Transcript ANOVA introduction Mike Tucker 27th April 2012

Introduction ANOVA
Mike Tucker
School of Psychology
B209 Portland Square
University of Plymouth
Drake Circus
Plymouth, PL4 8AA
Tel: +44 (0)1752 584860
Email: [email protected]
__________________________________
Relation to other Stats
Correlational
• Collect data and
describe
relationships.
• Can’t say much about
causal relations.
Experimental
• Manipulate conditions
and infer something
about their causal
effects.
• Can infer cause and
effect relations.
Relation to other Stats
• ANOVA is typically used to analyse data from
experimental research where conditions are
explicitly manipulated -allowing causal
inferences to be made.
• However bear in mind that what determines
whether it is legitimate to draw causal inferences
is how the data is obtained – not which test is
used to analyse it.
Simple & Factorial ANOVA
• Simple ANOVA designs: extension of TTest to more than two conditions.
• Factorial ANOVA: conditions are created
that are the result of crossing the levels of
two or more FACTORS.
Factors and Factor Levels
• A Factor is a property or dimension of the
experimental situation that is manipulated.
• The Levels of a factor are the particular
settings of the Factor used in the
experiment.
Factors and their Levels
•
•
•
•
Factors:
Stimulus Duration
Sex
Treatment
• Age Group
•
•
•
•
Levels:
100 ms, 200 ms …
Male, Female..
CBT, Drug,
Psychoanalytic
• 18-25, 26-40, one foot
in the grave
Typical Experimental Design
• An Exp. to determine
differences in three
antihistamine drugs on
driving performance.
• Performance
measured in a driving
simulator under two
difficulty levels (Hard &
Easy).
Design
• 3 x Drugs
• 2 x Difficulty Level
= 6 Drug by Difficulty
combinations:
Benadryl + Hard
Benadryl + Easy
Tylenol + Hard
….etc
•Factor 1 = Drug
•Factor 1 Levels =
Benadryl, Tylenol,
Chlortrimeton
Factor 2 = Task
Difficulty
Factor 2 Levels =
Hard, Easy.
Design
• This an example of a 2 x 3 design.
• Sometimes written in results sections as:
“…The data was subjected to a 3 x 2 between
subjects ANOVA with the factors Drug
(Benadryl, Tylenol, Chlortrimeton) and Task
Difficulty (Easy, Hard)….”
Layout
Task Difficulty
Easy
10 subs
Hard
10 subs
Tylenol
10 subs
Chlortrimeton 10 subs
10 subs
10 subs
Benadryl
Drug
Sixty adult drivers selected at random.
Each one randomly assigned to one of the six
experimental conditions (Between Subjects Design)
Between vs. Within Subjects
Between Subjects
Within Subjects
• Each participant only
undergoes testing in one
of the experimental
conditions (Factor
combinations)
• Avoids contaminating
effects but requires more
participants and design
more noisy (e.g.
individual differences)
• AKA repeated Measures
• Each participant
undergoes testing at all
levels of one or more of
the factors.
• More sensitive as
individual differences
controlled for but not
always possible (e.g.
Carry Over effects).
Between vs. Within
• Drug / Driving Simulation manipulations:
Classic case for a between subjects manipulation:
• Carry over effect from drugs
• Practice / order effects from driving simulation
• Case could be made for a within subjects manipulation
of the Driving Task (e.g. – each subject given only one
drug but tested under both driving difficulty levels). This
would depend on, for example, the time course of the
drug’s effects and the duration of the driving task.
How does ANOVA work?
• Recall the independent groups T-test:
(This a special case of the more general
ANOVA design where you have a single
factor with 2 levels – e.g. you are just
comparing 2 means, level 1 vs. level 2.)
How does ANOVA work?
Relation to T-test
• Two experimental conditions yield 2
samples of data (scores).
• Each has a mean x and variance s2
• The variance is just a measure of the
‘average difference between data points’.
• Main question: does the observed
difference between the two sample means
reflect a real difference (statistically
significant difference).
Mean and Variance
• Mean
• Easy to understand – just
the average score.
• = Sum of all scores
divided by the number of
scores:
SX / n
• Variance
• Just a measure of the
average spread or
variation of the scores.
