as ơ can be expected up to 4.4 fold of the calculated standard
deviation! However, if several sets of data can be combined (pooled), the overall
degress of freedom and thus the reliability are increased. In succh a case, only the
overall standard deviation should be reported. A prerequisite for such pooling of data
is that all data sets must have similiar ơ (if means are looked at, they must also
have the same true mean; for verification, see the discussion on precision level in
section 2.1.2). interestingly, a confidence interval is mentioned in the ICH guideline
(althought it is not clearly stated whether with respect to the standard deviation on
the mean) 1b
However, the author is not aware of any publiation on pharmaceutical validation
which reports it. Following the standard approach with six or more determinations for
a standard deviation, the confidence interval will not provide much additional
information, but the benefit could be to cause people to hesisate before reporting
standard deviations from three determinations only.
Significance tests
Confidence intervals are also the basis of statistical tests. In the case of significance
tests, the test hypothesis (H0) assumes, for example, no difference (zero) between
two mean results. This is fulfilled (or strictly, the hypothesis can not be rejected),
when the two confidence intervals overlap. However, as the confidence intervals
become tighter with increasing number of determinations, (theoretically) any
difference- however small- can be shown to be significant. For example, assuming a
standard deviation of 0,5 . a difference of 0.5 is significant with nine determinstions,
but even a difference of 0.1 will become significant when there are 200 values. Of
course, this is (usually) not of (practical) interest ( see accuracy, section 2.3.1)
Rebust parameters
The above-described parameters are based on the assumption of normal
distribution. If this prerequisite is not fulfilled, or disturbed, for example by a singledeviating result (outlier), the application of rebust parameters that are not based on
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specific assumptions. The analogue to the arithmetic mean is the median. The
middle value in an ordered sequence of results. A comparison between mean and
median may provide information about a possible disturbance in the data. However,
it is often a very complex matter to estimate confidence intervals or variabilities for
rebust parameters.
Another alternative to estimate description parameters of any distribution is the
(thousand fold) repeated calculation from an experimental set of data (re-sampling)
to achieve a simulated distribution, the so-called bootstrap, or the estimination of
variability from the noise of a single measurement using a probabbility theory named
the ‘function of mutual information’. However, these techniques are beyond the
scope of this book, and the reader is referred to spesialised literature.
Precision levels
Regarding an analytical procedure, each of the steps will contribute to the overall
variability. Therefore, the overall uncertainty can be estimated by summing up each
of contributing variabilities, the so-called bottom-up approach. However, this
approach is quite complex because each and every step has not only to be taken
into account, but also its variability must be known or determined.
Alternatively, the othee approach (top-down) usually applied in pharmaceutical
analysis combines groups of contributions obtained experimentally the precision
levels. Such a holistic approach is easier to apply, bacause each of the individual
contributinng.
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