Nested Designs

Lab 8 – Nested ANOVA
October 8 & 9, 2017
FANR 6750
Richard Chandler and Bob Cooper

Outline
1 Overview
2 Using aov
3 Using lme
Overview Using aov Using lme 2 / 16

Scenario
We subsample each experimental unit

Scenario
For example
• We count larvae at multiple subplots within a plot
• We weigh multiple chicks in a brood

Scenario
For example
• We count larvae at multiple subplots within a plot
• We weigh multiple chicks in a brood
We’re interested in treatment eﬀects at the experimental
(whole) unit level, not the subunit level

The additive model
yijk = µ + αi + βij + εijk

The additive model
Because we want our inferences to apply to all experimental
units, not just the ones in our sample, βij is random.

The additive model
Speciﬁcally:
βij ∼ Normal(0, σ2
B)

The additive model
Speciﬁcally:
βij ∼ Normal(0, σ2
B)
And as always,
εijk ∼ Normal(0, σ2
)

Hypotheses
Treatment eﬀects
H0 : α1 = · · · = αa = 0
Ha : at least one inequality

Hypotheses
Treatment eﬀects
H0 : α1 = · · · = αa = 0
Ha : at least one inequality
Random variation among experimental units
H0 : σ2
B = 0
Ha : σ2
B > 0

Example data
Import data
gypsyData <- read.csv("gypsyData.csv")
str(gypsyData)
## 'data.frame': 36 obs. of 3 variables:
## $ larvae : num 16 16 15.8 14.2 13.9 14.2 13.5 13.4 14 13.
## $ Treatment: Factor w/ 3 levels "Bt","Control",..: 1 1 1 1 1
## $ Plot : int 1 1 1 1 2 2 2 2 3 3 ...

Example data
Import data
gypsyData <- read.csv("gypsyData.csv")
str(gypsyData)
## 'data.frame': 36 obs. of 3 variables:
## $ larvae : num 16 16 15.8 14.2 13.9 14.2 13.5 13.4 14 13.
## $ Treatment: Factor w/ 3 levels "Bt","Control",..: 1 1 1 1 1
## $ Plot : int 1 1 1 1 2 2 2 2 3 3 ...
Convert Plot to a factor and then cross-tabulate
gypsyData$Plot <- factor(gypsyData$Plot)
table(gypsyData$Treatment, gypsyData$Plot)
##
## 1 2 3 4 5 6 7 8 9
## Bt 4 4 4 0 0 0 0 0 0
## Control 0 0 0 4 4 4 0 0 0
## Dimilin 0 0 0 0 0 0 4 4 4

Incorrect analysis
aov.wrong <- aov(larvae ~ Treatment + Plot,
data=gypsyData)

Incorrect analysis
data=gypsyData)
summary(aov.wrong)
## Df Sum Sq Mean Sq F value Pr(>F)
## Treatment 2 215.39 107.69 208.89 <2e-16 ***
## Plot 6 11.17 1.86 3.61 0.0093 **
## Residuals 27 13.92 0.52
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' '

Incorrect analysis
data=gypsyData)
summary(aov.wrong)
## Treatment 2 215.39 107.69 208.89 <2e-16 ***
## Plot 6 11.17 1.86 3.61 0.0093 **
## Residuals 27 13.92 0.52
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' '
The denominator degrees-of-freedom are wrong

Correct analysis
aov.correct <- aov(larvae ~ Treatment + Error(Plot),
data=gypsyData)

Correct analysis
aov.correct <- aov(larvae ~ Treatment + Error(Plot),
data=gypsyData)
summary(aov.correct)
##
## Error: Plot
## Treatment 2 215.39 107.69 57.87 0.00012 ***
## Residuals 6 11.17 1.86
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' '
##
## Error: Within
## Residuals 27 13.92 0.5156

What happens if we analyze plot-level means?
The aggregate function is similar to tapply but it works on
entire data.frames. Here we get averages for each whole plot.
plotData <- aggregate(formula=larvae ~ Treatment + Plot,
data=gypsyData, FUN=mean)

What happens if we analyze plot-level means?
The aggregate function is similar to tapply but it works on
entire data.frames. Here we get averages for each whole plot.
plotData <- aggregate(formula=larvae ~ Treatment + Plot,
data=gypsyData, FUN=mean)
plotData
## Treatment Plot larvae
## 1 Bt 1 15.50
## 2 Bt 2 13.75
## 3 Bt 3 14.00
## 4 Control 4 18.25
## 5 Control 5 18.75
## 6 Control 6 19.25
## 7 Dimilin 7 12.50
## 8 Dimilin 8 13.50
## 9 Dimilin 9 13.00

