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Two-Way ANOVA

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Two-way Analysis of Variance

Two-way ANOVA is applied to a situation in which you have twoindependent nominal-level variables and one interval or betterdependent variable

Each of the independent variables may have any number oflevels or conditions (e.g., Treatment 1, Treatment 2, Treatment3 No Treatment)

In a two-way ANOVA you will obtain 3 F ratios One of these will tell you if your first independent variable

has a significant main effecton the DV A second will tell you if your second independent variable has

a significant main effecton the DV The third will tell you if the interaction of the two independent

variables has a significant effect on the DV, that is, if theimpact of one IV depends on the level of the other

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The Three Effects in a Two-WayANOVA

Lets consider an example: What is the impact ofgender, ethnicity, and their interaction on annualincome? One of these will tell you if your first independent

variable has a significant main effecton the DV

What is the main effect of gender on income, regardlessof (across all levels of) ethnicity?

A second will tell you if your second independentvariable has a significant main effecton the DV What is the main effect of ethnicity on income,

regardless of (across all levels of) gender

The third will tell you if the interaction of the twoindependent variables has a significant effect on the DV What is the combined effect of gender and ethnicity on

income that could not be detected by considering thetwo IVs separately? (e.g., what is the interaction ofgender and ethnicity with respect to income; is theeffect of gender different for different categories of

ethnicity?

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The Null Hypotheses in a Two-WayANOVA

The null hypotheses in a two-wayANOVA are these:

The population means for the DV areequal across levels of the first factor

The population means for the DV areequal across levels of the second factor

The effects of the first and secondfactors on the DV are independent of oneanother

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An Interaction Effect in Two-WayAnalysis of Variance

What is the impact of genderand ethnicity on annualsalary, and how do theyinteract? In this example,there may not be much ofa main effect either forgender or ethicnity, butthere may be aninteraction effect: forexample, are females whoare Hispanic paid morethan males who areHispanic, while femaleswho are African-Americanare paid less than maleswho are African-American?

Female Male

Hispanic Salary

Average isHigh

Salary

Average isLow

African-American

SalaryAverage isLow

SalaryAverage isHigh

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Some Conventions to Know

For convenience purposes, one factor or IV is usuallycalled the column variable and the other the rowvariable

When describing your design in the opening

statement of a Method section you will refer to it as a2 X 2 design, or a 3 X 3 design, where the firstnumber refers to the number of levels of the rowvariable and the second number refers to the numberof levels of the column variable. When there aremore than two factors involved, in a multiple factor

ANOVA, you will see 4 X 2 X 4, which means thatthere are three factors in the design, the first withfour, the second with two, and the third with fourlevels of the factor. The order is usually arbitrary

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More Conventions to Know

An independent variable is called a factor, and itsseparate impact on the DV is called a main effect

The term between effect or between-groups effect inANOVA language refers to the differences in the DV

between or among levels of a factor and is the samething as the variables main effect (e.g., differences inthe DV between men and women, or between AfricanAmericans and Hispanics)

The term within effect or within-groups effect inANOVA language refers to the differences in the DVwithin a level of the factor (e.g., differences amongthe individuals within the female category or theAfrican-American category

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Various Estimates in Two-WayANOVA

Estimates for the main effects of the two independentvariables

The between estimate, or between mean square forthe row variable (for example, ethnicity) is based on

the deviation of each row mean of the DV (mean forHispanic, mean for African-American) from theoverall or grand mean of the DV

Similarly, the between estimate, or between meansquare for the column variable, gender, is based onthe deviation of each column mean of the DV (mean

for females, mean for males) from the overall orgrand mean of the DV

Each of these estimates is calculated as if the otherfactor did not exist

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Estimates in Anova

The estimate or mean square for theinteraction effect of gender and ethnicity isbased on the deviation of the cell means(mean on the DV from each of these

combinations: Hispanic/female;Hispanic/male; African-American/female;African-American/male) from the grandmean, after differences due to the twofactors (gender, ethnicity) acting

independently and the error variance(individual variability within the cells) havebeen accounted for or removed

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Within Estimate and FRatio, Two-Way ANOVA

The estimate or mean square for the within-cellvariance isbased on the deviation of each score on the DV from the meanof its own cell. It is usually called the errorterm (error beingwhatever you cant explain by factors and their interaction)

Whenever the independent variables are regarded as fixed,(levels are not randomly sampled) the Fratios for the twofactors (gender, ethnicity) and their interaction are calculatedby dividing the appropriate main effect or interaction effectestimate by the within estimate

