Theor Appl Genet (1986) 72:211-218
9 Springer-Verlag 1986
Theoretical studies on the necessary number of components in mixtures 3. Number of components and risk considerations M. Hiihn Institut for Pflanzenbau und Pflanzenzt~chtung, Universit~it Kiel, 01shausenstrasse 40, D-2300 Kiel, Federal Republic of Germany Received November 5, 1985; Accepted December 4, 1985 Communicated by P. M. A. Tigerstedt
Summary. Theoretical studies on the necessary numbers of components in mixtures (for example multiclonal varieties or mixtures of lines) have been performed according to risk considerations - using the binomial distribution and the Polya-distribution. The 'risk' r of a mixture has been defined as the probability of 'catastrophic' losses (catastrophe = decrease of productivity of q% or more by 'susceptibilities' of the components). Using 1) the binomial distribution and 2) its generalization, the Polya-distribution, and several simplifying assumptions, the risks r = r (x, a, q, n) have been calculated numerically (n = number of components in the mixture, a = parameter for the intensity of contagion and dispersion of 'susceptibilities' (for example: diseases and epidemics), x = probability of 'susceptibility'). The Polya-model reduces to the binomial case if a = 0. The main results are: 1. For each number n of components the risk r decreases markedly with decreasing x (for each q and for each a). 2. For x ~ q the risk r decreases with an increasing number n of components (for each a). 3. For each number n of components and x and q with x < q the risk r increases with increasing a. 4. For given q, x and a the functions r = r (n) are asymptotic for larger numbers n of components with n > n * . In spite of further increasing numbers of components in the mixture the risk remains almost constant. For all situations, where the risk decreases with increasing n these numbers n*, therefore, can be considered as necessary numbers of components in mixtures, n* depends on q, x and a. Nevertheless, a global and rough conclusion can be formulated: In many situations one obtains necessary numbers of 3 0 - 4 0 components for a :~ 0 and 2 0 - 3 0 components for a = 0. Key words: Mixtures - N u m b e r of components - Risk studies - Binomial distribution - Polya-distribution
Introduction and problem For varying reasons clonal forestry has been intensively discussed in the last years. Most forest tree breeders propose the development of multiclonal varieties (= mixtures of different clones artificially created with definite proportions) to maintain some genetic diversity in the stands. Although the following studies have been worked out with regard to this field of applications they will also be valid for agricultural crops 1) including multilines (= mixtures of isolines that differ by single, major genes for reaction to a pathogen 2) including mixtures of an arbitrary number of pure lines which are more different among each other than isolines and 3) including mixtures of an arbitrary number of any components. To provide a simultaneous discussion of these situations - multilines, multiclonal varieties, mixtures of pure lines and mixtures of any other components - we use the general terms 'mixture' and 'components'. In two preceding publications of this series the problems of necessary numbers of components in mixtures have been discussed 1) according to phenotypic yield stability (Hfihn 1985) and 2) according to yielding-ability (Ht~hn 1986). The aim of this paper is to give some statistical approaches and numerical results concerning the necessary number of components in mixtures using risk considerations based on different probability distributions (binomial-distribution and Polya-distribution). No successive generations shall be investigated. Only one period of rotation age from the initial composition of the mixture until the final harvest shall be analysed. Risk considerations have been already applied in special cases using the binomial distribution with simple numerical calculations by Kang (1982). But this approach has not been extended, generalized and systematically elaborated until now. The number of components in mixtures is closely connected with the "risk" of this mixture. A risk has the meaning of a (relative) frequency or probability. Therefore, risk considerations can be performed by using appropriate probability distributions. At first,
212 necessary numbers of components in mixtures with respect to a given risk level have been derived by using the binomial distribution. Afterwards an interesting and realistic generalization (Polya-distribution) will be considered.
Assumptions 1 - 5 together with the binomial probability give: r=
x y (1 - x) "-y
(1)
y=yo\Y] where:
Theoretical investigations and s o m e numerical results
The simplifying assumptions used to determine the probability of 'catastrophic' events are: 1. The total number of components which is available for the composition of mixtures can be divided into 'susceptible' and 'non-susceptible' components. This term 'susceptible' describes not only the effects of epidemics and diseases. The effects of all possible biotic and abiotic causes leading to a loss of productivity are summarized and described by this term 'susceptibility'. 2. Equal proportions of the n components in the initial composition of the mixture. 3. If the 'susceptibility' occur to a component, it will wipe-out the component completely. 4. q% or more loss of productivity due to 'susceptibility' at any given year is 'catastrophic' (for example with q = 0.20). At first we assume that this loss should be realized by the total loss of q% or more of all components in the mixture. 5. The probability that 'susceptibility' will occur in any component is the same for all the components, but the 'susceptibilities' are independent events. After these preliminiaries we may ask for the frequency of catastrophic losses. These frequencies are the risks in which we are interested in. The statistical facts needed are: suppose we have a population of members, each of which exhibits either some quality A or a complementary quality B (= not-A) with proportions x and 1 - x. If we take a random sample of n members from the population (putting back each chosen m e m b e r into the population before selecting the next member) the relative frequency of obtaining y members with A and n - y members with B can be expressed by the binomial probability:
(:) xy
y
We identify: population of totality of components which are members -~ available for the composition of mixtures quality A --* 'susceptibility' quality B --* 'non-susceptibility'.
r = n = y = Y0 =
probability of catastrophe, number of components in the mixture, number of components lost, qn catastrophe level, that is: Y0 -100 ' x -- probability of'susceptibilities'. I f we assume, for example, q = 20% we obtain: r=
(n/XY(1--x)n-Y.
~ y=n/5\Y!
(2)
The risk r can be calculated easily by using appropriate tables of the cumulative binomial distribution (for example: Wetzel et al. 1967). Numerical calculations have been performed for the following parameter-values: q -- 0.05, 0.10, 0.20, 0.30, 0.40, 0.50 x = 0 . 0 5 , 0.10, 0.15, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.90 n -- 2, 3 . . . . . 1,000. For practical applications only x _-__q will be of interest. This means that the probability of susceptibility of the components (= relative frequency of susceptible components in the totality of all components) should be equal or less than the loss-of-productivitypercentage q (= relative frequency of the really susceptible components under the n components used in the mixture). Some selected numerical results of r = r(n) for different q and x are given in Fig. 1 a - c . The main r-results can be summarized as follows: 1. The risk r decreases markedly with decreasing x. 2. For x ~ q the risk r decreases with an increasing number n of components. For x > q the risk r increases with an increasing number n of components. 3. For given q and x the functions r = r (n) are asymptotic for larger numbers n of components. Therefore, there exist numbers n* with only insignificant changes of r for n > n*. In spite of further increasing numbers of components in the mixture, the risk remains almost constant. These numbers n*, therefore, can be considered as necessary numbers of components in mixtures, n* depends on q and x. Of course, these n* can be calculated more exactly by stating a definite numerical condition: for example,
213 n = lfl*/,
0
x =0.05
'+ . . . . .
x =0.10
...........
x =0.15
........
c
q:30%
.~.., . . . .
x:
...........
