Two-Way Balanced Independent Samples ANOVA

Overview of Computations

Partitioning the SStotal• The total SS is divided into two sources

– Cells or Model SS– Error SS– The model is

ijkjkkjijk eY

Partitioning the SScells• The cells SS is divided into three sources

– SSA, representing the main effect of factor A– SSB, representing the main effect of factor B– SSAxB, representing the A x B interaction

• These sources will be orthogonal if the design is balanced (equal sample sizes)– They sum to SScells– Otherwise the analysis gets rather complicated.

Gender x Smoking History SMOKING HISTORY GENDER never < 1m 1 m - 2 y 2 y - 7 y 7 y - 12 y marginal Male 30 20 22 25 28 25 Female 60 30 35 45 50 44 marginal 45 25 28.5 35 39 34.5

Degrees of Freedom

• dftotal = N - 1• dfA = a - 1• dfB = b - 1• dfAxB = (a - 1)(b -1)• dferror = N - ab

Source TableSource SS df MS F p A-gender 9025 1 9025 75.84 <.001 B-smoking history 5140 4 1285 10.80 <.001 AxB interaction 1240 4 310 2.61 .041 Error 10,710 90 119 Total 26,115 99

Simple Main Effects of Gender

• MS = SS / df; F = MSeffect / MSE• MSE from omnibus model = 119 on 90 df

Smoking History SS Gender at never < 1m 1 m - 2 y 2 y - 7 y 7 y - 12 y F(1, 90) 37.82 4.20 7.10 16.81 20.34 p <.001 .043 .009 <.001 <.001

Interaction Plot

Simple Main Effects of Smoking

• Smoking 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.

Multiple Comparisons Involving A Simple Main Effect

• Smoking 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.

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:

.0125.405. pc

8785.4)10/110/1(119

• See Obtaining p values with SPSS

Never Smoked vs Quit 8785.4

)90( ji MMt

p Significant?

< 1 m (60-30) / 4.8785=6.149 < .001 yes 1 m - 2 y (60-35) / 4.8785=5.125 < .001 yes 2 y - 7 y (60-45) / 4.8785=3.075 .0028 yes 7 y - 12 y (60-50) / 4.8785=2.050 .0433 no

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.

Bonferroni Tests, Main Effect of Smoking

• c = 10, so adj. criterion = .05 / 10 = .005.• n’s are 20: 20 scores went into each mean.

L e v e l i v s j t = S i g n i f i c a n t ? 1 v s 2

80.54496.320)20/120/1(119/)2545(

y e s

1 v s 3 ( 4 5 - 2 8 . 5 ) / 3 . 4 4 9 6 = 4 . 7 8 y e s 1 v s 4 ( 4 5 - 3 5 ) / 3 . 4 4 9 6 = 2 . 9 0 y e s 1 v s 5 ( 4 5 - 3 9 ) / 3 . 4 4 9 6 = 1 . 7 4 n o 2 v s 3 ( 2 8 . 5 - 2 5 ) / 3 . 4 4 9 6 = 1 . 0 1 n o 2 v s 4 ( 3 5 - 2 5 ) / 3 . 4 4 9 6 = 2 . 9 0 y e s 2 v s 5 ( 3 9 - 2 5 ) / 3 . 4 4 9 6 = 4 . 0 6 y e s 3 v s 4 ( 3 5 - 2 8 . 5 ) / 3 . 4 4 9 6 = 1 . 8 8 n o 3 v s 5 ( 3 9 - 2 8 . 5 ) / 3 . 4 4 9 6 = 3 . 0 4 y e s 4 v s 5 ( 3 9 - 3 5 ) / 3 . 4 4 9 6 = 1 . 1 6 n o

Smoking History Mean

< 1 m 25.0A

1m - 2y 28.5AB

2y - 7y 35.0BC

7y - 12y 39.0CD

Never 45.0D

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

Results of Bonferroni Test

Eta-Squared

• For the interaction,

• For gender,

• For smoking history, 197.115,26

140,52

346.115,26025,92

047.115,26

12402

CI.90 Eta-Squared

• Compute the F that would be obtained were all other effects excluded from the model.

• For gender,

nInteractioSmokeError

Gender

MSMSF

752.51)4490/()1240514010710(

1/9025

CI.90 Eta-Squared

See Using SPSS to Obtain a Confidence Interval for R2 From Regression

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.

Partial Eta-Squared• Value of η2 can be affected by number and

magnitude of other effects in model.• For example, if our data were only from

women, SSTotal would not include SSGender and SSInteraction.

• This would increase η2.• Partial eta-squared estimates effect if were

not other effects in the model.

Partial Eta-Squared

• For the interaction,

• For gender,

• For smoking history, 324.710,10140,5

140,52

p

457.710,10025,9

025,92

p

104.714,10240,1

240,12

p

CI.90 on Partial Eta-Squared

• If you use the source table F-ratios and df with the NoncF script, it will return confidence intervals on partial eta-squared.

• Gender: [.33, .55]• Smoking: [.17, .41]• Interaction [.002, .18] note it excludes 0

CI.90 on Partial Eta-Squared

Omega-Squared

• For the interaction,

• For gender,

• For smoking history, 18.234,26

)119(4140,52

34.234,26

)119(1025,92

03.234,26

764119115,26

)119(4240,12

SAS EFFECTSIZE

• PROC 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.

2 or Partial 2 ?• I generally prefer 2

• Kline 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?

25.10025

*

2

ErrorBABA

Effect

SSSSSSSSSS

For every effect,

For every effect,

50.50252

ErrorEffect

Effectp SSSS

SSThese sum to 150%

Assumptions

• Normality within each cell• Homogeneity of variance across cells

Advantages of Factorial ANOVA

• Economy -- 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.

One-Way ANOVAConsider the partitioning of the sums of squares illustrated to the right.SSB = 15 and SSE = 85. Suppose there are two levels of B (an experimental manipulation) and a total of 20 cases.

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.

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.

Add Sex to the ModelLet us see what happens if we include sex and the interaction in the model. SSSex = 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.

Enhancement of PowerThe 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.

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, ηp

2 = .46, 90% CI [.33, .55], and smoking history, F(4, 90) = 10.80, p < .001, , ηp

2 = .33, 90% CI [.17, .41],as well as a significant interaction between gender and smoking history, F(4, 90) = 2.61, p = .041, ηp

2 = .10, 90% CI [.002, .18]. 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.

Table 1. Mean ability to detect the scent. Smoking History Gender < 1 m 1 m -2 y 2 y - 7 y 7 y - 12 y never marginal Male 20 22 25 28 30 25 Female 30 35 45 50 60 44 Marginal 25 28 35 39 45

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. For the women, smoking history had a significant effect on ability to detect the scent, F(4, 90) = 11.97, p < .001. 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.

Interaction Plot