Methodology of analysis of variance for three-factor field experiments in agronomy

Many agricultural scientists often study various biological patterns within the framework of three-factor experiments. Three-factor field experiments are complex, both in terms of their design, data collection, and experimental data processing, and it is even more difficult to interpret the obtained results, especially regarding the interaction of factors affecting the experimental outcomes.

For example, in a three-factor field experiment, the following is intended to be studied:

– for factor A – rates of sowing seed (3.0; 5.0; 7.0 million grains per 1 ha);

– for factor B – rates of mineral fertilizer (without fertilizer; N50P40K40; N80P60K60);

– for factor C – testing of rice cultivars (Liman, Zhemchuzhny, Regul, Rapan, and Kurchanka).

The experiment was established with 4 replications. Thus, the experiment was designed according to the scheme: 3╳3╳5╳4 (3 – rates of sowing seed; 3 – rates of mineral fertilizer; 5 – rice cultivars; 4 – replications). Therefore, rice yield needs to be studied in a three-factor field experiment, where experimental data from 180 plots must be processed. This is a complex and labor-intensive experiment. It should be processed without allowing any, even insignificant, errors. The quality of our research results depends on how correctly we choose the program and input the experimental data files without errors.

Table 240 – Matrix of results of three-factor analysis of variance for rice yield

 Number of degrees of freedom Variance type Sum of squares Mean square F-crit. F-tab. LSD 
 Total 3290.44 179 18.38 
 Replications 21.13 3 7.04 
 Variants 2988.94 44 67.93 2.07 1.45 32.0 1.4 
 Factor A 441.81 2 220.91 0.54 0.31 104.0 3.1 
 Factor B 832.47 2 416.23 0.54 0.31 196. 3.1 
 Factor C 1539.02 4 384.75 0.69 0.44 181.0 2.4 
 Interaction AB 1.53 4 0.38 0.93 0.62 0.2 2.4 
 AC 2.45 8 0.31 1.20 0.82 0.1 2.0 
 BC 161.84 8 20.23 1.20 0.82 9.5 2.0 
 ABC 9.81 16 0.61 2.07 1.45 0.3 1.7 
 Errors 280.38 132 2.12

Using the variance values for each type, it is necessary to determine the share of influence (contribution) to yield formation. To do this, subtract the error value (280.38) from each variance type value:

– total variance: 3290.4–280.38 = 3010.06;

– replication variance: 21.13–280.38 = –259.25;

– variant variance: 2988.9–280.38 = 2708.56;

– factor A variance: 441.81–280.38 = 161.43;

– factor B variance: 832.47–280.38 = 552.09;

– factor C variance: 1539.02–280.38 = 1258.64;

– interaction AB variance: 1.53–280.38 = –278.85;

– interaction AC variance: 2.45–280.38 = –277.93;

– interaction BC variance: 161.84–280.38 = –118.54;

– interaction ABC variance: 9.81–280.38 = –270.57.

Calculation of the proportional influence of factors on yield

The next arithmetic step is to sum all variance type values minus errors, resulting in a sum of 6485.58.

After this, one should determine the share of influence (contribution) of each type of variation in the formation of yield across the entire general population of variability based on ratios, similar to the calculations in two-factor analysis of variance. The total sum of variation values minus errors, which is 6485.58, is taken as 100% influence.

The share of contribution of total variation to rice yield is calculated as follows:

Ultimately, the sum of contributions of all variation types to the formation of rice grain yield should be 100%. Next, an analysis for each variation type is conducted, corresponding conclusions are formulated, and recommendations are provided.

In this experiment, the dominant contribution to the formation of rice cultivar yield is made by: total variation (46.41%), variants (41.76%), and factor C (cultivar genotypes, 19.40%). Other types of variance are either small or have a negative influence on the formation of rice yield.

Compilation of the final matrix of experimental results

Next, a final matrix of the results of the three-factor analysis of variance of rice yield is compiled, providing the complete experimental scheme for three gradations: factor A, factor B, and factor C.

• Average yield results for each variant, calculated from four replications, are presented.
• Average yield values for each factor and interaction AB, AC, BC, and ABC are entered, and interaction effects are provided.
• At the bottom of the matrix, the values of LSD05 (Least Significant Difference) for each gradation of factors and interactions should be shown.

Based on the LSD05 values for each type of gradation, an analysis of the efficiency of the conducted experiment is performed, and the best variants of the experiment are noted. In this experiment, interaction effects are not significant. They are either small or have negative values.

After compiling the table of results of the three-factor analysis of variance for statistical data on rice cultivar yield obtained under the influence of sowing seed rates and various mineral fertilizer rates, data from all variants are analyzed using the LSD05 for variants. We determine which cultivar and in which variant showed the highest yield.

Table 241 – Results of three-factor analysis of variance for rice yield, centners/ha (average across the experiment 63.23) Factor gradations Average by:

 A*) B C A B C AB AC BC variants 2 56.3 56.1 54.0 B1 3 62.1 61.6 59.5 4 65.2 64.6 63.0 5 63.0 63.7 61.5 1 61.4 63.2 61.4 62.9 60.8 2 57.6 56.0 A1 B2 3 63.8 62.3 4 66.6 65.0 5 65.0 63.0 1 65.9 63.9 66.4 64.3 2 61.1 58.5 B3 3 66.3 64.5 4 69.6 67.5 5 66.1 64.5 1 60.5 61.9 56.8 2 58.1 56.0 B1 3 63.8 61.5 4 66.8 64.5 5 65.0 63.8 1 63.1 63.1 62.8 2 57.5 A2 B2 3 63.5 4 66.5 5 65.3 1 65.8 66.3 2 60.8 B3 3 66.5 4 69.5 5 66.0 1 62.6 64.3 59.0 2 60.4 58.3 B1 3 65.8 63.8 4 68.8 66.3 5 66.8 65.8 1 65.2 65.0 65.3 2 59.3 A3 B2 3 65.5 4 68.3 5 66.8 1 62.2 68.0 68.8 2 58.3 63.8 B3 3 63.9 68.0 4 66.9 71.8 5 64.9 67.8 LSD05 0.54 0.54 0.69 0.93 1.20 1.20 2.07 )

* For names of factor gradations, see Table 239

*) For the names of factor gradations, see Table 239

These results must be compared with the control (fertilizer application rates) and the standard. Then, the results for each factor and interaction are analyzed in turn with the LSD05 (of factors and interaction). Afterward, the analysis of interaction effects is initiated. If the value of the interaction effect is higher than the significance of the similar interaction, then it is statistically significant. In this example, the BC interaction effect for the Liman cultivar against the background of N80P60K60 fertilizer at a sowing rate of 3.0 million seeds per 1 ha is 1.56. This means that a significant interaction with the fertilizer application rate has been established for this rice cultivar.

A three-factor experiment has very complex gradations, but after their correct analysis, it is possible to explain the share of the influence of each factor in the formation of rice yield for the studied cultivars.