Comparing Several Means:

One-way ANOVALesson 15

Analysis of Variance

or ANOVA Comparing 2 or more treatments

i.e., groups Simultaneously

H0: 1 = 2 = 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 = .05

Experimentwise probability of type 1 error P (1 or more Type 1 errors) ~

Experimentwise Error

H0: 1 = 2 = 3

Approximate experimentwise errorH0: 1 = 2 = .05H0: 1 = 3 = .05H0: 2 = 3 = .05

experimentwise .15 ANOVA: only one H0

= .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 timeSingle-factor design

with 3 levels

Caffeine dose

0 mg 50 mg 100 mg

1 rt M 2 rt M 3 rt M

3 x 2 Factorial design

Sexmale

female

0 mg 50 mg 100 mg

1 rt M 2 rt M 3 rt M

4 rt M 5 rt M 6 rt M

Test Statistic F ratio

ratio of 2 variances

(error) chanceby expected difference

means samplebetween differencet

(error) chance by expected e)(differenc variancemeans sample between es)(differenc variance

F

same concept as t tests

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 same as s2

pooled ~

Logic of ANOVA

Differences b/n groups (means) bigger than difference between individuals?

If H0 false then distance between groups should

be larger ~

W

B

MSMSF

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

WBT SSSSSS T

B

SSSSR 2

* % variance explained by IV

Calculating SS

TSS 2)( Grandi XX

BSS

WSS 2)( ki XX

2)( Grandk XX

Calculating MS

dfSSMS

Calculating MSW Same as s2

pooled for > 2 samples

WW df

SSSSSSMS 321 kNdfW

Calculating MSB

B

BB df

SSMS 1kdfB

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 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 1Males

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1 2 3 4

Detour Learning Trial

Mea

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Cagemates