Exercise 3.4

The effect of five different methods (A, B, C, D, E) on the time to assemble a device is being studied. The experimenter wants to investigate how long each method takes with each of 4 workers. Also, since workers might get tired or get more practiced, they might slow down or speed up near the end. Therefore, the order that each worker tries a method also matters. Then, the experimenter decides to run the experiment as a Latin square so that worker and order effects may be systematically controlled. The data was recorded below.

#import data
library(readr)
data <- read_csv("/Volumes/GoogleDrive/My Drive/NORATIKAH/EDA/coding/Exercise 3.4.csv")

── Column specification ────────────────────────────────────────────────────────────────────────────────────────────────────────────────
cols(
  Order = col_double(),
  Worker = col_double(),
  Treatment = col_character(),
  Time = col_double()
)

data

Treatment = as.factor(data$Treatment)
Row = as.factor(data$Order)
Column = as.factor(data$Worker)
results = aov(Time~Row+Column+Treatment,data)
summary(results)

            Df Sum Sq Mean Sq F value  Pr(>F)   
Row          3   18.5   6.167   3.524 0.08852 . 
Column       3   51.5  17.167   9.810 0.00993 **
Treatment    3   72.5  24.167  13.810 0.00421 **
Residuals    6   10.5   1.750                   
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

1. Nuisance factor : Worker and order
2. Treatment : Method (A,B, C, D , E), Response : Time
3. \(H_{0}\): All population means are equal @ no treatments effect
   \(H_{1}\): At least one of the population means is different @ there is treatment effects
   \(p-value=0.0042\)
   Since (\(p-value=0.0042\))\(<\)(\(\alpha=0.05\)), reject \(H_{0}\).
   At \(\alpha=0.05\), at least one of the population means is different @ there is treatment effects
   

TukeyHSD(results)

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

Fit: aov(formula = Time ~ Row + Column + Treatment, data = data)

$Row
     diff        lwr       upr     p adj
2-1  1.25 -1.9881345 4.4881345 0.5756823
3-1 -1.00 -4.2381345 2.2381345 0.7191024
4-1  1.75 -1.4881345 4.9881345 0.3304308
3-2 -2.25 -5.4881345 0.9881345 0.1761447
4-2  0.50 -2.7381345 3.7381345 0.9474067
4-3  2.75 -0.4881345 5.9881345 0.0924529

$Column
     diff        lwr         upr     p adj
2-1  5.00  1.7618655  8.23813449 0.0070204
3-1  2.25 -0.9881345  5.48813449 0.1761447
4-1  1.75 -1.4881345  4.98813449 0.3304308
3-2 -2.75 -5.9881345  0.48813449 0.0924529
4-2 -3.25 -6.4881345 -0.01186551 0.0492740
4-3 -0.50 -3.7381345  2.73813449 0.9474067

$Treatment
     diff        lwr       upr     p adj
B-A  1.75 -1.4881345 4.9881345 0.3304308
C-A  5.75  2.5118655 8.9881345 0.0034505
D-A  3.50  0.2618655 6.7381345 0.0363534
C-B  4.00  0.7618655 7.2381345 0.0202927
D-B  1.75 -1.4881345 4.9881345 0.3304308
D-C -2.25 -5.4881345 0.9881345 0.1761447

plot(TukeyHSD(results))

The sigificant par of treatments are : C-A, D-A, and C-B. The most significant treatment is the pair of method C and A.