Independent-measures analysis of

variance (Oneway ANOVA)

Repeated measures designs (t-tests)

Individual differences

Oneway ANOVA

WithinTotal

i

Between

i

iiiiWithin

Total

SSor SS N

G

n

TSS

group eachfor n

XXMXSS

N

GX

N

XXMXSS

i

22

2

22

22

2

22

)()(

)()(

Independent Samples

Compare two groups that are unrelated to each other

Numerator is difference between groups

Does not control for the impact of individual differences

Related Samples

Compare two measures from one person or one related pair of people

Numerator is difference within pair

Controls for the impact of individual differences

Null hypothesis: in the population, there are no

mean differences among the treatment groups

Alternate hypothesis states that there are

mean differences among the treatment groups.

...: 3210 H

H1: At least one treatment mean μ

is different from another

removed) sdifference individual(

effect, treatmentno withexpected es)(differenc variance

s treatmentbetween es)(differenc varianceF

F ratio based on variances

• Same structure as independent measures

• Variance due to individual differences is

not present

Participant characteristics that vary from one person to another.

• Not systematically present in any treatment group or by research design

Characteristics may influence measurements on the outcome variable

• Eliminated from the numerator by the research design

• Must be removed from the denominator statistically

Numerator of the F ratio includes

• Systematic differences caused by treatments

• Unsystematic differences caused by random factors (reduced because same individuals in all treatments)

Denominator estimates variance reasonable to expect from unsystematic factors

• Effect of individual differences is removed

• Residual (error) variance remains

Total Variability

Between Treatments

Within Treatments

Between Subjects

Error Variance

Repeated Measures ANOVA

(Equations follow)

First stage

• Identical to independent samples ANOVA

• Compute SSTotal =SSBetween treatments + SSWithin treatments

Second stage

• Removing the individual differences from the denominator

• Compute SSBetween subjects and subtract it from SSWithin treatments to find SSError

There is a computational equation for the Sum of Squared Deviations from the Mean:

With ANOVA, we sum groups of means

• G2 = Sum of all the scores squared

• T1 = sum of scores in Treatment Group 1

• P1 = sum of scores of Person 1 across time

• TA 1 and TB 1 = sums of scores across levels of a factor.

N

XX

22 )(

N

GXSStotal

22

treatment each insidetreatmentswithin SSSS

N

G

n

TSS treatmentsbetween

22

Note that this is the

Computational Formula

for SS

N

G

k

PSSbetween

22

subjects

subjectsbetweentreatments withinerror SSSSSS

dftotal = N – 1

dfwithin treatments = Σdfinside each treatment

dfbetween treatments = k – 1

dfbetween subjects = n – 1

dferror = dfwithin treatments – dfbetween subjects

error

errorerror

df

SSMS

treatments between

treatments betweenntstreatme between

df

SSMS

error

mentstreat between

MS

MSF

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Solve the ANOVA table problem

Try to finish in 10 minutes (1 minute per slot to complete)

SS df MS F

Between Treatments 10

Within Treatments

Between Subjects 26

Error

Total 110

A sample of n=10 individuals participate in a repeated measures study of 4 treatment conditions. Is there evidence of significant differences among the treatments? Hint: begin with df.

Subject Before 6 months 1 year

A 56 64 69B 79 91 89C 68 77 81D 59 69 71E 64 77 75F 74 88 86G 73 85 86H 47 64 69I 78 98 100J 61 77 85K 68 86 93L 64 77 87M 53 67 76N 71 85 95O 61 79 97P 57 77 89Q 49 65 83R 71 93 100S 61 83 94T 58 75 92U 58 74 92

21 subjects were in the study

The dependent variable was their Growth Mindset, the belief in the ability for intelligence to grow.

They received two months of coaching on how to grow problem solving intelligence.

Their Growth Mindset was measured on a 100-point scale at three times: • before the program,

• six months after it started,

• one year after it started

Much more output than you want

Need to ask for some Options to get SPSS to do as much of the work as possible.

• Descriptives

• Plots

• Multiple Comparisons

• Effect Size

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Descriptives

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Follow steps on diagram at right

Percentage of variance explained by the

treatment differences

Partial η2 is percentage of variability that has

not already been explained by other factors

error

treatments between

subjectsbetweentotal

treatments between

SS

SS

SSSS

SS

2

Determine exactly where significant

differences exist among more than two

treatment means

• Tukey’s HSD can be used (almost always

same number of subjects) or Scheffé if

dropouts mean unequal measures.

• Substitute SSerror and dferror in the formulas

Can people increase their actual problem-solving intelligence by

learning that their brains are able to change? Researchers found

the Growth Mindset of 21 people increased from just before they

began a Growth Mindset Coaching Program (M=63.3, s=8.9),

when measured six months (M=78.6, s=9.6) and one year after

starting the program (M=86.1, s=9.6). The differences were

significant (F(2,40)=121.89, p<.001). Post-hoc tests with

Bonferroni correction showed that self-efficacy at each time was

significantly improved from the one before. In fact, the impact of

the training program was large, with about 86% of the variability

in Problem Solving Growth Mindset related to the training

(2=.859). The coaching program seems to be an effective way

to help people tap into the potential of the Growth Mindset.

The observations within each treatment

condition must be independent.

The population distribution within each

treatment must be normal.

The variances of the population

distribution for each treatment should be

equivalent.

Decide if each of the following statements

is True or False.

• For the repeated-measures ANOVA, degrees of freedom for SSerror is (N–k) – (n–1).

T/F

• N is the number of scores and nis the number of participantsFalse