Statistics for Statistics for Education ResearchEducation Research

Lecture 7

Two-Way ANOVA & Two-Way ANCOVA

Instructor: Dr. Tung-hsien Instructor: Dr. Tung-hsien HeHe

the@tea.ntue.edu.twthe@tea.ntue.edu.tw

Conditions for Using Conditions for Using Two-Way ANOVATwo-Way ANOVA1. 1. Two independent variablesTwo independent variables with with at least at least

two levelstwo levels for for each independent variable.each independent variable.2. To explore 2. To explore main effectsmain effects and and interaction interaction

effectseffects of two independent factors on of two independent factors on one single dependent variable.one single dependent variable.

3. To test at least six means: 3. To test at least six means: two marginal two marginal meansmeans and and 4 cell means4 cell means..

AssumptionsAssumptions1. Independent, Random Samples1. Independent, Random Samples2. Normally Distributed Dependent 2. Normally Distributed Dependent

VariablesVariables

3. Homogeneity of Variance3. Homogeneity of Variance Factorial DesignsFactorial Designs::1. Explore main means of two factors 1. Explore main means of two factors

(independent variables](independent variables]2. Explore cell means of two factors2. Explore cell means of two factors3. See Table 15.1 for detail: (An 3. See Table 15.1 for detail: (An

example of two-way ANOVA, example of two-way ANOVA, 3X33X3 factorial design)factorial design)

Advantages of Factorial DesignsAdvantages of Factorial Designs1. 1. Main Effects by Comparing Marginal Main Effects by Comparing Marginal

MeansMeans: the effects of one of two : the effects of one of two variables without considering the other variables without considering the other factor; for a two-way ANOVA with 3 level factor; for a two-way ANOVA with 3 level for each factor, how many main effects for each factor, how many main effects should there be?should there be?

2. 2. Interaction Effects by Comparing Cell Interaction Effects by Comparing Cell MeansMeans: the effects caused by the two : the effects caused by the two factors simultaneously; for a two-way factors simultaneously; for a two-way ANOVA with 3 level for each factor, how ANOVA with 3 level for each factor, how many interaction effects should there many interaction effects should there be?be?

3. 3. Maintain Maintain level level after an Additional after an Additional Independent Variable are Included.Independent Variable are Included.

Components of VariancesComponents of Variances1. 1. Within-Cell VarianceWithin-Cell Variance (S (Sww

22):):2. Variance among 2. Variance among JJ row (S row (Sjj

22):):3. Variance among 3. Variance among KK column (S column (Skk

22):):4. Variance due to the interaction 4. Variance due to the interaction

between J & Kbetween J & K (S (Sjkjk22):):

Hypothesis Testing:Hypothesis Testing:1. Ho: u1. = u2. = . . . = uj; S1. Ho: u1. = u2. = . . . = uj; S jj

22/S/Sww22 = 1 = 1

2. Ho: u.1 = u.2 = . . . = u.k; S2. Ho: u.1 = u.2 = . . . = u.k; Skk22/S/Sww

22 = 1 = 13. Ho: all 3. Ho: all effects = 0 effects = 04. Fj = MSj/MSw4. Fj = MSj/MSw5. Fk = MSk/MSw5. Fk = MSk/MSw6. Fjk = MSjk/MSw6. Fjk = MSjk/MSw

Degree of FreedomDegree of Freedom1. Row: J-11. Row: J-12. Column: K-12. Column: K-13. Interactions: (J-1] * (k-1)3. Interactions: (J-1] * (k-1)4. Within-Cell: JK (n-1)4. Within-Cell: JK (n-1) Meanings of Significant F ratiosMeanings of Significant F ratios1. 1. Significant of Main EffectsSignificant of Main Effects: the : the

effects of levels of an independent effects of levels of an independent variable on the dependent variable variable on the dependent variable are not identical;are not identical;

2. 2. Significant of Interaction EffectsSignificant of Interaction Effects: : the effects of levels of an the effects of levels of an independent variable is not the same independent variable is not the same across the levels of the other across the levels of the other independent variable. independent variable.

