Biomathematics

Importance indice: los estimates and solution effectiveness on production. Technical Note

0000-0002-2928-3193Germano Leão Demolin-Leite¹ 

¹Instituto de Ciências Agrárias, Universidade Federal de Minas Gerais, Montes Claros, Minas Gerais State, Brazil.

Abstract

Frequencies, magnitudes, and distributions of occurrence can affect the events. The problem or the solution will be greater depending on the degrees of frequency and magnitude. Indices, hence, are used to assist in decision making on certain issues. The motivation of this study was to develop an indice able to identify the loss (L.S.) and solution (S.S.) sources, and their importances in terms of loss or income gain on system. The developed indice was: percentage of the importance indice (% I.I.) = [(ks ₁ x c ₁ x ds ₁ )/Σ (ks ₁ x c ₁ x ds ₁ ) + (ks ₂ x c ₂ x ds ₂ ) + (ks _n x c _n x ds _n )] x 100. The L.S.₁ and L.S.₇ were the most important on reducing production and S.S.₃ and S.S.₅ in increasing income gain on system. It is show the percentage of the importance indice (% I.I.), an indice, capable of detecting the key loss and solution sources on system, can be applied in some knowledge areas.

Key words: Abundance; agriculture; constancy; forestry production

Events (eg., agricultural pest) can have different magnitudes (numerical measurements), frequencies, and distributions (aggregate, random, or regular) of occurrence. In general, higher magnitude and frequency, with aggregated distribution, greater will be the problem or the solution (eg., natural enemies versus pests) on system (^{Da Silva et al. 2017}). Hence, indices are used to help on decision-making in certain questions and, many of them, determine key-factors in an event, on some knowledge areas, such as in agrarian and biological: Crop and Ecological Life Tables (^{Henderson and Southwood 2016} and ^{Da Silva et al. 2017}), among others. In general, these tools use abundance (magnitude), constancy and/or frequency of the events, which can be analyzed by correlation, factor analysis, frequency distribution, matrices, mean or t-tests, multiple or simple regression analysis (^{Henderson and Southwood 2016} and ^{Da Silva et al. 2017}). The objective of this study was to develop an indice, which can determine the loss and solution sources, classifying them according to their importance in terms of loss or income gain on system (eg. natural system = cerrado).

The data used were adapted (^{Leite et al. 2006}, ²⁰¹², ²⁰¹⁶, ²⁰¹⁷) and classified as loss source (L.S.) or solution source (S.S.) are not mentioned, and production, in 48 samples. Scientific names of herbivorous insects (L.S.) and natural enemies (S.S.) due to the importance indice can be used in other areas such as exotic mammals, plant diseases, weeds, versus production.

The type of distribution (aggregated, random, or regular) of L.S. or S.S. was defined by the Chi-square test using the BioDiversity Professional program, version 2 (^{Krebs 1989}). The data were subjected to simple regression analysis and theirs parameters were all significant (P< 0.05) using the statistical program System for Analysis Statistics and Genetics (^{SAEG 2007}), version 9.1 (table 1). Simple equations were selected by observing the criteria: i) distribution of data in the figures (linear or quadratic response), ii) the parameters used in these regressions were the most significant ones (P <0.05), iii) P < 0.05 and F of the Analysis of Variance of these regressions, and iv) the coefficient of determination of these equations (R²). Only loss sources and solution sources with P < 0.05 were showed in the table 1. It is necessary knowledge of the system to select the possible loss sources and solution sources.

Table 1 Aggregated, regular, or random distribution of the loss or solution sources; and simple regression equations with their coefficients of determination (R ²), significance (P) and F of the analysis of variance (ANOVA) of reductions of production (R.P.) by source of loss (L.S.) and reductions of loss sourcers (R.L.S.) due solution sources (S.S.). n = 48 

