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Anais da Escola Superior de Agricultura Luiz de Queiroz

Print version ISSN 0071-1276

An. Esc. Super. Agric. Luiz de Queiroz vol.4  Piracicaba  1947 

A determinação dos números de indivíduos mínimos necessários na experimentação genética



F. G. Brieger

Chefe da Seção Técnica de Genética da Escola Superior de Agricultura "Luiz de Queiroz" da Universidade de São Paulo




The main object of the present paper consists in giving formulas and methods which enable us to determine the minimum number of repetitions or of individuals necessary to garantee some extent the success of an experiment. The theoretical basis of all processes consists essentially in the following. Knowing the frequency of the desired p and of the non desired ovents q we may calculate the frequency of all possi- ble combinations, to be expected in n repetitions, by expanding the binomium (p-+q)n.
Determining which of these combinations we want to avoid we calculate their total frequency, selecting the value of the exponent n of the binomium in such a way that this total frequency is equal or smaller than the accepted limit of precision
n/pª{ 1/n1 (q/p)n + 1/(n-1)| (q/p)n-1 + 1/ 2!(n-2)| (q/p)n-2 + 1/3(n-3) (q/p)n-3... < Plim - -(1b)
There does not exist an absolute limit of precision since its value depends not only upon psychological factors in our judgement, but is at the same sime a function of the number of repetitions For this reasen y have proposed (1,56) two relative values, one equal to 1-5n as the lowest value of probability and the other equal to 1-10n as the highest value of improbability, leaving between them what may be called the "region of doubt However these formulas cannot be applied in our case since this number n is just the unknown quantity. Thus we have to use, instead of the more exact values of these two formulas, the conventional limits of P.lim equal to 0,05 (Precision 5%), equal to 0,01 (Precision 1%, and to 0,001 (Precision P, 1%).
The binominal formula as explained above (cf. formula 1, pg. 85), however is of rather limited applicability owing to the excessive calculus necessary, and we have thus to procure approximations as substitutes. We may use, without loss of precision, the following approximations: a) The normal or Gaussean distribution when the expected frequency p has any value between 0,1 and 0,9, and when n is at least superior to ten.
b) The Poisson distribution when the expected frequecy p is smaller than 0,1.
Tables V to VII show for some special cases that these approximations are very satisfactory.
The praticai solution of the following problems, stated in the introduction can now be given:
A) What is the minimum number of repititions necessary in order to avoid that any one of a treatments, varieties etc. may be accidentally always the best, on the best and second best, or the first, second, and third best or finally one of the n beat treatments, varieties etc. Using the first term of the binomium, we have the following equation for n:
n = log Riim / log (m:) = log Riim / log.m - log a --------------(5)
B) What is the minimun number of individuals necessary in 01der that a ceratin type, expected with the frequency p, may appaer at least in one, two, three or a=m+1 individuals.
1) For p between 0,1 and 0,9 and using the Gaussean approximation we have:

on - ó. p (1-p) n - a -1.m

b= δ. 1-p /p e c = m/p } -------------------(7)

n = b + b2 + 4 c/ 2

n´ = 1/p

n cor = n + n' ---------- (8)

We have to use the correction n' when p has a value between 0,25 and 0,75. The greek letters delta represents in the present esse the unilateral limits of the Gaussean distribution for the three conventional limits of precision : 1,64; 2,33; and 3,09 respectively.
h we are only interested in having at least one individual, and m becomes equal to zero, the formula reduces to :

c= m/p o para a = 1

a = { b + b2}2 = b2 = δ2 1- p /p }-----------------(9)

n = 1/p

n (cor) = n + n´

2) If p is smaller than 0,1 we may use table 1 in order to find the mean m of a Poisson distribution and determine.

n = m: p

C) Which is the minimun number of individuals necessary for distinguishing two frequencies p1 and p2?

1) When pl and p2 are values between 0,1 and 0,9 we have:

n = { δ p1 ( 1-pi) + p2) / p2 (1 - p2)

n= 1/p1-p2  }------------ (13)

n (cor)

We have again to use the unilateral limits of the Gaussean distribution. The correction n' should be used if at least one of the valors pl or p2 has a value between 0,25 and 0,75.
A more complicated formula may be used in cases where whe want to increase the precision :

n (p1 - p2) δ { p1 (1- p2 ) / n= m

δ = δ p1 ( 1 - p1) + p2 ( 1 - p2) c= m / p1 - p2

n = { b2 + 4 4 c }2 }--------- (14)

n = 1/ p1 - p2

2) When both pl and p2 are smaller than 0,1 we determine the quocient (pl-r-p2) and procure the corresponding number m2 of a Poisson distribution in table 2. The value n is found by the equation :

n = mg /p2 ------------- (15)

D) What is the minimun number necessary for distinguishing three or more frequencies, p2 p1 p3.

If the frequecies pl p2 p3 are values between 0,1 e 0,9 we have to solve the individual equations and sue the higest value of n thus determined :

n 1.2 = {δ p1 (1 - p1) / p1 - p2 }2 = Fiim

n 1.2 = { δ p1 ( 1 - p1) + p1 ( 1 - p1) }2 } -- (16)

Delta represents now the bilateral limits of the : Gaussean distrioution : 1,96-2,58-3,29.
2) No table was prepared for the relatively rare cases of a comparison of threes or more frequencies below 0,1 and in such cases extremely high numbers would be required.

E) A process is given which serves to solve two problemr of informatory nature : a) if a special type appears in n individuals with a frequency p(obs), what may be the corresponding ideal value of p(esp), or; b) if we study samples of n in diviuals and expect a certain type with a frequency p(esp) what may be the extreme limits of p(obs) in individual farmlies ?
I.) If we are dealing with values between 0,1 and 0,9 we may use table 3. To solve the first question we select the respective horizontal line for p(obs) and determine which column corresponds to our value of n and find the respective value of p(esp) by interpolating between columns.
In order to solve the second problem we start with the respective column for p(esp) and find the horizontal line for the given value of n either diretly or by approximation and by interpolation.
2) For frequencies smaller than 0,1 we have to use table 4 and transform the fractions p(esp) and p(obs) in numbers of Poisson series by multiplication with n.
Tn order to solve the first broblem, we verify in which line the lower Poisson limit is equal to m(obs) and transform the corresponding value of m into frequecy p(esp) by dividing through n. The observed frequency may thus be a chance deviate of any value between 0,0... and the values given by dividing the value of m in the table by n.
In the second case we transform first the expectation p(esp) into a value of m and procure in the horizontal line, corresponding to m(esp) the extreme values om m which than must be transformed, by dividing through n into values of p(obs).
F) Partial and progressive tests may be recomended in all cases where there is lack of material or where the loss of time is less importent than the cost of large scale experiments since in many cases the minimun number necessary to garantee the results within the limits of precision is rather large.
One should not forget that the minimun number really represents at the same time a maximun number, necessary only if one takes into consideration essentially the disfavorable variations, but smaller numbers may frequently already satisfactory results.
For instance, by definition, we know that a frequecy of p means that we expect one individual in every total o(f1-p). If there were no chance variations, this number (1- p) will be suficient. and if there were favorable variations a smaller number still may yield one individual of the desired type.
r.nus trusting to luck, one may start the experiment with numbers, smaller than the minimun calculated according to the formulas given above, and increase the total untill the desired result is obtained and this may well b ebefore the "minimum number" is reached.
Some concrete examples of this partial or progressive procedure are given from our genetical experiments with maize.



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