Two-Way Between-Groups ANOVA -...

|CHAPTER FOURTEEN|

Two-Way Between-GroupsANOVANOTE TO INSTRUCTORS

It is essential that students understand the distinction between different inde-pendent variables and levels of independent variables. For example, it wouldnot be unusual for a student to think that there are four independent variableswith a 2 × 2 ANOVA, when in fact there are only two independent variables,each with two levels. As you can probably imagine, if students do not thor-oughly understand the distinction, their ability to perform most statistical cal-culations will be severely hampered. Make sure you go through many exampleswith students, distinguishing between independent variables and levels of oneindependent variable. Another important concept to stress is the distinctionbetween a main effect and an interaction. Students often have a difficult timeunderstanding what an interaction actually tells the researcher and why inter-actions are important. Therefore, a number of supplementary exercises in thisinstructor’s manual have been created to help students master this distinction.

OUTLINE OF RESOURCES

I. Two-Way ANOVA■ Discussion Question 14-1 (p. 126)■ Discussion Question 14-2 (p. 127)■ Classroom Activity 14-1: Two-Way Analysis of Variance: Singers’

Height (p. 127)■ Discussion Question 14-3 (p. 127)■ Discussion Question 14-4 (p. 128)■ Classroom Activity 14-2: Make It Your Own: Gender Perception

(p. 128)

II. Understanding Interactions in ANOVA■ Discussion Question 14-5 (p. 128)■ Discussion Question 14-6 (p. 129)■ Classroom Activity 14-3: Interpreting Effects Visually (p. 130)

125

III. Conducting a Two-Way Between-Groups ANOVA■ Discussion Question 14-7 (p. 130)■ Classroom Activity 14-4: Make It Your Own: Learning and Memory

(p. 131)■ Discussion Question 14-8 (p. 131)■ Classroom Activity 14-5: Affecting Memory (p. 131)■ Classroom Activity 14-6: Two-Way ANOVAs in Context (p. 132)

IV. Next Steps: Variations on ANOVA■ Discussion Question 14-9 (p. 133)

IV. Handouts and Transparency Masters■ Handout 14-1: Interpreting Effects Visually (p. 134)■ Handout 14-2A: Affecting Memory (p. 136)■ Handout 14-2B: Affecting Memory (p. 137)■ Handout 14-2C: Affecting Memory (p. 138)■ Handout 14-3: Two-Way ANOVAs in Context (p. 139)

CHAPTER GUIDE

I. Two-Way ANOVA1. An interaction occurs in a factorial design when two or more inde-

pendent variables have an effect in combination that we do not see whenwe examine each independent variable on its own.

2. A two-way ANOVA is a hypothesis test that includes two nominalindependent variables, regardless of their numbers of levels, and aninterval dependent variable.

> Discussion Question 14-1How does a one-way ANOVA hypothesis test differ from a two-way ANOVA hypothesis test?

Your students’ answers should include: ■ A one-way ANOVA includes one nominal independent variable with

more than two levels and an interval dependent variable, whereas a two-way ANOVA includes two nominal independent variables with at leasttwo levels and an interval dependent variable.

3. Any ANOVA with at least two independent variables can be called afactorial analysis of variance (or multifactorial ANOVA). This termrefers to a statistical analysis used with one interval dependent variable and at least two nominal independent variables (also calledfactors).

4. A factor is another word used to describe an independent variable in astudy with more than one independent variable.

126 CHAPTER 14 ■ TWO-WAY BETWEEN-GROUPS ANOVA

5. Using a two-way ANOVA is advantageous over conducting two sepa-rate one-way ANOVAs because, with a two-way ANOVA, you have theopportunity to explore interactions as well as the isolated effect of thetwo independent variables on the dependent variable.

> Discussion Question 14-2Why is it better to conduct one two-way ANOVA rather than two separate one-way ANOVAs?

Your students’ answers should include: ■ Conducting a two-way ANOVA allows the researcher to compare the two

independent variables under the time, energy, and resources of one study. ■ A two-way ANOVA can study interactions between the variables as well

as the isolated effect of the two independent variables on the dependentvariable.

Classroom Activity 14-1Two-Way Analysis of Variance: Singers’ Height

http://lib.stat.cmu.edu/DASL/Datafiles/Singers.html■ Modify the data set found at the above URL, and create:

■ a new variable of voice (high voices are sopranos and tenors whilelow voices are altos and basses) and

■ a new variable of gender (females are altos and sopranos while malesare basses and tenors).

■ With this modification you can perform a 2 (voice) by 2 (gender) analysisof variance. Are deeper-voiced singers taller?

6. When we draw out the design of our study, we call each box of thedesign a cell, which depicts the unique combination of levels of theindependent variables in a factorial design.

