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Summary express
Conclusions of the test
General simplified information
about the average test
Practical example with comments
In statistics, the concept of hypothesis
test is far from an easy concept and requires a good mastery of the concepts of
statistical calculation. Yet it is a process that many mycologists believe they
understand and know use intuitively. They are duped by the misleading term
"confidence coefficient" which is a phrase that could not be more
ill-chosen as discussed below!.
Problem : you lay out a series of
measurements which you wish to compare with a standard data or other
measurements
Comparison of a sample to a value
ex:
comparing the results of a lot of measurements with the average of a possible
specie
to
enter :
- the number of measurements, their average and their standard deviation
-
enter the average of the supposed standard specie
Comparison
of two samples
ex:
comparing measurements of two different samples
- enter for each sample : the
number of measurements, their averages and their standard deviation
= > the lower window immediately posts the numerical results of
the test and the associated comments
2 possibilities:
The assumption H0 is rejected. The test is significant :
The differences observed are not due to the only chance and are "significantly different".The assumption H0 is not rejected. The test is not significant :
The differences observed are can be due randomly of sampling (but it should not be deduced in so far that the averages are the same ones!)
General
information simplified about the average test
For more details, one will
be able to refer usefully to the concepts of "Statistics for
Mycologue" which were posted on the Mycomètre forum.
The traditional problem of the initial mycologist is as follows:
"I have a sample of
N measurements of length of spores and I calculated the average and the
standard deviation of this sample (manually, calculator, etc...).
Should I say that these
measurements are the same ones as those obtained by the author? "
Don't let us dreaming :
except very particular case, the statistical tests never makes it possible to
affirm, as it is often heard, "than there is 95% of ' chances' so that
dimensions similar" or "lie between such and such terminal".
Statistical calculation makes it possible to calculate
only one risk.
There are two kinds of
risks:
a/ the risk to dismiss an assumption whereas it is true (risk "alpha")
b/ the risk to accept the contrary assumption, whereas first is true (risk "beta")
Contrary to a spread idea,
these two risks are absolutely not complementary (we will not develop this
delicate concept here).
It is important to know at
the beginning which risk one chooses. The risk which interests the
mycologist is the Beta risk (i.e. not to want a priori that the given
species is"the good one", without going further in the
investigations), which is precisely most difficult to calculate.
The Beta risk depends on
the alpha risk chosen at the beginning. To calculate the Beta risk, it is also
necessary to know as a preliminary the "rule of the game", i.e. the
law of distribution of the population. Unfortunately, one generally does
not know the law of distribution of the sample (to find it is even more
delicate than to seek the answer to the question put above).
On the other hand one knows
the distribution of the Average which is roughly, when the sample is rather
large, a law of Laplace-Gauss known as "Normal Law".
There is not, for as much,
the right to compare measurements with averages. It acts of two populations absolutely
different (the proof being that they do not have the same distribution's
law), even if the measuring units are identical, which is even more fallacious
(that is invited "to mix the towels and the cloths").
One thus can, fault of
being able to compare directly and easily the samples, to compare the averages
known between them. Also let us specify that the sampling of the
measured spores must be done "honestly", without any sorting of any
kind: one must systematically measure every spore which is present,
except those which are obviously mutilated.
Calculation of the beta
risk
We saw that the calculation
of the beta risk is not elementary for a non-statistician. Mycostats makes
it possible to obtain it without any effort, under certain assumptions (of
Normality for the moment). It is as to note as we do not know any other
software which provides this result.
Assumptions:
In the case of the average
test, it is necessary to know, in theory, the two averages to be compared and
their standard deviations. But, very often, one knows model population
only his average or even as an indicative interval.
The cases of figure which
can arise at the mycologist are:
a. one knows only the average of the standard
species
It
is the default of Mycostats ("Echant / Type")
b. only one standard
interval is known: one can try to take the risk (still one!) to compute for
average with the center of the interval and to proceed as at the a). If the
distribution of the standard species is symmetrical, the approximation can be
sufficient.
One is brought back to the
preceding case
particular
case : one knows the estimated standard deviation of the average : to
notch the corresponding option in the test "Echant / Type"
C. If the author of the harvest of reference well specified the average and
the thought standard deviation of the sample
One
will choose, in Mycostats, the "2 samples" test..
Practises examples with Mycostats[1]
Nb: Initially, one will keep the default options.
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Open
Mycomètre (version demo or pro) Press
the Mycostats button Choose
in the bar of small tests/average An
example of data is posted automatically. |
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(one will
be unaware of the data "column" who allows to enter automatically the
data resulting from measurement with Mycomètre).
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In this example, the sample of 20 measurements has for average me = 11,1 and a standard deviation s = 1,25. The average of the type is mt= 11,5 The estimated s the Type is not known here, Mycostats admits that it is roughly equal to that of the sample. |
It is known that me< mt : one thus will carry out a unilateral test (a
unilateral test is preferable for the tests which concern us. Mycostats holds
account of it automatically).
The assumptions to be
tested are:
H0 : "the samples have even average: me = mt "
H1 : "me is higher than mt"
The risk chosen by defect
for alpha is 5 %
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For A= 5%, the H0 assumption is not rejected. |
Moreover, Mycostats gives b= 59,74 % so approximately 60%
Conclusion
With the risk 5% to be
mistaken, one does not reject the assumption that "the averages are
identical". The differences observed are can be due to the risks of
sampling.
Moreover, the risk of 5%
being selected, there is 60% of risk to be mistaken if it is said that mE
< mT (different averages).
Attention not to reject H0 does not mean that it is
accepted!
(Options:
if one connait the estimated standard deviation of the average, one will notch
the corresponding option.)
Another test:
Let us choose a risk a = 10%.
With the risk a = 10% , the H0 assumption is
rejected. That means that assumption "the averages are the same ones"
is rejected, with a risk 10% to be mistaken (thus higher).
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The value of b comes below 50%.
But then, would it be enough to work with a risk a different to have an optimal value of b ?
NOT! it would be so simple,
and it is even the opposite, because if one observes the value of the power of
the test, it is better for a=10%.
With the data of this
example, it would be difficult to conclude effectively (rejection of the
equality or not, with a certain risk)
What to do then?
We can remake complementary
(and independent!) measurements to try to confirm one of the two possibilities.
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For example, for the same
data, a=10% and a lot of 100 measurements, the power
of the test goes up to 97 %, with b=3 %, which would lead us to
conclude that it is necessary to reject the assumption of equality. |
Caution ! One never should benefit from the
Mycostats'calculation facility to reason anyhow:
One must be given at the
beginning a risk (a=5 % or 10 % are current prices),
and one observes the value of b posted.
According to the value
obtained for b, it will be can be useful to remake
complementary measures. If the low value of b is confirmed,
it will be necessary can be to be solved to admit that the average of the
sample does not coincide with that of the type.
If a great
value of b is confirmed, one will be able to admit that
it is not necessary to reject the assumption of equality.
But it is not test of acceptance!
Copy the
result of the test : the
button "copy" let us copy the result of the test in the copyboard.
For the other tests of comparison, only the input differs. The
conclusions are interpreted as the same manner.
NB : selon le nombre de mesures, Mycostats effectue les calculs selon la loi
Normale ou la loi de Student-Fisher