Chapter 8: Factorial ANOVA - Heather C. Lench, Ph.D.

1 Chapter 8: Factorial ANOVA **This Chapter corresponds to Chapter 13 of your book (Two Too Many Factors) What it is: Factorial ANOVA is used to test the influence of two (or more) factors on an outcome. It is a variant of the one way ANOVA you learned about in Chapter 7 and is based on the same mathematical principles ( the partitioning of variance into different sources). A quick note on terminology: A factor is another word for an independent variable. Within a factor, there are levels or groupings. For example, gender is a single factor that has 2 levels (male, female). In a Factorial ANOVA , you have two (or more factors) influencing a single outcome. Those factors can have any number of levels within each. When to use it: Per the flow chart on page 221 of Salkind (2008), you would use a Factorial ANOVA when: (1) you are examining differences between groups (as opposed to examining the relationships between variables), (2) the participants in the study were only tested once (as opposed to repeatedly), (3) you are comparing more than two groups and (4) you are dealing with more than one factor.

2 Questions asked by a Factorial ANOVA : Factorial ANOVA is exciting because you can use it to ask three different questions in a single study using just this one analysis! When you use Factorial ANOVA you have at least two different factors that you think might predict the outcome. For example, imagine you want to examine the influence of both gender and smoking status (smoker vs. non-smoker) on depression. Using a Factorial ANOVA would allow you to ask the following three questions: 1. Does gender affect depression? 2. Does smoking status affect depression? These two effects are called main effects and refer to the effect of a single factor ignoring the levels of the other factor. 3. Do gender and smoking status interact to predict depression? You ll learn more about interactions throughout the Chapter , but an interaction occurs when the effect of one factor depends on the other factor. An example of an interaction we might see in this study would be if the effect of smoking status on depression depended on whether you were male or female.

3 For example, maybe being a smoker makes women more depressed but makes males less depressed. Using SPSS to Calculate an independent t test (dataset: Chapter8 ) Imagine you want to compare the effectiveness of 2 different diets (low carb vs. low fat). You also want to assess whether people lose more weight on either diet if they are already overweight vs. normal weight. It s also possible that the effectiveness of these two specific diets depends on whether or not participants are already at a normal weight or are overweight. To test your ideas, you recruit 40 (20 normal weight, 20 overweight) people to participate in a study. Within each weight group, you randomly assign half of the participants to eat a low carb diet for 2 months and the other half to eat a low fat diet for 2 months. That means you now have four different groups (or cells): normal weight on low carb diet, normal weight on low fat diet, overweight on low carb diet, and overweight on low fat diet.

4 You collect data on how much each participant loses and enter the data in SPSS. Now follow the Famous Eight Steps : 1. Statement of the null and research hypotheses. For this Factorial ANOVA , we actually have three different null hypotheses and three different research hypotheses that is, we have a set of hypotheses for each of the three questions we re asking. Question 1: Is there an effect of diet type on weight loss? Null H0: Lowcarb = LowFat The average amount of weight loss between a low carb and a low fat diet is equal. Research H1: XLowCarb XLowFat The average amount of weight loss between a low carb and a low fat diet is not equal. Question 2: Is there an effect of initial weight on weight loss? Null H0: NormalWeight = Overweight The average amount of weight loss between normal and overweight groups is equal. Research H1: XNormalWeight XOverweight The average amount of weight loss between normal and overweight groups is not equal.

5 Question 3: Is there an interaction between diet type and current weight? Null H0: = = = The average amount of weight loss is equal between all four groups. Research H1: The average amount of weight loss is not equal between all four groups. 2. Set the level of risk at p < .05 3. Selection of the appropriate test statistic Again using the flowchart on page 221 of Salkind (2008) we see that a Factorial ANOVA is the appropriate test statistic because we are comparing the means (of weight loss) of participants who are only tested once (each person has one weight loss score for the 2 months). We also have more than 2 groups (we have 4 groups) and we have more than one factor (we have two factors: diet type and current weight). Now open the data set . Take a moment to familiarize yourself with the data. Note how data for this type of analysis should be entered. 1) Each participant has one row in the data 2) Two columns indicate which level of each factor each participant was in.

