Two-Way Balanced Independent Samples

ANOVAComputations

ContrastsConfidence Intervals

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

• Cell n = 10, Y2 = 145,140

SMOKING HISTORY GENDER never < 1m 1 m - 2 y 2 y - 7 y 7 y - 12 y marginal Male 300 200 220 250 280 1,250 (25) Female 600 300 350 450 500 2,200 (44) marginal 900 (45) 500 (25) 570 (28.5) 700 (35) 780 (39) 3,450

025,119100

3450)( 22

NYCM

115,26025,119140,1452 CMYSSTotal

Computing Treatment SS

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

SScells and SSerror

SSerror is then SStotal minus SSCells = 26,115 ‑ 15,405 =

10,710.

SMOKING HISTORY GENDER never < 1m 1 m - 2 y 2 y - 7 y 7 y - 12 y marginal Male 300 200 220 250 280 1,250 (25) Female 600 300 350 450 500 2,200 (44) marginal 900 (45) 500 (25) 570 (28.5) 700 (35) 780 (39) 3,450

cellsSS

405,15

025,11910

500450350300600280250220200300 2222222222

SSgender and SSsmoke

SMOKING HISTORY GENDER never < 1m 1 m - 2 y 2 y - 7 y 7 y - 12 y marginal Male 300 200 220 250 280 1,250 (25) Female 600 300 350 450 500 2,200 (44) marginal 900 (45) 500 (25) 570 (28.5) 700 (35) 780 (39) 3,450

025,9025,11950

200,2250,1 22

GenderSS

140,5

025,11920

780700570500900 22222

SmokeSS

SSSmoke x Gender

SSinteraction = SSCells SSGender SSSmoke =

15,405 ‑ 9,025 ‑ 5,140 = 1,240.

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

SS Gender, never smoked

SS Gender, stopped < 1m

SS Gender, stopped 1m ‑ 2y

SMOKING HISTORY GENDER never < 1m 1 m - 2 y 2 y - 7 y 7 y - 12 y marginal Male 300 200 220 250 280 1,250 (25) Female 600 300 350 450 500 2,200 (44) marginal 900 (45) 500 (25) 570 (28.5) 700 (35) 780 (39) 3,450

500,420

90010

600300 222

50020

50010

300200 222

84520

57010

350220 222

Simple Main Effects of Gender

SS Gender, stopped 2y - 7y

SS Gender, stopped 7y ‑ 12 y

SMOKING HISTORY GENDER never < 1m 1 m - 2 y 2 y - 7 y 7 y - 12 y marginal Male 300 200 220 250 280 1,250 (25) Female 600 300 350 450 500 2,200 (44) marginal 900 (45) 500 (25) 570 (28.5) 700 (35) 780 (39) 3,450

000,220

70010

450250 222

420,220

78010

500280 222

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

• SS Smoking history for men

• SS Smoking history for women

• 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.

68050250,1

10280250220200300 222222

700,550200,2

10500450350300600 222222

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

Interaction Contrasts 2 x 2• Coefficients must be doubly centered• Sum to zero in every row and every column• Consider a 2 x 2, for which there is only one

interaction contrast.

• (A1B1 + A2B2) – (A1B2 + A2B1)– One diagonal versus the other.

Rearranging terms,• (A1B1 - A2B1) – (A1B2 - A2B2)

– Effect of A at B1 versus effect of A at B2

• (A1B1 - A1B2) – (A2B1 - A2B2)– Effect of B at A1 versus effect of B at A2

  Coefficients  B1 B2

A1 1 -1A2 -1 1

SAS Code

• AB cells (level of A, level of B) are 11, 12, 21, 22

• Proc GLM; Class Cell; Model Y=Cell;• CONTRAST 'A x B' Cell 1 -1 -1 1;• This will produce the interaction SS.

2 x 3 Two Interaction df

• Simple main effect of combined B1B2 versus B3 at A1 contrasted with simple main effect of B1B2 versus B3 at A2,, or (next slide)

  A x B12 vs 3

    B1 B2 B3

  A1 1 1 -2  A2 -1 -1 2

• Simple main effect of A (A1 versus A2) at combined B1B2 contrasted with simple main effect of A at B3

• That is, the A x B interaction with levels 1 and 2 of B combined.

