anova - wofford college...a one-way anova was conducted to examine the hypothesis that different...

ANOVA continued…

Chapter 10

Zettergren (2003)

School adjustment in adolescence for previously rejected, average, and popular children.

Effect of peer reputation on academic performance and school adjustment

IV or Factor = Peer reputation 3 levels: rejected, average, popular (based on…)

3rd and 4th grade students ranked every classmate (same gender) in the order they wanted them to stay with the class if they were to move to a smaller room and not everyone could go

DV = Academic ability (8th grade)

DV = Attitudes toward school (8th grade)

Zettergren (2003) results

ANOVA: Partitions the Variance

Total Variance

Between Treatment Variance

1. Treatment effects

2. Error

Within Treatment Variance

Error

Between variance ---------------------- Within variance

F =

ANOVA: Example summary table

Source df SS MS F

Between

Within

2

12

30

16

15

1.33

11.28*

Total 14 46

F (2, 12) = 11.28, p < .01 * Significant at .01 level

between

betweenbetween

df

SSMS

within

withinwithin

df

SSMS

within

between

MS

MSF

Anova: Definitional formulas

Between groups SS (sums of squares) Sum of squared deviations from each group’s mean from grand mean multiplied by the number of Ss in group

Within groups SS Sum of squared deviations of each score from group mean

Total SS Sum of squared deviations of each score from the grand mean

])[( 2nMM Gg

2)( gMX

2)( GMX

ANOVA formulas: Sum of Squares (SS)