The degrees of freedom (DF) associated with each of the Fratios (Factor 1 main effect, Factor 2 main effect, theirinteraction) are k-1 and j-1, respectively, for each of the maineffects, where k and j are the number of levels of the

respective factors; df for the interaction term is (k-1)(j-1);and the df for the error term is N-jk

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Two-Way ANOVA, Example ofFtests

Test of the impact of sex and race on socioeconomic status:Significant main effect for race (see red dots)No significant main effect for sex (see green dots)No significant interaction of race and sex (see blue dots)

Factors (main effects and interaction effect)

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Two-Way ANOVA, Example ofFTests, Contd

Levene's Test of Equality of Error Varia ncesa

Dependent Variable: Respondent Socioeconomic Inde

3.785 5 1413 .002

F df1 df2 Sig.

Tests the null hypothesis that the error variance of

the dependent variable is equ al across groups.

Design: Intercept+RACE+SEX+RACE * SEXa.

According to the Levene test the groupvariances are significantly different so we willuse the Tamhane post hoc test instead ofSheffe to see which group means aresignificantly different. We will only do a test

on the factors for which the main effect wassignificantAccording to theTamhane test themeans for blacksand whites inSocioeconomicstatus were

significantlydifferent, butneither groupwas significantlydifferent fromother

2. Racew of Respondent

Dependent Variable: Respondent Socioeconomi c Index

48.514 .536 47.464 49.565

40.099 1.526 37.107 43.092

45.197 2.355 40.577 49.817

Racew of Respondent

white

black

other

M ea n Std. Error L ower Boun d Up pe r Boun d

95% Confidence Interval

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Plots of Main Effects andInteraction Effect

Estimated Marginal Means of Respondent Soc

Racew of Respondent

otherblackwhite

50

48

46

44

42

40

38

Estimated Marginal Means of Respondent So

Respondent's Sex

FemaleMale

46.5

46.0

45.5

45.0

44.5

44.0

43.5

43.0

Estimated Marginal Means of Respondent Soc

Racew of Respondent

otherblackw hite

52

50

48

46

44

42

40

38

36

Respondent's Sex

Male

Female

Plot of interaction effect: Note that the lines formales (red) and females (green) are very similaralthough there is a tiny bit of an interaction effect inthe Other category where women are actually higherthan men

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Two-Way ANOVA, SPSS example

Suppose you hypothesized that the amount of time a personspent on the Internet each week was influenced by two factors,their educational level and their marital status. (This will be a 3X 2 design with three levels of education (high school only,some post-high school, and college degree or more), and twolevels of marital status (married/with partner or not

married/with partner). Your first hypothesis was that the more educated people are, the

more time they will spend on the net. Your second hypothesis was that the amount of time people spend

on the net is likely to be influenced by their marital status, suchthat persons without partners are more likely to spend time on thenet than those who are married/have a partner.

Your third hypothesis is that education level and marital status willinteract, but you dont predict the nature of the interaction

Main effects may be interpreted in a straightforward way(treated as independent of one another and interpretedindividually) only if there is no significant interaction present;otherwise the interpretation of the main effects must take theinteraction into account

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SPSS Output, Two-Way ANOVA:Tests of Main Effects of Marital Status and EducationalLevel and Their Interaction on Time Spent on the Net

Tests of Hypotheses:

(1) There is no significant main effect for education level (F(2, 58) = 1.685, p =.194, partial eta squared = .055)

(2) There is no significant main effect for marital status (F(1, 58) = .441, p =.509, partial eta squared = .008)

(3) There is a significant interaction effect of marital status and education level (F(2, 58) = 3.586, p = .034, partial eta squared = .110)

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Plots of Main Effects (non-significant) ofMarital Status and Education Level

Estimated Marginal Means of TIMENET

MarriedorNot

NotMarried/PartnerMarried/Partner

4.8

4.6

4.4

4.2

4.0

3.8

Estimated Marginal Means of TIMENET

CollegeorNot

CollegeorMoreSomePostHighHighSchool

6.0

5.5

5.0

4.5

4.0

3.5

3.0

2.5

Generally, although the results are not significant, it would appear that unmarriedor non-partnered people spend more time on the net, and net use peaks with thepost-high school group and declines for college grads

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Plots of Interaction Effect of Education Leveland Marital Status on Time Spent on the Net

Estimated Marginal Means of TIMENET

CollegeorNot

CollegeorMoreSomePostHighHighSchool

9

8

7

6

5

4

3

2

MarriedorNot

Married/Partner

NotMarried/Partner

Education Level is plotted alongthe horizontal axis and hoursspent on the net is plotted alongthe vertical axis. The red andgreen lines show how maritalstatus interacts with educationlevel. If marital status had the

same effect on time spent on thenet across all levels of education,the lines would be more or lessparallel. In an interaction effect,they cross or diverge fromparallel in some way. Here wenote that the general trend forsingle people to spend more timeon the net is very strong for thepost-high school group but isreversed for high school gradsand college grads, where marriedpeople spend more time Whatdo you think might explain this?

