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

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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?

• 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

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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).7

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

values9

Prerequisites

The data should ...

1) be independent

2) be normally distributed

3) have equal Variances

(homoscedasticity)

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Mathematical model

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• 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

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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) 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.)

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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

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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

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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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