The international standard ISO 112691 results an example of nickel concentration C (mg/kg) influence in soil on barley L (mm) [1, p.186, table 17.2]. The standard recommends to process results by statistical methods of Student or Dannet. Our methods [2, 3] have allowed to reveal biotechnical regularities of underlying factors.
Basic data
Basis of modelling in the standard is the arithmetic mean significances calculation (table 1). However same arithmetic mean significance can characterize a wide variety of statistical distributions.
Table 1. Results of the barley base test of in soil samples on the ISO 112691 [1, p.186, table 17.2]
Nickel concentration, mg/kg 
Base length , mm 
Mean base length, mm 
0 
98 99 105 100 100 101 102 103 97 101 104 102 97 100 99 96 102 105 
101 
50 
96 104 99 101 102 97 102 100 97 98 98 100 99 97 96 100 103 98 
99 
100 
85 91 93 92 88 84 90 84 89 90 87 93 88 87 83 86 82 84 
88 
500 
17 9 8 11 16 13 11 12 9 15 15 7 8 15 17 10 12 9 
12 
1000 
9 3 4 11 3 7 7 5 8 7 4 5 5 6 9 7 6 8 
6 
In table 1 each several 18 plants for same concentration of nickel are placed in a number on the principle unknown to us. Therefore it is impossible obviously to select the influencing factor. In this case we apply a technique of mediate base length ranking. In the beginning we shall show statistical models on the table 1.
Biotechnical regularities
On a table model from table 1 some formulas of nickel concentration influence on a change of barley base length, mm (fig. 1), are received:
 On the arithmetic mean base length
; (1)
 On the typical base length
; (2)
 On the maximum base length (upper limit of a confidence interval)
; (3)
 On the minimum base length (lower limit of a confidence interval)
. (4)


