Transcript Comparing Several Means: One-way ANOVA Lesson 15 Analysis of Variance or ANOVA  Comparing 2 or more treatments  i.e., groups  Simultaneously  H 0: m 1 = m.

Comparing
Several Means:
One-way ANOVA
Lesson 15
Analysis of Variance
or ANOVA
 Comparing 2 or more treatments

i.e., groups

Simultaneously
 H 0: m 1 = m 2 = m 3 …
 H1: at least one population different
from others ~
Experimentwise Error
Why can’t we just use t tests?
 Type 1 error: incorrectly rejecting H0
 each comparison a = .05
 Experimentwise probability of type 1 error
 P (1 or more Type 1 errors) ~

Experimentwise Error
H 0: m 1 = m 2 = m 3
 Approximate experimentwise error

H0: m1 = m2
H0: m1 = m3
H0: m2 = m3
experimentwise

a = .05
a = .05
a = .05
a  .15
ANOVA: only one H0
 a = .05 (or level you select) ~
Analysis of Variance: Terminology
Factor
 independent variable
 Single-Factor Design (One-way)
 single independent variable with 2 or
more levels
 levels: values of independent variable ~

Analysis of Variance: Terminology
Repeated Measures ANOVA
 Same logic as paired t test
 Factorial Design
 More than one independent variable
 Life is complex: interactions
 Mixed Factorial Design
 At least 1 between-groups & within
groups variable
 Focus on independent-measures ~

e.g., Effects of caffeine on reaction time
Single-factor design
with 3 levels
Caffeine dose
0 mg
50 mg
100 mg
rt  M1
rt  M 2
rt  M 3
0 mg
50 mg
100 mg
male
rt  M1
rt  M 2
rt  M 3
female
rt  M 4
rt  M 5
rt  M 6
3 x 2 Factorial design
Sex
Test Statistic

F ratio
 ratio of 2 variances
F

t

variance(differences) between samplemeans
variance(difference) expectedby chance (error)
same concept as t tests
difference betweensample means
difference expected by chance(error)
F = t2
 Only 2 groups ~
F ratio
MS: mean squared deviations = variance
 MSB = MS between treatments
 Textbook: MSM
 Average distance b/n sample means
 MSW = MS within treatments
 Textbook: MSR
 differences between individuals
2
 same as s pooled ~

Logic of ANOVA
MS B
F 
MSW
Differences b/n groups (means) bigger
than difference between individuals?
 If H0 false
 then distance between groups should
be larger ~

Partitioning SS
SST = total sums of squares
 total variability
 SSB = between-treatments sums of squares
 variability between groups
 SSW = within-treatments sums of squares
 variability between individuals

SST  SS B  SSW
SSB
R 
SST
2
* % variance explained by IV
Calculating SS
SST
  ( X i  X Grand )
SS B
  ( X k  X Grand )
SSW
 (Xi  X k )
2
2
2
Calculating MS
SS
MS 
df
Calculating MSW

Same as s2pooled for > 2 samples
SS1  SS2  SS3
MSW 
dfW
dfW  N  k
Calculating MSB
SS B
MS B 
df B
df B  k  1
SPSS One-way ANOVA


Menu
 Analyze
 Compare Means
 One-way ANOVA
Dialog box
 Dependent List (DV)
 Factor (IV)
 Options:
Descriptives, Homogeneity of Variance

Post Hoc ~
Interpreting ANOVA
Reject H0
 at least one sample different from
others
 do not know which one(s)
 Must use post hoc tests
 Post hoc: after the fact
 ONLY if rejected H0 for ANOVA
 Many post hoc tests
 Differ on how conservative ~

Post Hoc Test: Pairwise comparisons
Adjusted a levels
 LSD (Least Significant Difference)
 Basically t-test, no adjustment
 Tukey’s HSD
 Similar logic to t – test
 Scheffe Test
 F test with only 2 groups
 Differ on how conservative
 More conservative  bigger difference
required ~

Detour Learning Task
Prenatal exposure to
methamphetamine
 effects on learning?

FIGURE 1
Males
Mean Latency to Social Contact
350
300
Strangers
Cagemates
250
200
150
100
50
0
1
2
3
Detour Learning Trial
4