Answer Lab Report 3

1. Give TWO conditions of balanced incomplete block design :
   -any two treatments appear together an equal number of times
   -not all treatments runs in each block
   

2. Response variable : strength of the paper
   

3. Treatment=hardwood concentrations, list = 2,4,6,8,10,12,14
   

#import data
library(readr)
data <- read_csv("/Volumes/GoogleDrive/My Drive/NORATIKAH/EDA/Assessments/Lab Report/Lab Report 3/Lab Report 3 v2.csv")

── Column specification ───────────────────────────────────────────────────────────────────────────────
cols(
  Days = col_double(),
  Hardwood = col_double(),
  Strength = col_double()
)

data

4. ANOVA coding
   

Treatment = as.factor(data$Hardwood)
Block = as.factor(data$Days)
#for BIBD, block come first then treatment
results = aov(Strength~Block+Treatment,data)
summary(results)

            Df Sum Sq Mean Sq F value  Pr(>F)   
Block        6 1114.3  185.71   8.814 0.00358 **
Treatment    6 1317.4  219.57  10.420 0.00205 **
Residuals    8  168.6   21.07                   
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

\(H_{0}\): All population means are equal @ no treatments effect
\(H_{1}\): At least one of the population means is different @ there is treatment effects
\(p-value=0.0021\)
Since (\(p-value=0.0021\))\(<\)(\(\alpha=0.05\)), reject \(H_{0}\).
At \(\alpha=0.05\), at least one of the population means is different @ there is treatment effects

22. Further analysis : Since there is significant difference between the treatments, post-hoc test is necessary

library(DescTools)

Registered S3 method overwritten by 'data.table':
  method           from
  print.data.table     

ScheffeTest(x=results)

  Posthoc multiple comparisons of means: Scheffe Test 
    95% family-wise confidence level

$Block
             diff      lwr.ci    upr.ci   pval    
2-1  7.000000e+00 -16.5282445 30.528245 0.9731    
3-1 -5.000000e+00 -28.5282445 18.528245 0.9982    
4-1  1.800000e+01  -5.5282445 41.528245 0.1801    
5-1  4.263256e-14 -23.5282445 23.528245 1.0000    
6-1 -3.000000e+00 -26.5282445 20.528245 1.0000    
7-1 -1.000000e+00 -24.5282445 22.528245 1.0000    
3-2 -1.200000e+01 -35.5282445 11.528245 0.6113    
4-2  1.100000e+01 -12.5282445 34.528245 0.7084    
5-2 -7.000000e+00 -30.5282445 16.528245 0.9731    
6-2 -1.000000e+01 -33.5282445 13.528245 0.8000    
7-2 -8.000000e+00 -31.5282445 15.528245 0.9365    
4-3  2.300000e+01  -0.5282445 46.528245 0.0565 .  
5-3  5.000000e+00 -18.5282445 28.528245 0.9982    
6-3  2.000000e+00 -21.5282445 25.528245 1.0000    
7-3  4.000000e+00 -19.5282445 27.528245 0.9998    
5-4 -1.800000e+01 -41.5282445  5.528245 0.1801    
6-4 -2.100000e+01 -44.5282445  2.528245 0.0898 .  
7-4 -1.900000e+01 -42.5282445  4.528245 0.1430    
6-5 -3.000000e+00 -26.5282445 20.528245 1.0000    
7-5 -1.000000e+00 -24.5282445 22.528245 1.0000    
7-6  2.000000e+00 -21.5282445 25.528245 1.0000    