010
x =0.15
x : 0.20 x :0.30 x =0.50
x:O.05
........
x : O20 .......
x = 0.30 x = 0.50
.......
1.0
1.o.
i
0.I~
o.e
0.6
0.6 "'%'"'"'-.
......................................................................
0.4
O.L
~x ,LK K 0.2
0.2
io
io
so
8o
q :20"/,
,Pn
~6o
~2o
x: 0.05
......
x:010
...........
x : 0.15
........
~:~, 2b
":::-~ . . . . . . . . . 4'o
~o
8b
16o
,Itn
12o
x : 0.20 x:O.30
.......
changes of r = r(n) less than a given amount c; or mathematically expressed, n* = smallest number n with ] r ( n ' ) - r(n")[ < c for all n' and n" with n ' > n* and n" > n*. For c-- 0.1 we obtain from Fig. 1 a - c necessary numbers o f approximately 2 0 - 3 0 components for almost all the interesting parameter values. Extended and more exact numerical results are given in Hiihn (1984).
x : 0.50 1.0-
O.B
4. In the special situation x = q, the risks r are nearly constant for different n for each chosen q (exception: low n-values). For each q these risks are approximately r = 0.54 (see Fig. 1 a - c ) . For x = q or Y0 = x n the risk r depends neither on the number n nor on the loss q. This result is approximately valid with sufficient accuracy.
\ \
.\ ,\
0.4
'..
'\. \ .
Q2
~
"x '\.
'
~"
~0
~0
i',';; ..... ; ...... t .....
60
,
80
~60
, I- n
~20
Fig. 1 a - e. Numerical results for the risk r, dependent on the number n of components in the mixture, for q = 0.10, 0.20 and 0.30, respectively, and different x-values
The preceding considerations give no explicit and direct calculation o f the necessary number o f components. They permit conclusions from the computed function r = r (n) to necessary numbers n - for example by using the condition o f nearly unchanging r with increasing n. But, necessary numbers o f components in mixtures can also be calculated directly: With x' = 1 - x = probability o f 'non-susceptibilities', the probability of non-catastrophe should be
214 equal or larger than a given level Q, for example Q =0.95:
Table 1. Necessary number n of components in mixtures for Q = 0.95 and Q = 0.99 and different q and x (only for q > x) q
( ~ ) (x')Y (1 -- x')n-Y >-- Q.
Necessary number n of components
x
(3) for Q = 0.95
for Q = 0.99
0.10 0.20 0.30 0.40 0.50
3 5 8 15 37
4 7 12 25 67
0.10 0.20 0.30 0.40
4 7 13 36
5 10 22 65
y=n-y0+ 1
To solve this inequality (3) for n, several a p p r o x i m a tions have been developed (Sedcole 1977). A reasonably reliable a p p r o x i m a t e solution for (3) m a y be obtained from the normal distribution (Sedcole 1977):
0.65
n - {2 [(n - Y0 + 1) - •2 + z 2 (1 - x') + z
(4)
0.55
with z = 1.645 for Q = 0.95 and z = 2.326 for Q = 0.99. Solving for n gives the conditions:
0.45
0.10 0.20 0.30
5 11 32
7 18 57
1 +xz 2 n >_--2 ( q - x)
0.35
0.I0 0.20
8 26
12 45
0.25
0.10
17
28
9 VZ2(1
--
X') 2 + 4(1 -- X') [(n -- Y0 + 1) -- • } 2 x'
and
[ 2 n ( q - x ) - 1]2 = 4 z 2 " > n x ( 1 - x)
(5)
These expressions (5) are not defined for q = x, x = 1 and x = 0: The cases x = 1 and x = 0 are o f no interest for practical applications. The special case x = q, however, may be of some interest. But, x = q must be excluded from the following calculations, which are based on (5). x = q can be handled in the general way as described before. Some numerical results from (5) for Q = 0.95 and Q = 0.99 and different q and x are s u m m a r i z e d in Table 1 (only for q > x). F o r practical applications only small q and small x are o f interest. If we consider, for example, the situation: q _-_- 32 for Q = 0.95 and n _~ 57 for Q = 0.99. But a safety-requirement o f Q = 0.99 would be unrealistic increased. Therefore, for Q = 0.95 and the p a r a m e t e r intervals with q ___0.45, x _-< 0.30 and
event ( = s u s c e p t i b i l i t y ) increases the p r o b a b i l i t y of becoming susceptible. The statistical facts needed herewith are: suppose we have a population o f N m e m b e r s each of which exhibits either some quality A or a c o m p l e m e n t a r y quality B (= not-A) with absolute frequencies M and N - M. If we take a r a n d o m sample of n m e m b e r s from the population (with putting back each chosen m e m b e r and, additionally, S m e m b e r s of the same quality as the chosen m e m b e r into the population before selecting the next m e m b e r ) the relative frequency of obtaining y m e m b e r s with A and n - y m e m b e r s with B can be expressed by the Polyaprobability:
(~) M(M+S)...[M+(y-1)SI(N-M)(N-M+S)...[N-M+(n-y-1)S] N ( N + S ) . . . [N + ( n q - x >_-0.15 we m a y conclude a necessary n u m b e r of components in mixtures o f a p p r o x i m a t e l y 30 components. An essential i m p r o v e m e n t and generalization o f the preceding risk considerations can be obtained by using the Polya-distribution. This distribution m a y be applied to the quantitative description and simulation o f phenomena with characteristics like contagious or infectious diseases and epidemics. These situations are characterized by the fact that the occurrence o f an
f ,y,
+ a ) . . . Ix + ( y -
1) S]
(see, for example, Fisz 1966). We identify: population of totality of components which are members ~ available for the composition of mixtures quality A ~ 'susceptibility' quality B ~ 'non-susceptibility'. With x = M / N , 1-x=N-M/N and a = S / N Polya-probability is (see, for example, Fisz 1966)
1) a](1 - x)(1 - x + a ) . . . [1 - x + ( n -
1(1 + a ) . . . [ 1 + ( n -
1) al
y-
the
1) a]
(6)
215 where y= n= x= a=
number of components lost, number of components in the mixture, probability of 'susceptibilities', parameter, which measures the intensity of contagion and dispersion of susceptibility (for example, diseases and epidemics) (= relative frequency of susceptibility-increase in consequence of one occurrence of a susceptibility) (Fisz 1966).
