Analysis of RT distributions
with R
Emil Ratko-Dehnert
WS 2010/ 2011
Session 07 – 21.12.2010
Last time ...
• Introduced significance tests (most notably
the t-test)
– Test statistic and p-value
– Confusion matrix
– Prerequisites and necessary steps
– Students t-test
– Implementation in R
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ANALYSIS OF VARIANCES
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ANOVA
• The Analysis of Variance is a collection of statis-
tical test
• Their aim is to explain the variance of a DV
(metric) by one or more (categorial) factors/ IVs
• Each factor has different factor levels
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Main idea
• Are the means of different groups (by factors)
different from each other?
H 0 : 1   2    n
H1 :  i , j :  i   j
• Is the variance of a group bigger than of the
whole data?
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ANOVA designs
• One-way ANOVA is used to test for differences
among two or more independent groups.
• Typically, however, the one-way ANOVA is used to
test for differences among at least three groups,
since the two-group case can be done by a t-test
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ANOVA designs (cont.)
• Factorial ANOVA is used when the experimenter
wants to study the interaction effects among the
treatments.
• Repeated measures ANOVA is used when the
same subjects are used for each treatment (e.g.,
in a longitudinal study).
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ONE-WAY ANOVA
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One-way ANOVA
• A one-way ANOVA is a generalization of the t-test
for more than two independent samples
• Suppose we have k populations of interest
• From each we take a random sample, for the ith
sample, let Xi1, Xi2, ..., Xini designate the sample
values
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Prerequisites
The data should ...
1) be independent
2) be normally distributed
3) have equal Variances (homoscedasticity)
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Mathematical model
X ij  i   ij
• Xij =
dependant variable
• i
=
group (i in 1, ..., k)
• j
=
elements of group i (j in 1, ..., ni)
• ni =
sample size of group i
• εij =
error term; ε ~ N(0, σ)
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Hypotheses
• Suppose we have k independent, iid samples
from populations with N(μi, σ) distributions,
i = 1, ... k. A significance test of
H 0 : 1   2    k
H1 :  i , j :  i   j
• Under H0, F has the F-distribution with k-1 and
n-k degrees-of-freedom.
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Fk-1, n-k with
k = amount of
factors and
n = sample size
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Example
• Two groups of animals receive different diets
• The weights of animals after the diet are:
Group 1:
45, 23, 55, 32, 51, 91, 74, 53, 70, 84 (n1 = 10)
Group 2:
64, 75, 95, 56, 44, 130, 106, 80, 87, 115 (n2 = 10)
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Example (cont.)
• Do the different diets have an effect on the weight?
x1  57.8
1 n1
2
var1 
( x1i  x1 )  479.7

n1  1 1
x2  85.2
1 n2
2
var2 
( x2i  x2 )  728.6

n2  1 1
• Means differ, but this might be due to natural variance
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Example (cont.)
• Global variance
• Test statistic
n1 var1  n2 var2
varg 
n1  n2
n1n2 ( x1  x2 )
F
 6.21
(n1  n2 ) varg
2
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Example (cont.)
• To assess difference of means, we need to compare
this F-value with the one we would get for the
for alpha = 0.05
 F = 4.41
 6.21 > 4.41
 H0 can be rejected
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ANOVA (CONT.)
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Effect size η2
• The effect size describes the ratio of variance
explained in the dependant variable by a
predictor while controlling for other predictors
streatment
 
stotal
2
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Power Analysis
• is often applied in order to assess the probability
of successfully rejecting H0 for specific designs,
effect sizes, sample size and α-level.
• can assist in study design by determining what
sample size would be required in order to have a
reasonable chance of rejecting the H0 when H1 is
true.
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A priori vs. post hoc analysis
• A priori analysis (before data collection) is used to
determine the appropriate sample size to achieve
adequate power
• Post hoc analysis (after data collection) uses
obtained sample size and effect size to determine
power of the study
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Follow-up tests
• ANOVA only decides whether (at least) one pair of
means is different, one commonly conducts
follow-up tests to assess which groups are
different:
Bonferroni-Test
Scheffé-Test
Tuckey‘s Range Test
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Visualisation of ANOVAs
http://www.psych.utah.edu/stat/introstats/anov
aflash.html
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ANOVAS WITH R
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oneway.test()
• The R function oneway.test() will perform the
one-way ANOVA
• One can use the model notation
oneway.test(values ~ ind, data = data)
to assign values to groups
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aov()
• Alternatively one can use the more general
aov() command for the one-way ANOVA
fit <- aov(y ~ A, data = mydataframe)
plot(fit)
# diagnostic plots
summary(fit)
# display ANOVA table
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AND NOW TO
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