7. Two-way ANOVAs are often notated by the number of levels for eachvariable. For example, if we were studying the effects of two inde-pendent variables on the dependent variable and each independentvariable had two levels, we would say that we were conducting a 2 × 2ANOVA.

8. In addition, either the term between-groups or within-groups is alsoincluded, depending on the specific experimental design.

> Discussion Question 14-3How would you interpret the phrase “4 × 2 between-groups ANOVA”?

Your students’ answers should include: ■ A 4 × 2 between-groups ANOVA is a hypothesis test that includes two

independent variables, one with 4 levels and one with 2 levels, and adependent variable.

9. A main effect occurs in a factorial design when one of the independentvariables has an influence on the dependent variable.

CHAPTER 14 ■ TWO-WAY BETWEEN-GROUPS ANOVA 127

10. An interaction occurs when the effect of one independent variable onthe dependent variable depends on the particular level of the otherindependent variables.

> Discussion Question 14-4What is the difference between a main effect and an interaction?

Your students’ answers should include: ■ A main effect occurs when one of the independent variables has an influ-

ence on the dependent variable.■ An interaction in a factorial design occurs when the effect of one inde-

pendent variable on the dependent variable depends on the particularlevel of the other independent variables.

Classroom Activity 14-2Make It Your Own: Gender Perception

http://opl.apa.org/Experiments/About/AboutGenderPerception.aspxThis online experiment is a 3 × 2 design, so it is appropriate only for largerclasses or classes in which students can go out and find more participants.The experiment was created by the Online Psychology Laboratory.Explanations and references are provided online.

11. In a two-way ANOVA, we will be calculating three F statistics—one foreach main effect and one for the interaction. Each F statistic will haveits own between-groups sum of squares (SS), degrees of freedom (df ),mean square (MS), and critical value. However, each F statistic willshare the MSwithin.

II. Understanding interactions in ANOVA1. There are two different kinds of interactions: quantitative interac-

tions and qualitative interactions.2. A quantitative interaction is an interaction in which one independent

variable exhibits a strengthening or weakening of its effect at one ormore levels of the other independent variable, but the direction of theinitial effect does not change.

3. A qualitative interaction is a particular type of quantitative interaction oftwo (or more) independent variables in which one independent variablereverses its effect depending on the level of the other independent variable.

> Discussion Question 14-5What would be an example of a quantitative interaction? A qualitative interaction?

Your students’ answers should include: ■ An example of a quantitative interaction would be an overall increase in

sales as a result of advertising in the lifestyle section of the newspaper ver-sus advertising in the business section of the same newspaper. However,when advertising is placed in both the lifestyle and the business sectionsof the newspaper, sales are the highest.


■ An example of a qualitative interaction would be a change in grades formen and women between the first and second half of the semester, whereduring the first half of the semester grades are higher for men and duringthe second half of the semester grades are higher for women. (See the fig-ure below.)

4. Marginal means are the means of the rows and the columns in a tablethat show the cells of a study with a two-way ANOVA design.

5. The easiest way to understand the main effect is to make a smaller tablefor each main effect, with only the appropriate marginal means.Separate tables allow us to focus on one main effect at a time withoutbeing distracted by the means in the cells.

6. When trying to examine interactions, it is often useful to create a bargraph of the findings. Once we have a bar graph, we can connect eachset of bars with a line.

7. Perfectly parallel lines indicate the likely absence of an interaction, butwe almost never see perfectly parallel lines emerging from real-life datasets. Nonparallel lines may indicate an interaction, but we need to con-duct an ANOVA and compare the F statistic with the cutoff to be sure.

> Discussion Question 14-6What is a quick, visual way to see if there is a significant interaction?

Your students’ answers should include:


Grades

1st half of semester(before midterm)

2nd half of semester(after midterm)

Females

Males

Time of semester (independent variable)

Gender(Independentvariable)

A Qualitative Interaction Example

■ Producing a bar graph of the findings is a quick, visual way to see overallpatterns; then we connect each set of bars with a line. How close to ordivergent from parallel the lines are indicates overall interaction patterns.

8. The core idea of an interaction is summed up by the phrase “it depends.”

Classroom Activity 14-3Interpreting Effects Visually

In this activity, students will gain more confidence interpreting and under-standing main effects and interactions visually. ■ Divide the class into small groups. ■ Give each group copies of Handout 14-1, including Handout Figures 14-

1 and 14-2, found at the end of this chapter. Handout 14-1 includes theinstructions for this exercise.

■ Have students label what variables they are studying.■ Next, have them determine whether there is a main effect and/or interac-

tion. ■ In addition, have them develop ideas for an experiment and what it would

mean for the experiment to reveal such effects.

III. Conducting a Two-Way Between-Groups ANOVA1. When conducting a two-way ANOVA, we still use the same six steps of

hypothesis testing.2. The assumptions for a two-way ANOVA are the same as those for the

one-way ANOVA. The sample should be randomly selected, the popu-lations should be normally distributed, and the population variancesshould be equal.

