Two-Way Balanced Independent Samples ANOVA - PowerPoint Presentation

Slide1

Two-Way Balanced Independent Samples ANOVA

Computations

Confidence

IntervalsSlide2

Partitioning the

SS

total

The total SS is divided into two sourcesCells or Model SSError SSThe model is Slide3

Partitioning the SScells

The cells

SS is divided into three sources

SSA, representing the main effect of factor ASSB, representing the main effect of factor BSSAxB, representing the A x B interactionThese sources will be orthogonal if the design is balanced (equal sample sizes)They sum to SScellsOtherwise the analysis gets rather complicated.Slide4

Gender x Smoking HistoryCell

n

= 10, Y2 = 145,140Slide5

Computing Treatment SS

Square and then sum group totals, divide by the number of scores that went into each total, then subtract the CM

. Slide6

SScells and

SS

error

SSerror is then SStotal minus SS

Cells = 26,115 ‑ 15,405 = 10,710. Slide7

SSgender and

SS

smokeSlide8

SSSmoke x Gender

SS

interaction

= SSCells  SSGender  SSSmoke

= 15,405 ‑ 9,025 ‑ 5,140 = 1,240. Slide9

Degrees of Freedomdftotal =

N

- 1dfA = a - 1dfB = b - 1

dfAxB = (a - 1)(b -1)dferror = N - abSlide10

Source TableSlide11

Simple Main Effects of Gender

SS

Gender, never smokedSS Gender, stopped < 1mSS

Gender, stopped 1m ‑ 2y Slide12

Simple Main Effects of Gender

SS

Gender, stopped 2y - 7ySS Gender, stopped 7y ‑ 12 y Slide13

Simple Main Effects of GenderMS = SS / df; F = MS

effect

/ MSEMSE from omnibus model = 119 on 90

df Slide14

Interaction PlotSlide15

Simple Main Effects of Smoking

SS

Smoking history for men

SS Smoking history for womenSmoking history had a significant simple main effect for women, F

(4, 90) = 11.97, p < .001, but not for men, F(4, 90) = 1.43,

p

=.23.

Slide16

Multiple Comparisons Involving A Simple Main EffectSmoking had a significant simple main effect for women.

There are 5 smoking groups.

We could make 10 pairwise comparisons.Instead, we shall make only 4 comparisons.We compare each group of ex-smokers with those who never smoked.Slide17

Female Ex-Smokersvs. Never Smokers

There is a special procedure to compare each treatment mean with a control group mean (Dunnett).

I’ll use a Bonferroni procedure instead.The denominator for each

t will be:Slide18

See Obtaining

p

values with SPSSSlide19

Multiple Comparisons Involving a Main Effect

Usually done only if the main effect is significant and not involved in any significant interaction.

For pedagogical purposes, I shall make pairwise comparisons among the marginal means for smoking.Here I use Bonferroni

, usually I would use REGWQ.Slide20

Bonferroni Tests, Main Effect of Smoking

c

= 10, so adj. criterion = .05 / 10 = .005.n’

s are 20: 20 scores went into each mean. Slide21

Smoking History

Mean

< 1 m

25.0

A

1m - 2y

28.5

AB

2y - 7y

35.0

BC

7y - 12y

39.0

CD

Never

45.0

D

Means sharing a superscript do not differ from one another at the .05 level.

Results of

Bonferroni

TestSlide22

(Semipartial) Eta-Squared



2

= SSEffect  SSTotalUsing our smoking history data,

For the interaction, For gender, For smoking history,

Slide23

CI.90 Eta-Squared

Compute the

F that would be obtained were all other effects added to the error term.

For gender,Slide24

CI.90 Eta-Squared

Use that

F with my Conf-Interval-R2-Regr.sas

F= 51.752 ;df_num = 1 ;df_den = 98 ;

eta_squared

eta2_lower

eta2_upper

0.34558

0.22117

0.44897

Get the CI with SPSS or RSlide25

CI.90 Eta-Squared

90

% CI [.22, .45] for gender[.005, .15] for smoking[.000, .17] for the interaction

Yikes, 0 in the CI for a significant effect!The MSE in the ANOVA excluded variance due to other effects, that for the CI did not.Slide26

Partial Eta-Squared

The value of

η2 can be affected by the number and magnitude of other effects contributing to variance in the outcome variable.