F and p values are the same as before
aov.plot <- aov(larvae ~ Treatment, data=plotData)
summary(aov.plot)
## Treatment 2 53.85 26.924 57.87 0.00012 ***
## Residuals 6 2.79 0.465
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

F and p values are the same as before
aov.plot <- aov(larvae ~ Treatment, data=plotData)
summary(aov.plot)
## Treatment 2 53.85 26.924 57.87 0.00012 ***
## Residuals 6 2.79 0.465
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(aov.correct)
##
## Error: Plot
## Treatment 2 215.39 107.69 57.87 0.00012 ***
## Residuals 6 11.17 1.86
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Error: Within
## Residuals 27 13.92 0.5156

Issues
When using using aov with Error term:
• You can’t use TukeyHSD
• You don’t get a direct estimate of σ2
B
• Doesn’t handle unbalanced designs well
• But, you can use model.tables and se.contrast

Issues
B
An alternative is to use lme function in nlme package

Issues
B
• Possible to get direct estimates of σ2
B and other variance
parameters
• Handles very complex models and unbalanced designs
• Possible to do multiple comparisons and contrasts using the
the glht function in the multcomp package.

Issues
B
• Possible to get direct estimates of σ2
B and other variance
parameters
• Handles very complex models and unbalanced designs
• Possible to do multiple comparisons and contrasts using the
the glht function in the multcomp package.
• But. . .
• Only works if there random eﬀects
• ANOVA tables aren’t as complete as aov

Using the lme function
library(nlme)
library(multcomp)
lme1 <- lme(larvae ~ Treatment, random=~1|Plot,
data=gypsyData)

Using the lme function
library(nlme)
library(multcomp)
lme1 <- lme(larvae ~ Treatment, random=~1|Plot,
data=gypsyData)
anova(lme1, Terms="Treatment")
## F-test for: Treatment
## numDF denDF F-value p-value
## 1 2 6 57.86567 1e-04

Variance parameter estimates
The ﬁrst row shows the estimates of σ2
B and σB. The second row shows
the estimates of σ2
and σ
VarCorr(lme1)
## Plot = pdLogChol(1)
## Variance StdDev
## (Intercept) 0.3363889 0.5799904
## Residual 0.5155556 0.7180220

Variance parameter estimates
The ﬁrst row shows the estimates of σ2
B and σB. The second row shows
the estimates of σ2
and σ
VarCorr(lme1)
## Plot = pdLogChol(1)
## Variance StdDev
## (Intercept) 0.3363889 0.5799904
## Residual 0.5155556 0.7180220
There is more random variation within whole units than among
whole units (after accounting for treatment eﬀects)

Extract the plot-level random effects
These are the βij’s
round(ranef(lme1), 2)
## (Intercept)
## 1 0.78
## 2 -0.48
## 3 -0.30
## 4 -0.36
## 5 0.00
## 6 0.36
## 7 -0.36
## 8 0.36
## 9 0.00

Multiple comparisons
tuk <- glht(lme1, linfct=mcp(Treatment="Tukey"))

Multiple comparisons
tuk <- glht(lme1, linfct=mcp(Treatment="Tukey"))
summary(tuk)
##
## Simultaneous Tests for General Linear Hypotheses
##
## Multiple Comparisons of Means: Tukey Contrasts
##
##
## Fit: lme.formula(fixed = larvae ~ Treatment, data = gypsyData, random = ~1 |
## Plot)
##
## Linear Hypotheses:
## Estimate Std. Error z value Pr(>|z|)
## Control - Bt == 0 4.3333 0.5569 7.781 <0.001 ***
## Dimilin - Bt == 0 -1.4167 0.5569 -2.544 0.0295 *
## Dimilin - Control == 0 -5.7500 0.5569 -10.324 <0.001 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## (Adjusted p values reported -- single-step method)

Assignment
To determine if salinity affects adult fish reproductive performance, a
researcher places one pregnant female in a tank with one of three salinity
levels: low, medium, and high, or a control tank. A week after birth, two
offspring (fry) are measured.
Run a nested ANOVA using aov and lme on the fishData.csv
dataset. Answer the following questions:
(1) What are the null and alternative hypotheses?
(2) Does salinity affect fry growth?
(3) If so, which salinity levels differ?
(4) Is there more random variation among or within experimental units?
Upload your self-contained R script to ELC at least one day before
your next lab

Nested Designs

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