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### Session 3

Two-way ANOVA

One-way ANOVA

- Imagine that we have administered a placement test to a large groups of Ss from five different schools. We want to know if the performance differs across the five schools. The design box for this study would look like:
- IV=schools; DV=test scores
- One independent variable with five levels: One-way ANOVA (One-factor ANOVA)

Two-way ANOVA

- If we want to compare the Ss scores on the test not only by school but also by sex, the design would be:
- The comparison is two-directional, this is a two-way ANOVA (factorial design). A one-way ANOVA can no longer be used.

k-way ANOVA

- It’s possible that as an administrator of an ESL program, we might want breakdown of scores by L1 membership (say four major language groups). The comparison will be like:
- One DV

Two-way ANOVA

- One DV, two or more IVs: factorial design
- Unrelated factorial design: several IVs and each has been measured using different subjects (between groups)
- Related factorial design: several IVs have been measured, but the same subjects have been used in all conditions (repeated measures)
- Mixed factorial design: several IVs have been measured; some have been measured with different subjects whereas others used the same subjects. This is an especially useful design in that it allows us to test different groups of people (e.g., b1, and b2,) under different conditions (e.g., a1, and a2) or to track the different groups over time.

Two-way ANOVA

- Main effect
- Interaction

Two-way ANOVA

- Variations of Factorial Designs
- A two-way factorial (two independent variables) contains 2 x 2 or 4 different conditions (combinations)
- A three-way (three independent variables) contains 2 x 2 x 2 or 8 different conditions
- A four-way (four independent variables) contains 2 x 2 x 2 x 2 or 16 different conditions
- A five-way (five independent variables) contains 2 x 2 x 2 x2 x2 or 32 different conditions, and so on.

Two-way ANOVA

- Strength of Effect
- Statistical significance differs from strength of effect.
- Statistical significance relates to the likelihood of obtaining a particular statistical outcome (e.g., an F ratio of a given value) by chance given that there is no true relationship in the population.
- Strength of effect indexes the amount of dependent variable variance accounted for by the independent variable(s). It is very important to differentiate between these concepts.
- Index of Strength of Effect: Eta squared

Example 1: Main Effects Not Significant, Significant Interaction

- Language students in a particular program take a course in reading as well as one in writing. To explore the type of teaching method that might be most effective for these courses, students are randomly assigned to either the reading or the writing course during a given semester and are instructed through either a lecture or a discussion format.
- A between-subjects design
- DV: their performance on a final exam. The highest possible standardized score is 100.
- IV: teaching methods

Example 1: Main Effects Not Significant, Significant Interaction

- Recognizing an Interaction

Example 1: Main Effects Not Significant, Significant Interaction

- Graphing an Interaction

Example 1: Main Effects Not Significant, Significant Interaction

- Interpreting an Interaction
- Select a variable to narrate.
- Narrate each level separately.
- Smoothly and grammatically integrate the sentences.
- Under the lecture method, students performed well in reading but only moderately in writing; under the discussion method, students performed moderately in reading but well in writing.

Example 1: Main Effects Not Significant, Significant Interaction

- Interactions supersede main effects
- If a significant interaction is obtained, it means that a different relationship is seen for different levels of an independent variable. For example, we saw one relationship between reading and writing under the lecture method and a different relationship between reading and writing under the discussion method.

Example 1: Main Effects Not Significant, Significant Interaction

- Interpreting an Interaction
- One implication of obtaining a significant interaction is that a statement of each main effect will not fully capture the results of the study. In this case, to state that the overall means of the students taking reading and writing were equal is true but really misses the point.
- The general rule is that when an interaction effect is present, the information it supplies is more enriched -- more complete -- than the information contained in the outcome of the main effects of those variables composing it. Sometimes, a main effect is moderately representative of the results. Other times, the main effects paint a non-representative picture of the study's outcome.
- In the present situation, if we examined only the main effects and ignored the interaction, we would improperly conclude that:
- the method used for teaching the courses did not make a difference, and
- that students in the two courses performed comparably.

