Two Factor Ind Anova - Psyc Statistics with Professor Canan Ipek. These are all the lecture notes from PDF

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Psyc Statistics with Professor Canan Ipek. These are all the lecture notes from the semester, with step by step instructions. ...

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PSYC 274 Two Factor Independent Measures Anova

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TWO FACTOR INDEPENDENT MEASURES ANOVA So far, we have dealt with research designs where we had one, single, independent variable with two or more levels. However, in most research situations the goal is to examine how two (or more) independent variables interact. For instance, researchers might be interested in understanding the effect of a new drug developed to treat migraine. To understand whether the newly developed drug has an effect on reducing the migraine pains, they might have two groups of participants. One group might be given the drug (treatment) and the other group the placebo (control). In addition, the researchers might be interested in understanding whether the drug has a different effect on males and females. In other words, the researchers are trying to understand if both variables; the drug and gender, interact with each other. To observe how/if one variable interacts with another, it is necessary to study both variables simultaneously in one study. Two-factor ANOVA provides the statistical tools to be able to do that. Two understand the process of two factor ANOVA analysis, we will use a sample study. The research question is: - Would the presence of a TV camera affect your performance on an exam? This research question would give us one independent variable, let’s call it audience. So, our first independent variable(IV) is audience. Researchers in the same study also divide the participants into two groups on the basis of personality: those high in self-esteem and those low in self-esteem. So, this is our second independent variable; let’s call it self-esteem (beware that this is a quasiindependent variable as you cannot randomly assign people to low and high selfesteem groups.) Our dependent variable in this study is the number of errors students make in the test. Before moving onto the details, a little bit of terminology is necessary. When a research study involves one independent variable, it is called single-factor research design; single for “one” and factor for “independent variable.” Following this, when a research study involves more than one factor (independent or quasi-independent variable), it is called a factorial design. In this handout, we examine ANOVA as it applies to research studies with exactly two factors. In addition, for each treatment condition, we will have separate participants. Therefore, our focus will be on two-factor independent measures ANOVA. We will refer to our first factor (Self-Esteem) as Factor A; and our second factor (Audience) as Factor B.

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A factorial design is usually presented as a matrix (Beware that I put Factor A in the rows and Factor B in the column): Table 1. Two-by-two matrix showing four different combinations of the variables, Producing four different conditions with two levels for each factor. Factor B: Audience Condition No Audience

Factor A: Self-Esteem

Audience

Low

Scores for participants with Low self-esteem tested with noaudience

Scores for participants with Low self-esteem tested with audience

High

Scores for participants with High self-esteem tested with noaudience

Scores for participants with High self-esteem tested with audience

Two-factor ANOVA evaluates three sets of mean differences: 1. Mean difference for Factor A 2. Mean difference for Factor B 3. Mean difference of specific combination of cells. Therefore, for a two factor ANOVA, we conduct three separate hypothesis test, in other words, we calculate three separate F-ratios. The goal is to evaluate the mean differences that may be produced by either of these factors acting independently, which we will call main effects, or by the two factors acting together, which we will call interactions. Main Effects: The mean differences among the levels of one factor are referred to as the main effect of that factor. So, the mean difference between Low and High self-esteem is referred to as the main effect of Factor A. Likewise, the mean difference between Audience and No-Audience is referred to as the main effect of Factor B.

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Table 2. Factor B: Audience Condition No Audience

Main%effect%of%Factor%A%

Factor A: Self-Esteem

Audience

Low

High

Main%effect%of%Factor%B%

Interaction An interaction between two factors occur 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. For instance, we might find that participants in noaudience group performed better in the test compared to participants in audience group (Main effect of Factor B); however we might find that the presence of audience effects only participants with low self-esteem but does not have an effect on the participants with high self-esteem. In such a case, two independent variables are interacting; effect of one factor (audience) depends on the other factor (self-esteem). Any “extra” mean differences that are not explained by the main effects are called interaction, or interaction between factors. Look at the following example:

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Table 3. Factor B: Audience Condition No Audience