For the special case a = 0, expression (6) gives the binomial probability. Therefore, the Polya-model turns out to be a generalization of the binomial model. r= ~
f(y) = Polya-risk
(7)
Y= Yo
with r = probability o f catastrophe, Y0 = catastrophe level, that is: Y0 = q n, q = catastrophic loss of productivity. Some selected numerical results of these risk-calculations are presented in Figs. 2 - 9 using the following numerical parameter values: a = 0.48 a = 0.24 a = 0.048 a = 0.016 a=0
q=50%; q=40%; q=30%; q=20%;
x--0.40: x=0.30: x=0.20: x=0.10:
Fig. Fig. Fig. Fig.
x= x x x x
0.05 0.10 0.20 0.30 0.40
q=20%; q=20%;
a=0.08: a=0.24:
Fig. 6 Fig. 7
x= x= x= x= x=
0.10 0.20 0.30 0.40 0.50
q=40%; q=40%;
a=0.08: a=0.24:
Fig. 8 Fig. 9
2 3 4 5
These numerical values for a have been chosen by the following considerations: We assume a mosaic-arrangement of all the N components, which are available for the composition of mixtures. Each component is represented once and occupies one region exlusively. If a component A becomes 'susceptible' the intensity of contagion and dispersion may be described quantitatively by the number S of additionally infected components. For the intensity of this contagion and dispersion process we distinguish the following situations: S = 8 (infection of one row of surrounding neighbouring components), S = 24 (infection of two rows of surrounding neighbouring components) and S = 48 (infection of three
rows of surrounding neighbouring components) (see Fig. 10). For .the total number N of components we used: N = 100, N = 500 and N = 1,000. Therefore, we obtain for a = S/N the following numerical values:
S••
100
500
1,000
8 24 48
0.080 0.240 0.480
0.016 0.048 0.096
0.008 0.024 0.048
For reasons of clearness only four different a's are considered in Figs. 2 - 5 . Additionally, the results of the binomial model with a = 0 have been included for reference and comparison purposes. Because of the same arguments as in the binomial model for practical applications only x ~ q will be of interest. The main r-results can be summarized as follows (see Figs. 2 - 9): 1. The risk r decreases markedly with decreasing x (for each q and for each a). 2. For x ~ q the risk r decreases with an increasing number n of components (for each a). For x > q: r decreases with increasing n for large avalues, while for all other a's the risk r increases with increasing n. Those a's, for which r decreases with increasing shift to larger a's for increasing x. 3. For given q, x and a the functions r = r ( n ) are asymptotic for larger numbers n of components. Therefore, there exist numbers n* with only insignificant changes of r for n > n*. In spite of further increasing numbers of components in the mixture the risk remains almost constant. For all situations, where the risk decreases with increasing n, these numbers n* can be considered as necessary numbers of components in mixtures. The risk functions r = r(n) depend on q, x and a. Therefore, in deriving the necessary numbers n* from the asymptotic properties of r = r(n) we have to consider this dependence on q, x and a too. Consequently, n* depends on q, x and a. But this dependence turns out to be not too much distinct. Therefore, a global and rough conclusion can be formulated: In many situations we obtain necessary numbers of 3 0 - 4 0 components for a :~ 0 and 2 0 - 3 0 components for a = 0. In some situations the necessary numbers n* are larger than these values, but, on the other side, very often the n* are much lower than these numbers. Extended numerical results are given in Hfihn 1984. 4. For x and q with x < q: For each number n of components the risk r increases with increasing a. Therefore, the most favourable situation--lowest risk
216
q =40%
q :50%
1.0-
o: --
x
0.89
-
Q4.0
0.6
0.6-
0.4
0.4-
0.2-
0.2-
.
. . 40
2 1.0-
.
. 80
""T'-,";---,--,120 160
,~n 200
q =30%
40
3
80
120
160
a=
x=0.20
x- 0.10
0.~,-
0.6-
0.6-
0.4.
0.4
-.. .....
. . . . . . . . 0.04 8
- ....
160
120
40
200
80
120
0.24 048
160
q = 20 %
q - 20% a = 0.08
a-
x~
~
0.8
0.8-
0,6
0.6-
0.4.-
0.4
0.2.
0.2
200
......... 0.05 ~ 0 . 1 0 .... 0.20 ..... 0.30 - - 0.40
0.24
1.o
1.0" f
0.016
0.2
.
80
40
- - 0
....
0.8-
,~"... ~.
200
q= 2 0 %
1.0-
--
0.2
0
--0.016 ........ 0.048 - 0.24 .... 0.48
x:O.30
0.8.
--
" 4 .
,
,
,
,
,
80
40
6
.
120
.
.
.
.
in
160
q-40%
.
.
r
80
0=0,24
c = 0.08
,
120
,
,
,
160
q=40%
1,0_r
0,8
0.8
0.4 9
.
40
7
1.0-
0.6-
.
200
,i-
n
200
X=
0.10
....
0.20
....
0.30
----
0/.0 0,50
.....
0.6
\
o,z, "\
I
x
0.2 9
~ i .
.
.
40
.
.
.
80
,
.
