psychology statistics
fadi
PsychologicalStatisticsRealFactorialDesigns1.pptxPsychological Statistics
Factorial Designs
(Two Way ANOVA)
(Two Factor Analysis of Variance
Experimental Factorial Designs
Independent Variable = Factor
Factorial Design = More than one Factor
Factor #1
Factor #2
Factor #3
Dependent Variable
Notation System
Uses the # of levels of the Factor
A 3 x 2 x 4 factorial design involves:
1 factor with 3 levels
1 factor with 2 levels
1 factor with 4 levels
A 2 x 3 factorial design involves:
1 factor with 2 levels
1 factor with 3 levels
Advantages of factorial designs
Combining variables permits researchers to examine how changes in one variable can influence the effects of another.
Factor #1
Interaction
Factor #2
Dependent Variable
1
2
3
Main Effects & Interactions
Each factor yields one Main Effect
Main Effect: Difference between the levels of a factor
Alcohol
-alcohol
-no alcohol
Caffeine
-no caff.
-200 mg
-400 mg
Reaction Time
1
2
The mean differences among the levels of one factor are referred to as the main effect of that factor.
When the design of the research study is represented as a matrix with one factor determining the rows and the second factor determining the columns, then the mean differences among the rows describe the main effect of one factor, and the mean differences among the columns describe the main effect for the second factor.
Main Effects
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The evaluation of main effects accounts for two of the three hypothesis tests in a two- factor ANOVA.
We state hypotheses concerning the main effect of factor A and the main effect of factor B and then calculate two separate F-ratios to evaluate the hypotheses.
In symbols,
H0: mA1 = mA2 H0: mA1 ≠ mA2
H0: mB1 = mB2 H0: mB1 ≠ mB2
Main Effects (cont’d.)
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Each of the main effects hypothesis tests in a two-factor ANOVA will have its own F-ratio and each F-ratio has the same basic structure:
variance (differences) between the means for factor A (row means)
variance (differences) expected if there is no treatment effect
and
variance (differences) between the means for factor B (column means)
variance (differences) expected if there is no treatment effect
Main Effects (cont’d.)
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Hypothetical data showing the treatment means for a two-factor study examining how different combinations of alcohol and caffeine affect reaction time
| No Caffeine | 200 Mg Caffeine | 400 Mg Caffeine |
| M = 225 | M = 200 | M = 175 |
| M = 275 | M = 250 | M = 225 |
No Alcohol
Alcohol
Overall
M = 250
Overall
M = 225
Overall
M = 200
Overall M = 200
Overall M = 250
Click to edit Master text styles
Second level
Third level
Fourth level
Fifth level
Interaction
An interaction occurs when one factor modifies the effects of another factor.
If the effects of one factor simply add onto the effects of another factor, then the two factors are independent (no interaction)
An interaction between two factors occurs whenever the mean differences between individual treatment conditions, or cells, are different from what would be predicted from the overall main effects of the factors.
The null hypothesis is that there is no interaction
H0: There is no interaction between factors A and B. The mean differences between treatment conditions are explained by the main effects of the two factors.
Interactions
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The alternative hypothesis is that there is an interaction between the two factors:
H1: There is an interaction between factors.
To evaluate the interaction, the two-factor ANOVA first identifies mean differences that are not explained by the main effects.
The extra mean differences are then evaluated by an F-ratio with the following structure:
variance (mean differences) not explained by main effects
variance (differences) expected if there are no treatment effects
Interactions (cont’d.)
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If the two factors are independent, so that one factor does not influence the effect of the other, then there is no interaction.
On the other hand, when the two factors are not independent, so that the effect of one factor depends on the other, then there is an interaction.
When the effect of one factor depends on the different levels of a second factor, then there is an interaction between the factors.
Interactions (cont’d.)
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The concept of an interaction can also be defined in terms of the pattern displayed in the graph.
When the results of a two-factor study are presented in a graph, the existence of nonparallel lines (lines that cross or converge) indicates an interaction between the two factors.
Interactions (cont’d.)