on the arithmetic mean base length 
on the typical base length 


upper limit of a confidence interval 
lower limit of a confidence interval 
Fig. 1. Graphs of nickel concentration influence on the barley base growth
All four equations show limits of growth on condition that . But the parameters of models of means that is equations (1) and (2) within pollutant concentration change differ from each other. It proves that the statistical numbers including 18 significances of base length don¢t follow the GaussianLaplace law that is known as normal distribution law [2, 3]. Therefore arithmetic mean method gives an error on a comparison with the typical approach.
Modelling conception
In a fig. 2 the schematic diagram of pollutant influence on plant base growth is represented. With that all statistical ensemble consisting of 5 x 18 = 90 members (table 2) is considered. The greatest and least significances of base length derive a variation of plant development and growth in changing soil conditions. The transition from a level in occurs under the death distribution. The line of transition divides area of behaviour of a plant set on two parts:
а) from above graph in a fig. 2 is plant base stasis effect;
б) from below graph is stasis effect by plants;
All equations are obtained on a line of stasis effect by plant in growth process.
Then biotechnical regularities of stasis by plants are received under the general formula:
, (5)
where  required model parameter.
Fig. 2. The schematic diagram of pollutant influence
And the effect of plant stasis will be identified by biotechnical regularity of a kind:
. (6)
Under the standard the longest base of a plant is measured.
Table model
The formulas (5) and (6) are applied to samplings and we shall receive table 2.
Table 2. Test result ranking
C, mg/kg 
L, mm 
Rank r 
L_{max}, mm 
L_{max} L mm 
C, mg/kg 
L, mm 
Rank r 
L_{max}, mm 
L_{max} L mm 
C, mg/kg 
L, mm 
Rank r 
L_{max}, mm 
L_{max} L mm 
0 
98 
7 
105 
7 
50 
99 
5 
104 
5 
500 
11 
5 
17 
6 
0 
99 
6 
105 
6 
50 
97 
7 
104 
7 
500 
12 
4 
17 
5 
0 
105 
0 
105 
0 
50 
96 
8 
104 
8 
500 
9 
7 
17 
8 
0 
100 
5 
105 
5 
50 
100 
4 
104 
4 
500 
15 
2 
17 
2 
0 
100 
5 
105 
5 
50 
103 
1 
104 
1 
500 
15 
2 
17 
2 
0 
101 
4 
105 
4 
50 
98 
6 
104 
6 
500 
7 
9 
17 
10 
0 
102 
3 
105 
3 
100 
85 
8 
93 
8 
500 
8 
8 
17 
9 
0 
103 
2 
105 
2 
100 
91 
2 
93 
2 
500 
15 
2 
17 
2 
0 
97 
8 
105 
8 
100 
93 
0 
93 
0 
500 
17 
0 
17 
0 
0 
101 
4 
105 
4 
100 
92 
1 
93 
1 
500 
10 
6 
17 
7 
0 
104 
1 
105 
1 
100 
88 
5 
93 
5 
500 
12 
4 
17 
5 
0 
102 
3 
105 
3 
100 
84 
9 
93 
9 
500 
9 
7 
17 
8 
0 
97 
8 
105 
8 
100 
90 
3 
93 
3 
1000 
9 
1 
11 
2 
0 
100 
5 
105 
5 
100 
84 
9 
93 
9 
1000 
3 
7 
11 
8 
0 
99 
6 
105 
6 
100 
89 
4 
93 
4 
1000 
4 
6 
11 
7 
0 
96 
9 
105 
9 
100 
90 
3 
93 
3 
1000 
11 
0 
11 
0 
0 
102 
3 
105 
3 
100 
87 
6 
93 
6 
1000 
3 
7 
11 
8 
0 
105 
0 
105 
0 
100 
93 
0 
93 
0 
1000 
7 
3 
11 
4 
50 
96 
8 
104 
8 
100 
88 
5 
93 
5 
1000 
7 
3 
11 
4 
50 
104 
0 
104 
0 
100 
87 
6 
93 
6 
1000 
5 
5 
11 
6 
50 
99 
5 
104 
5 
100 
83 
10 
93 
10 
1000 
8 
2 
11 
3 
50 
101 
3 
104 
3 
100 
86 
7 
93 
7 
1000 
7 
3 
11 
4 
50 
102 
2 
104 
2 
100 
82 
11 
93 
11 
1000 
4 
6 
11 
7 
50 
97 
7 
104 
7 
100 
84 
9 
93 
9 
1000 
5 
5 
11 
6 
50 
102 
2 
104 
2 
500 
17 
0 
17 
0 
1000 
5 
5 
11 
6 
50 
100 
4 
104 
4 
500 
9 
7 
17 
8 
1000 
6 
4 
11 
5 
50 
97 
7 
104 
7 
500 
8 
8 
17 
9 
1000 
9 
1 
11 
2 
50 
98 
6 
104 
6 
500 
11 
5 
17 
6 
1000 
7 
3 
11 
4 
50 
98 
6 
104 
6 
500 
16 
1 
17 
1 
1000 
6 
4 
11 
5 
50 
100 
4 
104 
4 
500 
13 
3 
17 
4 
1000 
8 
2 
11 
3 
Here means a maximum of base length in each sampling for same significance of an explanatory variable.
Number ranking
On vectorial orientation «it is better → worse» we shall give ranks r = 0,1,2,... to each significance of length in five samplings. After death distribution identification the biotechnical regularities (fig. 3) were obtained.
Fig. 3. Graphs of unknown factor influence
The high equation correlation coefficients (in a right upper graph angle) show availability of underlying factor or even groups of underlying factors in experiment results.
The residuals after the equations in a fig. 3 equal to less than 0,5 mm, that is there are below measuring precision of length of the longest plant base.
The twofactor image is shown in a fig. 4.
The unknown factor exerts noticeable influence for significant concentration of pollutant.
On rank length, that is on a maximum rank , it is possible to judge about attempts of a plant to resist to harmful pollutant influence.
Fig. 4. Disposition of base length
Confidence interval
It is determined more precisely on data of table 3, when curculating members of general statistical sampling consisting from 90 members as the formulas (fig. 5) come into account:
 on the upper limit of confidence interval of base length
; (7)
 on the under limit of confidence interval of base length
. (8)
Table 3. Limits of confidence interval
upper 
under 

0 
105 
0 
96 
0 
105 
50 
96 
50 
104 
50 
96 
100 
93 
100 
82 
100 
93 
500 
7 
500 
17 
1000 
3 
500 
17 
1000 
3 
1000 
11 


These binomial equations contain two stable distribution laws. First component is the death distribution offered by us [2, 3], in which in difference from the Laplace law intensity of death (degree of an explanatory variable) is entered. Second component indicating highstress excitation of test plants from pollutant effect is the biotechnical law of prof. P.М. Mazurkin. Besides first component is a special case of the biotechnical law. Therefore all regularities are received from one formula.
Fig. 5. Limits of confidence plant behaviour interval on the length of longest base in each plant (axis of ordinates) depending on pollutant concentration (abscissa)
Table 4. Scatter of length and stasis of plant base, mm
C, mg/kg 
Maximum 
Minimum 
Scatter ΔL 
Stasis 