Two-Way ANCOVATwo-Way ANCOVA1. All the conditions used for Two-Way ANOVA1. All the conditions used for Two-Way ANOVA2. 2. At lease one covariateAt lease one covariate is included. is included.3. The 3. The covariatecovariate is assumed to is assumed to affect the depeaffect the depe

ndent variablendent variable, but it is , but it is unrelated tounrelated to the the indeindependent variables pendent variables (the (the linearitylinearity assumption). assumption).

3. Assumptions and reasons for including cova3. Assumptions and reasons for including covariate(s) are the same for One-Way ANCOVA.riate(s) are the same for One-Way ANCOVA.

4. The requirements for the covariate(s) are id4. The requirements for the covariate(s) are identical fro One-Way ANCOVA and Two-Way Aentical fro One-Way ANCOVA and Two-Way ANCOVA.NCOVA.

5. The use of the covariate(s) in Two-Way AN5. The use of the covariate(s) in Two-Way ANCOVA can COVA can NOTNOT replace random selection. replace random selection.

Example of the Demo File:Example of the Demo File:1. Scenario: A researcher is interested 1. Scenario: A researcher is interested

in effects of the length of an exercise in effects of the length of an exercise program. The exercise program is program. The exercise program is divided into one-week, two-week, divided into one-week, two-week, and three-week programs. 24 and three-week programs. 24 females and 24 males are randomly females and 24 males are randomly selected and assigned into the three selected and assigned into the three exercise programs. The researcher exercise programs. The researcher measures every subject’s flexibility. measures every subject’s flexibility.

The researcher wants to know: (a) The researcher wants to know: (a) whether the three exercise programs whether the three exercise programs exercise significant and different exercise significant and different effects on flexibilityeffects on flexibility; (b) ; (b) whether the whether the genders exercise significant and genders exercise significant and different effects on flexibilitydifferent effects on flexibility; and ; and (3) (3) whether different genders who whether different genders who participate in different lengths of participate in different lengths of programs will reveal different programs will reveal different degrees of flexibilitydegrees of flexibility??

2. Conditions: 2. Conditions: a. a. 2 Independent Variables2 Independent Variables: : GenderGender has has 2 levels2 levels and and Exercise ProgramExercise Program has has 3 levels3 levelsb. One b. One Dependent VariableDependent Variable: : FlexibilityFlexibility c. c. Two Main EffectsTwo Main Effects; ; 6 Interaction 6 Interaction EffectsEffectsd. A 2X3 Factorial Designd. A 2X3 Factorial Design

3. Statistic Technique: Two-Way ANOVA3. Statistic Technique: Two-Way ANOVA4. SPSS procedures:4. SPSS procedures:

Example of the Demo File:Example of the Demo File:1. Scenario: A researcher is interested in 1. Scenario: A researcher is interested in

effects of the length of an exercise program. effects of the length of an exercise program. The exercise program is divided into one-The exercise program is divided into one-week, two-week, and three-week programs. week, two-week, and three-week programs. 24 females and 24 males are randomly 24 females and 24 males are randomly selected and assigned into the three selected and assigned into the three exercise programs. The researcher measures exercise programs. The researcher measures every subject’s flexibility. Also, every subject’s flexibility. Also, the the researcher suspectsresearcher suspects that participants’ that participants’ previous health conditionsprevious health conditions will will affect their affect their flexibilityflexibility. However, the researcher did not . However, the researcher did not include it as an independent variable. Thus, include it as an independent variable. Thus, the researcher decides to include it as a the researcher decides to include it as a covariatecovariate. .

The researcher wants to know that The researcher wants to know that when when controlling for previous health controlling for previous health conditionsconditions that are measured that are measured dichotomouslydichotomously ( (either 0 or 1either 0 or 1): (a) ): (a) whether the three exercise programs whether the three exercise programs exercise significant and different exercise significant and different effects on flexibility; (b) whether the effects on flexibility; (b) whether the genders exercise significant and genders exercise significant and different effects on flexibility; and (3) different effects on flexibility; and (3) whether different genders who whether different genders who participate in different lengths of participate in different lengths of programs will reveal different degrees programs will reveal different degrees of flexibility?of flexibility?