  A x B12 vs 3

    B1 B2 B3

  A1 1 1 -2  A2 -1 -1 2

• Simple main effect of A at B1 contrasted with simple main effect of A at B2

• Simple main effect of B12 (B1 versus B2 ignoring B3) at A1 contrasted with same effect at A2

• That is, the A x B interaction ignoring level 3 of B.

• Proc GLM; Class Cell; Model Y=Cell;• CONTRAST 'B12vs3' Cell 1 1 -2 -1 -1 2;

CONTRAST 'B1vs2' Cell 1 -1 0 -1 1 0;Contrast DF Contrast

SSMean

SquareF Value Pr > F

A x B12vs3 1 644.03 644.033 91.83 <.0001

A x B1vs2 1 152.10 152.10 21.69 <.0001

SSAxB = 644.03 + 152.10 = 769.13

Standardized Contrasts• Give the contrast in standard deviation units• Should the standardizer (the denominator of

estimated d) include or exclude variance due to other factors in the design?

• Kline argues that the standardizer should include all variance in the outcome variable.

• I generally agree.

Therapy x Sex of Patient, 2 x 2• You want to estimate d for the main effect of

therapy. • Should the denominator of estimated d

include variance due to sex?• The MSE excludes variance due to sex, but• In the population, sex may account for some

of the variance in the outcome variable.• The SQRT(MSE) would under-estimate the

population standard deviation.

• Kline says one should include in the standardizer variance due to any factor that is naturally variable in the population (like sex or type of therapy).

• But is the distribution of the factor in the research the same as it is in the population?

• If not, then the factor may account for more or less variance in the research than in the population.

Obtaining a Pooled Standardizer

• You decide to include in the standardizer, for the effect of therapy, variance due to all other effects.

• Could pool the SSwithin-cells, SSsex, and SSinteraction to form an appropriate standardizer.

• Or just drop Sex and (Therapy x Sex) from the model, run the ANOVA, and use the SQRT of the resulting MSE as the standardizer.

Simple Main Effects• Effect of therapy in men vs. in women.• Should the standardizer be computed

within-sex? If so, that for men might differ from that for women.

• Do you want to estimate d in a single-sex population or in a way that the estimate for men can be compared to that for women without considering differences in the denominators?

(Semipartial) Eta-Squared• 2 = SSEffect SSTotal

• Using our smoking history data,• 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 added to the error term.

• For gender,

dfF 98 1, on 752.5139.174

025.9

39.174199

025,9115,26

GenderTotal

GenderTotal

GxSHSHE

GxSHSHE

dfdfSSSS

dfdfdfSSSSSSMSE

CI.90 Eta-Squared• Use that F with my Conf-Interval-R2-

Regr.sas• 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• 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.

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

104.710,10240,1

240,12

p

ErrorEffect

Effectp SSSS

SS

2

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 0

Partial 2 and F For Interaction

104.

710,10240,1240,12

EGxSH

GxSHp SSSS

SS

61.2

90710,104240,1

EE

GxSHGxSH

dfSSdfSSF

Notice that the denominator of both includes error but excludes the effects of Gender and Smoking History.

Semi-Partial 2

047.710,10140,5025,9240,1

240,1

2

ESHGGxSH

GxSH

SSSSSSSSSS

Notice that, unlike partial 2 and F, the denominator of semi-partial 2 includes the effects of Gender and Smoking History.

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%

2 for Simple Main Effects

• For the women, SStotal = 11,055• and SSsmoking = 5,700• 2 = 5,700/11,055 = .52• To construct confidence interval, need

compute an F using data from women only.

• The SSE is 11,055 (total) – 5,700 (smoking) = 5,355.

• 90% CI [.29, .60]• For the men, 2 = .11, 90% CI [0, .20]

97.1145/53554/700,5 45) (4, F

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 Model

Let 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 ResultsParticipants 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.

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, 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.

Interaction Plot