---
title: "ADA2: Class 07, Ch 05a Paired Experiments and Randomized Block Experiments: Randomized complete block design (RCBD)"
author: Your Name
date: last-modified
description: |
[Advanced Data Analysis 2](https://StatAcumen.com/teach/ada2),
Stat 428/528, Spring 2023, Prof. Erik Erhardt, UNM
format:
html:
theme: litera
highlight-style: atom-one
page-layout: full # article, full # https://quarto.org/docs/output-formats/page-layout.html
toc: true
toc-location: body # body, left, right
number-sections: false
self-contained: false # !!! this can cause a render error
code-overflow: scroll # scroll, wrap
code-block-bg: true
code-block-border-left: "#30B0E0"
code-copy: false # true, false, hover a copy buttom in top-right of code block
fig-width: 6
fig-height: 4
fig-align: center # default, left, right, or center
execute: # https://quarto.org/docs/computations/execution-options.html, https://quarto.org/docs/computations/r.html
cache: false # false, true
echo: true # true, false Include the source code in output
warning: true # true, false Include warnings in the output.
error: true # true, false Include errors in the output (note that this implies that errors executing code will not halt processing of the document).
---
# Tire wear experiment
A fleet manager wishes to compare the wearability of 4 brands of tire: A, B, C,
and D. Four cars are available for the experiment and 4 tires of each brand are
available, for a total of 16 tires. The idea is to mount 4 tires on each of the
4 cars, ask the driver of each of the cars to drive his/her car for 20,000
miles and then to measure tread loss. We will measure tread loss in mils (0.001
inches). We will designate the 4 cars as cars c1, c2, c3, and c4.
We consider 3 experimental designs.
# Design 1: Brands are randomized to Cars
Naive design.
```
Car
c1 c2 c3 c4
-----------
Brand A B C D
A B C D
A B C D
A B C D
-----------
```
## __(1 p)__ What is the [obvious flaw](https://en.wikipedia.org/wiki/Confounding#Types_of_confounding) in this design?
### Solution
[answer, 1 sentence describing the problem.]
# Design 2: Completely randomized design (CRD)
Another design that negates the confounding is to use a completely randomized
design (CRD). This entails numbering the 16 tires, drawing at random the
numbers and assigning tires to cars in a completely random manner. The
following table illustrates a possible result of doing this.
```
Car
c1 c2 c3 c4 c1 c2 c3 c4
----------- -----------
Brand C A C A 12 14 10 13
A A D D 17 13 11 9
D B B B 13 14 14 8
D C B C 11 12 13 9
----------- -----------
```
Bring this data in R.
```{R}
library(erikmisc)
library(tidyverse)
# Plan:
# Read the data values and the treatment labels,
# reshape both into long format and combine.
d2_dat <- read.table(text="
c1 c2 c3 c4
12 14 10 13
17 13 11 9
13 14 14 8
11 12 13 9
", header = TRUE)
d2_trt <- read.table(text="
c1 c2 c3 c4
C A C A
A A D D
D B B B
D C B C
", header = TRUE, as.is = TRUE)
d2_dat_long <-
d2_dat %>%
pivot_longer(
cols = everything()
, names_to = "Car"
, values_to = "Wear"
) %>%
mutate(
Car = factor(Car)
)
d2_trt_long <-
d2_trt %>%
pivot_longer(
cols = everything()
, names_to = "Car"
, values_to = "Brand"
) %>%
mutate(
Car = factor(Car)
, Brand = factor(Brand)
)
d2_all <-
bind_cols(
d2_dat_long
, d2_trt_long %>% select(Brand)
#, Brand = d2_trt_long$Brand
)
str(d2_all)
```
The appropriate analysis for this experiment is the one-way ANOVA.
```{R}
# Group means
m_d2 <-
d2_all %>%
group_by(Brand) %>%
summarise(
m = mean(Wear)
) %>%
ungroup()
m_d2
```
```{R, fig.height = 5, fig.width = 6}
# Plot the data using ggplot
library(ggplot2)
p <- ggplot(d2_all, aes(x = Brand, y = Wear))
# plot a reference line for the global mean (assuming no groups)
p <- p + geom_hline(yintercept = mean(d2_all$Wear),
colour = "black", linetype = "dashed", size = 0.3, alpha = 0.5)
# boxplot, size=.75 to stand out behind CI
#p <- p + geom_boxplot(size = 0.75, alpha = 0.5)
# points for observed data
p <- p + geom_point(aes(shape = Car, colour = Car), position = position_jitter(w = 0.2, h = 0), alpha = 1, size = 2)
# diamond at mean for each group
p <- p + stat_summary(fun = mean, geom = "point", shape = 18, size = 6,
colour = "red", alpha = 0.8)
# confidence limits based on normal distribution
#p <- p + stat_summary(fun.data = "mean_cl_normal", geom = "errorbar",
# width = .2, colour = "red", alpha = 0.8)
p <- p + labs(title = "Design 2: Tire Wear") + ylab("Wear (mil)")
print(p)
```
## Fit model
```{R}
fit_d2 <- lm(Wear ~ Brand, data = d2_all)
```
## __(1 p)__ Are the assumptions for the one-way ANOVA met?
```{R, fig.height = 3, fig.width = 10}
# plot diagnostics
e_plot_lm_diagostics(fit_d2, sw_plot_set = "simpleAV")
library(nortest)
ad.test(fit_d2$residuals)
```
### Solution
[answer]
## __(1 p)__ Can we infer a difference in mean wear levels between the 4 Brands?
```{R}
library(car)
Anova(fit_d2, type=3)
```
### Solution
[answer]
# Design 3: Randomized Complete Block Design (RCBD)
In this case, each car tests all four brands. Thus one tire from each brand is
selected at random and randomly allocated to the 4 wheels of car c1. Then one
tire from each brand is selected and the four are randomly allocated to car c2,
and so forth. Here are the results of that design.