Total SS: sum of squared deviations from grand mean

Within SS = SS = sum of squared deviations from grp

mean

Between SS = sum of squared deviations of grp mean

from grand mean

SStotal = SSbetween + SSwithin

N

GXSSTOTAL

22

N

G

n

TSSbetween

22

)(

Where G = grand (overall) mean

Where N = total number of scores

Where T = sum of scores for group

Where n = number of scores in group

))(

(

2

2

g

g

gwithinn

XXSS

ANOVA formulas: SS

Total SS =

= 106 – 302/15 = 46

Within SS = SS

= 6 + 6 + 4 = 16

Between SS =

= 52/5 + 202/5 + 52/5 – 302/15

= 30

SStotal = SSbetween + SSwithin

46 = 30 + 16

N

GXSS

22 Temp Cond

1 2 3

0 4 1 X2=

1 3 2 106

3 6 2 G=30

1 3 0 N=15

0 4 0 k=3

T1=5 T2=20 T3=5

SS1=6 SS2=6 SS3=4

n1=5 n2=5 n3=5

M1=1 M2=4 M3=1

N

G

n

TSSbetween

22

)(

Vitamin C

Study: Year 1 = # of cold symptoms, Year 2 = # cold

symptoms with treatment

Factor: Group

Placebo

Low dose of Vitamin C

High dose of Vitamin C

Dependent variable: Difference in cold symptoms

from year 1 to year 2

Hypotheses

Boxplot of Vitamin C data

101010N =

Vitamin C Treatment

hilowpl acebo

DIF

F

20

10

0

-10

-20

3

1

Vitamin C: Data

Report

DIFF

3.50 10 4.143 35

-2.10 10 4.067 -21

-2.00 10 5.477 -20

-.20 30 5.182 -6

Vitamin C Treatment

placebo

low

hi

Total

Mean N Std. Dev iation Sum

Report

DIFF2

27.7000 10 47.32406 277.00

19.3000 10 24.23977 193.00

31.0000 10 12.75408 310.00

26.0000 30 30.87014 780.00

Vitamin C Treatment

placebo

low

hi

Total


X2

T1= T2= T3= G=

SS1= SS2= SS3= x2=

n1= n2= n3= N=

M1= M2= M3= k=

Vitamin C: Data

Report

DIFF

3.50 10 4.143 35

-2.10 10 4.067 -21

-2.00 10 5.477 -20

-.20 30 5.182 -6

Vitamin C Treatment

placebo

low

hi

Total


Report

DIFF2

27.7000 10 47.32406 277.00

19.3000 10 24.23977 193.00

31.0000 10 12.75408 310.00

26.0000 30 30.87014 780.00

Vitamin C Treatment

placebo

low

hi

Total


T1= 35 T2= -21 T3= -20 G= -6

SS1= SS2= SS3= x2=

n1= 10 n2= 10 n3= 10 N= 30

M1=

3.5

M2=

-2.1

M3=

-2.0

k= 3

Vitamin C: Data

Report

DIFF

3.50 10 4.143 35

-2.10 10 4.067 -21

-2.00 10 5.477 -20

-.20 30 5.182 -6

Vitamin C Treatment

placebo

low

hi

Total


Report

DIFF2

27.7000 10 47.32406 277.00

19.3000 10 24.23977 193.00

31.0000 10 12.75408 310.00

26.0000 30 30.87014 780.00

Vitamin C Treatment

placebo

low

hi

Total


T1= 35 T2= -21 T3= -20 G = -6

SS1=

154.5

SS2=

148.9

SS3=

270

x2=

780

n1= 10 n2= 10 n3= 10 N= 30

M1= 3.5 M2= -2.1 M3= -2.0 k= 3

g

g

gn

XXSS

2

2 )(

10

)35(277

2

SS

10

)21(193

2SS

10

)20(310

2SS

Within SS

Vitamin C: Data

T1= 35 T2= -21 T3= -20 G = -6

SS1= 154.5 SS2= 148.9 SS3= 270 x2= 780

n1= 10 n2= 10 n3= 10 N= 30

M1= 3.5 M2= -2.1 M3= -2.0 k= 3

N

GXSS

22

30

)6(780

2SS

Total SS:

Within SS:

Between SS:

SS: 154.5 + 148.9 + 270 = 573.4

= 778.8

N

G

n

TSSbetween

22

30

6)

10

20

10

21

10

35(

2222

betweenSS = 205.4

Vitamin C: Data

T1= 35 T2= -21 T3= -20 G = -6

SS1= 154.5 SS2= 148.9 SS3= 270 x2= 780

n1= 10 n2= 10 n3= 10 N= 30

M1= 3.5 M2= -2.1 M3= -2.0 k= 3

between

betweenbetween

df

SSMS

within

withinwithin

df

SSMS

within

between

MS

MSF 7.102

2

4.205betweenMS

237.2127

4.573withinMS

836.4237.21

7.102F

dfbetween = k – 1

dfwithin = N - k

Critical F @ .05 = 3.35, @ .01 = 5.49

ANOVA summary table

SPSS v. Write-up

Source df SS MS F

Between

Within

2

27

205.4

573.4

102.7

21.2

4.84*

Total 30 778.8

ONEWAY ANOVA

DIFF

205.400 2 102.700 4.836 .016

573.400 27 21.237

778.800 29

Between Groups

Within Groups

Total

Sum of

Squares df Mean Square F Signif icance

* Significant at the .02 level

Vitamin C: Conclusions

A one-way ANOVA was conducted to

examine the hypothesis that different types of

vitamin C treatment have a differential effect

on cold symptoms compared to prior years

without the treatment.

It was found that the number of colds were

significantly different for the placebo (M =

3.5), low dose (M = -2.1), and high dose (M =

-2.0) groups, F(2, 27) = 4.8, p < .05.

Post Hoc Tests

If find a “significant” F: there is at least 1 mean that is different

Post-tests examine which means are and are not significantly different

Compare 2 means at a time (pair-wise comparisons)

Type I error: divide alpha among all tests need to do Planned comparisons: based on predictions

Tukey’s HSD

Scheffe test (numerator is for MSbetween for only the two treatments you want to compare)

Bonferroni

Effect size

How much of the variability in DV is attributed

to IV?

