Features of the Method of Iteration of Means in Studying Populations

Abstract

This article considers the features of the mean iteration method when studying populations the quantitative indicators of which have different statistical distributions. The studies were conducted in the central part of the Northwest Caucasus. The published and field data used by the authors came from eight sample plots laid out in the lower mountain, forest–steppe, and steppe zones of the central part of the region, where various indicators were studied for seven plant species. The sample size in the sample plots was 122–485 individuals. Statistical data processing was carried out using the Stadia8.0 and Microsoft Excel for Windows programs. The values of inter-iteration means and gradations were established using known and original methods. It was revealed that, with a normal statistical distribution, the inter-iteration means have close values (difference of 0–4.23%) with the values of the means increased by 0.5–2 standard deviations. To correct extreme inter-iteration values, where the sample is insignificant (1–4 observations), it is recommended to use forecast models, which should be calculated separately for each option. The allocation of gradations of quantitative traits in populations by the iteration of means method ensures an adequate distribution of indicators in three or five gradations compared to methods focused on the average increased by the value of the standard deviation or dividing the indicators into equal values.

INTRODUCTION

The study of populations, as well as the selection on this basis of the organisms most adapted to specific conditions, which is especially important for arid ecosystems, is largely associated with quantitative traits and their division into certain gradations (Kryuchkov and Stolnov, 2018; Singh and Samant, 2020). In this case, various methodological developments are used. In their selection, with special requirements, the researcher uses strictly defined ones (Zlobin, 2009; Metody…, 2015; Prikaz…, 2016), and in their absence chooses according to individual preferences, based on the availability of generally accepted or original methods of the author (Biganova et al., 2022; Zuhua et al., 2022; Hadson et al., 2023). The use of various methodological approaches is an important requirement in predictive multimodeling (Parrish et al., 2012; Cameron et al., 2022; Hadson et al., 2023). Depending on the goals and objectives, a significant part of the methods both in the study of populations and in the selection of promising highly productive individuals use three- or five-point gradations. They are distinguished on the basis of various approaches, among which an important place is occupied by the method of equal gradation value (Biganova et al., 2013; Programma…, 1999; Metody…, 2015; Prikaz…, 2016) or the value of the average, increased (decreased) by several values of the standard deviation (Tsarev et al., 2010). There are also other approaches, which include the method of iteration of means, which ensures the selection of the required set of objects regardless of the statistical distribution of the studied features (Sukhorukikh and Biganova, 2023). It uses the principle of universality of the mean (Shmoilova et al., 2004), the features of the standard deviation as a measure of variability of features (Lakin, 1990), and inter-iteration means as an element of a generalized model. In the literature, there is only one report on its application for the selection of a promising gene pool. Other features of the method have not been studied. Considering the importance of understanding the features of various methodological developments for population studies (Zlobin, 2009; Metody…, 2015), identifying the necessary gene pool to increase landscape productivity (Prikaz…, 2016; Kryuchkov and Stolnov, 2018; Kryuchkov et al., 2023), creating forecast models (Parrish et al., 2012; Cameron et al., 2022), the features of the above method require separate study.

The purpose of this work is to identify the features of the iteration method of means in the study and division into gradations of quantitative indicators of populations. To achieve this goal, the problems of calculating the inter-iteration averages of the studied indicators were solved; their comparison with the values of the arithmetic mean, increased (decreased) by 0.5–2.5 standard deviations, and forecast models were calculated to adjust the inter-iteration averages, and the quantitative indicators were divided into equal values.

