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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. - PowerPoint PPT Presentation
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Analysis of RT distributionswith R
Emil Ratko-DehnertWS 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 R2
ANALYSIS OF VARIANCES
3
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
4
Main idea• Are the means of different groups (by factors)
different from each other?
• Is the variance of a group bigger than of the
whole data?5
ji
n
jiH
H
:,:
:
1
210
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
6
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).7
ONE-WAY ANOVA
8
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
values9
Prerequisites
The data should ...
1) be independent
2) be normally distributed
3) have equal Variances
(homoscedasticity)
10
Mathematical model
11
• 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)
ijiijX
Hypotheses
• Suppose we have k independent, iid samples
from populations with N(μi, σ) distributions,
i = 1, ... k. A significance test of
• Under H0, F has the F-distribution with k-1 and
n-k degrees-of-freedom. 12
ji
k
jiH
H
:,:
:
1
210
Fk-1, n-k with
k = amount of
factors and
n = sample size
13
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) 14
Example (cont.)
• Do the different diets have an effect on the weight?
• Means differ, but this might be due to natural variance
15
8.571x
2.852 x
1
1
211
11 7.479)(
1
1var
n
i xxn
2
1
222
22 6.728)(
1
1var
n
i xxn
Example (cont.)
• Global variance
• Test statistic
16
21
2211 varvarvar
nn
nng
21.6var)(
)(
21
22121
gnn
xxnnF
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 rejected17
ANOVA (CONT.)
18
Effect size η2
• The effect size describes the ratio of variance
explained in the dependant variable by a
predictor while controlling for other
predictors
19
total
treatment
s
s2
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. 20
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 study21
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 Test22
Visualisation of ANOVAs
http://www.psych.utah.edu/stat/introstats/anovaflash.html
23
ANOVAS WITH R
24
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
25
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
26
AND NOW TO
27