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Step-by-step Two-Way ANOVA inSPSS

First, download the socialsurveysmall.sav data file We are going to test the hypotheses that

Sex of respondent has a significant main effect on hours per dayspent watching TV

Home ownership has a significant main effect on hours per day

spent watching TV Sex of respondent and home ownership have a significantinteraction effect on hours per day spent watching TV

Go to Analyze/General Linear Model/ Univariate Move the variables Respondents Sex and OwnsOwnHome into

the Fixed Factor window Move the Hours per Day Watching TV variable into the

Dependent Variables window Click on Model, select Full Factorial, and Continue Ignore the Contrasts Button for now

http://www-rcf.usc.edu/~mmclaugh/550x/DataFiles/socialsurveysmall.savhttp://www-rcf.usc.edu/~mmclaugh/550x/DataFiles/socialsurveysmall.sav

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Step-by-Step Two-Way ANOVA inSPSS

Next, we are going click on the Plotsbutton to select the plots we want.

First we get plots for the main effects Move the Sex factor into the Horizonal

Axis window and click the Add button Move the Homeown factor into the

Horizontal Axis window and click the Addbutton

Next we will get plots for the interactioneffect Move the Sex factor into the Horizontal

Axis window and the Homeown factorinto the Separate Lines window and clickthe Add button

Move the Homeown factor into theHorizontal Axis window and the Sexfactor into the Separate Lines windowand click the Add button

Click Continue

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Step-by-step Two-Way ANOVA inSPSS

We will skip the post-hoc tests button this timebecause our variables only have two levels each andthe post-hoc tests are only performed when there aremore than two levels. Otherwise you do the post hoctests just as you did for one-way ANOVA by moving

the factors you want to test into the Post Hoc Testsbox and selecting Sheffe and Tamhane tests Click on Options and move all of the Factors (overall,

Sex, Homeown, and Sex*Homeown) into the DisplayMeans for box

Check Compare Main Effects, Descriptives, Estimates

of Effect Size, Observed Power, and HomogeneityTests, and set the confidence interval to 95%

Click Continue and then OK Compare your output to the next several slides

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Your SPSS Output for Two-WayANOVA

1. Sex of respondent has a significant main effect on hours per day spent watching TV

2. Home ownership has a significant main effect on hours per day spent watching TV3. Sex of respondent and home ownership have a significant interaction effect on hours per dayspent watching TVNow write a paragraph in which you report the results of the significance tests! Remember thatthe interpretation of the main effects in a straightforward way is complicated by the significantinteraction We also need to be a bit skeptical since the partial eta squares are very low and asyou will see on the next slide there is a very large SD in one of the conditions

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Examining the Main Effects of Sexand Homeownership

As you can see in the table ofmeans, there is a trend forfemales to watch more TVthan males and for non-homeowners to watch moreTV than homeowners, butthere is a particularlypronounced trend for femalenon-homeowners to watchmore TV than everybodyelse.

Descriptive Statistics

Dependent Variab le: Hours Per Day Watching T V

2.77 2.197 305

2.93 2.249 146

2.82 2.213 451

2.63 1.790 353

3.69 3.173 201

3.02 2.436 554

2.70 1.989 658

3.37 2.842 347

2.93 2.339 1005

OwnsOwnHome

Owns Own Home

Doesn't Own Home

Total

Owns Own Home

Doesn't Own Home

Total

Owns Own Home

Doesn't Own Home

Total

Respondent's Sex

Male

Female

Total

Mean Std. Deviation N

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Examining the Interaction Effect ofSex and Homeownership

Although the interaction effect is not extremely strong, there is a trend forthe relationship between homeownership and hours spent watching TV tobe different for men than women; women who dont own homes are muchmore likely to spend more time watching tv than owners, compared tomen, for whom homeownership makes less of a difference