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Hypothetical data showing the treatment means for a two-factor study examining how different combinations of alcohol and caffeine affect reaction time
| No Caffeine | 200 Mg Caffeine | 400 Mg Caffeine |
| M = 210 | M = 200 | M = 190 |
| M = 290 | M = 250 | M = 210 |
No Alcohol
Alcohol
Overall
M = 250
Overall
M = 225
Overall
M = 200
Overall M = 200
Overall M = 250
Click to edit Master text styles
Second level
Third level
Fourth level
Fifth level
Interpreting Interactions
| M = 30 | M = 30 |
| M = 30 | M = 10 |
No Drug
Drug
Empty
Stomach
Full
Stomach
Pain
No Drug
Drug
30
10
20
30
20
30
20
Empty Stomach
Full Stomach
Fig 11.4 (a) Data showing how an overall main effect can be distorted by an interaction
Interpreting Interactions
| M = 30 | M = 40 |
| M = 30 | M = 20 |
No Drug
Drug
Empty
Stomach
Full
Stomach
Pain
No Drug
Drug
40
10
30
35
25
30
30
20
Empty Stomach
Full Stomach
Fig 11.4 (b) Data showing how an interaction can disguise main effects
Main Effect for Factor A; no interaction
40
30
20
10
Factor B
Two levels of Factor A
Main Effect for Factor A & Factor B; no interaction
40
30
20
10
Factor B
Two levels of Factor A
No Main Effect for either factor; but an interaction
40
30
20
10
Factor B
Two levels of Factor A
Mixed Designs
| Mean recall Mean = 70 | Mean recall Mean = 23 |
| Mean recall Mean = 48 | Mean recall Mean = 35 |
Happy
Sad
Positive Words
Negative Words
Higher-Order Factorial Designs
A x B x C
3 Main effects
A
B
C
3 (2-Way) Interactions
A x B
B x C
A X C
1 (3-way) Interaction
A x B x C
Higher-Order Example
A study is designed to look at the effects of teaching methods and gender on academic performance, for 1st grade and 2nd grade students
Factor 1 = Teaching method
Factor 2 = Gender
Factor 3 = Grade level
Results
Main effects
Main effect of Teaching
Main effect of Gender
Main effect of Grade level
Interactions
Teaching by Gender
Gender by Grade level
Teaching by Grade level
Teaching by gender by grade level (two teaching methods are equally effective for boys & girls in the first grade; but one method works better in the 2nd)
In the context of ANOVA, an independent variable (or a quasi-independent variable) is called a factor, and research studies with two factors are called factorial designs or simply two-factor designs.
The two factors are identified as A and B, and the structure of a two-factor design can be represented as a matrix with the levels of factor A determining the rows and the levels of factor B determining the columns.
An Overview of the Two-Factor Independent-Measures ANOVA
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The two-factor ANOVA allows us to examine three types of mean differences within one analysis.
Traditionally, the two independent variables in a two-factor experiment are identified as factor A and factor B.
For the study presented in Table 14.1, gender is factor A, and the level of violence in the game is factor B.
An Overview of the Two-Factor Independent-Measures ANOVA (cont’d.)
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The goal of the study is to evaluate the mean differences that may be produced by either of these factors acting independently or by the two factors acting together.
An Overview of the Two-Factor Independent-Measures ANOVA (cont’d.)
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A Two-Factor Experiment as a Matrix
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TABLE 14.1 The structure of a two-factor experiment presented as a matrix. The two factors are gender and level of violence in a video game, with two levels for each factor.
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Hypothetical Data for an Experiment
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TABLE 14.2 Hypothetical data for an experiment examining the effect of violence in a video game on the aggressive behavior of males and females
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Modified Hypothetical Data for an Experiment
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TABLE 14.3 Hypothetical data for an experiment examining the effect of violence in a video game on the aggressive behavior of males and females. The data show the same main effects as the values in Table 14.2 but the individual treatment means have been modified to create an interaction.
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The Treatment Means
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FIGURE 14.1 (a) Graph showing the treatment means for Table 14.2, for which there is no interaction. (b) Graph for Table 14.3, for which there is an interaction.
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The Results from a Two-Factor Study
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FIGURE 14.2 Two graphs showing the results from a two-factor study. A line graph is shown in (a) and a bar graph (b)
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The two-factor ANOVA consists of three hypothesis tests, each evaluating specific mean differences.
These are three separate tests, but they are also independent of each other.
The outcome for any one of the three tests is totally unrelated to the outcome for either of the other two.
Thus, it is possible for data from a two-factor study to display any possible combination of significant and/or not significant main effects and interactions.
Independence of Main Effects and Interactions
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Three Sets of Data
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TABLE 14.4 Three sets of data showing different combinations of main effects and interaction for a two-factor study. (The numerical value in each cell of the matrices represents the mean value obtained for the sample in that treatment condition.)
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Structure of the Analysis for a Two-Factor ANOVA
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FIGURE 14.3 Structure of the analysis for a two-factor ANOVA.
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Structure of the Analysis for a Two-Factor ANOVA
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FIGURE 14.3 Structure of the analysis for a two-factor ANOVA.
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Hypothetical Alcohol & Caffeine Data
150
175
200
225
250
275
300
No Caff200 Mg400 Mg
Caffeine
Reaction Time in Msec
No Alcohol
Alcohol
Chart2
| No Caff | No Caff |
| 200 Mg | 200 Mg |
| 400 Mg | 400 Mg |
Sheet1
| No Caff | 200 Mg | 400 Mg | |
| No Alcohol | 225 | 200 | 175 |
| Alcohol | 275 | 250 | 225 |
Sheet1
Sheet2
Sheet3
Hypothetical data showing the treatment means for a
two-factor study examining how different combinations
of alcohol and caffeine affect reaction time
100
150
200
250
300
123
Amount of Caffeine (in Mg)
Reation Time (in
Msec)
No Alcohol
Alcohol
Chart4
| 210 | 290 |
| 200 | 250 |
| 190 | 210 |
Sheet1
| 210 | 200 | 190 |
| 290 | 250 | 210 |
