1. Instructor: Dr. Tung-hsien He the@tea.ntue.edu.tw
Statistics for Education Research Lecture 7 Two-Way ANOVA & Two-Way ANCOVA
Instructor: Dr. Tung-hsien He

2. Two-Way ANOVA Conditions for Using Two-Way ANOVA
1. Two independent variables with at least two levels for each independent variable.
2. To explore main effects and interaction effects of two independent factors on one single dependent variable.
3. To test at least six means: two marginal means and 4 cell means.
Assumptions
1. Independent, Random Samples
2. Normally Distributed Dependent Variables

3. Two-Way ANOVA 3. Homogeneity of Variance Factorial Designs:
1. Explore main means of two factors (independent variables]
2. Explore cell means of two factors
3. See Table 15.1 for detail: (An example of two-way ANOVA, 3X3 factorial design)

4. Two-Way ANOVA Advantages of Factorial Designs
1. Main Effects by Comparing Marginal Means: the effects of one of two variables without considering the other factor; for a two-way ANOVA with 3 level for each factor, how many main effects should there be?
2. Interaction Effects by Comparing Cell Means: the effects caused by the two factors simultaneously; for a two-way ANOVA with 3 level for each factor, how many interaction effects should there be?

5. Two-Way ANOVA
3. Maintain  level after an Additional Independent Variable are Included.
Components of Variances
1. Within-Cell Variance (Sw2):
2. Variance among J row (Sj2):
3. Variance among K column (Sk2):
4. Variance due to the interaction between J & K (Sjk2):

6. Two-Way ANOVA Hypothesis Testing:
1. Ho: u1. = u2. = = uj; Sj2/Sw2 = 1
2. Ho: u.1 = u.2 = = u.k; Sk2/Sw2 = 1
3. Ho: all  effects = 0
4. Fj = MSj/MSw
5. Fk = MSk/MSw
6. Fjk = MSjk/MSw

7. Two-Way ANOVA Degree of Freedom 1. Row: J-1 2. Column: K-1
3. Interactions: (J-1] * (k-1)
4. Within-Cell: JK (n-1)
Meanings of Significant F ratios
1. Significant of Main Effects: the effects of levels of an independent variable on the dependent variable are not identical;

8. Two-Way ANOVA
2. Significant of Interaction Effects: the effects of levels of an independent variable is not the same across the levels of the other independent variable.

9. Two-Way ANCOVA Two-Way ANCOVA
1. All the conditions used for Two-Way ANOVA
2. At lease one covariate is included.
3. The covariate is assumed to affect the dependent variable, but it is unrelated to the independent variables (the linearity assumption).
3. Assumptions and reasons for including covariate(s) are the same for One-Way ANCOVA.
4. The requirements for the covariate(s) are identical fro One-Way ANCOVA and Two-Way ANCOVA.

10. Two-Way ANCOVA
5. The use of the covariate(s) in Two-Way ANCOVA can NOT replace random selection.

11. Demo of Two-Way ANOVA Exercise
Example of the Demo File:
1. Scenario: A researcher is interested in effects of the length of an exercise program. The exercise program is divided into one-week, two-week, and three-week programs. 24 females and 24 males are randomly selected and assigned into the three exercise programs. The researcher measures every subject’s flexibility.

12. Demo of Two-Way ANOVA Exercise
The researcher wants to know: (a) whether the three exercise programs exercise significant and different effects on flexibility; (b) whether the genders exercise significant and different effects on flexibility; and (3) whether different genders who participate in different lengths of programs will reveal different degrees of flexibility?

13. Demo of Two-Way ANOVA Exercise
2. Conditions:
a. 2 Independent Variables: Gender has 2 levels and Exercise Program has 3 levels
b. One Dependent Variable: Flexibility
c. Two Main Effects; 6 Interaction Effects
d. A 2X3 Factorial Design
3. Statistic Technique: Two-Way ANOVA
4. SPSS procedures:

14. Demo of Two-Way ANCOVA Exercise
Example of the Demo File:
1. Scenario: A researcher is interested in effects of the length of an exercise program. The exercise program is divided into one-week, two-week, and three-week programs. 24 females and 24 males are randomly selected and assigned into the three exercise programs. The researcher measures every subject’s flexibility. Also, the researcher suspects that participants’ previous health conditions will affect their flexibility. However, the researcher did not include it as an independent variable. Thus, the researcher decides to include it as a covariate.

15. Demo of Two-Way ANCOVA Exercise
The researcher wants to know that when controlling for previous health conditions that are measured dichotomously (either 0 or 1): (a) whether the three exercise programs exercise significant and different effects on flexibility; (b) whether the genders exercise significant and different effects on flexibility; and (3) whether different genders who participate in different lengths of programs will reveal different degrees of flexibility?

16. Demo of Two-Way ANCOVA Exercise
2. Conditions:
a. 2 Independent Variables: Gender has 2 levels and Exercise Program has 3 levels
b. 1 Dependent Variable: Flexibility
c. 1 Covariate: Previous Health Conditions
c. Two Main Effects; 6 Interaction Effects
d. A 2X3 Factorial Design
3. Statistic Technique: Two-Way ANCOVA
4. SPSS procedures:

17. Hsieh's Study 3P
PP: Different teaching methods may exercise significant effects on EFL learning.
IP: However, researchers have different opinions about the effectiveness of distinctive methods, including phonic instruction, KK phonetic symbol instruction, “All English” instruction, and “Bilingual” instruction on: (a) letter-recognition, letter-sound discrimination, spelling, pronunciation, conversation tests.
SP: Different methods should exercise different effects on these aspects.

18. Hsieh's Study Literature Reviews: Criticisms:
1. Are the studies reviewed related to 3 P?
2. Does the literature review section highlight the links?
Study Design:
1. Experimental Study
2. Subjects: graders divided into 4 groups
3. Instruments: Letter recognition, (5 aspects in IP)

19. Hsieh's Study
4. Analyses: One-Way, Two-Way ANOVAs, Pearson r, chi-square.
Criticisms:
1. Is this an experimental study? What happens to the control of extraneous variables as the study lasted two years long?
2. Are participants randomly selected? What happens if they are not randomly selected?

20. Hsieh's Study
3. Is there any detailed information about how participants’ performances on the five instruments are transformed into numeric data?
4. Why One-Way ANOVA? What is the dependent variable and independent variable? (NOTE: For One-Way ANOVA, there should be only one independent variable with K levels and with one dependent variable!). So, what happens as the author argued “for the pre-test including letter recognition and basic conversation, . . One-way ANOVA results show there was no . . .” (2nd line on p. 105).

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