Description of the experimental area

This municipality is located 17 km south of Toluca. The municipal seat is located at 19° 09’ 30’’north latitude and 99° 37’17’’ west longitude, at 2 650 meters above sea level. Its subhumid temperate climate has an annual average of 12.8 °C, minimum and maximum of -4 and 26 °C. Its Andosol soils are derived from volcanic ash, acidic pH (between 3.8 and 6) and rest on a volcanic substrate of loose pumicite (tepojal), between 40 and 100 cm. It is made up of San Diego La Huerta (SD), Cabecera Municipal (CM), San Lorenzo Cuauhtenco (SL), Santa María Nativitas (SM), Zaragoza de Guadalupe (ZG), San Marcos de la Cruz (SMC), San Bartolito, San Andrés Ocotlán and Concepción Coatipa (^{Ramos and Gerón, 1998}; ^{González et al., 1999}; ^{González et al., 2006}).

Genetic material

In this study, the 30 producers were considered shown in Table 1.

Preliminary sample size

It was considered that: a) the value of N is known, but there is no information on n; b) random sampling without replacement will be applied to record simple statistics in r quantitative variables; c) the six locations are representative of that municipality; and d) the choice of plants with complete competence was made in 400 m² of the center of one ha.

The first methodology was proposed by ^{Rendon and Cervantes (1991)} and is based on:

(i = 1, 2, 3 ,…, r)

Where: N and n are farmers and plants and corn considered, bi fluctuates from zero to one and is the proportion of the standard deviation (σ_i) for each of r variables (the smaller the more reliable, but the demand for more time and human resources and financial). This equation does not depend on unknown parameters and because of its simplicity it can be used to generate tables of n at different values of N and bi; its application leads to:

a) For N= 2 350 farmers (^{González et al., 1999}) and bi= 0.2

b) In commercial planting of 50 000 ha^-1 plants

N and n were approximate to 30 farmers and 50 plants and corn per farmer (almost 21 000 data) and the bi values used in the preliminary sampling were 0.05, 0.075 and 0.1.

Preliminary calculation of n with InfoStat

In this, the value is obtained to estimate a population average or proportion with a confidence of precision determined by the user. It also allows calculating n to detect, in a variance analysis of a fixed effects model in monofactorial experiments, a difference between means of groups or populations as small as specified by the user, as well as the sample size to estimate the difference between two populations (^{Balzarini et al., 2008}; ^{Di Rienzo et al., 2008}). To estimate n the following is used.

Where: σ is the standard deviation, c is the amplitude required for the confidence interval (1-α) % for the population mean, c can be chosen arbitrarily or expressed as a fraction ‘f ’ of the sample mean (c= Ȳf) . In this study f took values of 0.05, 0.075 or 0.1.

Data register

They were quantified (^{CIMMYT, 1999}; ^{González et al., 2011}): female flowering (FF; days), plant and corn heights (AP, AM; cm), stem diameter (DT; cm), length (LR), diameter (DR) and rachis weight (PR) (cm, cm, g), length (LE; cm) and corn diameter (DE; cm), rows of grain (NH), corn weight (PE; g) and quality leaf for wrapping tamales (NHT).

Statistical analysis

Simple statistics were obtained with the statistical analysis system (SAS) version 6.03 for Windows and with ^{InfoStat (2008)}. The procedures are described in Gómez and Gómez (1984). An analysis of main components was also made with 30 farmers and 12 variables and another with six locations and those variables (^{Sánchez, 1995}; ^{González et al., 2010}).

Simple statisticians

The female flowering (FF) varied from 85 to 125 days, its mean and range were 103.47 and 40 days. Male flowering and FF are positively and significantly correlated and so are with physiological maturity; they are a reliable indicator of the biological cycle of Cacahuacintle but differ in part from that published by ^{Arellano et al. (2010)}; ^{González et al. (2006}, ²⁰⁰⁸).

