Problem statement

An experiment was conducted in RCBD to study the comparative performance of fodder sorghum lines under rain fed conditions. Data is furnished below. Are all lines same? If not carry out LSD test and Duncan test to compare the lines.

library(readxl)
RBD <- read_excel("D:/MARV BS MATH/Marv 4th year, 1st sem/Regression Analysis/RBD.xlsx", 
    col_types = c("text", "text", "numeric"))
View(RBD)

Fitting of linear model

Null Hypothesis : African tall=Co-11=FS-1=K-7=Co-24.

Alternative Hypothesis : Atleast one variety is different.

model <- lm(RBD$Yield~ RBD$Replication+RBD$Variety)

Obtain ANOVA

anova <-anova(model)
anova
Analysis of Variance Table

Response: RBD$Yield
                Df Sum Sq Mean Sq F value  Pr(>F)  
RBD$Replication  3  80.80  26.934  0.9205 0.46033  
RBD$Variety      4 520.53 130.133  4.4476 0.01958 *
Residuals       12 351.11  29.259                  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Interpretation

The replication was non-significant so the mean yield for all the four replications is same. The treatment was significant i.e. yield of at least on variety is different from the rest. As treatment is significant we should switch to multiple mean comparison test like LSD or DNMRT test.

Below codes are used to obtain plots of fitted vs Residuals and Normal QQ plots

par(mfrow=c(1,2))
plot(model, which=1)
plot(model, which=2)

The assumptions of ANOVA are not violated.

library(agricolae)
Warning: package 'agricolae' was built under R version 4.2.2

LSD test

library(agricolae)
LSD <-LSD.test(RBD$Yield, RBD$Variety,12,29.259)
LSD
$statistics
  MSerror Df   Mean       CV  t.value      LSD
   29.259 12 31.275 17.29547 2.178813 8.333639

$parameters
        test p.ajusted      name.t ntr alpha
  Fisher-LSD      none RBD$Variety   5  0.05

$means
             RBD$Yield      std r      LCL      UCL  Min  Max    Q25   Q50
African tall    30.450 7.403378 4 24.55723 36.34277 22.9 39.1 25.150 29.90
Co-11           31.200 2.762849 4 25.30723 37.09277 29.5 35.3 29.575 30.00
Co-24           25.550 5.674798 4 19.65723 31.44277 20.4 31.8 20.925 25.00
FS-1            28.475 3.155287 4 22.58223 34.36777 24.4 32.1 27.550 28.70
K-7             40.700 6.274286 4 34.80723 46.59277 32.1 47.0 38.700 41.85
                Q75
African tall 35.200
Co-11        31.625
Co-24        29.625
FS-1         29.625
K-7          43.850

$comparison
NULL

$groups
             RBD$Yield groups
K-7             40.700      a
Co-11           31.200      b
African tall    30.450      b
FS-1            28.475      b
Co-24           25.550      b

attr(,"class")
[1] "group"

Interpretation

The variety K-7 gives highest yield with is significantly different from the rest of the varieties. The performance of variety Co-11 was statistically at par with African tall, FS-1 and Co-24.

Duncan test

library(agricolae)
DNMRT <-duncan.test(RBD$Yield, RBD$Variety, 12,29.259)
DNMRT
$statistics
  MSerror Df   Mean       CV
   29.259 12 31.275 17.29547

$parameters
    test      name.t ntr alpha
  Duncan RBD$Variety   5  0.05

$duncan
     Table CriticalRange
2 3.081307      8.333639
3 3.225244      8.722927
4 3.312453      8.958792
5 3.370172      9.114897

$means
             RBD$Yield      std r  Min  Max    Q25   Q50    Q75
African tall    30.450 7.403378 4 22.9 39.1 25.150 29.90 35.200
Co-11           31.200 2.762849 4 29.5 35.3 29.575 30.00 31.625
Co-24           25.550 5.674798 4 20.4 31.8 20.925 25.00 29.625
FS-1            28.475 3.155287 4 24.4 32.1 27.550 28.70 29.625
K-7             40.700 6.274286 4 32.1 47.0 38.700 41.85 43.850

$comparison
NULL

$groups
             RBD$Yield groups
K-7             40.700      a
Co-11           31.200      b
African tall    30.450      b
FS-1            28.475      b
Co-24           25.550      b

attr(,"class")
[1] "group"

Interpretation

Same as the LSD test as both shows same set of sequence of letters.

Save the file in txt

sink("RBD.txt")
print(anova)
Analysis of Variance Table

Response: RBD$Yield
                Df Sum Sq Mean Sq F value  Pr(>F)  
RBD$Replication  3  80.80  26.934  0.9205 0.46033  
RBD$Variety      4 520.53 130.133  4.4476 0.01958 *
Residuals       12 351.11  29.259                  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
print("DNMRT Result")
[1] "DNMRT Result"
print(DNMRT$statistics)
  MSerror Df   Mean       CV
   29.259 12 31.275 17.29547
print(DNMRT$groups)
             RBD$Yield groups
K-7             40.700      a
Co-11           31.200      b
African tall    30.450      b
FS-1            28.475      b
Co-24           25.550      b
print("LSD Result")
[1] "LSD Result"
print(LSD$statistics)
  MSerror Df   Mean       CV  t.value      LSD
   29.259 12 31.275 17.29547 2.178813 8.333639
print(LSD$groups)
             RBD$Yield groups
K-7             40.700      a
Co-11           31.200      b
African tall    30.450      b
FS-1            28.475      b
Co-24           25.550      b
sink()

Script Prepared by Raj Popat, PhD scholar, Department of Agricultural Statistics, Anand Agricultural University, Anand