Source	Qui-square test
Loss	Variance	Mean	Chi-square	d.f.	P	Distribution
1	177.45	16.5	505.45	47	0.000	Aggregated
2	93.45	20.54	213.81	47	0.000	Aggregated
3	0.25	0.46	26.00	47	0.994	Regular
4	0.33	0.58	26.86	47	0.992	Regular
5	1050.97	37.08	1332.02	47	0.000	Aggregated
6	19.38	1.67	546.40	47	0.000	Aggregated
7	4936.34	29.00	8000.28	47	0.000	Aggregated
Solution
1	57.66	11.71	231.45	47	0.000	Aggregated
2	1.53	1.50	48.00	47	0.432	Random
3	50.21	7.50	314.67	47	0.000	Aggregated
4	0.55	0.71	36.59	47	0.863	Random
5	1.57	1.04	70.96	47	0.014	Aggregated
6	3.77	0.75	236.00	47	0.000	Aggregated
7	0.20	0.13	74.00	47	0.007	Aggregated
8	140.50	7.58	870.81	47	0.000	Aggregated
9	193.33	6.83	1329.76	47	0.000	Aggregated
Simple regression analysis				ANOVA
				R^2	P	F
R.P. = - 39.43 + 33.26 x L.S.1 - 0.80 x L.S.1 ^2				0.61	0.0000	35.25
R.P. = 50.85 + 1404.77 x L.S.7 - 2242.16 x L.S.7 ^2				0.20	0.0060	5.75
R.L.S.1 = - 0.46 + 5.13 x S.S.3 - 0.21 x S.S.3 ^2				0.99	0.0000	7312.19
R.L.S.7 = 0.13 + 0.46 x S.S.4 - 0.18 x S.S.4 ^2				0.39	0.0000	14.15
R.L.S.7 = 0.11 + 0.26 x S.S.5 - 0.04S.S.5 ^2				0.53	0.0000	25.63
R.L.S.7 = 0.21 + 0.16 x S.S.6 - 0.01 x S.S.6 ^2				0.27	0.0007	8.50
R.L.S.7 = 0.10 + 0.04 x S.S.8 - 0.0006 x S.S.8 ^2				0.71	0.0000	55.10
R.L.S.7 = 0.15 + 2.94 x S.S.9 - 3.71 x S.S.9 ^2				0.44	0.0000	17.89

The developed indice was:

$\% I.I.\mathrm{ }=\left\{\right(k{s}_1\mathrm{ }\mathrm{x}\mathrm{ }{c}_1\mathrm{ }\mathrm{x}\mathrm{ }{ds}_1)/\mathrm{\Sigma }\mathrm{ }({ks}_1\mathrm{ }\mathrm{x}\mathrm{ }{c}_1\mathrm{ }\mathrm{x}\mathrm{ }{ds}_1)\mathrm{ }+\mathrm{ }({ks}_2\mathrm{ }\mathrm{x}\mathrm{ }{c}_2\mathrm{ }\mathrm{x}\mathrm{ }d{s}_2)\mathrm{ }+\mathrm{ }(k{s}_n\mathrm{ }\mathrm{x}\mathrm{ }{c}_n\mathrm{ }\mathrm{x}\mathrm{ }{ds}_n\left)\right\}\mathrm{x}100$

where,

i) key source (ks) is:

$Ks=\mathrm{ }\&\#091;\left({\mathbf{R}}^2\mathrm{ }\mathrm{x}\mathrm{ }\right(1\mathrm{ }-\mathrm{ }\mathbf{P})\&\#093;/\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{ }n\mathrm{ }\mathrm{o}\mathrm{n}\mathrm{ }\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{ }\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s},$

where,

R² = determination coefficient and P = significance of ANOVA, of the simple regression equation of the loss source (L.S.) or solution source (S.S.).

In the case of L.S. is:

$ks\mathrm{ }=\mathrm{ }\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{ }\mathrm{o}\mathrm{n}\mathrm{ }\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{ }(R.P.)/n,$

where,

R.P. = [R ² x (1 - P)]/total n of the L.S. on the samples,

In the case of S.S. is:

$ks\mathrm{ }=\mathrm{ }\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{ }\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{ }(E.S.)/n,$

where,

E.S. = [R ² x (1 - P)]/total n of the S.S. on the samples.