120
160
,
,
200
=n
~n
40
9
80
120
160
200
Figs. 2 - 9 . N u m e r i c a l results for t h e risk r, d e p e n d e n t o n the n u m b e r n o f c o m p o n e n t s in the mixture, for different v a l u e s o f q, x a n d a
217 O
Q
O
O
e
0
0
decreasing importance in determining the risk r if the loss-of-productivity-percentage q increases. Furthermore, for x = q all the possible numerical risk values show an extremely restricted variability: Some numerical results are given in Table 2. Therefore, we m a y conclude the following rough result: F o r practical applications and x = q the possible risks r are between 0.5 and 0.6. e
o
e
o
l
e
e
Fig. 10. Schematic representation of the contagion and dispersion process sta~ing ~om a component A
Table 2. Numerical r-intervals for different values of a, n, x and q
x=q=0.10 x = q = 0.20 x = q = 0.30 x = q = 0.40 x = q = 0.50
a -~ 0.24 n ___200
a ~ 0.048 n ~ 50
0.34 0.41 0.45 0.48 0.50
0.47 0.49 0.51 0.52 0.53
~ r_~ 0.65 _~ r _~ 0.62 < r ~ 0.62 _~ r ~_ 0.62 ~ r ~ 0.62
~ r_~ ~ r~ _~ r _ ~ r~ ~ r~
0.65 0.62 0.62 0.62 0.62
shows the binomial m o d e l with a = 0. An increasing intensity of contagion and dispersion of susceptibility (for example, o f diseases and epidemics) increases the risk. That's, of course, a very obvious result. (Exception: Tripels (n, x, a) with n = low, x = low and a = very large. But, such situations are o f no relevance for practical applications). For x and q with x >- q: Here we obtain the reverse conclusion: F o r each n u m b e r n o f components the risk r decreases with increasing a. The most unfavourable situation, therefore, must be the b i n o m i a l m o d e l with a = 0. But this condition x - q is o f no interest for practical applications. 5. In the following three cases the graphs o f r = r(n) for different a's are located very close together: a) low q, large x b) low x, large q c) equal values for x and q. F o r a), b) and c) the risks r are hardly affected by different intensities o f contagion and dispersion of diseases and epidemics. F o r a) the risks are near to r = 1 and for b) they are near to r = 0. Both situations are o f no special interest for practical applications. But c) m a y be o f interest: 6. For x = q: The graphs o f r = r(n) for different a's are of an increasing coincidence for increasing q. That means: The numerical value o f a and, therefore, the intensity o f contagion and dispersion o f susceptibility (for example, of diseases and epidemics) is o f a
Discussion A critical discussion of the numerical results on necessary numbers of components in mixtures, which have been derived by risk considerations p r i m a r i l y should be deal with an investigation o f the numerous simplifying assumptions and their possible importance and resulting restrictions. The main simplifying assumptions are: 1. Equal proportions o f the components in the initial composition of the mixture. 2. Static division of the components in 'susceptible' and 'non-susceptible' ones. 3. Occurrence of 'susceptibility' to a component will wipe-out the component completely. 4. q% or more loss of productivity should be realized by the total loss of q% or more of all components in the mixture. 5. Equal p r o b a b i l i t y that 'susceptibility' will occur for all the components. 6. 'Susceptibilities' o f the different components are independent events. Some of these strong restrictions and simplifications can be overcome by a m o d i f i e d definition and reinterpretation of the parameters involved. F o r example: The previous definition o f x has been: x = p r o b a b i l i t y of 'susceptibility' with the resulting total loss o f a susceptible component. If we define: x = p r o b a b i l i t y o f the occurrence o f a certain decrease o f productivity. Such a m o d i f i e d x-value together with similarly m o d i fied y0-values will describe more realistic situations in the field of practical applications. Therefore, the numerical results on necessary numbers o f components in mixtures, which have been obtained under very simplifying assumptions are nevertheless o f a realistic and extended validity and relevance for practical applications. Not only by these modified definitions of the parameters but, furthermore, by the transition from the binomial distribution to the Polya-distribution some restrictions and simplifying assumptions can be abolished. Further improvements may be obtained by considering and incorporating phytopathological and epidemiological facts concerning the origin, dispersion and quantitative amount of damage of epidemics and diseases. Several publications deal with theoretical models
218 and approaches of a quantitative description of different aspects of epidemics and diseases in mixtures (see, for example, Trenbath 1977; Kampmeijer and Zadoks 1977; Barrett 1978; Marshall and Pryor 1978 and 1979; Barret 1980; Marshall and Burdon 1981; Barrett 1981; ~stergaard 1983). An integration and combination of these studies and results with the risk considerations based on probability distributions of this paper will provide essential improvements of the previous results.
References Barrett JA (1978) A model of epidemic development in variety mixtures. In: Scott PR, Bainbridge A (eds) Plant disease epidemiology. Selected papers of the conference on plant disease epidemiology and dispersal of plant parasites. Blackwell Scientific, Oxford, 129-137 Barrett JA (1980) Pathogen evolution in multilines and variety mixtures. Z Pflanzenkr Pflanzenschutz 87: 383- 396 Barrett JA (1981) Disease progress curves and dispersal gradients in multilines. Phytopath Z 100:361-365 Fisz M (1966) Wahrscheinlichkeitsrechnung und Mathematische Statistik, 4 Aufl. HochschulMicher for Mathematik, Band 40. VEB Deutscher Verlag der Wissenschaften, Berlin HiJhn M (1984) Theoretische Untersuchungen zur Problematik der Mindestklonzahl in Mehrklonsorten. 5. Festlegung von optimalen Klonanzahlen fiber Risikobetrachtungen anhand von bestimmten Wahrscheinlichkeitsverteilungen. Hess Min Landesentwicklung, Umwelt, Landwirtschaft und Forsten, Wiesbaden, pp 1-57
Hahn M (1985) Theoretical studies on the necessary number of components in mixtures. 1. Number of components and yield stability. Theor Appl Genet 70: 383- 389 Hiihn M (1986) Theoretical studies on the necessary number of components in mixtures. 2. Number of components and yielding-ability. Theor Appl Genet 71:622-630 Kampmeijer P, Zadoks JC (1977) EPIMUL, a simulator of foci and epidemics in mixtures of resistant and susceptible plants, mosaics and multilines. Simulation monographs. Centre for Agricultural Publishing and Documentation, Wageningen Kang H (1982) Probability of catastrophe (_~ 20% loss of entire crop at any given year) (personal communication) Marshall DR, Pryor AJ (1978) Multiline varieties and disease control. 1. The 'dirty crop' approach with each component carrying a unique single resistance gene. Theor Appl Genet 51:177-184 Marshall DR, Pryor AJ (1979) Multiline varieties and diesase control. 2. The 'dirty crop' approach with components carrying two or more genes for resistance. Euphytica 28: 145-159 Marshall DR, Burdon JJ (1981) Multiline varieties and disease control. 3. Combined use of overlapping and disjoint gene sets. Aus J Biol Sci 34:81-95 Ostergaard H (1983) Predicting development of epidemics on cultivar mixtures. Phytopathology 73:166-172 Sedcole JR (1977) Number of plants necessary to recover a trait. Crop Sci 17:667-668 Trenbath BR (1977) Interactions among diverse hosts and diverse parasites. Ann NY Acad Sci 287:124-150 Wetzel W, Jrhnk MD, Naeve P (1967) Statistische Tabellen. de Gruyter, Berlin
9 Springer-Verlag 1986
Theoretical studies on the necessary number of components in mixtures 3. Number of components and risk considerations M. Hiihn Institut for Pflanzenbau und Pflanzenzt~chtung, Universit~it Kiel, 01shausenstrasse 40, D-2300 Kiel, Federal Republic of Germany Received November 5, 1985; Accepted December 4, 1985 Communicated by P. M. A. Tigerstedt
