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Comparing Groups – Part 2
Wilcoxon Signed Rank (again)
H0: θ = 0
HA: θ ≠ 0
θ represents the population mean or median
n
R
R   rank _ AbsDiff


j
j 1
or in colors… add up the rank_AbsDiff for
the green
negative scores
n
R
R   rank _ AbsDiff


j
j 1
or in colors… add up the rank_AbsDiff for
the blue
positive scores
S  (R  R ) / 2
V  (n * (n  1) * (2n  1)) / 24
T
S * n 1
n*V  S
2
n
R
R    rank _ AbsDiff  j
j 1
or in colors… add up the rank_AbsDiff for
the green
negative scores
R- = 1.5 + 3 + 4.5 + 4.5+ 7.5 +
7.5+7.5+14.5+18.5 = 64.5
n
R
R   rank _ AbsDiff


j
j 1
or in colors… add up the rank_AbsDiff for
the blue
positive scores
R+ = 1.5 + 7.5 + 11 +11 +11 +14.5 +14.5
+14.5+18.5+18.5+18.5+21+22= 188.5
S  (R  R ) / 2
188.5-64.5/2=62
One-way ANOVA in R
• In the last lecture I gave you a
demonstration of using SAS to conduct a
one-way ANOVA. Many people think that
R is a superior tool for doing analyses like
ANOVA. While R is not critical for simple
models like you have seen, it is invaluable
for graphics to describe more complicated
models.
R and Anxiety Disorders
• Start R and then type library(Rcmdr)
• Load the dataset using the Import Data
option on the Data menu.
• Load the Generalized Anxiety Disorder
(GAD) data. It has Hamilton Rating Scale
for Anxiety data for people taking placebo,
high or low dose drug.
Summarize the Data
Show me the means.
• You want to see if there is a
difference in means in the 3rd
variable for the levels of the
2nd variable. You can
request the mean plot:
PB
27
28
The Plot of the Design
25
24
23
LO
22
mean of HAMA
26
I really like
boxplots to see
the variability
around these
means.
HI
DOSEGRP
Factors
Set Factor Order
Side-by-side Boxplot
Boxplot Reordered
Gad <- na.omit(Gad)
plot.design(Gad[, c(2,3)])
attach(Gad)
thePlot = (boxplot(HAMA~DOSEGRP, ylab="HAMA", xlab="DOSEGRP"))
where = seq(thePlot$n)
theMeans = tapply(HAMA, DOSEGRP, mean)
points(where, theMeans, col = "red", pch = 18)
20
25
HAMA
30
35
I want means.
PB
LO
DOSEGRP
HI
Testing for Differences
• Once the data is set with the placebo
group as the baseline, it is easy to ask for
tests for the differences vs. the baseline
level.
General Linear Models
• It is important for you to know that
ANOVAs are more than stand-alone
analyses that are not related to modeling.
ANOVA fits into a class of statistics called
General Linear Models. They are well
implemented in R and Rcmdr.
Notice you now have a working
model.
Checking the Model
• With the working model you can get lots of
summary information, including diagnostic
plots.
More Advanced Contrasts
• There is a plethora of methods for dealing
with multiple comparisons. You can search
CRAN for specific methods you see in
textbooks. For example, Walker uses
Dunnett’s T test for contrasting the placebo
vs. the other levels.
• I start search CRAN with RSiteSearch(“blah”)
Nasty code….
• I eventually found the method implemented
in a package called multcomp:
library(multcomp)
gad.aov = aov(HAMA ~ DOSEGRP, data = Gad)
summary(glht(gad.aov, linfct = mcp(DOSEGRP="Dunnett")))
confint(glht(gad.aov, linfct = mcp(DOSEGRP="Dunnett")))
Two-Way ANOVA
• When you have two or more predictors, you
want to know if the variables impact the outcome
means by themselves and also if they interact. If
variables interact, it means that together they do
things to the outcome variable beyond what you
would expect from looking at each variable
alone. For example, smoking and eating too
much both hurt longevity but the combination of
the two factors may not be as bad as expected
or the combination may be especially lethal.
Anemia
• People with Cervical (C), Prostate(P) or
Colorectal (R) cancers with chemotherapyinduced anemia were treated either with a
drug or placebo and the changes in their
hemoglobin levels were assessed. The
question of interest is, does the drug
reduce anemia?
Box Plots of Course
1
HGBCH
0
1
-1
0
-2
-1
-2
HGBCH
2
2
3
3
• With code or R commander get boxplots
for the levels of the predictors:
ACT
PBO
TRT
C
P
TYPE
R
Design Plots
• You have analysis variables in columns 1,
2 and 4. So specify the design plot like
this:
C
1.0
1.4
ACT
PBO
0.6
mean of HGBCH
P
TRT
R
TYPE
Factors
Interaction Plots
• In addition to the main effects for the
predictors, you want to see if the drug
behaves differently in the people with the
three different cancer groups. Perhaps it
helps increase hemoglobin in one group
and decreases it another.
Type your outcome variable last.
• It looks like there are differences between
the cancer types and the drug seems to
increase the hemoglobin relative to the
placebo.
1.0
1.5
P
C
R
0.5
mean of HGBCH
2.0
TYPE
ACT
PBO
TRT
Modeling
Anova with a capital A is part of the car package
Results
To turn off the * stuff,
type this code before you model:
options(show.signif.stars=FALSE)
ANOVA as a Model
• You can build this ANOVA as a linear
model.
interaction only
main effect
and interaction
Main effect
used for nesting
y ~ A/B
y ~ A + A:B
y ~ A + B % in % A
Set interaction limits
(A+B+C) ^ 2 is equal to
A*B*C - A:B:C
Remove an
explanatory effect.
SAS EG example
• See the parallel information in the SAS EG
project.