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I’ll upload one data in R.

install.packages("readr")
library (readr)
github="https://raw.githubusercontent.com/agronomy4future/r_code/main/corn_grain_yield.csv"
dataA=data.frame(read_csv(url(github),show_col_types = FALSE))

    variety reps nitrogen grain_yield
1       CV1    1       N1       16.59
2       CV1    2       N1       16.59
3       CV1    3       N1       15.91
4       CV1    4       N1       17.57
5       CV1    5       N1       16.40
6       CV1    6       N1       16.20
7       CV1    7       N1       18.45
8       CV1    8       N1       14.35
9       CV1    9       N1       13.27
10      CV1   10       N1       16.15
.
.
.

I have 10 corn varieties and want to analyze the impact of nitrogen treatments (N0 and N1), variety, and their interaction on grain yield. Since replicates are considered as blocks, I will conduct a two-way ANOVA analysis with blocks as the statistical model.

The statistical model for two-way ANOVA with blocks is as follows:

yijk = μ  + αi + βj + δij + γk + εijk

where
yijk: observed values for treatment (ij; i = factor 1, j = factor 2) and replicates (k)
μ: grand mean of observed values
αi: treatment effect of factor 1
βj: treatment effect of factor 2
δij: interaction effect between factor 1 (i) and factor 2 (j)
γk: block effect
εijk: residuals

Based on the statistical model for two-way ANOVA with blocks, we can conduct the analysis using R. Below is an example code:

anova2way= aov (grain_yield ~ nitrogen + variety +  nitrogen:variety +  factor(reps), data=dataA)
summary(anova2way)

                  Df   Sum Sq  Mean Sq  F value   Pr(>F)    
nitrogen           1   41.93   41.93    50.352    3.27e-11 ***
variety            9   134.23  14.91    17.912    < 2e-16 ***
factor(reps)       9   9.38    1.04     1.252     0.267    
nitrogen:variety   9   150.41  16.71    20.072    < 2e-16 ***
Residuals        171   142.38   0.83                     
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Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

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