Blocking and Two-way ANOVA
6.1 Randomized Complete
Block (RCB) Design
(and it’s observational relative)
Prof Beckman
Agenda
- 6.1 Discuss randomized complete block design (RCB)
Announcements
- Announcement about Research Study in Canvas
- discussion about class on 4/8
Spring 2024 MLR Test Scores
mlrTest <- c(21, 23, 28, 28.5, 30.5, 36, 37.5, 37.5, 40, 46, 46, 46, 46.5, 47.5,
48, 49, 49.5, 51, 52, 52.5, 53.5, 54, 55, 55, 56, 56, 58.5, 59, 59)
mlrData <- tibble(mlrScore = mlrTest, mlrPercent = (mlrTest/60 * 100))
# summary statistics
# favstats(~ mlrScore, data = mlrData)
favstats(~ mlrPercent, data = mlrData)
## min Q1 median Q3 max mean sd n missing
## 35 62.5 80 90 98.33 75.98 18.39 29 0
# score density (%)
mlrData %>%
ggplot(aes(x = mlrPercent)) +
geom_density() +
geom_rug()
Example (FranticFingers Data)
Scientists Scott and Chen published research that compared the
effects of caffeine with those of theobromine (a similar chemical found
in chocolate) and with those of a placebo. Their experiment used four
human subjects and took place over several days. Each day each subject
swallowed a tablet containing one of caffeine, theobromine, or the
placebo. Two hours later they were timed while tapping a finger in a
specified manner (that they had practiced earlier, to control for
learning effects). The response is the number of taps in a fixed time
interval.
Discussion
Questions: In this scenario, what are the…
- Treatments?
- Experimental Units?
- Responses?
- Parameter of Interest?
- What would your data set look like (i.e., in R)?
data("FranticFingers", package = "Stat2Data")
glimpse(FranticFingers)
## Rows: 12
## Columns: 4
## $ ID <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
## $ Rate <int> 11, 56, 15, 6, 26, 83, 34, 13, 20, 71, 41, 32
## $ Subj <fct> A, B, C, D, A, B, C, D, A, B, C, D
## $ Drug <fct> Pl, Pl, Pl, Pl, Ca, Ca, Ca, Ca, Th, Th, Th, Th
FranticFingers %>%
pivot_wider(id_cols = "Drug", names_from = "Subj", values_from = "Rate")
## # A tibble: 3 × 5
## Drug A B C D
## <fct> <int> <int> <int> <int>
## 1 Pl 11 56 15 6
## 2 Ca 26 83 34 13
## 3 Th 20 71 41 32
Additional terminology
- factors and levels: the
subjects and drugs are factors, that
is, categorical predictor variables
- each category of a factor is a
level
- the factor subjects has four levels: A, B,
C, and D (the people)
- the factor drug has three levels: placebo,
caffeine, and theobromine
- crossed factors: We say that the two factors in
FranticFingers are crossed because we have
data for every combination of subjects and drugs
Experiments vs. Observational
Experiment
If the treatment is assigned at random to the units in each block (a
unique treatment for each unit) we call it a randomized complete
block (RCB) design.
Example: Twin study with random assignment
Observational
A complete block design for observational data has a unique treatment
for each unit of the block but no random assignment.
Example: Five students (blocks) each taking four exams
(treatments)
Examples [Practice Rmd]
- What are the experimental treatments and what is the blocking factor
in each scenario?
Frantic Fingers Data
Four subjects were given a placebo, caffeine, or theobromine (in a
random order), and a finger tap rate was recorded after each drug.
- we expect subject to subject difference (nuisance)
- we want to investigate treatment differences {placebo, caffeine,
theobromine}
- RCB experiment–we randomly assigned the order of Drugs within each
subject
River Iron Data
Iron concentrations (ppm) were measured in three different (upstream,
midstream, and downstream) in four different rivers.
- we expect river to river differences (nuisance)
- we want to investigate the location differences {upstream,
midstream, downstream}
- observational data–we can’t “randomly assign” the order of
up/mid/downstream status within each river
Additional terminology
Two Way ANOVA Model
\[Y = \mu + \alpha_i + \beta_j +
\epsilon\]
- \(\epsilon \overset{iid}{\sim}
N(0,\sigma)\)
- \(\alpha_i\) is the treatment
effect for group \(i \in \{1, ... ,
G\}\)
- \(\beta_j\) is the effect for block
\(j \in \{1, ... , B\}\)
ANOVA with NO block
adjustment
NoBlocks <- aov(Rate ~ Drug, data = FranticFingers)
summary(NoBlocks)
## Df Sum Sq Mean Sq F value Pr(>F)
## Drug 2 872 436 0.68 0.53
## Residuals 9 5810 646
ANOVA with randomized
complete blocks (RCB)
- RCB experiment–we randomly assigned the order of Drug within each
Subject
- we expect subject to subject difference (nuisance; we don’t care
about the p-value)
- we want to investigate treatment differences {placebo, caffeine,
theobromine}
- Blocking adjusts for subject differences so we have more
statistical power to detect Treatment differences!
rcbModel <- aov(Rate ~ Drug + Subj, data = FranticFingers)
summary(rcbModel)
## Df Sum Sq Mean Sq F value Pr(>F)
## Drug 2 872 436 7.88 0.0210 *
## Subj 3 5478 1826 33.00 0.0004 ***
## Residuals 6 332 55
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Essential R Today
Disclaimer: Intended to recap some of the key R functions used
today, but generally will not include everything. The “Notes” and
“Practice” Rmd files accompanying class discussions generally include
more detail and show how these tools are used in context.
# analysis of variable model
myModel <- aov(Response ~ Treatments + Blocks, data = DataSetName)
summary(myModel)
# important: "Treatments" & "Blocks" must be modeled as a "factor" data type
# the following alternative can solve simple problems with variable type
myModel <- aov(Response ~ factor(Treatments) + factor(Blocks), data = DataSetName)
summary(myModel)
# Plot of treatment effect after adjusting for blocks
DataSetName %>%
mutate(GrandMean = mean(Response)) %>% # overall mean response
group_by(Block) %>% # group by blocks for calculations
mutate(BlockMean = mean(Response)) %>% # mean response for each block
mutate(BlockEffect = BlockMean - GrandMean) %>%
ungroup() %>% # finished grouping by blocks
gf_point(Response - BlockEffect ~ Treatment, data = . )