When a S.S. acts on more than one L.S., theirs E.S. are summed. E.S. or R.P. = 0 when E.S. or R.P. is non-significative on the L.S. or R.P., respectively, and

ii) constancy (c) is:

$c\mathrm{ }=\mathrm{ }\mathrm{\Sigma }\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{o}\mathrm{c}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{L}.\mathrm{S}.\mathrm{ }\mathrm{o}\mathrm{r}\mathrm{ }\mathrm{S}.\mathrm{S}.\mathrm{ }\mathrm{o}\mathrm{n}\mathrm{ }\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{ }\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s},$

where,

absence = 0 or presence = 1, and

iii) distribution source (ds) is:

$ds =\mathrm{ }1\mathrm{ }-\mathrm{ }\mathbf{P}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{ }\mathrm{C}\mathrm{h}\mathrm{i}-\mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{ }\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{L}.\mathrm{S}.\mathrm{ }\mathrm{o}\mathrm{r}\mathrm{ }\mathrm{S}.\mathrm{S}.\mathrm{ }\mathrm{o}\mathrm{n}\mathrm{ }\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{ }\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s}.$

Percentage of loss of production per loss source (% L.P.L.S.) is:

$\% L.P.L.S. =\mathrm{ }(L.P.L.S./P.)\mathrm{ }\mathrm{x}\mathrm{ }100,$

where,

P. = total production on the system,

and

$L.P.L.S.\mathrm{ }=\mathrm{ }(n\mathrm{ }\mathrm{x}\mathrm{ }R.P.L.S.)/\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{ }\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s},$

where,

R.P.L.S. = {R ² x (1 - P)]/total n of L.S. on the samples.

Percentagem of loss of production per loss source (% L.P.L.S.) per soluction source (S.S.) is:

$\% L.P.L.S.\mathrm{ }=\mathrm{ }\left\{\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{ }\mathrm{g}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{ }\right(I.G.\left)\mathrm{ }\mathrm{x}\mathrm{ }100\right\}/\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{ }\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{ }\mathrm{b}\mathrm{y}\mathrm{ }\mathrm{L}.\mathrm{S}.,$

where,

I.G. = {total production (P.) x reduction of the L.S. by S.S. (R.L.S.)] x total n of the S.S on the samples,

and

$R.L.S.\mathrm{ }=\mathrm{ }\&\#091;\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{ }n\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{ }\mathrm{S}.\mathrm{S}.\mathrm{ }\mathrm{x}\mathrm{ }ks\mathrm{ }\left\{{\mathbf{R}}^2\mathrm{ }\mathrm{x}\mathrm{ }\right(1\mathrm{ }-\mathrm{ }\mathbf{P}\left)\right\}\&\#093;/\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{ }\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s}.$

The ks of the S.S. are separeted per L.S..

Interaction between two or more sources of loss or solution may be added as a treatment to be tested together with the other sources. If not, the interaction, as a treatment, may apply the following:

1. ks of the interaction = [(R ² x (1 - P)]/total n on the samples, R ² = determination coefficient and P = significance of ANOVA of the interaction, of the simple regression equation of the loss source (L.S.) or solution source (S.S.) of the interaction. But the new n of the interaction will be obtained from the means of this parameter isolated from the two or more sources of loss or solution,

2. c and ds of the interaction will be obtained from the means of these parameters isolated from the two or more sources of loss or solution, and

3. all calculations are made separately for the interaction and at the end it is compared with the other sources of loss or solution.

The loss source (L.S.) L.S.₁ and L.S.₇ showed, among the seven L.S., the % I.I. (85.06 and 14.94%, respectively) significatives on production reduction (5.89 and 3.37%, respectively), on system (tables 2, 3).

Table 2 Total number (n), reduction on production (R.P.), effectiveness of the solution (E.S.), key-source (ks), constancy (c), distribution source (ds), number of importance indice (n. I.I.), sum of n. I.I. (Σ n. I.I.), and percentage of I.I. by loss source (L.S.) or solution source (S.S.) by L.S 

