Assessment of the significance of differences in field experiments using Student's t-test

To determine the significance of values between two mean trait values, Student's t-test is used. Let the number of grains per ear of the winter wheat cultivar Pobeda-50 in the N50P50K50 variant be 33.7±1.28, and without fertilizer be 27.3 ± 1.16. It is necessary to determine the significance of the differences of this trait in the two experimental variants. For this purpose, one can use the formula: x1 - x 2 t,

S2  S2 х1 х2 where: t is the Student's t-test; x 1 and x 2 are the mean values of the number of grains per ear obtained in the two variants;

S x 1 and S x 2 are the errors of the mean values of the number of grains per ear obtained in the two variants.

Substituting the values into the formula:

33.7 - 27.3 6.4 6.4 6.4 3.7

2 2 1.64 + 1.35 2.99 1.73

The calculated value t = 3.7 was obtained. Using the table (which is present in all books on biometric statistics and in Appendix 2), we find the "t-test value" at the 5% significance level. For the analysis, we took 25 plants of the Pobeda-50 cultivar from each variant, which is n = 25. The number of degrees of freedom is determined by n-1. In our experiment, 25–1 = 24. With 24 degrees of freedom at the 5% significance level, t is equal to 2.06. In our experiment, the actual or calculated t is equal to 3.7. If the actual t is greater than the tabular t, then the differences between the two mean values of the plants regarding the number of grains per ear are statistically significant. This result can be verified using the method of one-way analysis of variance.

If several cultivars, lines, and hybrids grown on different backgrounds of mineral fertilizers were studied in the experiment, or if other factors were used (plant density in the plot, growth regulators, etc.), then to determine the significance of differences between traits, one can use Student's t-test. The final table may have the following form. If t fact > t tabl. – the differences between the variants for the cultivar are significant. And, conversely, if t fact < t tabl. – the differences are not significant.

Table 232 – Biometric indicators of rice cultivars grown at different rates of mineral fertilizers Mineral ferti- Plant height, cm Number of pan- Number of grains lizer rate, icles in the main in the main pan- Cultivar kg/ha x t panicle, pcs. icle, pcs. x t x t Liman N50P25K25 91.4 9.7*) 130.1 7.1 119.2 0.45 Liman N25P10K10 76.4 86.6 115.4 Lider N50P25K25 102.6 1.4 108.8 0.1 96.8 0.1 Lider N25P10K10 97.5 108.0 96.4 Druzhny N50P25K25 102.4 6.1 161.5 5.9 139.4 7.3 Druzhny N25P10K10 88.0 109.5 91.7 Khazar N50P25K25 95.0 3.5 146.1 4.4 137.6 5.8 Khazar N25P10K10 89.2 113.2 101.2 *) t tabl. = 2.09

1012 12.4.7. Analysis of variance

The main purpose of the analysis of variance is to study the significance of the difference between the mean values of traits. If only the mean values in two samples are compared, the analysis of variance will yield the same result as the standard Student's t-test for independent samples. If two independent groups of traits or observations are compared, then analysis of variance should be used. This is because, when investigating the statistical significance of the difference between the means of two (or several) groups, we are actually comparing (i.e., analyzing) sample variances. The fundamental concept of the analysis of variance was proposed in 1920 by R.A. Fisher. Perhaps a more natural term would be "sum of squares analysis" or "variation analysis," but by tradition, the term "analysis of variance" is used.

For a sample of volume n, the sample variance is calculated as the sum of squares of deviations from the sample mean, divided by n-1 (sample size minus one). Thus, at a fixed sample volume n, the variance is a function of the sum of squares (of deviations), abbreviated as SS from the English words "sum of squares". Further on, we often omit the word "sample," keeping in mind that we are considering a sample variance or an estimate of variance. The basis of the analysis of variance is the decomposition of variance into parts or components.

SS of errors and SS of the effect. Within-group variability (SS) is usually called the residual component or error variance. This means that, usually, during an experiment, it cannot be predicted or explained. On the other hand, the SS of the effect can be explained by the difference between the mean values in the groups, since they possess different mean values.

One can perform a significance test, which in the analysis of variance is based on comparing the variance component caused by intergroup spread, called the mean square of the effect or MS of the effect, and the variance component caused by within-group spread, called the mean square of the error or MS of the error.

If the null hypothesis (equality of means in two populations or cultivars) is true, then one can expect a relatively small difference in sample means due to purely random variability. Therefore, under the null hypothesis, the within-group variance will practically coincide with the total variance calculated without regard to group membership. The calculated within-group variances can be compared using the F-test, which checks whether the ratio of the variances is significantly greater than 1.

The goal of analysis of variance (ANOVA) is to test the statistical significance of the difference between means (for cultivars or agricultural practices). This test is performed by partitioning the sum of squares into components, that is, by decomposing the total variance into parts, one of which is due to random error and the other is related to the difference in mean values across experimental variants. The latter variance component is then used to analyze the statistical significance of the difference between the mean values. If this difference is significant, the null hypothesis is rejected and the alternative hypothesis regarding the existence of a difference between the means is accepted.

To ensure that field experiment results are not accidental, its structure must be properly planned. Any parameters you measure in the field — for example, plant height or the number of grains per ear — are referred to by statistics as dependent variables. Their reliability depends entirely on the number of variants and replications you have incorporated into the research design.

How to design an experimental scheme for different tasks

For simple tasks, where only one technological solution needs to be evaluated, single-factor experiments are used. Depending on the research objective, the tillage depth, doses of active ingredients, or the number of tested cultivars vary. All key parameters of such experiments are summarized in the table.

When it is necessary to study the interaction of several practices, multi-factor experiments are set up. The most common example is a two-factor experiment, where the influence of nutrition and genetics is evaluated simultaneously. For the first factor (rates of mineral or organic fertilizers), at least three variants should be provided with 3–4-fold replication. For the second factor (cultivars), 5 to 10 variants are used with at least three-fold replication.

• Replication of single-factor experiments — 3–4-fold
• Number of cultivars in a two-factor experiment — from 5 to 10 variants
• Variants for each factor — at least 3

For complex field research, three-factor schemes are applied. For example, the first scheme may include mineral fertilizer rates (at least three variants), seed sowing rates (three variants), and cultivars (5 cultivars with at least three-fold replication). A second popular scheme combines predecessor, tillage method, and mineral fertilizer rates.