This report captures work done for the 11th flipped assignment, studying the effects of Randomized Complete Block Design.

Setting things up: 

# setup Libraries
library(knitr)
library(dplyr)
library(tidyr)
library(GAD)
# READ IN FILE
setwd("D:/R Files/")
dat <- read.csv("chemicals.csv",header=TRUE)

# PIVOT COLUMNS
dat <- pivot_longer(dat,c(Bolt1,Bolt2,Bolt3,Bolt4,Bolt5))

# SET UP VARIABLES
solution <- as.fixed(dat$Chemical)
days <- as.random(dat$name)
obs <- as.integer(dat$value)

My_Factor <- solution
My_Response <- obs
My_Factor_Name = "Chemical"
My_Response_Name = "Strength"


Question 1

We are studying the effect of the four chemical agents on the strength of a particular type of cloth. And, because there might be variability from one bolt to another, the chemist decided to use a randomized block design, with the bolts of cloth considered as blocks.

The Linear Effects for this Model are:
\(\quad y_{ij}\) = \(\mu\) + \(\tau_i\) + \(\beta_j\) + \(\epsilon_{ij}\)

Where:
\(\tau_i\) is the effect of the \(\ i^{th}\) treatment of chemical agent used
\(\beta_j\) is the effect of the \(\ j^{th}\) block of cloth bolt used, and
\(\epsilon_{ij}\) is the random error term.

The Hypotheses we will test are:
\(\quad H_0\) : \(\tau_i\) = 0 \(\forall\) i
\(\quad H_a\) : \(\tau_i \neq\) 0for at least one \(\ i\)
\(\quad\)at a significance level of \(\alpha\) = 0.15

By running a General ANOVA Design on the linear model of chemical agents and cloth bolts, we get the following results: 

## Analysis of Variance Table
## 
## Response: obs
##          Df Sum Sq Mean Sq F value    Pr(>F)    
## solution  3  12.95   4.317  2.3761    0.1211    
## days      4 157.00  39.250 21.6055 2.059e-05 ***
## Residual 12  21.80   1.817                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

With a p-value of 0.1211 and a significance level of 0.15, we reject the Null hypothesis. There is enough evidence to support the claim of a significant effect of the chemical agent used.

Question 2

We now assume that the researcher did not block on Bolt, but rather ran the experiment at a completely randomized design on random pieces of cloth. In this case we will run a one-way ANOVA.

The Linear Effects for this Model are:
\(\quad y_{ij}\) = \(\mu\) + \(\tau_i\) + \(\epsilon_{ij}\)

Where:
\(\tau_i\) is the effect of the \(\ i^{th}\) treatment of chemical agent used
\(\epsilon_{ij}\) is the random error term.

The Hypotheses we will test are:
\(\quad H_0\) : \(\tau_i\) = 0 \(\forall\) i
\(\quad H_a\) : \(\tau_i \neq\) 0 for at least one \(\ i\)
\(\quad\)at a significance level of \(\alpha\) = 0.15

By running a one-way ANOVA Design on the linear model of chemical agents and we get the following results: 

##                           Df Sum Sq Mean Sq F value Pr(>F)
## My_ANOVA_Input_Table[[1]]  3  12.95   4.317   0.386  0.764
## Residuals                 16 178.80  11.175

With a p-value of 0.764 and a significance level of 0.15, we fail to reject the Null hypothesis. There is not enough evidence to support the claim of a significant effect of the chemical agent used.

Question 3

The differences between the blocked design and the non-blocked was startling, the blocked having a p-value of 0.12 with the non-blocked having a a p-value of 0.76. This clearly shows that the block (bolt type) represents a significant amount of nuisance variablity.