Loss source
L.S.	n	R.P.	ks	c	ds	n. I.I.	Σ n. I.I.	% I.I.
1	792	0.6100	0.000770202	38	1.000	0.029267677	0.034409056	85.058
2	986	0.0000	0.000000000	48	1.000	0.000000000	0.034409056	0.000
3	22	0.0000	0.000000000	22	0.006	0.000000000	0.034409056	0.000
4	28	0.0000	0.000000000	26	0.008	0.000000000	0.034409056	0.000
5	1780	0.0000	0.000000000	46	1.000	0.000000000	0.034409056	0.000
6	80	0.0000	0.000000000	10	1.000	0.000000000	0.034409056	0.000
7	1392	0.1988	0.000142816	36	1.000	0.005141379	0.034409056	14.942
Solution source
S.S. not associated with any L.S. or associated with L.S.2-6
S.S.	n	E.S.	ks	c	ds	n. I.I.	Σ n. I.I.	% I.I.
1	562	0.000	0.000000000	48	1.000	0.000000000	0.000000000	0.000
2	72	0.000	0.000000000	38	0.568	0.000000000	0.000000000	0.000
7	7	0.000	0.000000000	8	0.993	0.000000000	0.000000000	0.000
L.S.1
3	360	0.990	0.002750000	38	1.000	0.104500000	0.104500000	100.00
L.S.7
4	34	0.39	0.011470588	26	0.134	0.040726564	0.529809273	7.687
5	51	0.53	0.010392157	28	0.986	0.287031585	0.529809273	54.176
6	36	0.270	0.007494750	14	1.000	0.104926500	0.529809273	19.805
8	365	0.710	0.001945205	32	1.000	0.062246575	0.529809273	11.749
9	328	0.440	0.001341463	26	1.000	0.034878049	0.529809273	6.583

I.I. = ks x c x ds. ks = R.P./n or E.S./n. R.P. or E.S. = R ² x (1 - P), R ² = determination coefficient and P = significance of ANOVA, of the simple regression equation. c = Σ of occurrence of L.S. or S.S. on each sample, 0 = absence or 1 = presence. ds = 1 - P of Chi-square test of the L.S. or S.S.. When a S.S. operates in more than one L.S., its E.S. are summed. R.P. or E.S. = 0 when R.P. or S.S. non-significant with reduction on production or of the L.S.

Table 3 Total number (n) and reduction on production per loss source (R.P.L.S.), total samples (Sa.), loss of production (L.P.) by loss source (L.P.L.S.) and production per sample (P.), and % of L.P.L.S. per sample; and total number (n) and ks of the solution source (S.S.), reduction of L.S. (R.L.S.), income gain (I.G.) and its %, and % of R.P.L.S. by S.S 

Loss of production by loss source
L.S.			n	R.P.L.S.	Sa.		L.P.L.S.	P.	% L.P.L.S.
1			792	0.61	48		10.07	171	5.89
7			1392	0.1988	48		5.77	171	3.37
Reduction on production per loss source and total
L.S.1
S.S.	n	ks	Sa.	R.L.S.	L.P.	P.	I.G.	% I.G.	% R.P.L.S.
3	360	0.99	48	7.425	10.07	171	0.208	0.122	2.063
Σa	---	---	---	---	---	---	---	0.122	2.063
L.S.7
4	34	0.39	48	0.276	5.77	171	0.047	0.027	0.813
5	51	0.53	48	0.563	5.77	171	0.064	0.037	1.104
6	36	0.27	48	0.202	5.77	171	0.032	0.019	0.562
8	365	0.71	48	5.399	5.77	171	0.085	0.050	1.479
9	328	0.44	48	3.007	5.77	171	0.053	0.031	0.917
Σb	---	---	---	---	---	---	---	0.165	4.875
Σa+b	---	---	---	---	---	---	---	0.287	6.934

L.P.L.S. = (n x R.P.L.S.)/Sa. % L.P.L.S. = (L.P.L.S./P.) x 100. R.L.S. = (n x ks)/Sa.. I.G. = (P. x R.L.S.) x n. S.S.. % I.G. = (I.G. x 100)/P.. % R.P.L.S. = (I.G. x 100)/L.P. Ks of S.S. are separated by L.S.