In Table 2 shows the heights of plant and corn (AP and AM) ranged from 1.26 to 3.9 m and from 1 to 2.9 m, their averages were 2.67 and 1.61 m. Both averages are similar to those recorded by ^{González et al. (2006)} and by ^{Arellano et al. (2010)}. The stem diameters (DT) varied from 2.5 to 4.6 cm, their average was 3.07 cm, these are primary components of the grain and forage yield in maize of the Mexican center (^{Franco et al., 2015}). In creoles, positive and significant correlations have been observed between them, but with higher plants there is more acame and cob rot and less grain yield (^{González et al., 2007}; ^{Arellano et al., 2010}).

The number of rows (NH) varied from 6 to 26, with an average of 12.08. In other NH studies, it ranged from 8 to 20, with an average of 14. Cacahuacintle cobs with six rows had not been detected and with only 26 in Palomero Toluqueño (^{González et al., 2006}; ^{González et al., 2008}). In the first case perhaps sterility or insects damaged two rows of grain.

The diameter of corn with leaves (DCH) varied from 4.60 to 9.70 cm, with an average of 7.16 cm. The corn diameters (DM) were located between 3.6 and 7.7 cm, with an average of 5.81 cm. Both are components of the corn’s visual quality; the best ones will be commercialized between $0.50 and just over $2.00 per piece (^{Aguilar et al., 2006}; ^{González et al., 2006}).

The number of grains in a row of corn (NG) ranged from 11 to 40, with an average of 24.75. Grains and cobs per plant and grain weight are reliable indicators of the productivity of a cultivar (^{Andrade et al., 2000}). ^{Arellano et al. (2010)} recorded between 0.63 and 1.28 cobs per plant in Cacahuacintle, with weights of 100 seeds between 33 and 67 g. With 12 rows per cob there would be between 132 and 480 grains. If 80% of 50 000 ha^-1 plants produce cob there would be between 5 280 000 and 19 200 000 grains, if all the grains of the cob are used as seed, they could be planted between 88 and 320 ha.

The number of quality leaves for wrapping tamales (NHT) ranged from 0 to 7, with an average of 2.92. In times of greatest demand, a profit of up to $3.00 per chosen plant is obtained. In this and other surrounding municipalities there are people specialized in exploiting this activity to supply the local markets of the Toluca Valley. There are no studies in this breed to determine whether NHT could be considered as an indirect selection criterion.

Spinal diameters (DR) ranged between 1.5 and 2.1 cm, with an average of 1.85 cm. The length of the spine (LR) varied from 3 to 30 cm, with an average of 13.09 cm. Spine weights (PR) differed from 5 to 90 g, with a central value of 27.2 g. It would be desirable to define whether these are important in the translocation of assimilated leaves and stems to the corn or the cob to use them as criteria for indirect selection in plant breeding and seed production.

The length of the corn (LE) ranged from 7.1 to 30 cm, its average was 20.38 cm. Corn weights with totomoxtle (PE) varied from 100 to 999 g, with an average of 530.44 g (Table 2). These results are mainly attributed to the efficient empirical improvement that farmers have made and the application of better technological packages. Both are very important to obtain a better price, in other regions of Mexico, a corn bought in the field costs from $1.00 to $2.50 at the time of greatest demand (^{Aguilar et al., 2006}; ^{Arellano et al., 2010}; ^{Coutiño et al., 2010}). Long cobs, 8 and 12 rows, would be used for the production of pozole and those between 14 and 16 would be marketed as corn (^{Aguilar et al., 1999}; ^{González et al., 2006}), but the corn of the Jala breed should be considered up to 60 cm (^{Aguilar et al., 2006}). In corn diameter, with and without totomoxtle, its dimensions are larger than those recorded in the other four races of High Valleys of Central Mexico (^{González et al., 2007}; ^{González et al., 2008}).

Principal component analysis

The results observed in Figure 1 confirm that in the municipality of Calimaya there are very heterogeneous environmental conditions; there are altitude gradients from 2,610 to more than 3 000 m (^{Aguilar et al., 1999}; ^{González et al., 2006}; ^{Arellano et al., 2010}). These differences are not observable in the Tenango del Valle-Toluca-Atlacomulco region, except in its mountainous parts (^{González et al., 2008}, ²⁰¹¹). Calimaya’s agriculture is rainfed, so temperature, humidity, hail and frost, together with the heterogeneity of its soils, have contributed to its differentiation (^{Aguilar et al., 1999}; ^{González et al., 1999}). Although there are few, little genetic variability has been estimated in this breed, which supports the previous results (^{González et al., 2006}; ^{González et al., 2008}).