6 A. One column indicates which group the participant was in on the diet factor. Here we ve used 1 s to indicate the participant was in the low carb diet and 2 s to indicate the participant was in the low fat diet. b. One column indicates which group the participant was in on the weight factor. Here we ve used 1 s to indicate the participant was normal weight and 2 s to indicate the participant was overweight. 3) Another column indicates each participant s score on the outcome measure here it s how many pounds the person lost at the end of the study. The data should look like this in SPSS: 4. Computation of the test statistic. SPSS will calculate an obtained value (and find the associated p-value) for each of the three possible effects. To do this, click on the Analyze drop-down menu, highlight General Linear Model , and then click on Univariate , as pictured below. The following pop-up window will appear: Next, highlight the dependent variable (PoundsLost) and click it over to the dependent variable box.

7 Then, one at a time, highlight each of your independent variables (DietType, Weight) and click them over to the fixed factor(s) box. Your screen should look like the picture below: Whenever you do a Factorial ANOVA , also tell SPSS to produce the descriptive statistics for each group. This will help you interpret any significant effects you find. To do this, click on the box that says options . Click the box next to Descriptive Statistics in the display box. Your window should look like this: Click continue and then OK. Now navigate to the output window. Your output will look like this: Between-Subjects Factors Value Label N DietType Low Carb Low Fat 10 Weight Normal Weight Overweight 10 Descriptive Statistics Dependent Variable:PoundsLost DietType Weight Mean Std. Deviation N Low Carb Normal Weight .6000 .659555 Overweight Fat Normal Weight .936485 Total Normal Weight of Between-Subjects Effects Dependent Variable:PoundsLost Source Type III Sum of Squares df Mean Square F Sig.

8 Corrected Model Intercept DietType Weight DietType * Weight Error Total Corrected Total a. R Squared = .950 (Adjusted R Squared = .940) This box tells you how many participants were in each group of each factor. For example, there were 10 people on the low carb diet and 10 people on te low fat diet. Note that if you added these N s together that would equal double our actual sample size this is because each person is counted twice (once for the diet group they were in and once for the weight group). This box gives you the obtained values and p values for each of the three possible effects. The Source Column identifies the different sources of variance. The df column tells you the degrees of freedom. The Mean Square for each source is provided. Recall, the mean square is equal to the Sum of Squares divided by the degrees of freedom. The F and sig. columns give you the exact obtained and p values for each source. Just like in one way ANOVA , the F for each source is equal to the mean square of that source divided by the mean square for the error.

9 This part of the output gives you all the means and standard deviations you need to interpret any significant effects you find. If you have an interaction, you ll want to use the separate means for each of the four groups. Following, the output, these means have been graphed in excel you can do the same or just do this on a piece of paper. Alternatively, if you have a main effect (or 2 main effects) you ll want to use these numbers. These give you the collapsed means that are important for a main effect ( , the average pounds lost for low carb dieters, regardless of weight). Plot of means (from excel) If you have an interaction and would like to plot the means in excel, enter them as seen below, then use the insert line graph function. low carb low fat normal Interpreting the Output The output for a Factorial ANOVA has a lot of components. We want to know which of our three potential effects were significant (if any). To determine this, you look at the Test of Between Subjects Effects table.

10 Identify the rows of the table that are associated with each of our three effects (the two main effects and the interaction). We see that there is a significant effect of weight AND a significant interaction. Looking at the graph above, we see: The main effect of weight: people who are overweight lose more weight (regardless of diet type) than people who are normal weight. The interaction effect between weight and diet type: overweight people lost more weight on a low carb diet, whereas normal weight people lost more weight on a low fat diet. o This means that the effect of diet type DEPENDS on your current weight this is the telltale sign of an interaction. Now, back to the eight steps: 5. Determination of the value needed for rejection of the null hypothesis Factorial ANOVA is too complex for us to calculate it by hand but the critical values are the same as the ones we would use in a one-way ANOVA and can be found in your book on pages 336-338.