Solution source (S.S.) S.S.₃ (% I.I. = 100) reduced the loss per L.S.₁; and S.S.₅ (% I.I. = 54.18), S.S.₆ (% I.I. = 19.81), S.S.₈ (% I.I. = 11.75), S.S.₄ (% I.I. = 7.69), and S.S.₉ (% I.I. = 6.58) that of L.S.₇ on system production. The possible solution sources S.S.₁, S.S.₂, and S.S.₇ showed % I.I. = 0.00% due to non-significative effect on the reduction of losses by important L.S. or due to reduced the L.S. which did not correlate with production loss on system. The S.S.₃ reduced production loss (2.06%) per L.S.₁ increasing in income gain (0.12%) on system production. The loss of production per L.S.₇ was reduced by the S.S.₈ (1.48%), S.S.₅ (1.10%), S.S.₉ (0.92%), S.S.₄ (0.81%), and S.S.₅ (0.56%), totaling 4.88%. The loss reduction per L.S.₇ due to the soluction factors S.S.₈, S.S.₅, S.S.₄, S.S.₉, and S.S.₆, increasing in income gain (0.05, 0.04, 0.03, 0.03, and 0.02%, respectively), totaling 0.17%. The total reduction in production loss due to loss sources (L.S.₁ and L.S.₇) was 6.93%, with an increase on system productivity of 0.29% due to solution sources cited above (tables 2, 3).

The percentage of importance indice (% I.I.) was effective in identifying of loss sources on system (eg., reduction on production), being simpler than a Crop Life Table (^{Da Silva et al. 2017}), but this indice does not replace a Crop Life Table. The use of % I.I. is for cases (eg. natural system, cerrado) in which it is not possible to evaluate all flowers and fruits of all plants in the experimental useful plot, identifying the factors of plant loss, as done by Crop Life Table (^{Da Silva et al. 2017}). Parameters of Life Table supply reliable information, eg. reproductive potential and mortality factors of species (^{Henderson and Southwood 2016}). Fruit production and arthropods (leaves, flowers, and fruits) data, used to test % I.I., were obtained on Caryocar brasiliense Camb. (Caryocaraceae) trees, over 3 m high, randomly, in cerrado areas, in three years, monthly (^{Leite et al. 2006}, ²⁰¹², ²⁰¹⁶, ²⁰¹⁷). Flowers and fruits were evaluated on some tree branches and then estimated the total per tree (^{Leite et al. 2006}), thus, the use of this indice is for cases where it is not possible to use a Crop Life Table.

The % I.I. was, also, effective in identifying solution sources on system (eg., increasing production), similar to an Ecological Life Table (^{Henderson and Southwood 2016}). The % I.I. does not replace an Ecological Life Table (Henderson and Southwood 2016). The use of % I.I. is for cases (eg. natural system, cerrado) in which it is not able to mark and monitor the animal (eg., pest insects), identifying the cause of its mortality, as done by Ecological Life Table (^{Henderson and Southwood 2016}). Insect pest rearing, detailed field studies, time and researchers trained to identify and quantify the control of natural factors daily until the insect pest life cycle is complete, are the major steps to determine the parameters of a Life Table of pest insects (^{Da Silva et al. 2017}). The evaluation of herbivorous insects and their natural enemies, including spiders, on C. brasiliense trees, was not individually during their lives (^{Leite et al. 2012}, ²⁰¹⁶, ²⁰¹⁷), nor would it be possible due to the height of these plants in cerrado areas. But, with the application of this indice, it was possible to determine the effects of these natural enemies on herbivores and fruit production per tree on natural system.

The % I.I. separated the loss sources (eg., L.S._{1 =} 85.06%) on production reduction (eg., 5.89%) and the solution sources (eg., S.S.₅ = 54.18%) with total income gain (eg., 0.29%) on system, with the possibility to calculate, monetarily, these losses or effectiveness of the solutions. The % I.I. can help, as example, to determine which pests, eg. exotic mammals, insects, plant diseases, and weeds, cause the biggest problems in plant production and the best control methods (eg., biological control) are more harmful or effective on system (eg., crops) and how much money is lost or saved. Here it is shown the percentage of I.I. is an indice to detect the loss or solution key-sources on a system, doing it possible to obtain of loss and income gain on some knowledge areas.

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Received: November 25, 2020; Accepted: April 15, 2021

Email: [e-mail]

Conflicts of interest: The author declares that there are no conflicts of interests

Author contributions: Original idea, experimental design, conducting the experiment, writing the manuscript