library(readxl)
LSD_Exam <- read_excel("D:/MARV BS MATH/Marv 4th year, 1st sem/Regression Analysis/LSD Exam.xlsx")
LSD_Exam
# A tibble: 25 × 4
   Batch   Day Ingredients Hours
   <dbl> <dbl> <chr>       <dbl>
 1     1     1 A               8
 2     1     2 B               7
 3     1     3 D               1
 4     1     4 C               7
 5     1     5 E               3
 6     2     1 C              11
 7     2     2 E               2
 8     2     3 A               7
 9     2     4 D               3
10     2     5 B               8
# … with 15 more rows
head(LSD_Exam)
# A tibble: 6 × 4
  Batch   Day Ingredients Hours
  <dbl> <dbl> <chr>       <dbl>
1     1     1 A               8
2     1     2 B               7
3     1     3 D               1
4     1     4 C               7
5     1     5 E               3
6     2     1 C              11
str(LSD_Exam)
tibble [25 × 4] (S3: tbl_df/tbl/data.frame)
 $ Batch      : num [1:25] 1 1 1 1 1 2 2 2 2 2 ...
 $ Day        : num [1:25] 1 2 3 4 5 1 2 3 4 5 ...
 $ Ingredients: chr [1:25] "A" "B" "D" "C" ...
 $ Hours      : num [1:25] 8 7 1 7 3 11 2 7 3 8 ...

Changing variables structure into factors

LSD_Exam$Batch <- as.factor(LSD_Exam$Batch)
LSD_Exam$Day <- as.factor(LSD_Exam$Day)
LSD_Exam$Ingredients <- as.factor(LSD_Exam$Ingredients)
str(LSD_Exam)
tibble [25 × 4] (S3: tbl_df/tbl/data.frame)
 $ Batch      : Factor w/ 5 levels "1","2","3","4",..: 1 1 1 1 1 2 2 2 2 2 ...
 $ Day        : Factor w/ 5 levels "1","2","3","4",..: 1 2 3 4 5 1 2 3 4 5 ...
 $ Ingredients: Factor w/ 5 levels "A","B","C","D",..: 1 2 4 3 5 3 5 1 4 2 ...
 $ Hours      : num [1:25] 8 7 1 7 3 11 2 7 3 8 ...
attach(LSD_Exam)

Applying analysis of variance model

model <- lm(Hours ~ Batch+Day+Ingredients)
anova(model)
Analysis of Variance Table

Response: Hours
            Df Sum Sq Mean Sq F value    Pr(>F)    
Batch        4  15.44   3.860  1.2345 0.3476182    
Day          4  12.24   3.060  0.9787 0.4550143    
Ingredients  4 141.44  35.360 11.3092 0.0004877 ***
Residuals   12  37.52   3.127                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The ingredient was significant i.e. reaction time of at least one ingredient is different from the rest. As ingredient is significant we should switch to multiple mean comparison test like LSD test.

Mean comparison test

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

LSD test

LSD.test(y = model,
         trt = "Ingredients",
         DFerror = model$df.residual,
         MSerror = deviance(model)/model$df.residual,
         alpha = 0.05,
         group = TRUE,
         console = TRUE)

Study: model ~ "Ingredients"

LSD t Test for Hours 

Mean Square Error:  3.126667 

Ingredients,  means and individual ( 95 %) CI

  Hours      std r      LCL       UCL Min Max
A   8.4 1.140175 5 6.677038 10.122962   7  10
B   5.6 2.073644 5 3.877038  7.322962   3   8
C   8.8 1.643168 5 7.077038 10.522962   7  11
D   3.4 2.073644 5 1.677038  5.122962   1   6
E   3.2 1.923538 5 1.477038  4.922962   1   6

Alpha: 0.05 ; DF Error: 12
Critical Value of t: 2.178813 

least Significant Difference: 2.436636 

Treatments with the same letter are not significantly different.

  Hours groups
C   8.8      a
A   8.4      a
B   5.6      b
D   3.4      b
E   3.2      b

The results showed that varieties C and A were statistically at par and reaction time of a chemical process significantly differ more than ingredients B, D and E.

SNK test

SNK.test(y = Hours,
         trt = Ingredients,
         DFerror = model$df.residual,
         MSerror = deviance(model)/model$df.residual,
         alpha = 0.05,
         group = TRUE,
         console = TRUE)

Study: Hours ~ Ingredients

Student Newman Keuls Test
for Hours 

Mean Square Error:  3.126667 

Ingredients,  means

  Hours      std r Min Max
A   8.4 1.140175 5   7  10
B   5.6 2.073644 5   3   8
C   8.8 1.643168 5   7  11
D   3.4 2.073644 5   1   6
E   3.2 1.923538 5   1   6

Alpha: 0.05 ; DF Error: 12 

Critical Range
       2        3        4        5 
2.436636 2.983558 3.320217 3.564608 

Means with the same letter are not significantly different.

  Hours groups
C   8.8      a
A   8.4      a
B   5.6      b
D   3.4      b
E   3.2      b