3. In a two-way ANOVA, we will have three sets of hypotheses for each ofour F statistics. The null hypothesis for the main effects is the same asthe one for the one-way ANOVA. For the interaction, we state the nullhypothesis in words: the effect of one independent variable does notdepend on the levels of the other independent variable. The researchhypothesis is that the effect of one independent variable depends on thelevel of the other independent variable.

> Discussion Question 14-7What are the null hypotheses in a two-way ANOVA?

Your students’ answers should include: ■ The three null hypotheses correspond to the three elements of the

ANOVA, the two main effects, and the interaction. ■ The null hypotheses for the main effects are that there are no differences

between groups.■ The null hypothesis for the interaction is that the effect of one independ-

ent variable is not dependent on the levels of the other independent variable.


Classroom Activity 14-4Make It Your Own: Learning and Memory

http://opl.apa.org/Experiments/About/AboutLearningandMemory.aspxThis online learning experiment studies traverse patterning of arbitrary associ-ations. The design is a 2 × 3. The experiment was created by the OnlinePsychology Laboratory. Explanations and references are provided online.

4. The degrees of freedom for each independent variable are the same asbefore. To calculate the between-groups degrees of freedom for theinteraction, we multiply the degrees of freedom for the two maineffects. The within-groups degrees of freedom are calculated the sameas those calculated for a one-way between-groups ANOVA, by takingthe sum of the degrees of freedom in each of the cells.

5. The critical values are obtained the same way as those obtained in theone-way ANOVA, except there will be three cutoffs for each of thethree F statistics.

6. Calculating the F statistic for each of the main effects is the same as calcu-lating the F statistic in the one-way ANOVA. Only the F statistic for theinteraction is different. To calculate the F statistic for the interaction,obtain the between-groups sum of squares for the interaction (SSaxr). Theformula for SSaxr is: SSaxr = SStotal – (SSa + SSr + SSwithin).

> Discussion Question 14-8How do we calculate the between-groups sum of squares for the interaction?

Your students’ answers should include: ■ The between-groups sum of squares for the interaction is essentially what

is left over when the main effects and within-groups sum of squares areaccounted for.

■ The formula for the between-groups sum of squares for the interactionis: SSaxr = SSaxr = SStotal – (SSa + SSr + SSwithin).

7. The last step is to compare the obtained F statistic with the cutoff. Thisprocess is the same as that used for the one-way ANOVA.

Classroom Activity 14-5Affecting Memory

In this activity, students will gain an opportunity to calculate a 2 × 2ANOVA using a hands-on activity. ■ First, divide the class into two groups. One group will receive Handout

14-2A listing a series of common four-letter words. The second groupwill receive Handout 14-2B containing a series of nonsense four-letterwords. Both handouts can be found at the end of this chapter.

■ Optional: These handout can be used as transparencies. You may displaythe handouts on an overhead projector by photocopying them ontoacetate, or you may use them with PowerPoint by scanning them into


your computer. To use this optional method, you will most likely have toadminister the tests separately.

■ Next, teach only half the class a mnemonic device (such as the peg-wordsystem or the method of loci or some other device). The other half willnot learn any mnemonic technique.

■ Now, have the students try to memorize their word lists in a set amountof time.

■ After the time is up, have students write down on a blank sheet of paperthe number of words that they remember.

■ Then, have them check their work to determine how many words theyremembered correctly.

■ The number of correct words will serve as the dependent variable, andstudents will use this information to calculate a two-way ANOVA.Students should receive Handout 14-2C, found at the end of this chapter,to help them complete the ANOVA.

Classroom Activity 14-6Two-Way ANOVAs in Context

In this activity, students will read and interpret findings from the article:Levav, J., and Fitzsimons, G. J. (2006). The role of ease of representation.Psychological Science, 17(3), 207–213. (To view or purchase this article, go to your local library or visit Blackwell Publishing online athttp://www.blackwellpublishing.com.)■ Have students identify the two-way ANOVAs discussed in the article.■ Then have the students interpret what the findings from the analyses

mean using Handout 14-3, found at the end of this chapter.

8. We calculate R2 as our measure of effect size for a two-way ANOVA justas we did for a one-way ANOVA.

9. The effect size for the main effects are calculated the same way they arecalculated for the one-way ANOVA.

10. We must now also calculate the effect size for the interaction,R2

interaction. The effect size for the interaction is calculated with the for-mula: R2

interaction = SSinteraction/(SStotal – SSrows – SScolumns).

IV. Next Steps: Variations on ANOVA1. There are four other more complicated designs involving ANOVAs.

These include: a mixed-design ANOVA, a multivariate analysis of vari-ance (MANOVA), an analysis of covariance (ANCOVA), and a multi-variate analysis of covariance (MANCOVA).