$Treatment
            diff     lwr.ci    upr.ci   pval    
4-2     3.000000 -20.528245 26.528245 1.0000    
6-2    11.666667 -11.861578 35.194911 0.6438    
8-2    18.000000  -5.528245 41.528245 0.1801    
10-2   20.333333  -3.194911 43.861578 0.1049    
12-2    5.666667 -17.861578 29.194911 0.9946    
14-2    9.000000 -14.528245 32.528245 0.8781    
6-4     8.666667 -14.861578 32.194911 0.9000    
8-4    15.000000  -8.528245 38.528245 0.3491    
10-4   17.333333  -6.194911 40.861578 0.2096    
12-4    2.666667 -20.861578 26.194911 1.0000    
14-4    6.000000 -17.528245 29.528245 0.9915    
8-6     6.333333 -17.194911 29.861578 0.9870    
10-6    8.666667 -14.861578 32.194911 0.9000    
12-6   -6.000000 -29.528245 17.528245 0.9915    
14-6   -2.666667 -26.194911 20.861578 1.0000    
10-8    2.333333 -21.194911 25.861578 1.0000    
12-8  -12.333333 -35.861578 11.194911 0.5790    
14-8   -9.000000 -32.528245 14.528245 0.8781    
12-10 -14.666667 -38.194911  8.861578 0.3740    
14-10 -11.333333 -34.861578 12.194911 0.6763    
14-12   3.333333 -20.194911 26.861578 1.0000    

---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

TukeyHSD(results)

  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = Strength ~ Block + Treatment, data = data)

$Block
             diff        lwr       upr     p adj
2-1  7.000000e+00  -7.309004 21.309004 0.5452172
3-1 -5.000000e+00 -19.309004  9.309004 0.8200632
4-1  1.800000e+01   3.690996 32.309004 0.0145450
5-1  4.263256e-14 -14.309004 14.309004 1.0000000
6-1 -3.000000e+00 -17.309004 11.309004 0.9782633
7-1 -1.000000e+00 -15.309004 13.309004 0.9999467
3-2 -1.200000e+01 -26.309004  2.309004 0.1115313
4-2  1.100000e+01  -3.309004 25.309004 0.1575900
5-2 -7.000000e+00 -21.309004  7.309004 0.5452172
6-2 -1.000000e+01 -24.309004  4.309004 0.2210906
7-2 -8.000000e+00 -22.309004  6.309004 0.4149066
4-3  2.300000e+01   8.690996 37.309004 0.0032020
5-3  5.000000e+00  -9.309004 19.309004 0.8200632
6-3  2.000000e+00 -12.309004 16.309004 0.9972844
7-3  4.000000e+00 -10.309004 18.309004 0.9217599
5-4 -1.800000e+01 -32.309004 -3.690996 0.0145450
6-4 -2.100000e+01 -35.309004 -6.690996 0.0057301
7-4 -1.900000e+01 -33.309004 -4.690996 0.0105800
6-5 -3.000000e+00 -17.309004 11.309004 0.9782633
7-5 -1.000000e+00 -15.309004 13.309004 0.9999467
7-6  2.000000e+00 -12.309004 16.309004 0.9972844

$Treatment
            diff         lwr        upr     p adj
4-2     3.000000 -11.3090041 17.3090041 0.9782633
6-2    11.666667  -2.6423374 25.9756707 0.1252127
8-2    18.000000   3.6909959 32.3090041 0.0145450
10-2   20.333333   6.0243293 34.6423374 0.0070052
12-2    5.666667  -8.6423374 19.9756707 0.7332911
14-2    9.000000  -5.3090041 23.3090041 0.3060975
6-4     8.666667  -5.6423374 22.9756707 0.3397085
8-4    15.000000   0.6909959 29.3090041 0.0394390
10-4   17.333333   3.0243293 31.6423374 0.0180585
12-4    2.666667 -11.6423374 16.9756707 0.9877953
14-4    6.000000  -8.3090041 20.3090041 0.6867666
8-6     6.333333  -7.9756707 20.6423374 0.6393680
10-6    8.666667  -5.6423374 22.9756707 0.3397085
12-6   -6.000000 -20.3090041  8.3090041 0.6867666
14-6   -2.666667 -16.9756707 11.6423374 0.9877953
10-8    2.333333 -11.9756707 16.6423374 0.9938365
12-8  -12.333333 -26.6423374  1.9756707 0.0993180
14-8   -9.000000 -23.3090041  5.3090041 0.3060975
12-10 -14.666667 -28.9756707 -0.3576626 0.0442095
14-10 -11.333333 -25.6423374  2.9756707 0.1405131
14-12   3.333333 -10.9756707 17.6423374 0.9644345

plot(TukeyHSD(results))

The sigficant pair of treatments are : 8-2, 10-2, 8-4, 10-4, 12-10 since p-value is samller than alpha=0.05. The most significant treatment is 10-2.