Summary. Theoretical studies on the necessary numbers of components in mixtures (for example multiclonal varieties or mixtures of lines) have been performed according to risk considerations - using the binomial distribution and the Polya-distribution. The 'risk' r of a mixture has been defined as the probability of 'catastrophic' losses (catastrophe = decrease of productivity of q% or more by 'susceptibilities' of the components). Using 1) the binomial distribution and 2) its generalization, the Polya-distribution, and several simplifying assumptions, the risks r = r (x, a, q, n) have been calculated numerically (n = number of components in the mixture, a = parameter for the intensity of contagion and dispersion of 'susceptibilities' (for example: diseases and epidemics), x = probability of 'susceptibility'). The Polya-model reduces to the binomial case if a = 0. The main results are: 1. For each number n of components the risk r decreases markedly with decreasing x (for each q and for each a). 2. For x ~ q the risk r decreases with an increasing number n of components (for each a). 3. For each number n of components and x and q with x < q the risk r increases with increasing a. 4. For given q, x and a the functions r = r (n) are asymptotic for larger numbers n of components with n > n * . In spite of further increasing numbers of components in the mixture the risk remains almost constant. For all situations, where the risk decreases with increasing n these numbers n*, therefore, can be considered as necessary numbers of components in mixtures, n* depends on q, x and a. Nevertheless, a global and rough conclusion can be formulated: In many situations one obtains necessary numbers of 3 0 - 4 0 components for a :~ 0 and 2 0 - 3 0 components for a = 0. Key words: Mixtures - N u m b e r of components - Risk studies - Binomial distribution - Polya-distribution
Introduction and problem For varying reasons clonal forestry has been intensively discussed in the last years. Most forest tree breeders propose the development of multiclonal varieties (= mixtures of different clones artificially created with definite proportions) to maintain some genetic diversity in the stands. Although the following studies have been worked out with regard to this field of applications they will also be valid for agricultural crops 1) including multilines (= mixtures of isolines that differ by single, major genes for reaction to a pathogen 2) including mixtures of an arbitrary number of pure lines which are more different among each other than isolines and 3) including mixtures of an arbitrary number of any components. To provide a simultaneous discussion of these situations - multilines, multiclonal varieties, mixtures of pure lines and mixtures of any other components - we use the general terms 'mixture' and 'components'. In two preceding publications of this series the problems of necessary numbers of components in mixtures have been discussed 1) according to phenotypic yield stability (Hfihn 1985) and 2) according to yielding-ability (Ht~hn 1986). The aim of this paper is to give some statistical approaches and numerical results concerning the necessary number of components in mixtures using risk considerations based on different probability distributions (binomial-distribution and Polya-distribution). No successive generations shall be investigated. Only one period of rotation age from the initial composition of the mixture until the final harvest shall be analysed. Risk considerations have been already applied in special cases using the binomial distribution with simple numerical calculations by Kang (1982). But this approach has not been extended, generalized and systematically elaborated until now. The number of components in mixtures is closely connected with the "risk" of this mixture. A risk has the meaning of a (relative) frequency or probability. Therefore, risk considerations can be performed by using appropriate probability distributions. At first,
212 necessary numbers of components in mixtures with respect to a given risk level have been derived by using the binomial distribution. Afterwards an interesting and realistic generalization (Polya-distribution) will be considered.
Assumptions 1 - 5 together with the binomial probability give: r=
x y (1 - x) "-y
(1)
y=yo\Y] where:
Theoretical investigations and s o m e numerical results
The simplifying assumptions used to determine the probability of 'catastrophic' events are: 1. The total number of components which is available for the composition of mixtures can be divided into 'susceptible' and 'non-susceptible' components. This term 'susceptible' describes not only the effects of epidemics and diseases. The effects of all possible biotic and abiotic causes leading to a loss of productivity are summarized and described by this term 'susceptibility'. 2. Equal proportions of the n components in the initial composition of the mixture. 3. If the 'susceptibility' occur to a component, it will wipe-out the component completely. 4. q% or more loss of productivity due to 'susceptibility' at any given year is 'catastrophic' (for example with q = 0.20). At first we assume that this loss should be realized by the total loss of q% or more of all components in the mixture. 5. The probability that 'susceptibility' will occur in any component is the same for all the components, but the 'susceptibilities' are independent events. After these preliminiaries we may ask for the frequency of catastrophic losses. These frequencies are the risks in which we are interested in. The statistical facts needed are: suppose we have a population of members, each of which exhibits either some quality A or a complementary quality B (= not-A) with proportions x and 1 - x. If we take a random sample of n members from the population (putting back each chosen m e m b e r into the population before selecting the next member) the relative frequency of obtaining y members with A and n - y members with B can be expressed by the binomial probability:
(:) xy
y
We identify: population of totality of components which are members -~ available for the composition of mixtures quality A --* 'susceptibility' quality B --* 'non-susceptibility'.
r = n = y = Y0 =
probability of catastrophe, number of components in the mixture, number of components lost, qn catastrophe level, that is: Y0 -100 ' x -- probability of'susceptibilities'. I f we assume, for example, q = 20% we obtain: r=
(n/XY(1--x)n-Y.
~ y=n/5\Y!
(2)
The risk r can be calculated easily by using appropriate tables of the cumulative binomial distribution (for example: Wetzel et al. 1967). Numerical calculations have been performed for the following parameter-values: q -- 0.05, 0.10, 0.20, 0.30, 0.40, 0.50 x = 0 . 0 5 , 0.10, 0.15, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.90 n -- 2, 3 . . . . . 1,000. For practical applications only x _-__q will be of interest. This means that the probability of susceptibility of the components (= relative frequency of susceptible components in the totality of all components) should be equal or less than the loss-of-productivitypercentage q (= relative frequency of the really susceptible components under the n components used in the mixture). Some selected numerical results of r = r(n) for different q and x are given in Fig. 1 a - c . The main r-results can be summarized as follows: 1. The risk r decreases markedly with decreasing x. 2. For x ~ q the risk r decreases with an increasing number n of components. For x > q the risk r increases with an increasing number n of components. 3. For given q and x the functions r = r (n) are asymptotic for larger numbers n of components. Therefore, there exist numbers n* with only insignificant changes of r for n > n*. In spite of further increasing numbers of components in the mixture, the risk remains almost constant. These numbers n*, therefore, can be considered as necessary numbers of components in mixtures, n* depends on q and x. Of course, these n* can be calculated more exactly by stating a definite numerical condition: for example,
213 n = lfl*/,
0
x =0.05
'+ . . . . .
x =0.10
...........
x =0.15
........
c
q:30%
.~.., . . . .
x:
...........