```
Car
c1 c2 c3 c4 c1 c2 c3 c4
----------- -----------
Brand B D A C 14 11 13 9
C C B D 12 12 13 9
A B D B 17 14 11 8
D A C A 13 14 10 13
----------- -----------
```
Read in the data.
```{R}
d3_all <- read.table(text="
Car Wear Brand
c1 14 B
c1 12 C
c1 17 A
c1 13 D
c2 11 D
c2 12 C
c2 14 B
c2 14 A
c3 13 A
c3 13 B
c3 11 D
c3 10 C
c4 9 C
c4 9 D
c4 8 B
c4 13 A
", header = TRUE) %>%
mutate(
Car = factor(Car)
, Brand = factor(Brand)
)
str(d3_all)
```
Means and plots by Brand and by Car.
```{R}
# Group means
m_d3_b <-
d3_all %>%
group_by(Brand) %>%
summarise(
m = mean(Wear)
) %>%
ungroup()
m_d3_c <-
d3_all %>%
group_by(Car) %>%
summarise(
m = mean(Wear)
) %>%
ungroup()
m_d3_b
m_d3_c
```
```{R, fig.height = 4, fig.width = 6}
par(mfrow=c(1,2))
boxplot(split(d3_all$Wear, d3_all$Brand))
boxplot(split(d3_all$Wear, d3_all$Car))
par(mfrow=c(1,1))
```
```{R, fig.height = 5, fig.width = 6}
# Plot the data using ggplot
library(ggplot2)
p <- ggplot(d3_all, aes(x = Brand, y = Wear))
# plot a reference line for the global mean (assuming no groups)
p <- p + geom_hline(yintercept = mean(d3_all$Wear),
colour = "black", linetype = "dashed", size = 0.3, alpha = 0.5)
# boxplot, size=.75 to stand out behind CI
#p <- p + geom_boxplot(size = 0.75, alpha = 0.5)
# points for observed data
p <- p + geom_point(aes(shape = Car, colour = Car), position = position_jitter(w = 0.2, h = 0), alpha = 1, size = 2)
# colored line for each Care
p <- p + geom_line(aes(group = Car, colour = Car), alpha = 0.5)
# diamond at mean for each group
p <- p + stat_summary(fun = mean, geom = "point", shape = 18, size = 6,
colour = "red", alpha = 0.8)
# confidence limits based on normal distribution
#p <- p + stat_summary(fun.data = "mean_cl_normal", geom = "errorbar",
# width = .2, colour = "red", alpha = 0.8)
p <- p + labs(title = "Design 3: Tire Wear") + ylab("Wear (mil)")
print(p)
```
## __(2 p)__ Briefly, what relationships are there between Wear and Brand or Car?
Refer to the numerical and graphical summaries above.
### Solution
[answer]
## Fit model
```{R}
fit_d3 <- lm(Wear ~ Brand + Car, data = d3_all)
```
## __(1 p)__ Are the assumptions for the RCBD met?
```{R, fig.height = 3, fig.width = 10}
# plot diagnostics
e_plot_lm_diagostics(fit_d3, sw_plot_set = "simpleAV")
library(nortest)
ad.test(fit_d3$residuals)
```
### Solution
[answer]
## __(1 p)__ Can we infer a difference in mean wear levels between the 4 brands?
It is appropriate to test whether there are differences between
Brands controlling for the effect of Car.
This is the additive model.
__Note that because the Car blocks are part of the experimental design,
they should remain in the model regardless of whether the block is significant or not.__
```{R}
library(car)
Anova(fit_d3, type=3)
```
### Solution
[answer]
## __(2 p)__ Perform the pairwise comparisons and summarize numerically and graphically.
```{R}
# Contrasts to perform pairwise comparisons
cont_d3_b <-
emmeans::emmeans(
fit_d3
, specs = "Brand"
)
# Means and CIs
cont_d3_b
# Pairwise comparisons
cont_d3_b %>% pairs()
```
__EMM plot interpretation__
This __EMM plot (Estimated Marginal Means, aka Least-Squares Means)__
is only available when conditioning on one variable.
The __blue bars__ are confidence intervals for the EMMs;
don't ever use confidence intervals for
EMMs to perform comparisons -- they can be very misleading.
The __red arrows__ are for the comparisons among means;
the degree to which the "comparison arrows" overlap reflects as much as
possible the significance of the comparison of the two estimates.
If an arrow from one mean overlaps an arrow from
another group, the difference is not significant, based on the adjust setting
(which defaults to "tukey").
```{R, fig.height = 3, fig.width = 6}
# Plot means and contrasts
p <- plot(cont_d3_b, comparisons = TRUE)
p <- p + labs(title = "Tukey-adjusted Treatment contrasts")
p <- p + theme_bw()
print(p)
```
Summarize the results in a table like this, where the effect of the Brands are
sorted and the bars indicate pairs that are not statistically different. Then
summarize in words.
```
Example:
Brand: Z Y W X
-------
-------
```
### Solution
[answer]
# Design 4: Your idea!
## __(1 p)__ How can this experiment be further improved by design?
There are further factors that we haven't yet considered inherent in this experiment.
Bonus if you can name this experimental design.
### Solution
[answer]