2. A mixed-design ANOVA is used to analyze the data from a study withat least two nominal independent variables and an interval dependentvariable; at least one independent variable must be within-groups and atleast one independent variable must be between-groups.


3. We can use a multivariate analysis of variance (MANOVA), a formof ANOVA, when there is more than one dependent variable.Multivariate refers to the number of dependent variables, not the num-ber of independent variables.

4. Sometimes we suspect that a third variable might be affecting ourdependent variable. Such a variable would be known as a covariate, or aninterval variable that we suspect associates, or covaries, with our inde-pendent variable of interest. An analysis of covariance (ANCOVA) is atype of ANOVA in which a covariate is included so that statistical findingsreflect effects after an interval variable has been statistically removed.

5. A multivariate analysis of covariance (MANCOVA) combines aspectsof the MANOVA with the ANCOVA. Specifically, it is an ANOVA withmultiple dependent variables and the inclusion of a covariate.

> Discussion Question 14-9What are the differences between a MANOVA, an ANCOVA, and a MANCOVA?

Your students’ answers should include: ■ Since multivariate refers to the number of dependent variables; in a

MANOVA, a form of an ANOVA, there is more than one dependent vari-able.

■ In an ANCOVA, a covariate—an interval variable that we suspect associ-ates, or covaries, with our independent variable of interest—is included sothat statistical findings reflect effects after an interval variable has beenstatistically removed.

■ A MANCOVA is an ANOVA with multiple dependent variables andincludes a covariate.


HANDOUT 14-1: INTERPRETING EFFECTS VISUALLY

Directions: In the space below each graph, label the number of main effects and/or interactions.Based on each figure, develop an idea for an experiment based on the number of levels of independ-ent variables and discuss what the significant effects would mean for each variable. The variablesthat you use for your experiment can be repeated across figures.

Is there a main effect? ______________________________________________________

Is there an interaction? _______________________________________________________

List some possible ideas for independent and dependent variables in the spaces below.

____________________________________________________________________________

____________________________________________________________________________

____________________________________________________________________________

____________________________________________________________________________


Perceivedprobability

of beingsexually

harrassed

Short Mid-length Long

Hair length

Men

Women

Gender of rater

Handout Figure 14-1

HANDOUT 14-1: (continued )

Is there a main effect? ______________________________________________________

Is there an interaction? _______________________________________________________

List some possible ideas for independent and dependent variables in the spaces below.

____________________________________________________________________________

____________________________________________________________________________

____________________________________________________________________________

____________________________________________________________________________


Perceivedprobability

of beingsexually

harrassed

Short Mid-length Long

Hair length

Men

Women

Gender of rater

Handout Figure 14-2

HANDOUT 14-2A: AFFECTING MEMORY

Directions: In the time allotted, try to remember as many of the following four-letter words as possible.

Tree Snap Chat Book Ball Wind

Safe Glad Task Rant Snap Lint

Damp Hush Fact Doll Blue Fish

Fast Pest Mark Gasp Lamp Wake

Yard Sent Bold Moon Give Nose

Heal Limb Pass Gnat Rock Kale

Well Peel Rose Sand Hail Risk

Take Joke Able Fall Note Vent

Pill Tone Chip Root Bull Dime


HANDOUT 14-2B: AFFECTING MEMORY

Directions: In the time allotted, try to remember as many of the following four-letter words as possible.

Erte Pnsa Ahtc Obok Blal Nwdi

Efsa Lgad Taks Artn Nsap Tiln

Pdma Hhus Aftc Ldol Ubel Fihs

Afts Epst Rkma Spag Pmal Eawk

Dray Nset Odlb Oomn Vgie Eson

Lhae Mbil Psas Angt Rkoc Akel

Wlle Epel Reos Dans Ilah Sirk

Atek Kjoe Lbea Lafl Eton Vtne

Lpil Oent Phci Ootr Ulbl Emid


HANDOUT 14-2C: AFFECTING MEMORY

Directions: Answer each of the following questions below regarding the memory test.

1. In the space below, draw the design of the study and label each of the cells.

2. With the data collected from the class, calculate the two-way ANOVA in the spacebelow. Be sure to draw the appropriate source table.


HANDOUT 14-3: TWO-WAY ANOVAS IN CONTEXT

Directions: Answer each of the questions below using the article: Levav, J., and Fitzsimons, G. J.(2006). The role of ease of representation. Psychological Science, 17(3), 207–213.

1. Summarize the main points of the article. What were the authors’ hypotheses? Whatdid the researchers do? What did they conclude?

2. Is there a two-way ANOVA in this article? How many? How do you know?

3. Interpret the two-way ANOVA(s). What do the findings from the analyses tell usabout the experiment?


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