010
x =0.15
x : 0.20 x :0.30 x =0.50
x:O.05
........
x : O20 .......
x = 0.30 x = 0.50
.......
1.0
1.o.
i
0.I~
o.e
0.6
0.6 "'%'"'"'-.
......................................................................
0.4
O.L
~x ,LK K 0.2
0.2
io
io
so
8o
q :20"/,
,Pn
~6o
~2o
x: 0.05
......
x:010
...........
x : 0.15
........
~:~, 2b
":::-~ . . . . . . . . . 4'o
~o
8b
16o
,Itn
12o
x : 0.20 x:O.30
.......
changes of r = r(n) less than a given amount c; or mathematically expressed, n* = smallest number n with ] r ( n ' ) - r(n")[ < c for all n' and n" with n ' > n* and n" > n*. For c-- 0.1 we obtain from Fig. 1 a - c necessary numbers o f approximately 2 0 - 3 0 components for almost all the interesting parameter values. Extended and more exact numerical results are given in Hiihn (1984).
x : 0.50 1.0-
O.B
4. In the special situation x = q, the risks r are nearly constant for different n for each chosen q (exception: low n-values). For each q these risks are approximately r = 0.54 (see Fig. 1 a - c ) . For x = q or Y0 = x n the risk r depends neither on the number n nor on the loss q. This result is approximately valid with sufficient accuracy.
\ \
.\ ,\
0.4
'..
'\. \ .
Q2
~
"x '\.
'
~"
~0
~0
i',';; ..... ; ...... t .....
60
,
80
~60
, I- n
~20
Fig. 1 a - e. Numerical results for the risk r, dependent on the number n of components in the mixture, for q = 0.10, 0.20 and 0.30, respectively, and different x-values
The preceding considerations give no explicit and direct calculation o f the necessary number o f components. They permit conclusions from the computed function r = r (n) to necessary numbers n - for example by using the condition o f nearly unchanging r with increasing n. But, necessary numbers o f components in mixtures can also be calculated directly: With x' = 1 - x = probability o f 'non-susceptibilities', the probability of non-catastrophe should be
214 equal or larger than a given level Q, for example Q =0.95:
Table 1. Necessary number n of components in mixtures for Q = 0.95 and Q = 0.99 and different q and x (only for q > x) q
( ~ ) (x')Y (1 -- x')n-Y >-- Q.
Necessary number n of components
x
(3) for Q = 0.95
for Q = 0.99
0.10 0.20 0.30 0.40 0.50
3 5 8 15 37
4 7 12 25 67
0.10 0.20 0.30 0.40
4 7 13 36
5 10 22 65
y=n-y0+ 1
To solve this inequality (3) for n, several a p p r o x i m a tions have been developed (Sedcole 1977). A reasonably reliable a p p r o x i m a t e solution for (3) m a y be obtained from the normal distribution (Sedcole 1977):
0.65
n - {2 [(n - Y0 + 1) - •2 + z 2 (1 - x') + z
(4)
0.55
with z = 1.645 for Q = 0.95 and z = 2.326 for Q = 0.99. Solving for n gives the conditions:
0.45
0.10 0.20 0.30
5 11 32
7 18 57
1 +xz 2 n >_--2 ( q - x)
0.35
0.I0 0.20
8 26
12 45
0.25
0.10
17
28
9 VZ2(1
--
X') 2 + 4(1 -- X') [(n -- Y0 + 1) -- • } 2 x'
and
[ 2 n ( q - x ) - 1]2 = 4 z 2 " > n x ( 1 - x)
(5)
These expressions (5) are not defined for q = x, x = 1 and x = 0: The cases x = 1 and x = 0 are o f no interest for practical applications. The special case x = q, however, may be of some interest. But, x = q must be excluded from the following calculations, which are based on (5). x = q can be handled in the general way as described before. Some numerical results from (5) for Q = 0.95 and Q = 0.99 and different q and x are s u m m a r i z e d in Table 1 (only for q > x). F o r practical applications only small q and small x are o f interest. If we consider, for example, the situation: q _-_- 32 for Q = 0.95 and n _~ 57 for Q = 0.99. But a safety-requirement o f Q = 0.99 would be unrealistic increased. Therefore, for Q = 0.95 and the p a r a m e t e r intervals with q ___0.45, x _-< 0.30 and
event ( = s u s c e p t i b i l i t y ) increases the p r o b a b i l i t y of becoming susceptible. The statistical facts needed herewith are: suppose we have a population o f N m e m b e r s each of which exhibits either some quality A or a c o m p l e m e n t a r y quality B (= not-A) with absolute frequencies M and N - M. If we take a r a n d o m sample of n m e m b e r s from the population (with putting back each chosen m e m b e r and, additionally, S m e m b e r s of the same quality as the chosen m e m b e r into the population before selecting the next m e m b e r ) the relative frequency of obtaining y m e m b e r s with A and n - y m e m b e r s with B can be expressed by the Polyaprobability:
(~) M(M+S)...[M+(y-1)SI(N-M)(N-M+S)...[N-M+(n-y-1)S] N ( N + S ) . . . [N + ( n q - x >_-0.15 we m a y conclude a necessary n u m b e r of components in mixtures o f a p p r o x i m a t e l y 30 components. An essential i m p r o v e m e n t and generalization o f the preceding risk considerations can be obtained by using the Polya-distribution. This distribution m a y be applied to the quantitative description and simulation o f phenomena with characteristics like contagious or infectious diseases and epidemics. These situations are characterized by the fact that the occurrence o f an
f ,y,
+ a ) . . . Ix + ( y -
1) S]
(see, for example, Fisz 1966). We identify: population of totality of components which are members ~ available for the composition of mixtures quality A ~ 'susceptibility' quality B ~ 'non-susceptibility'. With x = M / N , 1-x=N-M/N and a = S / N Polya-probability is (see, for example, Fisz 1966)
1) a](1 - x)(1 - x + a ) . . . [1 - x + ( n -
1(1 + a ) . . . [ 1 + ( n -
1) al
y-
the
1) a]
(6)
215 where y= n= x= a=
number of components lost, number of components in the mixture, probability of 'susceptibilities', parameter, which measures the intensity of contagion and dispersion of susceptibility (for example, diseases and epidemics) (= relative frequency of susceptibility-increase in consequence of one occurrence of a susceptibility) (Fisz 1966).
