New Statistical Algorithms for the Analysis of Mass - FU Berlin, FB MI ...

70 CHAPTER 4. (BIO-)MEDICAL APPLICATIONS
be subjected to calculations known as linear regression. This works as a foundation
for correlation analysis and to various other tests of statistical significance.
Despite their mathematical preciseness and undoubtedly correctness,
inferences drawn from these results may be incorrect, thoroughly misleading,
or failing to call attention to the basic insufficiency of the experiment. This
usually has two main reasons:
� assumptions underlying the statistical procedure are not fulfilled
� problems connected to the data were of a completely different type from
those for which the particular statistical methods provide useful information.
Indeed, most data sets provide some useful information, but this is no guarantee
that the information actually desired has been obtained.
In most cases the goal of statistical analyses is to draw inferences from the
particular to the general and often people are not familiar with the problems
of inductive inference, which is closely tied to this. R. A. Fisher has pointed
to a basic and most important difference between the results of induction and
deduction (Fisher, 1959) which we recall briefly and illustrate by two small
examples. By using deduction, conclusion based on correct partial information
are always correct, despite the incompleteness of the premises. Let us use a
well-known theorem from geometry as an example: the sum of the angles
of a plane triangle always equals to 180 degrees. This does not necessitate
information as to whether it is isosceles or not. If any information of this
type is subsequently added, it cannot possibly alter the fact expressed by the
theorem.
As an counterexample, inferences drawn by induction from incomplete
information may be entirely wrong, even when the information given is unquestionably
correct. Let us use a simple example from physics. Suppose we
were given the data of Table 4.2.3 on the pressure and volume of a fixed mass
of gas. Analyzing the data one might infer (by induction) that the pressure
Molar volume (liters) Pressure (atmospheres)
0.182 54.5
0.201 60.0
0.216 64.5
0.232 68.5
0.243 72.5
Table 4.2.1: Volume-Pressure Relation for a gas, an apparently proportional relationship
of a gas is proportional to its volume (of course a completely erroneous statement).
What went wrong ? The answer is simply that another important
item of information was omitted, namely that each pair of measurements was
obtained at a different temperature, as indicated in Table 4.2.3. Of course,
this example is artificially constructed and extreme but it emphasizes the basic
problem in inductive reasoning: the data not only has to be correct but also
complete to enable correct inference. In this simple example the missing piece
of information was easily identifiable because we have a good understanding
of the physical background. What researchers need to be aware of is that