MATERIALS AND METHODS

This work is a continuation of the studies previously published by the authors. It uses field materials and individual previous literature data from seven trial plots (TP 1—TP 5, TP 7, and TP 8) where the traits had a normal distribution (Sukhorukikh and Biganova, 2023). The main indicators have been calculated and published for the first time. The trial plots were located on the territory of the Republic of Adygea and adjacent areas under various conditions. In the mountain forest zone, on TP 1, the overall fruit quality score was studied for 485 bushes; on TP 2, the average fruit weight was studied for 217 plants; on TP 6, the shell thickness was studied for 138 individuals of common hazel (Corylus avellana L); on TP 7, the height was studied for 167; and on TP 8, the growth was studied for 163 three-year-old walnut seedlings. In the forest–steppe zone, the average height was determined in shelterbelts, at TP 3 for 124 lanceolate ash trees (Fraxinus lanceolata B.), at TP 4 for 122 ramets of common oak (Quércus róbur L.), and in the steppe zone at TP 5, of the trunk circumference at a height of 1.3 m for 159 black walnut trees (Juglans nigra L.) (Sukhorukikh and Biganova, 2023). Since various terms are used in known methods for population studies, the allocation of a promising gene pool (large—small, high—low, etc.), to exclude discrepancies when dividing sets of indicators, we used the term “gradation.”

The distribution of the data array into three gradations using different methods is as follows.

Method 1. The values were established taking into account the arithmetic mean (average) (Х_mean1), standard deviation (σ), and maximum (X_max) and minimum (X_min) values of the feature for the entire array (Tsarev et al., 2010). Gradation A (towards decreasing feature)—(Х_mean1 – σ; X_min), gradation B (average)—(Х_mean1 ± σ), gradation C (towards increasing feature) (Х_mean1 + σ; X_max).

Method 2. The values of inter-iteration means were taken into account towards increasing (Х_{mean k}); decreasing feature () at iteration steps (Sukhorukikh and Biganova, 2023). Gradation A—less (( + )/2), Gradation B—(( + )/2; (Х_mean2 + Х_mean3)/2), Gradation C—more ((Х_mean2 + Х_mean3)/2).

Method 3. Based on the uniform division of the difference between the minimum and maximum values of the indicators into equal values (R) in gradations (Zlobin, 2009; Metody…, 2015). For three gradations, the difference is divided by three, and for five, into five parts (n), respectively R = (X_max – X_min)/n.

When dividing into five gradations for method 1, we took into account the change in average indicators in the population within (Х_mean1 ± σ) and the features of the allocation of a promising gene pool, according to which its value should be equal to or more than (Х_mean1 ± 2σ) (Tsarev et al., 2010). For method 2, the average values are the same as for three gradations and the allocated promising gene pool is more than or equal to (Х_mean4 + Х_mean5)/2 (Sukhorukikh and Biganova, 2023). Accordingly, for methods 1 and 2, gradations A and C are divided into two (indices “a” and “b”) parts.

Method 1. In the direction of decreasing the trait, gradation Aa is less than or equal to [Х_mean1 – 2σ], gradation Ab is [less than Х_mean1 – σ; more than Х_mean1 – 2σ]. Towards an increase in the feature, the Cb gradation is [more than Х_mean1 + σ; less than Х_mean1 + 2σ], the Ca gradation is equal to or greater than [Х_mean1 + 2σ].

Method 2. Towards a decrease in the feature, the Aa gradation is less than or equal to , the Ab gradation is more than ; less than . Towards an increase in the feature, the Cb gradation is more than [(Х_mean2 + Х_mean3)/2]; less than [(Х_mean4 + Х_mean5)/2], and the Ca gradation is more than or equal to [(Х_mean4 + Х_mean5)/2].

Statistical data processing was performed using the Stadia 8.0 program for Windows and Microsoft Excel.

RESULTS AND DISCUSSION

Based on the iteration of means, the corresponding inter-iteration means for TP 1–5 were calculated. Their values are presented in Table 1.

Table 1.  . Values of inter-iteration average indicators on TP 1–5

TP	Mean/ standard deviation	Inter-iteration means in the direction of increase/decrease in the values of the feature
1
2
3
4
5
Number of inter-iteration averages (In) in the direction of increasing/decreasing the value of the feature
	0	1/–1	2/–2	3/–3	4/–4	5/–5

Columns 2, 6, 8 contain extracts from the authors’ work (Sukhorukikh and Biganova, 2023).

As follows from the data obtained (Table 1), the inter-iteration means increase (decrease) with an increase in iterations and do not exceed the maximum (minimum) value of the features. With the sample sizes of TP 3 (124) and TP 4 (122 frames), it was impossible to calculate the mean for the extreme iteration, since this data array contained the only maximum value of the indicator.