For the special case a = 0, expression (6) gives the binomial probability. Therefore, the Polya-model turns out to be a generalization of the binomial model. r= ~
f(y) = Polya-risk
(7)
Y= Yo
with r = probability o f catastrophe, Y0 = catastrophe level, that is: Y0 = q n, q = catastrophic loss of productivity. Some selected numerical results of these risk-calculations are presented in Figs. 2 - 9 using the following numerical parameter values: a = 0.48 a = 0.24 a = 0.048 a = 0.016 a=0
q=50%; q=40%; q=30%; q=20%;
x--0.40: x=0.30: x=0.20: x=0.10:
Fig. Fig. Fig. Fig.
x= x x x x
0.05 0.10 0.20 0.30 0.40
q=20%; q=20%;
a=0.08: a=0.24:
Fig. 6 Fig. 7
x= x= x= x= x=
0.10 0.20 0.30 0.40 0.50
q=40%; q=40%;
a=0.08: a=0.24:
Fig. 8 Fig. 9
2 3 4 5
These numerical values for a have been chosen by the following considerations: We assume a mosaic-arrangement of all the N components, which are available for the composition of mixtures. Each component is represented once and occupies one region exlusively. If a component A becomes 'susceptible' the intensity of contagion and dispersion may be described quantitatively by the number S of additionally infected components. For the intensity of this contagion and dispersion process we distinguish the following situations: S = 8 (infection of one row of surrounding neighbouring components), S = 24 (infection of two rows of surrounding neighbouring components) and S = 48 (infection of three
rows of surrounding neighbouring components) (see Fig. 10). For .the total number N of components we used: N = 100, N = 500 and N = 1,000. Therefore, we obtain for a = S/N the following numerical values:
S••
100
500
1,000
8 24 48
0.080 0.240 0.480
0.016 0.048 0.096
0.008 0.024 0.048
For reasons of clearness only four different a's are considered in Figs. 2 - 5 . Additionally, the results of the binomial model with a = 0 have been included for reference and comparison purposes. Because of the same arguments as in the binomial model for practical applications only x ~ q will be of interest. The main r-results can be summarized as follows (see Figs. 2 - 9): 1. The risk r decreases markedly with decreasing x (for each q and for each a). 2. For x ~ q the risk r decreases with an increasing number n of components (for each a). For x > q: r decreases with increasing n for large avalues, while for all other a's the risk r increases with increasing n. Those a's, for which r decreases with increasing shift to larger a's for increasing x. 3. For given q, x and a the functions r = r ( n ) are asymptotic for larger numbers n of components. Therefore, there exist numbers n* with only insignificant changes of r for n > n*. In spite of further increasing numbers of components in the mixture the risk remains almost constant. For all situations, where the risk decreases with increasing n, these numbers n* can be considered as necessary numbers of components in mixtures. The risk functions r = r(n) depend on q, x and a. Therefore, in deriving the necessary numbers n* from the asymptotic properties of r = r(n) we have to consider this dependence on q, x and a too. Consequently, n* depends on q, x and a. But this dependence turns out to be not too much distinct. Therefore, a global and rough conclusion can be formulated: In many situations we obtain necessary numbers of 3 0 - 4 0 components for a :~ 0 and 2 0 - 3 0 components for a = 0. In some situations the necessary numbers n* are larger than these values, but, on the other side, very often the n* are much lower than these numbers. Extended numerical results are given in Hfihn 1984. 4. For x and q with x < q: For each number n of components the risk r increases with increasing a. Therefore, the most favourable situation--lowest risk
216
q =40%
q :50%
1.0-
o: --
x
0.89
-
Q4.0
0.6
0.6-
0.4
0.4-
0.2-
0.2-
.
. . 40
2 1.0-
.
. 80
""T'-,";---,--,120 160
,~n 200
q =30%
40
3
80
120
160
a=
x=0.20
x- 0.10
0.~,-
0.6-
0.6-
0.4.
0.4
-.. .....
. . . . . . . . 0.04 8
- ....
160
120
40
200
80
120
0.24 048
160
q = 20 %
q - 20% a = 0.08
a-
x~
~
0.8
0.8-
0,6
0.6-
0.4.-
0.4
0.2.
0.2
200
......... 0.05 ~ 0 . 1 0 .... 0.20 ..... 0.30 - - 0.40
0.24
1.o
1.0" f
0.016
0.2
.
80
40
- - 0
....
0.8-
,~"... ~.
200
q= 2 0 %
1.0-
--
0.2
0
--0.016 ........ 0.048 - 0.24 .... 0.48
x:O.30
0.8.
--
" 4 .
,
,
,
,
,
80
40
6
.
120
.
.
.
.
in
160
q-40%
.
.
r
80
0=0,24
c = 0.08
,
120
,
,
,
160
q=40%
1,0_r
0,8
0.8
0.4 9
.
40
7
1.0-
0.6-
.
200
,i-
n
200
X=
0.10
....
0.20
....
0.30
----
0/.0 0,50
.....
0.6
\
o,z, "\
I
x
0.2 9
~ i .
.
.
40
.
.
.
80
,
.