rank r_{min} 
L_{max} 
rank r_{max} 
L_{min} 
L_{Cmax} 
L_{Cmin} 

0 
0 
105 
9 
96 
9 
0 
0 
50 
0 
104 
8 
96 
8 
1 
0 
100 
0 
93 
11 
82 
11 
12 
14 
500 
0 
17 
9 
7 
10 
87 
89 
1000 
0 
11 
7 
3 
8 
94 
93 
Small nickel concentrations up to 30 ... 40 mg/kg exert positive influence to robust plant development and growth. And depauperate plants fast reduce their growth, but according to increase of nickel concentration in an interval 10 ... 250 мг/kg depauperate plants receive death highstress excitation. These conclusions are preliminary, as the scale of explanatory variable, accepted in the standard, is small. The additional researches by the rules of developmental experiments are necessary.
Scatter of length of base and stasis
The composite indexes obtained on the basis of table 1 data disregarding recurrings of the members of statistical sampling, are represented in table 4.
The interval between lines of the upper and under limits of confidence interval shows scatter of length of plant base ΔL. This parameter is equal to . Stasis L_{C} will be defined on a difference of a variable length L from greatest base length L_{0} for zero concentration of pollutant in soil that is from expression .
The change of a maximum rank will be defined under the biotechnical formula (fig. 6)
. (9)
Fig. 6. Graph of maximum rank of base length (axis of ordinates) from pollutant concentration (abscissa)
The disposition of points concerning contour shows insufficiency of observations in an interval 100 ... 500 mg/kg.
Scatter of base length (fig. 7) shows even more intuitive intense sparsity of nickel concentration significances:
. (10)
Fig. 7. Graph of scatter of base length ( axis of ordinates ) from pollutant concentration (abscissa)
The residuals after the formula (10) are capable to give wave component (fig. 8) because of that they is significant more error of measurements in ± 0,5 mm.
(11)
Fig. 8. Wave component graph of model of base length scatter (axis of ordinates) from pollutant concentration (abscissa)
Thus, for small concentrations pollutant offers vibrational perturbation in plant behavior. Therefore influencing on development and growth of barley base can be both positive, and negative depending on a soil nickel concentration change scale, accepted in experiments.
Then the general model of barley base length scatter change, depending on nickel concentration in soil, after repeated parametric identification in a software envelope CurveExpert1.3 will be defined by the trinomial equation of kind:
. (12)
Stasis of robust species
They are present in each subgroup (fig. 9) and are determined by the tendency (trend) of difference kind and wave component on formula:


a) 
b) 
Fig. 9. Behavior of robust barley species for stasis: a  on a trend with first two not wave components; b  on third wave component
. (13)
Dynamic factor is significant on a model (13) in an interval 0 ... 500 mg/ kg.
Stasis of depauperate species
With even greater dynamism perturbation influence of nickel concentration in soil (fig. 10) on growth depauperate barley species is exhibited:
. (14)


a) 
b) 
Fig. 10. Behavior of depauperate species for stasis on two and three components
Comparison of the graphs in a fig. 9b and the fig. 10b shows that depauperate species confront own death longer, than robust barley plant species. But behavior dynamism of robust species is greater. It is visible for comparison of the graphs under both full formulas (13) and (14) in a fig. 11.


a) 
b) 
Fig. 11. Adaptation of robust (a) and depauperate (b) plants to polluted soil
Stasis in plant subgroups
The subgroups from 18 plants conduct itself variously. The makers of an example from ISO 112691 have arranged results of monitoring so, that on significances of soil nickel concentration regularities were obtained by us for each plant subgroup under the formula (6), that is under the expression .
For first three significances of nickel concentration in soil 0, 50 and 100 mg/kg under the formulas of biotechnical regularities in a fig. 3 functional accuracy , where r  rank of distribution r =0,1,2,... was received.
The concentration levels in 500 (fig. 12a) and 1000 (fig. 12b) mg/kg have changed proportionality of small levels of the nickel contents in soil under the formulas:
; . (16)


a) 
b) 
Fig. 12. Stasis for nickel concentration 500 (a) и 1000 (b) mg/kg in soil
The plants for C =500 confront and change growth under the biotechnical law, and at a level C = 1000 mg/kg the aggravation of base growth occurs under the exponential law.
References
 Fomin, G.S. Soil. Quality and ecological safety control under the international standards. The directory.  М.: Publishing house «Protector», 2001.  304 p.
 Mazurkin, P.М. Statistical modelling. Heuristic mathematical approach / P.М. Mazurkin.  YoshrarOla: MarGTU, 2001.  100 p.
 Mazurkin, P.М. Geoecology: Regularities of modern natural sciences / P.М. Mazurkin.  YoshrarOla: MarGTU, 2006.  336 p.