Data

library(readxl)
## Warning: package 'readxl' was built under R version 4.1.3
data2 <- read_excel("D:/data2.xlsx")
## New names:
## * `` -> `...3`
## * `` -> `...4`
## * `` -> `...5`
## * `` -> `...6`
## * `` -> `...7`
## * `` -> `...8`
## * `` -> `...9`
## * `` -> `...10`
## * `` -> `...11`
## * `` -> `...12`
## * `` -> `...13`
## * `` -> `...14`
## * `` -> `...15`
## * `` -> `...16`
## * `` -> `...17`
View(data2)
data2
## # A tibble: 60 x 17
##    Treatment  DPPH ...3  ...4    ...5  ...6  ...7  ...8  ...9  ...10 ...11 ...12
##    <chr>     <dbl> <lgl> <chr>   <chr> <chr> <chr> <chr> <chr> <chr> <lgl> <chr>
##  1 T0+       0.052 NA    <NA>    <NA>  <NA>  <NA>  <NA>  <NA>  <NA>  NA    <NA> 
##  2 T0+       0.05  NA    <NA>    <NA>  <NA>  <NA>  <NA>  <NA>  <NA>  NA    <NA> 
##  3 T0+       0.05  NA    <NA>    <NA>  <NA>  <NA>  <NA>  <NA>  <NA>  NA    <NA> 
##  4 T0+       0.054 NA    raw da~ <NA>  <NA>  <NA>  <NA>  <NA>  <NA>  NA    <NA> 
##  5 T0+       0.055 NA    <NA>    <NA>  T0+   T0-   T1    T2    T3    NA    <NA> 
##  6 T0+       0.054 NA    <NA>    R1    5.19~ 4.80~ 0.113 0.129 0.13~ NA    <NA> 
##  7 T0+       0.059 NA    <NA>    <NA>  0.05  4.49~ 0.14~ 0.14~ 0.127 NA    <NA> 
##  8 T0+       0.057 NA    <NA>    <NA>  0.05  4.39~ 0.161 0.16~ 0.16  NA    <NA> 
##  9 T0+       0.055 NA    <NA>    <NA>  5.39~ 4.29~ 0.191 0.19~ 0.191 NA    <NA> 
## 10 T0+       0.053 NA    <NA>    Mean  5.19~ 4.49~ 0.152 0.15~ 0.153 NA    <NA> 
## # ... with 50 more rows, and 5 more variables: ...13 <chr>, ...14 <chr>,
## #   ...15 <lgl>, ...16 <chr>, ...17 <chr>
data2<-data2[, 1:2]
data2
## # A tibble: 60 x 2
##    Treatment  DPPH
##    <chr>     <dbl>
##  1 T0+       0.052
##  2 T0+       0.05 
##  3 T0+       0.05 
##  4 T0+       0.054
##  5 T0+       0.055
##  6 T0+       0.054
##  7 T0+       0.059
##  8 T0+       0.057
##  9 T0+       0.055
## 10 T0+       0.053
## # ... with 50 more rows
library(tidyverse)
## Warning: package 'tidyverse' was built under R version 4.1.3
## -- Attaching packages --------------------------------------- tidyverse 1.3.1 --
## v ggplot2 3.3.6     v purrr   0.3.4
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## v tidyr   1.2.0     v stringr 1.4.0
## v readr   2.1.2     v forcats 0.5.1
## Warning: package 'ggplot2' was built under R version 4.1.3
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## -- Conflicts ------------------------------------------ tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
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library(ggpubr)
## Warning: package 'ggpubr' was built under R version 4.1.3
library(rstatix)
## Warning: package 'rstatix' was built under R version 4.1.3
## 
## Attaching package: 'rstatix'
## The following object is masked from 'package:stats':
## 
##     filter
library(dplyr)
data2 <- data2 %>%
  reorder_levels(Treatment, order = c("T0+", "T0-", "T1", "T2", "T3"))
library(dplyr)
library(rstatix)
data2%>%
  group_by(Treatment)%>%
  get_summary_stats(DPPH, type="common")
## # A tibble: 5 x 11
##   Treatment variable     n   min   max median   iqr  mean    sd    se    ci
##   <fct>     <chr>    <dbl> <dbl> <dbl>  <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 T0+       DPPH        12 0.05  0.059  0.054 0.003 0.054 0.003 0.001 0.002
## 2 T0-       DPPH        12 0.043 0.048  0.045 0.001 0.045 0.001 0     0.001
## 3 T1        DPPH        12 0.111 0.194  0.152 0.04  0.153 0.031 0.009 0.02 
## 4 T2        DPPH        12 0.124 0.195  0.154 0.039 0.158 0.027 0.008 0.017
## 5 T3        DPPH        12 0.111 0.195  0.146 0.04  0.15  0.031 0.009 0.02

Interpretation

  The mean for treatments are T0+ is 0.054, T0- is 0.045, T1 is 0.153, T2 is 0.158 and T3 is 0.150.
    
library(ggplot2)
ggplot(data2) +
  aes(x = Treatment, y = DPPH, color = Treatment) +
  geom_jitter() +
  theme(legend.position = "none")

Interpretation

  The above graph shows the plotting of data by treatment, which contains five treatments such as T0+, T0-, T1, T2, and T3.
library(tidyverse)
library(ggpubr)
library(rstatix)
ggboxplot(data2, x = "Treatment", y = "DPPH", fill="Treatment")

Interpretation

  It clearly shows that there are differences between the five treatments T0+, T0-, T1, T2 and T3.