2. Conditions: 2. Conditions: a. a. 2 Independent Variables2 Independent Variables: : GenderGender has has 2 levels2 levels and and Exercise ProgramExercise Program has has 3 3 levelslevelsb. b. 1 Dependent Variable1 Dependent Variable: : FlexibilityFlexibility c. c. 1 Covariate1 Covariate: : Previous Health Previous Health ConditionsConditionsc. c. Two Main EffectsTwo Main Effects; ; 6 Interaction Effects6 Interaction Effectsd. d. AA 2X3 Factorial Design2X3 Factorial Design

3. Statistic Technique: Two-Way ANCOVA3. Statistic Technique: Two-Way ANCOVA4. SPSS procedures:4. SPSS procedures:

3P3PPP: Different teaching methods may PP: Different teaching methods may

exercise significant effects on EFL exercise significant effects on EFL learning. learning.

IP: However, researchers have different IP: However, researchers have different opinions about the effectiveness of opinions about the effectiveness of distinctive methods, including phonic distinctive methods, including phonic instruction, KK phonetic symbol instruction, KK phonetic symbol instruction, “All English” instruction, and instruction, “All English” instruction, and “Bilingual” instruction on: (a) letter-“Bilingual” instruction on: (a) letter-recognition, letter-sound discrimination, recognition, letter-sound discrimination, spelling, pronunciation, conversation spelling, pronunciation, conversation tests.tests.

SP: Different methods should exercise SP: Different methods should exercise different effects on these aspects. different effects on these aspects.

Literature Reviews:Literature Reviews:Criticisms:Criticisms:1. Are the studies reviewed related to 3 1. Are the studies reviewed related to 3

P?P?2. Does the literature review section 2. Does the literature review section

highlight the links?highlight the links? Study Design:Study Design:1. Experimental Study1. Experimental Study2. Subjects: 162 5-graders divided into 4 2. Subjects: 162 5-graders divided into 4

groupsgroups3. Instruments: Letter recognition, . . . (5 3. Instruments: Letter recognition, . . . (5

aspects in IP)aspects in IP)

4. Analyses: One-Way, Two-Way 4. Analyses: One-Way, Two-Way ANOVAs, Pearson r, chi-square.ANOVAs, Pearson r, chi-square.

Criticisms:Criticisms:1. Is this an experimental study? What 1. Is this an experimental study? What

happens to the control of extraneous happens to the control of extraneous variables as the study lasted two variables as the study lasted two years long?years long?

2. Are participants randomly selected? 2. Are participants randomly selected? What happens if they are not What happens if they are not randomly selected?randomly selected?

3. Is there any detailed information about 3. Is there any detailed information about how participants’ performances on the five how participants’ performances on the five instruments are transformed into numeric instruments are transformed into numeric data?data?

4. Why One-Way ANOVA? What is the 4. Why One-Way ANOVA? What is the dependent variable and independent dependent variable and independent variable? (NOTE: For One-Way ANOVA, variable? (NOTE: For One-Way ANOVA, there should be only one independent there should be only one independent variable with K levels and with one variable with K levels and with one dependent variable!). So, what happens as dependent variable!). So, what happens as the author argued “for the pre-test the author argued “for the pre-test including letter recognition and basic including letter recognition and basic conversation, . . One-way ANOVA results conversation, . . One-way ANOVA results show there was no . . .” (2show there was no . . .” (2ndnd line on p. 105). line on p. 105).

5. Why Two-Way ANOVA? (Refer to Table 3, p.5. Why Two-Way ANOVA? (Refer to Table 3, p. 105). Two-Way ANOVA is used when there 105). Two-Way ANOVA is used when there are two independent variables. So, what hare two independent variables. So, what happens to this study? What are the two indappens to this study? What are the two independent variables? (Hint: Look at the df=ependent variables? (Hint: Look at the df=4 for Variable A (tests)). Is Two-Way ANOVA 4 for Variable A (tests)). Is Two-Way ANOVA a proper technique for this study?a proper technique for this study?