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2
test, Gosset discovered the Student's test (Student is Gosset's pseudonym), and
Fisher the
F
test. Fisher, who was a great mathematical statistician [222], was also
a great geneticist, he gave a statistical interpretation to natural selection, by means
of the
Natural selection theorem
showing that, under suitable hypotheses, the force
driving evolution pushes a biological population toward the maximum of possible
fitness (the maximum possible demographic rate reachable in a given context).
The use of correlation indexes and of tests of significance, according to specific
methods, provide effective tools in
statistical regression
techniques, which, in very
general terms, are aimed at finding the best analytical forms which underly time
series of observed data, an important field of statistical application in the same line
of Gauss' methods of minimum least squares (by means of which Gauss solved an
astronomical puzzle by determining the orbit of the asteroid Ceres).
It is reported [228] that Henry Poincare proved the dishonesty of a baker, by
discovering that the weights of the daily loaves of bread fluctuated without follow-
ing the gaussian distribution (clearly, a guilty defect with respect to the declared
weight). This is a simple example of using a law of chance for finding the exis-
tence of a non-accidental motivation providing an observed effect. In other words,
Poincare's experiment answered the question:
Is there some cause which is not due
only to chance, affecting the weights of loaves of bread?
.
Hypothesis testing
is largely the product of Ronald Fisher, Jerzy Neyman, Karl
Pearson, and Egon Pearson (Karl's son), developed in the early 20th century. Fisher
coined the term
tests of significance
for denoting the essence of hypothesis testing.
In general, such tests provide tools for evaluating data coming from sampling popu-
lations or from analytical models for comparing them, in order to discover whether
data of distinct sources significantly differ with respect to casual differences. The
distinction between casual and causal differences is the basis for: i) recognizing
the existence of cause providing some observed effects; ii) establishing that some
observed effects can be associated to a given cause acting on individuals of a pop-
ulation, or iii) evaluating if a model explains some observed data, apart from some
approximation errors of casual nature.
For example, in a population of people affected by a disease, does the treatment
with a drug have a positive effect against some pathology? Let us assume that we
want to observe the possible effects of a specific drug on the blood pressure in
persons affected by blood hypertension. Comparing two samples, one of which was
exposed to the drug treatment, we would deduce whether the effects we observe
could be associated to the treatment or are simply casual fluctuations.
The basic approach of a statistical test consists in the comparison of the casual
hypothesis
H
0
, claiming that the observed effects are casual, against the opposite
hypothesis
H
1
, according to which the observed effects are due to a phenomenon
acting on the population. In this way then we can conclude, on a statistical basis, in
favor of a proved effect caused by the phenomenon in question. If the test concerns
only a parameter of a population, and no comparison is performed with a population
of reference, we can only conclude that a cause is acting on the parameter, apart from
the influence of chance (the nature of this cause has to be deduced by using other
kinds of information, as in the case of Poincare's experiment).
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