120
160
,
,
200
=n
~n
40
9
80
120
160
200
Figs. 2 - 9 . N u m e r i c a l results for t h e risk r, d e p e n d e n t o n the n u m b e r n o f c o m p o n e n t s in the mixture, for different v a l u e s o f q, x a n d a
217 O
Q
O
O
e
0
0
decreasing importance in determining the risk r if the loss-of-productivity-percentage q increases. Furthermore, for x = q all the possible numerical risk values show an extremely restricted variability: Some numerical results are given in Table 2. Therefore, we m a y conclude the following rough result: F o r practical applications and x = q the possible risks r are between 0.5 and 0.6. e
o
e
o
l
e
e
Fig. 10. Schematic representation of the contagion and dispersion process sta~ing ~om a component A
Table 2. Numerical r-intervals for different values of a, n, x and q
x=q=0.10 x = q = 0.20 x = q = 0.30 x = q = 0.40 x = q = 0.50
a -~ 0.24 n ___200
a ~ 0.048 n ~ 50
0.34 0.41 0.45 0.48 0.50
0.47 0.49 0.51 0.52 0.53
~ r_~ 0.65 _~ r _~ 0.62 < r ~ 0.62 _~ r ~_ 0.62 ~ r ~ 0.62
~ r_~ ~ r~ _~ r _ ~ r~ ~ r~
0.65 0.62 0.62 0.62 0.62
shows the binomial m o d e l with a = 0. An increasing intensity of contagion and dispersion of susceptibility (for example, o f diseases and epidemics) increases the risk. That's, of course, a very obvious result. (Exception: Tripels (n, x, a) with n = low, x = low and a = very large. But, such situations are o f no relevance for practical applications). For x and q with x >- q: Here we obtain the reverse conclusion: F o r each n u m b e r n o f components the risk r decreases with increasing a. The most unfavourable situation, therefore, must be the b i n o m i a l m o d e l with a = 0. But this condition x - q is o f no interest for practical applications. 5. In the following three cases the graphs o f r = r(n) for different a's are located very close together: a) low q, large x b) low x, large q c) equal values for x and q. F o r a), b) and c) the risks r are hardly affected by different intensities o f contagion and dispersion of diseases and epidemics. F o r a) the risks are near to r = 1 and for b) they are near to r = 0. Both situations are o f no special interest for practical applications. But c) m a y be o f interest: 6. For x = q: The graphs o f r = r(n) for different a's are of an increasing coincidence for increasing q. That means: The numerical value o f a and, therefore, the intensity o f contagion and dispersion o f susceptibility (for example, of diseases and epidemics) is o f a
Discussion A critical discussion of the numerical results on necessary numbers of components in mixtures, which have been derived by risk considerations p r i m a r i l y should be deal with an investigation o f the numerous simplifying assumptions and their possible importance and resulting restrictions. The main simplifying assumptions are: 1. Equal proportions o f the components in the initial composition of the mixture. 2. Static division of the components in 'susceptible' and 'non-susceptible' ones. 3. Occurrence of 'susceptibility' to a component will wipe-out the component completely. 4. q% or more loss of productivity should be realized by the total loss of q% or more of all components in the mixture. 5. Equal p r o b a b i l i t y that 'susceptibility' will occur for all the components. 6. 'Susceptibilities' o f the different components are independent events. Some of these strong restrictions and simplifications can be overcome by a m o d i f i e d definition and reinterpretation of the parameters involved. F o r example: The previous definition o f x has been: x = p r o b a b i l i t y of 'susceptibility' with the resulting total loss o f a susceptible component. If we define: x = p r o b a b i l i t y o f the occurrence o f a certain decrease o f productivity. Such a m o d i f i e d x-value together with similarly m o d i fied y0-values will describe more realistic situations in the field of practical applications. Therefore, the numerical results on necessary numbers o f components in mixtures, which have been obtained under very simplifying assumptions are nevertheless o f a realistic and extended validity and relevance for practical applications. Not only by these modified definitions of the parameters but, furthermore, by the transition from the binomial distribution to the Polya-distribution some restrictions and simplifying assumptions can be abolished. Further improvements may be obtained by considering and incorporating phytopathological and epidemiological facts concerning the origin, dispersion and quantitative amount of damage of epidemics and diseases. Several publications deal with theoretical models
218 and approaches of a quantitative description of different aspects of epidemics and diseases in mixtures (see, for example, Trenbath 1977; Kampmeijer and Zadoks 1977; Barrett 1978; Marshall and Pryor 1978 and 1979; Barret 1980; Marshall and Burdon 1981; Barrett 1981; ~stergaard 1983). An integration and combination of these studies and results with the risk considerations based on probability distributions of this paper will provide essential improvements of the previous results.
References Barrett JA (1978) A model of epidemic development in variety mixtures. In: Scott PR, Bainbridge A (eds) Plant disease epidemiology. Selected papers of the conference on plant disease epidemiology and dispersal of plant parasites. Blackwell Scientific, Oxford, 129-137 Barrett JA (1980) Pathogen evolution in multilines and variety mixtures. Z Pflanzenkr Pflanzenschutz 87: 383- 396 Barrett JA (1981) Disease progress curves and dispersal gradients in multilines. Phytopath Z 100:361-365 Fisz M (1966) Wahrscheinlichkeitsrechnung und Mathematische Statistik, 4 Aufl. HochschulMicher for Mathematik, Band 40. VEB Deutscher Verlag der Wissenschaften, Berlin HiJhn M (1984) Theoretische Untersuchungen zur Problematik der Mindestklonzahl in Mehrklonsorten. 5. Festlegung von optimalen Klonanzahlen fiber Risikobetrachtungen anhand von bestimmten Wahrscheinlichkeitsverteilungen. Hess Min Landesentwicklung, Umwelt, Landwirtschaft und Forsten, Wiesbaden, pp 1-57
Hahn M (1985) Theoretical studies on the necessary number of components in mixtures. 1. Number of components and yield stability. Theor Appl Genet 70: 383- 389 Hiihn M (1986) Theoretical studies on the necessary number of components in mixtures. 2. Number of components and yielding-ability. Theor Appl Genet 71:622-630 Kampmeijer P, Zadoks JC (1977) EPIMUL, a simulator of foci and epidemics in mixtures of resistant and susceptible plants, mosaics and multilines. Simulation monographs. Centre for Agricultural Publishing and Documentation, Wageningen Kang H (1982) Probability of catastrophe (_~ 20% loss of entire crop at any given year) (personal communication) Marshall DR, Pryor AJ (1978) Multiline varieties and disease control. 1. The 'dirty crop' approach with each component carrying a unique single resistance gene. Theor Appl Genet 51:177-184 Marshall DR, Pryor AJ (1979) Multiline varieties and diesase control. 2. The 'dirty crop' approach with components carrying two or more genes for resistance. Euphytica 28: 145-159 Marshall DR, Burdon JJ (1981) Multiline varieties and disease control. 3. Combined use of overlapping and disjoint gene sets. Aus J Biol Sci 34:81-95 Ostergaard H (1983) Predicting development of epidemics on cultivar mixtures. Phytopathology 73:166-172 Sedcole JR (1977) Number of plants necessary to recover a trait. Crop Sci 17:667-668 Trenbath BR (1977) Interactions among diverse hosts and diverse parasites. Ann NY Acad Sci 287:124-150 Wetzel W, Jrhnk MD, Naeve P (1967) Statistische Tabellen. de Gruyter, Berlin