Factor A: Self-Esteem

Audience

Low

High

If we examine the main effects in Table 3., we will see that there is no main effect of factor A; whether people have low or high self-esteem does not affect their test score (both group has a mean of 20). We also see that there is a main effect of factor B; people who take the test with no audience make less errors on average (15) compared to those who took the test in the presence of audience (25). The point you need to understand here is that just by looking at the main effect of factor B (that 10 point mean difference; 15 vs 25), can you conclude that students who take the test with no audience make less error on average compared to those who take the test in the presence of audience? Well, NO. And you can see why NO if you compare mean differences between no-audience and audience across the levels of the other factor (self-esteem). Here is how you do it: - We found that participants who take the exam with no audience make less errors on average (main effect of Factor B). Now the question you ask is: - Is this true for participants with low self-esteem AND participants with high self-esteem. - In other words, is it case that participants with low self-esteem AND high self-esteem both make less error with no audience. - If you look at the cell mean differences, you will see that this is not the case. People with low-esteem do make less error with no audience (10 vs 30), however whether there is audience or not does not make any difference for people with high self-esteem. THIS is interaction; the two independent variables are interacting; i.e. the effect of one factor (audience/no audience) depends on the other factor (self-esteem).

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Various combinations of main effect and interaction are possible. Refer to your course book to see possible combinations. Now, let’s get back to hypothesis testing. As mentioned before, in a two-factor design we will run three separate hypothesis test which means we will compute three F-ratios. The process involves two main stages; similar to one-way ANOVA, we will first partition total variance into between-treatments variance and within-treatments variance. The second stage involves partitioning between treatments variance into Factor A variance, Factor B variance, and Interaction variance.

Total!! variability!

Within-treatments! variance!

Betweentreatments!

Factor!A!! variance!

Factor!B!! variance!

Interaction! variance!

Hypothesis Testing in a Two-Factor Design Evaluating the main effect of Factor A - To evaluate the main effect of Factor A, we set our hypotheses: H0 : μA1 = μA2 H1 : μA1 ≠ μA2 - And to make a decision, we calculate F-ratio that compares the mean differences betwee the levels of Factor A. F=

variance between means for Factor A variance expected if there is no treatment effect

Evaluating the main effect of Factor B - To evaluate the main effect of Factor B, we set our hypotheses:

PSYC 274 Two Factor Independent Measures Anova H0 : μB1 = μB2 H1 : μB1 ≠ μB2 - And to make a decision, we calculate F-ratio that compares the mean differences betwee the levels of Factor B. F=

variance between means for Factor B variance expected if there is no treatment effect

Evaluating interaction - Mean differences that are not explained by the main effects are an indication between the two factors. The null and the alternative hypothesis for evaluating interaction is: H0: There is no interaction between factors A and B. All of the mean differences between treatment conditions are explained by the main effects of the two factors. H1: There is an interaction between factors.

- F-ratio for evaluating interaction: F=

variance not explained by mean effects variance expected if there are no treatment effects

Now, let’s see how to calculate each F-ratio to test main effects and interaction. Stage 1 Like we did in one-way ANOVA analysis, we will first partion Total into Between and Within-treatments Total:

dftotal = N-1

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PSYC 274 Two Factor Independent Measures Anova

Within: SSwithin = ΣSSinside each treatment dfwithin = Σ(n-1) = Σdfin each treatment

Between: SSbetween = SStotal - SSwithin OR

dfbetween = number of cells - 1

Stage 2 Partitioning between treatments into FactorA, FactorB, and Interaction . 1. Factor A (rows in Table 1/2/3) - evaluates the mean differences between the levels of factor A.

dfA = number of rows - 1 2. Factor B (columns in Table 1/2/3) - evaluates the mean difference between the levels of factor B.

dfB = number of columns - 1

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3. The A´B interaction

Stage 3 Now, we will calculate the denominator (MSwithin) we will use in all of our F-ratio calculations.

MSwithin =

SSwithin dfwithin

Stage 4 Calculate MS for each main effect and interaction MSA = SSA dfA MSB = SSB dfB

MSA´B = SSA´B dfA´B

Stage 5 Calculate F-ratio for each main effect and interaction FA =

MSA MSwithin

FB =

MSB MSwithin

FA´B = MSA´B MSwithin

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Stage 6 We will determine critical F values to evaluate the null hypothesis for each factor and interaction. We will use the F-distribution table. The relevant df values to determine critical values are: - For Factor A: (dfA, dfwithin) - For Factor B: (dfB, dfwithin) - For A´B: (dfA´B, dfwithin)

INTERPRETING THE RESULTS FROM A TWO-FACTOR ANOVA Because the two-factor anova involves three separate tests, you should be careful while interpreting the results; most importantly when there is a significant interaction, you should be cautious about accepting the main effects at face value. Even though you may find significant main effect for Factor A or Factor B or both, if interaction comes out as significant, that means there is no systematic main effect of Factor A or Factor B. In other words, the effect of one Factor depends on the other Factor....