• The ‘average’ of the
squared distances of the
individual scores from the
mean. (The squaring is
simply to stop the
negative and positive
distances cancelling out):
S(X-X)2 / n
How it works -3 samples of 4 (red dots) taken from 3 populations
The 3 sample means are the orange dots.
H0 True
H0 False
POPULATIONS
Distribution of
SAMPLE MEANS
OF SIZE N drawn
from the 3 populations
H0 – all drawn from same population
Vs. H1 – some drawn from different populations
•ANOVA works because if H0 is true the variation of the
sample means should be only due to sampling variation.
•If H0 isn’t true and there are real differences in the
populations from which the samples are drawn
(experimentally this just means the different conditions)
then the variation of the sample means will be larger than
expected from the amount of variation within each
population (condition).
•ANOVA works simply by giving the probability of obtaining
the observed amount of variation between condition
means if it was true that each condition represented a
random sample from a single parent ‘population’.
• A good estimate of the variation of
individual scores within the ‘parent
population’ can be obtained simply by
using an average the variance within each
condition.
What is the variance within a condition?
Suppose we have data from an experiment with 3 conditions:
+23
-18
The red vertical lines represent the condition averages
The red horizontal lines represent the deviations of individual scores from
their associated condition means. The variance of the scores within a
condition is the sum of these values (after squaring) divided by the
number of observations in the condition minus 1 (the condition D.F.)
This estimate isn’t affected by real differences between conditions
However the variation between condition means is
affected.
Recall:
Can also work out
the SD of the raw
scores from the SD
of the sample
means = 2.2 x
sqrt(10) =7
Raw scores SD= 7”
7”
4’0”
m
5’5”
7’0”
2.2”
Samples of size 10
SD of the sample means
= 7/sqrt(10)
= 7/3.16 = 2.2
4’0”
5’5”
7’0”
Two estimates of the population variance
• They both agree if and only if there is no real
difference between conditions
• In ANOVA the F ratio is used to compare these
two estimates of the population variance.
• The F ratio is just the estimate based on the
variation between the condition means
(MSBetween) divided by the estimate based on the
variation of the data within each condition
(MSwithin)
Note MS = mean square and is just another term used in ANOVA for a variance
F Ratio in ANOVA
• If the two estimates agree then this F ratio
should be approximately 1. If there are real
differences between conditions (i.e. the ‘parent
populations’ from which they are drawn are not
the same) then this estimate will be larger than
1.
• If it is larger than 1 by a sufficient amount the p
value will be <.05 and you conclude that there is
evidence for real differences between thee
conditions.
Layout
Task Difficulty
Easy
10 subs
Hard
10 subs
Tylenol
10 subs
Chlortrimeton 10 subs
10 subs
10 subs
Benadryl
Drug
Sixty adult drivers selected at random.
Each one randomly assigned to one of the six
experimental conditions (Between Subjects Design)
Main effects and Interactions
• In factorial designs you can test for main
effects of Factors and Interactions
between them.
• A significant main effect of drug would
indicate that there are genuine differences
between the three drug groups (ignoring or
averaging across task difficulty)
Drug Main effect
Task Difficulty
Easy
Hard
Benadryl
10 subs
10 subs
Tylenol
10 subs
10 subs
Chlortrimeton
10 subs
10 subs
Drug
25
20
15
Series1
10
5
0
Benadryl
Tylenol
Drug
Main effect plot for Drug
Chlortrimeton
Task Main effect
Task Difficulty
Easy
Hard
Benadryl
10 subs
10 subs
Tylenol
10 subs
10 subs
Chlortrimeton
10 subs
10 subs
Drug
14
12
10
8
Series1
6
4
2
0
Easy
Difficult
Main effect plot for Task Difficulty
25
20
15
Easy
Hard
10
5
0
Benadryl
Tylenol
Chlortrimeton
Interaction plot– Drug by Task Difficulty
No Interaction just drug main effect
25
20
15
Easy
Hard
10
5
0
Benadryl
Tylenol
Chlortrimeton
Interaction plot – Drug by Task Difficulty
Interaction no main effects
60
50
40
Easy
30
Hard
20
10
0
Benadryl
Tylenol
Chlortrimeton
Interaction plot – Drug by Task Difficulty
Interaction and main effect of Task but no
main effect of drug