RA Fisher and Statistics

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RA Fisher and Statistics
Paul R. Earl pearl_at_dsi.uanl.mx Facultad de
Ciencias Biológicas Universidad Autónoma de Nuevo
León San Nicolás, NL, 66450, Mexico
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A view of the famous biometricianSir Ronald
Aylmer Fisher (1890-1962) is the architect of
multivariate analysis. BIOMETRICS of June
1964, Vol 20, No 2 is dedicated to him. Several
books are now strongly recommended An
Introduction to Probability Theory and its
Applicaions by William Feller, Time Series
Analysis, Forecasting and Control by George EP
Box and Gwilym M Jenkins. See JP Benzecri (1982)
Bordas, Paris. This much information is enough
for a start. Be certain to read R. A. Fisher The
Life of a Scientist. Wiley, New York by Joan
Fisher Box, 1978. Another enjoyable book is
Choice and Chance by WA Whitworth, 1901 reprinted
in 1942 by GE Stechert, New York.
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Regardless, statistics poses a severe problem for
many Latin students. The fault can be poor or
little teaching of arithmetic and algebra.
Students that fear mathematics cannot succeed in
probability, statistics and so forth. On the
other hand, students in engineering likely do
very well with internet and math. Those students
want to learn ! Regardless, the original public
school problem seems to be slow reading.The
first problem is data management. The students
may not know how to enter variables X1, X2, X3...
They might use 10 columns (cols) for sex which is
1 male, 2 female using up ONLY one col. Suppose
X5 in letters takes 10-25 cols. Assign a number.
Then this variable will need only 1-3 cols. This
seems all very simple--but it's not !
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Biostatistics lecture referencesSee
http//www.pitt.edu/super1/lecture/lec25191/001.h
tm on analysis of variance. See
http//statgen.iop.kcl.ac.uk/bgim/mle/sslike_1.htm
l on maximum likelihood,
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Analysis of variance (Fisher's ANOVA)Three
conceptual models of ANOVA are 1)
Fixed-effects model,
2) Random-effects model and
3) Mixed effects,
depending most on the number of treatments and
levels of the experiment. Oneway gives
differences among independent sets. Factorial can
help to explain the effects of 2 or more
treatment variables. often using 2 X 2 design.
Multivariate analysis (MANOVA) is used with more
than one dependent variables.
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The total sum of squares is partioned into
components related into the effects of the model.
The number of degrees of freedom (df) can also be
partioned and specifies the chi-squared
distribution which describes the associated sum
of squares.Source Sum of
df Mean F squares
Squares-----------------------
--------------------------------------------------
----Factor A SSA a-1 MSA MSA/MSEFactor
B SSB b-1 MSB MSB/MSEError SSE
ab(r-1) MSE-------------------------------------
-----------------------------------------Total S
ST ab(r-1) r repetitions
Is the F ratio an error term ? Let's add
probability p. What does p 0.01 mean ? The
idea here is merely to give you a start.
Simplicity is a great blessing ! What do you
know about quality control, loss functions and
experimentation ?
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Maximum likelihood estimation (MLE)MLE is a
method of fitting statistical models to observed
data. Assuming  that each observation is
statistically independent, the joint probability
of the observed data is given by the product of
the individual probabilitiesWhen considered
as a function of the model parameter(s), this is
called the likelihood function of the observed
data. The MLEs of the model parameter(s) maximize
the likelihood function or, equivalently,
maximize the log-likelihood function  and can
be calculated by any suitable method.
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PAMLPAML is a package of programs for
phylogenetic analyses of DNA or protein sequences
using MLE. It is maintained and distributed for
academic use free of charge by Ziheng Yang. ANSI
C source codes are distributed for UNIX/Linux/MAC
OS X and executables are provided for MS Windows.
PAML may be useful if you are interested in the
process of sequence evolution.
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Some other biostatisticsThere are 2 kinds of
statistical error depending on which hypothesis
has been identified as the true state of
nature.A null hypothesis is a speculation to be
nullified or supported. An alternate hypothesis
may prevail. The null hypothesis is presumed true
until tested to be otherwise. A null hypothesis
is a speculation to be nullified or supported. An
alternate hypothesis may prevail. The null
hypothesis is presumed true until tested to be
otherwise.
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A chi-squared test is any hypothesis test where
the test statistic has a chi-squared distribution
when the null hypothesis is true, or any in which
the probability distribution of the test
statistic (assuming the null hypothesis is true)
can be made to approximate a chi-squared
distribution as closely as desired by MAKING THE
SAMPLE SIZE LARGE ENOUGH. Important ! ! Another
popular small-sample test is Student's T-test.
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Type I error or error of the first kind or alpha
error (? error) is a
false positive. The null hypothesis was rejected
when it was actually true. Type II error or
error of the second kind or beta error (? error)
is a false negative. The error was not rejecting
the null hypothesis when the alternate hypothesis
is true.
Actual condition
True False
Test result  Positive
True Positive
(i.e. correct result) False
Positive (Type I)