Figure 1 Biplot for cooperating producers (in number) of the municipality of Calimaya and variables evaluated (in letters) in Cacahuacintle corn. 

The other components that contribute to this differentiation are the empirical genetic improvement that the farmers have made, the diversity of technological packages that they apply and the different agronomic management they provide to their plots (^{Aguilar et al., 1999}; ^{González et al., 1999}). In trials conducted on farmers’ lands it has been estimated that the genetic variability in Cacahuacintle varied from 15.9 to 50.2% in plant and cob heights, length and rows of the cob and grain yield, but in cob diameter, pot weights and there were negative estimates of grain per plant and volumetric (^{González et al., 2006}).

Regarding the variables, several groups (G’s) were detected: in G1 FF and NH were identified, in G2 the stem, corn and corn leaf diameters were observed; in G3 the rachis and corn weights appeared. In G4, rachis length and diameter stood out, number of quality leaves for wrapping tamales, number of grains per row and length of corn, in G5, plant and cob heights were associated (Figure 1). ^{González et al. (2006)} concluded that grain yield had a greater relationship with plant and cob heights, and length, diameter, and weights of grain and pot per cob.

The previous results also suggest stratifying the sampling area to reduce the environmental differences that exist between locations, as ^{González et al. (2006)}; ^{González et al. (2008)}; ^{Arellano et al. (2010)}; ^{Romero et al. (2018)}.

The interrelations shown in Figure 2 have been verified in the Cacahuacintle creole evaluation experiments in Calimaya (^{González et al., 2006}; ^{González et al., 2008a}, ^b), but there are no antecedents for the dimensions of the spine. ^{González et al. (2011)} concluded that the dimensions of the spine were positively and significantly correlated with grain yield in maize of the Conical breed. These results also suggest that the collection of creoles where Cacahuacintle predominates could be used to form a balanced compound that could be used to conserve, increase and use its genetic diversity in situ, or cross these compounds with other breeds of flour grain, with subsequent improvement by selection and hybridization.

Figure 2 Biplot between communities of the municipality of Calimaya and variables evaluated in Cacahuacintle corn. 

Sample sizes proposed

With method 1, n= 396 when bi= 0.05; yes bi= 0.075 or 0.1, n= 175 or 100. With method 2, when bi= 0.05, n varies from 45 (FF) to 7588 (DR), but with bi= 0.075 or 0.1 should be evaluated between 20 and 3 372 or between 11 and 1897 plants or corn (Table 3). The previous results allow us to verify that method 1 is simpler. With it, the same number of plants and corn is sampled in each variable at the same value of bi, nor are population means and variances required, among other simple statistics (^{Rendón and Cervantes, 1991}; ^{Cochran, 1998}; ^{Pérez, 2005}).

However, for the characteristics that present little variability, it could lead to choosing very large sample sizes, such as in FF, AM, AP, NH, DCH, DM, and NG, since with the methodology 2 a value of n smaller is proposed to 60. If there are budgetary, time or qualified personnel restrictions, the values of n provided by both methodologies could be combined. Consideration should also be given to the fact that in that municipality the environmental conditions, genetic variability, technological packages and the different agronomic management that farmers apply will lead to variability, regardless of the method and technique of sampling (^{González et al., 2006}; ^{González et al., 2008}; ^{Arellano et al., 2010}).

The variables were defined in Table 1. E1, E2, E3, are the maximum permissible errors (%) for method 1; F1, F2, F3, are the products between the arithmetic mean with the values of bi, at 0.05, 0.075 and 0.1 respectively and correspond to method 2.

The calculation of sample sizes with other methodologies, based on the preliminary values presented in this study, could provide better unbiased estimates of population parameters, particularly if a stratification of the communities in the upper part is performed and considering the different technological packages that producers use. The use of genetic and experimental designs, for better control of environmental heterogeneity and to divide genetic variability, would also be very useful.