Effect size for ANOVA: eta-squared (ή2)

Total

Between

SS

SS2

Self-esteem study: Self-Esteem Descriptor (SED) at 5, 7, 9, 11, 13

2525252525N =

Self-esteem at age 1



Self esteem at age 7


160

140

120

100

80

60

40

20

0

-20

6

3

8

115

3

11

3

125

11

Self-esteem: Between subject

ONEWAY Descriptives

SED

25 33.8800 27.91702 5.58340 22.3564 45.4036 2.00 106.00

25 27.6000 35.35180 7.07036 13.0075 42.1925 1.00 138.00

25 29.6000 31.49206 6.29841 16.6007 42.5993 3.00 127.00

25 29.9600 34.86340 6.97268 15.5691 44.3509 1.00 125.00

25 16.0800 16.95071 3.39014 9.0831 23.0769 1.00 66.00

125 27.4240 30.20181 2.70133 22.0773 32.7707 1.00 138.00

1.00

2.00

3.00

4.00

5.00

Total

N Mean Std. Dev iation Std. Error Lower Bound Upper Bound

95% Conf idence Interval for

Mean

Minimum Maximum

ONEWAY ANOVA

SED

4539.088 4 1134.772 1.254 .292

108567.4 120 904.729

113106.5 124

Between Groups

Within Groups

Total

Sum of

Squares df Mean Square F Signif icance

Self-esteem: Within subject

Descriptive Statistics

33.88 27.917 25

27.60 35.352 25

29.60 31.492 25

29.96 34.863 25

16.08 16.951 25

Self -esteem at age 5





Mean Std. Dev iation N

Tests of Within-Subjects Effects

Measure: MEASURE_1

4539.088 4 1134.772 4.771 .001

4539.088 2.989 1518.796 4.771 .004

4539.088 3.461 1311.589 4.771 .003

4539.088 1.000 4539.088 4.771 .039

22834.512 96 237.859

22834.512 71.727 318.355

22834.512 83.058 274.922

22834.512 24.000 951.438

Sphericity Assumed

Greenhouse-Geisser

Huy nh-Feldt

Lower-bound

Sphericity Assumed

Greenhouse-Geisser

Huy nh-Feldt

Lower-bound

Source

AGE

Error(AGE)

Ty pe III Sum

of Squares df Mean Square F Sig.

Self-esteem: Planned contrasts

Paired Samples Test

6.28 18.311 3.662 -1.28 13.84 1.715 24 .099

4.28 22.868 4.574 -5.16 13.72 .936 24 .359

3.92 22.546 4.509 -5.39 13.23 .869 24 .393

17.80 20.738 4.148 9.24 26.36 4.292 24 .000

Self -esteem at age 5 -


Pair

1



Pair

2



Pair

3



Pair

4

Mean Std. Dev iation

Std. Error

Mean Lower Upper

95% Conf idence

Interv al of the

Dif f erence

Paired Dif ferences

t df Sig. (2-tailed)

Paired Samples Statistics

33.88 25 27.917 5.583

27.60 25 35.352 7.070

33.88 25 27.917 5.583

29.60 25 31.492 6.298

33.88 25 27.917 5.583

29.96 25 34.863 6.973

33.88 25 27.917 5.583

16.08 25 16.951 3.390



Pair

1



Pair

2



Pair

3



Pair

4

Mean N Std. Dev iation

Std. Error

Mean

Self-esteem write-up

Within-subject design

A longitudinal study was conducted on self-esteem. A repeated-measures ANOVA was conducted over five time periods; five years old (M = 33.88, SD = 27.92), seven years old (M = 27.60, SD = 35.35), nine years old (M = 29.60, SD = 31.49), 11 years old (M = 29.96, SD = 34.86), and 13 years old (M = 16.08, SD = 16.95). A significant effect of age was found, F (4, 96) = 4.77, p = .001.

Post-hoc tests were performed comparing the youngest age (five years old) with each of the other ages (7, 9, 11, and 13 years). One significant result was found. Self-esteem at age five (M = 33.88, SD = 27.92) was significantly different compared to self-esteem at age 13 (M = 16.08, SD = 16.95), t(24) = 4.29, p < .001. This suggests that self-esteem remains stable from age five until age 11, and then declines at age 13.

Repeated-measures ANOVA

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anova - wofford college...a one-way anova was conducted to examine the hypothesis that different...

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