(i.e. wrong result) Negative 
False Negative (Type II)
(i.e. wrong result)
True Negative

(i.e. correct result)Type I is usually set
at 0.05 or 0.01. Refer to the F ratio in ANOVA.
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Fisher informationThe Fisher information is the
variance of the score. It is the amount of
information that a random variable X carries
above an unobservable variable (parameter) ? on
which the likelihood function of X, L (?) F (X,
?) depends. The likelihood function is the joint
probability of the data. As the expectation of
the score is zero, the variance is the second
moment of the score, the derivative of the log of
the likelihood function with respect to ?.
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The following modified biography of Fisher mostly
by PC Mahalanobis appeared in Sankhy, 4,
1958.Fisher was born on the 17th of February in
1890 in East Finchley, one of the northern
suburbs of London. Love of mathematics dominated
his education. Harrow was a model school
available to Fisher since his family was rich.
The only difficulty was very bad eyesight. This
kept him out of World War I.He entered Gonville
and Caius College, Cambridge in 1909, and passed
the Mathematical Tripos Part II in 1912 as a
Wrangler. Under the negative influence of Bateson
against Darwinism, Fisher became keenly
interested in Mendelism. He thus wrote The
General Theory of Natural Selection by 1930
containing the survival of Darwinism..
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FundamentalsExact measurements compactly tabled
should make up the the experiment. Replication,
randomization and block division are fundamental
statistical elements. Accuracy is the deviation
between the experimental and true results.
Precision is related to the spread of the sample.
This dispertion is given by the standard
deviation. Replication is essential because it
is the sole source of the estimate of error,
while randomization is necessary to guarantee the
validity of the estimate, i. e., to ensure that
the estimate will be unbiased.
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The general theory of natural selectionFisher
had been working for a long time on Mendelism and
genetics. His book on Genetical Theory of Natural
Selection was published in 1930 and constituted a
milestone. The 3 men that founded the theory of
population genetics and thereby NeoDarwinism were
Sewall Wright, JBS Haldane and RA Fisher.
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More biographyHe was appointed Galton professor
in the University of London on the retirement of
Karl Pearson in 1933. Fisher was always
interested in eugenics--human genetics--associated
with the Eugenics Society first as Honorary
Secretary and later as Vice-President. He also
took over from Karl Pearson the editorial charge
of the Annals of Eugenics in 1933.In 1935,
Fisher published Design of Experiments. He
visited the USA in 1936 and received an honorary
degree from Harvard. In 1937, he accepted a
fellowship from the Indian Statistical
Institute.
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The main currents of Fisher's
work are shown next.Fisher's
work falls naturally into
3 main streams-Contributions to the
mathematical theory of statistics-Application of
statistical theory to agriculture and the design
of experiments-Contributions to genetics.
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The theory of sampling distributionThe idea of
the random sampling distribution of statistics is
fundamenal. Introduce the algebra of point sets.
The probability of the entire sample space is
unity, or P 1. The earliest example of the
modern type of distribution was that of ?2 (Chi
squared) by Karl Pearson in 1900. Several years
later Student gave the correct distribution of
the sample variance and his now famous
T-statistic. Student was the pename of William S
Gossett..
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The theory of estimates and statistical
inferenceFierce controvercy has raged over this
subject since 1768 of Bayes' An Essey towards
Solving a Problem in the Doctrine of Chance (Phil
Trans liii, p 370) in which he proposed to solve
this problem with the help of the principle of
equal distribution of ignorance. Fisher
enlightened the probllem in his notable memoir On
the Mathematical Foundation of Theoretical
Statistics (1921). This work laid the foundations
of statistical inference by emphasizing the
importance of exact solutions of sampling
problems.
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Human blood groups by RR Race of the Lister
Institute, London in BiometricsOne outcome of
Fisher's long interest in human genetics was his
setting up in 1935 a blood grouping department
in the Galton Laboratory. This was made possible
by a grant from the Rockefeller Foundation and by
the willingness of the late Dr GL Taylor, an
expert serologist, to leave the Department of
Pathology at Cambridge and devote himself to
blood groups, reviewed by RR Race in the 1964
memoirial issue of Biometry.
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Fisher saw in the blood groups a hopeful tool for
the improvement of knowledge of human genetics.
WL Stevens and DJ Finney were then at the Galton
Lab. Stevens applied Fisher's method of maximum
liklihood to calculations of the A1A2BO and MN
frequencies (Edwards AWF (1972) Likelihood.
Cambridge University Press and Sham PC (1998)
Statistics in Human Genetics. Arnold, London ).
Finney extended and applied Fisher's u-statistics
to linkage tests involving blood groups.