For example, if our data were only from women, SSTotal would not include SSGender and SSInteraction.This would increase η2.Partial eta-squared estimates what the effect would be if the other effects were all zero.Slide27

Partial Eta-Squared

For the interaction,

For gender,

For smoking history, Slide28

CI.90 on Partial Eta-Squared

If you use the source table

F-ratios and df with my Conf-Interval-R2-Regr.sas, it will return confidence intervals on partial eta-squared.Gender: [.33, .55]

Smoking: [.17, .41]Interaction [.002, .18] note that it excludes 0Slide29

Partial 2

and

F For Interaction

Notice that the denominator of both includes error but

excludes

the effects of Gender and Smoking History

.Slide30

Semi-Partial 

2

Notice that, unlike partial

2 and

F

,

the denominator of semi-partial



2

includes

the effects of Gender and Smoking History

.Slide31

Omega-Squared

For the interaction,

For gender,

For smoking history, Slide32

2 for Simple Main Effects

For the women, SStotal = 11,055and SS

smoking = 5,7002 = 5,700/11,055 = .52To construct confidence interval, need compute an F using data from women only.The SSE is 11,055 (total) – 5,700 (smoking) = 5,355. Slide33

90% CI [.29, .60]For the men, 

2

= .11, 90% CI [0, .20] Slide34

SAS EFFECTSIZEPROC

GLM; CLASS Age Condition; MODEL Items=Age|Condition / EFFECTSIZE alpha=

0.1;This will give you eta-squared, partial eta-squared, omega-squared, and confidence intervals for each.Slide35

Presenting the Results

Participants were given a test of their ability to detect the scent of a chemical thought to have

pheromonal

properties in humans. Each participant had been classified into one of five groups based on his or her smoking history. A 2 x 5, Gender x Smoking History, ANOVA was employed, using a .05 criterion of statistical significance and a MSE of 119 for all effects tested. There were significant main effects of gender,

F(1, 90) = 75.84, p < .001, 2 = .346, 90% CI [.22, .45], and smoking history,

F

(4, 90) = 10.80,

p

< .001, 

2

= .197, 90% CI [.005, .15], as well as a significant interaction between gender and smoking history,

F

(4, 90) = 2.61,

p

= .041, 

2

= .047, 90% CI [.00, .17

].

As shown in Table 1, women were better able to detect this scent than were men, and smoking reduced ability to detect the scent, with recovery of function being greater the longer the period since the participant had last smoked.Slide36
Slide37

The

significant interaction was further investigated with tests of the simple main effect of smoking history. For the men, the effect of smoking history fell short of statistical significance,

F

(4, 90) = 1.43, p = .23, 2 = .113, 90% CI [.00, .20]. For the women, smoking history had a significant effect on ability to detect the scent,

F(4, 90) = 11.97, p < .001,



2

= .516, 90% CI [.29, .60]. This significant simple main effect was followed by a set of four contrasts. Each group of female ex-smokers was compared with the group of women who had never smoked. The

Bonferroni

inequality was employed to cap the familywise error rate at .05 for this family of four comparisons. It was found that the women who had never smoked had a significantly better ability to detect the scent than did women who had quit smoking one month to seven years earlier, but the difference between those who never smoked and those who had stopped smoking more than seven years ago was too small to be statistically significant

.Slide38

Interaction PlotSlide39

2 or Partial 

2

?

I generally prefer 2Kline says you should exclude an effect from standardizer only if it does not exist in the natural population.Values of partial 2 can sum to greater than 100%. Can one really account for more than all of the variance in the outcome variable?Slide40

For every

effect,Slide41

For every

effect,

These sum to 150%Slide42

AssumptionsNormality within each cellHomogeneity of variance across cellsSlide43

Advantages of Factorial ANOVAEconomy -- study the effects of two factors for (almost) the price of one.

Power -- removing from the error term the effects of Factor B and the interaction gives a more powerful test of Factor A.

Interaction -- see if effect of A varies across levels of B.Slide44

One-Way ANOVA

Consider the partitioning of the sums of squares illustrated to the right.

SS

B = 15 and SSE = 85. Suppose there are two levels of B (an experimental manipulation) and a total of 20 cases. Slide45

Treatment Not Significant

MSB

= 15,

MSE = 85/18 = 4.722. The F(1, 18) = 15/4.72 = 3.176, p

= .092. Woe to us, the effect of our experimental treatment has fallen short of statistical significance. Slide46

Sex Not Included in the Model

Now suppose that the subjects here consist of both men and women and that the sexes differ on the dependent variable.

Since sex is not included in the model, variance due to sex is error variance, as is variance due to any interaction between sex and the experimental treatment.Slide47

Add Sex to the Model

Let us see what happens if we include sex and the interaction in the model.

SS

Sex = 25, SSB = 15,

SSSex*B

= 10, and

SSE

= 50. Notice that the

SSE

has been reduced by removing from it the effects of sex and the interaction.

Slide48

Enhancement of Power

The

MSB

is still 15, but the MSE is now 50/16 = 3.125 and the F(1, 16) = 15/3.125 = 4.80,

p = .044. Notice that excluding the variance due to sex and the interaction has reduced the error variance enough that now the main effect of the experimental treatment is significant.