ANOVA

res_aov <- aov(DPPH ~ Treatment, data = data2)

Normality Test

par(mfrow = c(1, 2)) # combine plots
# histogram
hist(res_aov$residuals)
# QQ-plot
library(car)
## Warning: package 'car' was built under R version 4.1.3
## Loading required package: carData
## Warning: package 'carData' was built under R version 4.1.3
## 
## Attaching package: 'car'
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## 
##     recode
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##     some
qqPlot(res_aov$residuals,
       id = FALSE # id = FALSE to remove point identification
)

Interpretation

   The histogram resembles a bell curve as seen above, means that the residuals have a normal distribution. Moreover, the points in the QQ-plots roughly follow the straight line, with the majority of them falling within the confidence bands. This also indicates that residuals have a normal distribution.

Normality Test

shapiro.test(res_aov$residuals)
## 
##  Shapiro-Wilk normality test
## 
## data:  res_aov$residuals
## W = 0.93435, p-value = 0.003052

Interpretation

  The Shapiro-Wilk p-value = 0.003052 on the residuals is less than the usual significance level of 0.05. Thus, we reject the hypothesis that residuals have a normal distribution.

Equality of Variances Test

library(car)
leveneTest(DPPH ~ Treatment,
  data = data2)
## Levene's Test for Homogeneity of Variance (center = median)
##       Df F value    Pr(>F)    
## group  4  16.451 6.403e-09 ***
##       55                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Interpretation

  The p-value is greater than the 0.05 level of significance.
res.kruskal <- data2 %>% kruskal_test(DPPH ~ Treatment)
res.kruskal
## # A tibble: 1 x 6
##   .y.       n statistic    df             p method        
## * <chr> <int>     <dbl> <int>         <dbl> <chr>         
## 1 DPPH     60      45.8     4 0.00000000277 Kruskal-Wallis

Interpretation

  There was a significant difference between the five treatments by the p-value.

Effect size

The effect size values normally interpreted as 0.01- < 0.06 (small effect), 0.06 – < 0.14 (moderate effect) and >= 0.14 (large effect).

data2 %>% kruskal_effsize(DPPH ~ Treatment)
## # A tibble: 1 x 5
##   .y.       n effsize method  magnitude
## * <chr> <int>   <dbl> <chr>   <ord>    
## 1 DPPH     60   0.759 eta2[H] large

Interpretation

  The size of the effect is substantial. This means that even with a small sample size, we can easily identify significant differences.

Pairwise Comparison

res1<- data2 %>%
  dunn_test(DPPH ~ Treatment, p.adjust.method = "bonferroni")
res1
## # A tibble: 10 x 9
##    .y.   group1 group2    n1    n2 statistic            p     p.adj p.adj.signif
##  * <chr> <chr>  <chr>  <int> <int>     <dbl>        <dbl>     <dbl> <chr>       
##  1 DPPH  T0+    T0-       12    12    -1.68  0.0922         9.22e-1 ns          
##  2 DPPH  T0+    T1        12    12     3.34  0.000841       8.41e-3 **          
##  3 DPPH  T0+    T2        12    12     3.69  0.000225       2.25e-3 **          
##  4 DPPH  T0+    T3        12    12     3.08  0.00210        2.10e-2 *           
##  5 DPPH  T0-    T1        12    12     5.02  0.000000509    5.09e-6 ****        
##  6 DPPH  T0-    T2        12    12     5.37  0.0000000771   7.71e-7 ****        
##  7 DPPH  T0-    T3        12    12     4.76  0.00000194     1.94e-5 ****        
##  8 DPPH  T1     T2        12    12     0.351 0.726          1   e+0 ns          
##  9 DPPH  T1     T3        12    12    -0.263 0.792          1   e+0 ns          
## 10 DPPH  T2     T3        12    12    -0.614 0.539          1   e+0 ns

Interpretation

  Based on the pairwise comparison, there is a significant difference observed in T0+, T0-, T1, T2, and T3.
res1 <- res1 %>% add_xy_position(x = "Treatment")
ggboxplot(data2, x = "Treatment", y = "DPPH") +
  stat_pvalue_manual(res1, hide.ns = TRUE) +
  labs(
    subtitle = get_test_label(res.kruskal, detailed = TRUE),
    caption = get_pwc_label(res1))