The value of the mean, increased (decreased) by 0.5–2.5 standard deviations for TPs 1–5 is presented in Table 2.

Table 2.  . Values of the average indicators of TP1–5, increased (decreased) by 0.5–2.5 standard deviations

TP	Values of Xmean1, increased/decreased by 0.5–2.5 σ
1
2
3
4
5
	Number of mean Xmean1, increased/decreased by 0.5–2.5 standard deviations
	1/–1	2/–2	3/–3	4/–4	5/–5

From Table 2 it follows that the calculated minimum values of Х_mean1 – 2.5σ for TP 2 and TP 3 are equal to the actual minimum of the samples. In TP 4, the calculated Х_mean1 + 2.5σ exceeds the recorded maximum by 0.86%, which may be due to the small sample size, 122 observations, or rounding errors. The remaining calculated values do not go beyond the recorded maximum (minimum) value of the indicators, which is consistent with the theory of normal distribution of features.

Based on the data from Tables 1 and 2, a comparison was made of the inter-iteration means with the corresponding values of the average Х_mean1, increased (decreased) by 0.5–2.5 σ. The results are presented in Table 3.

Table 3.  . Comparison of the values of inter-iteration average indicators TP1–TP5 with X_mean1, increased (decreased) by 0.5–2.5 standard deviations

TP	Differences of inter-iteration averages from the average, increased/decreased by the standard deviation, %
1
2
3
4
5

Numerator S₁ — ((X_mean1 + X_mean2)/2)/(X_mean1 + 0.5σ) … S₅ – ((X_mean5 + X_mean6)/2)/(X_mean1 + 2.5σ); denominator D₁ — ((X_mean1 + )/2)/(X_mean1 + 0.5σ) … D₅ — ((X_{mean 5} + )/2)/(X_mean1 + 2.5σ).

As follows from the data in Table 3, the difference between the values of inter-iteration means from (Х_mean1 + Х_mean2)/2 to (Х_mean4 + Х_mean5)/2 from the corresponding values of the mean, increased (decreased) by a different number of the standard deviation by the modulus in the direction of increasing indicators changed from 0 to 2.76, and in the direction of decreasing, 0.12–4.23%. This indicates the great consistency of methods 1 and 2 and the possibility of using method 2 without adjustment to identify gradations within these limits. For inter-iteration means (Х_mean5 + Х_mean6)/2, corresponding to 2.5σ, the difference was 0.99—18.39%. This is due to the small number of individuals (2–4 pcs.) in this interval. Therefore, to identify these gradations, it is necessary to adjust the values based on the forecast models, which are calculated from the available values of the previous inter-iteration means.

The models calculated from them for the objects under study have the following forms.

1

2

3

4

5

where y is the value of the inter-iteration averages and I_n is the number of the inter-iteration average.

The missing value calculated using the models for TP 3 (Х_mean5 + Х_mean6)/2 (Table 1) was 21.56 m, and the difference from the calculated Х_mean1 + 2.5σ was 0.97%; for TP 4 , it was 12.03 m, and the difference from the calculated Х_mean1 – 2.5 σ was 3.73%. The obtained data indicate the adequacy of the forecast method for calculating missing data in extreme gradations.

Pairwise statistical analysis of the slope coefficients of forecast models 1–5 did not reveal any similarity between them. This indicates that such models should be calculated separately for each case.

To study the features of the iteration method of means, the division of various indicators (PP6–8) into three and five gradations was considered (Tables 4, 5). These sample plots have a statistical distribution that differs from the normal one and are characterized by the following statistical indicators. For TP 6 (Х_mean1, 1.26 ± 0.02 mm; σ, 0.24 mm; V, 19.29%; As, 0.19), TP 7 (Х_mean1, 135.75 ± 3.45 cm; σ, 44.57 cm; V, 32.83%; As, –0.61), TP 8 (Х_mean1, 20.30 ± 1.31 cm; σ, 16.68 cm; V, 82.15%; As, 1.25).

