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

Importing the Dataset

library(readxl)
RBD <- read_excel("C:/1. School/School/4th Year/Experimental Design/Assignment/Finals/Seatwork 7 RCBD/RBD.xlsx")
RBD
## # A tibble: 20 x 3
##    Replication Variety      Yield
##          <dbl> <chr>        <dbl>
##  1           1 African tall  22.9
##  2           1 Co-11         29.5
##  3           1 FS-1          28.8
##  4           1 K-7           47  
##  5           1 Co-24         28.9
##  6           2 African tall  25.9
##  7           2 Co-11         30.4
##  8           2 FS-1          24.4
##  9           2 K-7           40.9
## 10           2 Co-24         20.4
## 11           3 African tall  39.1
## 12           3 Co-11         35.3
## 13           3 FS-1          32.1
## 14           3 K-7           42.8
## 15           3 Co-24         21.1
## 16           4 African tall  33.9
## 17           4 Co-11         29.6
## 18           4 FS-1          28.6
## 19           4 K-7           32.1
## 20           4 Co-24         31.8

Fitting of linear model \(H_o\): African tall=Co-11=FS-1=K-7=Co-24, \(H_a\): At least one variety is different

model <- lm(RBD$Yield~ RBD$Replication+RBD$Variety)
model
## 
## Call:
## lm(formula = RBD$Yield ~ RBD$Replication + RBD$Variety)
## 
## Coefficients:
##      (Intercept)   RBD$Replication  RBD$VarietyCo-11  RBD$VarietyCo-24  
##           29.195             0.502             0.750            -4.900  
##  RBD$VarietyFS-1    RBD$VarietyK-7  
##           -1.975            10.250

Obtain ANOVA

anova <-anova(model)
anova
## Analysis of Variance Table
## 
## Response: RBD$Yield
##                 Df Sum Sq Mean Sq F value  Pr(>F)  
## RBD$Replication  1   6.30   6.300  0.2072 0.65592  
## RBD$Variety      4 520.53 130.133  4.2806 0.01808 *
## Residuals       14 425.61  30.401                  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Discussion: From the result of the anova, the varieties are significantly different from one another at 0.05 level of significance. This further means that we reject the null hypothesis and accept the alternative hypothesis.

#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)

Discussion: As shown in the Normal Q-Q plot, almost of the observations are on the line or near the line. Thus normality is fulfilled.

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"

LSD test

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"
#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  1   6.30   6.300  0.2072 0.65592  
## RBD$Variety      4 520.53 130.133  4.2806 0.01808 *
## Residuals       14 425.61  30.401                  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
print("DNMRT Result")
## [1] "DNMRT Result"
print(DNMRT$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()