Example 3: Significant MainEffects and a Nonsignificant Interaction

- This involves an evaluation of a 4-week program to teach pragmatics knowledge to first-grade children. A standardized score representing pragmatic awareness is developed. Higher scores indicate greater sensitivity; the highest possible score is 250. A baseline measure is first obtained for the children. The instructional program is then administered, and another measure is obtained. To determine if there are any long-term effects of the program, a follow-up measurement is secured 2 months following the end of the program. Boys and girls are tracked separately.

Example 3: Significant MainEffects and a Nonsignificant Interaction

- 2 x 3 simple mixed design
- IV: The gender of the child (between-subjects variable)
- DV: measurement in three different times (within-subjects variable)

Example 3: Significant MainEffects and a Nonsignificant Interaction

Two-Way ANOVA post hoc analysis

SPSS Examples

This experiment involves presenting a group of students with a lecture on an syntax. One week later, the students are given a test on the lecture material. To manipulate the conditions at the time of learning, some students receive the lecture in a large classroom, and some hear the lecture in a small classroom. For those students who were lectured in the large room, one half are tested in the same large room, and the others are changed to the small room for testing. Similarly, one-half of the students who were lectured in the small room are tested in the small room, and the other half are tested in the large room. Thus the experiment involves four groups of subjects in a two-factor design. The score for each subject is the number of correct answers on the test.

SPSS Examples

- Hypotheses
- There are three hypotheses, one each for the two IVs and one for the interaction.
- Assumptions
- The assumptions that should be checked in the data are the NORMALITY and HOMOGENEITY OF VARIANCE assumptions.
- The normality assumption is generally not a cause for concern when the sample size is reasonably large
- HOMOGENEITY OF VARIANCE assumption is relatively important.

SPSS Examples

- Between Subjects Factors
- Descriptive Statistics
- Levene's Test of Equality of Error Variance
- Tests of Between Subjects Effects
- Demo

SPSS Examples

- Report tasks
- Display Ns, means, sds.
- Construct the Summary Table
- Comment upon the homogeneity of variance assumption.
- Conduct the appropriate post hoc tests and interpret.
- Graph the interaction.
- Write approximately 100 words on what the analysis has revealed to you about the research question.

SPSS Examples

- Report tasks
- Display Ns, means, sds.

SPSS Examples

- Report tasks
- Display Ns, means, sds.
- Construct the Summary Table

SPSS Examples

- Report tasks
- Display Ns, means, sds.
- Construct the Summary Table
- Comment upon the homogeneity of variance assumption.
- The homogeneity of variance tests indicate that this assumption was not violated.

SPSS Examples

- Report tasks
- Graph the interaction.
- The plot of the cell means clearly indicates a significant interaction with no main effects. The number of correct responses in the test depended on both the room the information was delivered and the room in which the testing occurred. The lowest number of correct responses occurred when the testing was in a room different to the one in which the material was learnt. Conversely, the greatest number correct occurred when the testing was in the same size room as where the material was learnt.

SPSS Examples

- Report tasks
- Write approximately 100 words on what the analysis has revealed to you about the research question.
- The experiment showed that students showed significantly greater recall (F(1,16)=92.6, p<.001) of unfamiliar material when tested in the same room as that in which the material was presented. This suggests that conditioned place learning has taken place such that the environmental cues associated with the acquisition of the material to be learned aided in the recall of that material one week later. The association between the environmental cues and the acquisition and recall of material was most likely inadvertent or non-conscious although this remains to be examined experimentally.

Practice

- A linguist was interested in the effects of alcohol on speaking fluency of a foreign language. Her rationale was that after alcohol had been consumed, speaking became more fluent. She was also interested in whether this effect was different for men and women. She picked 48 students: 24 male and 24 female. She then took groups of 8 subjects to a night-club and gave them either no alcohol (subjects received placebo drinks of alcohol-free lager), 2 pints of strong lager, or 4 pints of strong lager. At the end of the evening she chatted with the students in English. She then got a pool of independent judges to assess their fluency based on established rating criteria (out of 100). The data are in shown as follows:

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