Importing the data needed

library(readxl)
RBD <- read_excel("C:/Users/USER1/Desktop/darwin 4th year files/experimental design R/Yield.xlsx")
View(RBD)

Determining the null and alternative hypothesis:

\(H_o\): There is no significant difference on the mean yield of between the varieties of fodder sorghum under rain fed conditions. \

\(H_A\): There is a significant difference on the mean yield of between the varieties of fodder sorghum under rain fed conditions.

Fitting of linear model, we have

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$Replication  1   6.30   6.300  0.2072 0.65592  
## RBD$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

Generating the plots for fitted vs Residuals and Normal QQ plots

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

Doing the Duncan Test (DNMRT):

library("agricolae")
duncan <-duncan.test(RBD$Yield,RBD$Variety,12,29.259)
duncan
## $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"

Doing the 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"

Interpretation of Results:

From the data given, the plots show that assumptions to execute ANOVA was complied. Results of ANOVA shows that the replications has no significant difference on their mean yield, so we fail to reject the null hypothesis because \(p=0.65592>0.05\). The treatment was significant i.e. yield of at least one variety is different from the rest with \(p=0.01808<0.05\). As treatment is significant we should switch to multiple mean comparison test like LSD or DNMRT test.

Both of these comparison tests shows that 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.