Table 4.  . Number and values of indicators TP 6–8 when divided into three gradations

Method	Gradation A	Gradation B	Gradation C
Hazelnut shell thickness (TP 6)
1
2
3
Height of 3-year-old walnut seedlings (TP 7)
1
2
3
Annual growth of walnut shoots (TP 8)
1
2
3

Table 5.  . Number and values of indicators TP 6–8 when divided into five gradations

Method	Gradation Aa	Gradation Ab	Gradation B	Gradation Cb	Gradation Ca
Hazelnut shell thickness (TP 6)
1
2
3
Height of 3-year-old walnut seedlings (TP 7)
1					–
2
3
Annual growth of walnut shoots (TP 8)
1	–
2
3

Analysis of the data from Tables 4 and 5 shows that when dividing into three gradations using method 1, the average value of the indicators is located in the corresponding gradation B. At the same time, the deviation of the average in this gradation from the average for the entire sample increased from 1.59 to 26.35% with an increase in asymmetry from 0.19 to 1.25 and variation coefficients within 19.29–82.15%. This indicates the instability of the method. When dividing into five gradations, an increase in the asymmetry and the variation coefficient on the TP was accompanied by the impossibility of identifying some gradations. Thus, on TP 7 with left-sided asymmetry, gradation Ca was absent and all indicators in the direction of increasing the indicator were in gradation Cb, and the calculated minimum value of gradation Ca had unrealistic values for this population of 224.89 cm or more (actual max, 195 cm). The absence of real values was also observed on TP 8, which has a right-sided asymmetry, where the calculated value for gradation 1c was negative, 13.06 cm or less. The results obtained for method 1 indicate its inadequacy in identifying five gradations.

Method 2, with an increase in asymmetry and variation coefficients on PP 6–8, demonstrated an insignificant deviation of the average gradation B from the average of the entire sample within 1.59–2.36%, which indicates its stability. In addition, in all cases, it ensured the allocation of the corresponding gradations.

Method 3 was characterized by instability of the sample mean with an increasing variation coefficient and asymmetry. When allocating three gradations on TP 6, the mean of gradation B differed from the sample mean by 6.35%, and on TP 7, by 51.47%, but the data were in the corresponding gradation B. On TP 8, the mean of gradation B (20.3 cm) shifted to the neighboring gradation A (limit 28 cm and less). When divided into five gradations, the mean of the TP 6 sample was in the corresponding gradation B, but its mean differed from the mean of the entire sample by 6.67%. With the increase of asymmetry and the coefficient of variation on TP 7, the sample mean (135.75 cm) shifted to the Cb gradation (limits 126–160 cm), and on TP 8, the sample mean (20.3 cm) shifted to the lower Ab gradation (limits 17.7–33.2 cm). This indicates the inadequacy of method 3 for identifying three or five gradations with significant asymmetry and the coefficient of variation.

The use of methods 1–3 demonstrated different numbers of individuals in the same gradations, with the exception of cases on TP 6, gradations C, Aa, and Ca. The greatest difference in the number of individuals was observed on TP 8, where in gradation Ab the difference between methods 2 and 3 was 11.75 times.

CONCLUSIONS

(1) The values of inter-iteration means do not exceed the maximum (minimum) values in the studied populations.

(2) With a normal statistical distribution, inter-iteration means in comparison with the arithmetic mean, increased (decreased) by 0.5–2.5 standard deviations have close values of the indicators.

(3) Forecast models for determining the values of inter-iteration means must be calculated separately for each data array.

(4) When identifying gradations and selecting by values of required objects in populations the statistical distribution of indicators of which differs from normal, the use of the iteration method of means gives more adequate results compared to methods oriented towards the arithmetic mean increased (decreased) by the values of the standard deviation or dividing the indicators into equal parts of the population being studied.

FUNDING

This work was carried out within the state assignment of Ministry of Science and Higher Education of the Russian Federation.

ETHICS APPROVAL AND CONSENT TO PARTICIPATE

This work does not contain any studies involving human or animal subjects.

CONFLICT OF INTEREST

The authors of this work declare that they have no conflicts of interest.

Translated by A. Deryabina

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