Analysis of Variance (ANOVA) is a statistical test used to identify the effects of independent variables on the outcome of an experiment. In this lesson, learn about main effects and interaction effects and how ANOVA can be used to test for both.
Dependent & Independent Variables
Jamal works for a company that has developed a new drug to treat a certain type of cancer. It's Jamal's job to design a study that will determine which doses of the drug are most effective and if the effectiveness depends on the initial size of the tumor. How can he do this?
In Jamal's study, there are two independent variables: original tumor size and drug dosage. These are called independent variables because they are the things that he can control and change.
He would need to divide the patients into groups based on both tumor size and drug dosage. If there are three different tumor sizes and four different drug doses being tested, there will be 12 different groups of patients.
number of groups (N) = 3 x 4 = 12
The next thing to consider is what to measure at the end of the study. This is known as the dependent variable, and in this case, Jamal would probably want to measure the size of each patient's tumor at the end of the study. Then he could calculate the percentage of the original tumor that remains after treatment. This percentage would be the dependent variable, and Jamal could use it to determine if there are any differences that are due to either of the independent variables or to an interaction between them.
To determine if drug dosage or original tumor size affect the final tumor size, Jamal needs to test for three different effects:
Effect #1: Drug dosage - Are there any differences in the final tumor size that can be attributed to the drug dosage?
Effect #2: Original tumor size - Are there any differences in the final tumor size that can be attributed to the original tumor size?
Effect #3: Interaction between drug dosage and original tumor size - Is there an interaction between drug dosage and the original tumor size in determining the final tumor size?
Effects #1 and #2 are known as main effects because they are exclusively due to one factor or the other. In statistics, a main effect is the effect of just one of the independent variables on the dependent variable. The first step in determining if the main effect results in statistically significant differences in the dependent variable is calculating the marginal mean of each group. To find the marginal mean, average the means of the individual groups. For example, in the table below, the marginal mean for the 250 mg/kg treatment group is found by adding all the means in that column (88%, 92%, and 105%) and dividing by three to get 95%.
The main effect for each factor is determined by comparing marginal means. For example, to see if there are differences due to the drug concentration, Jamal should compare the marginal means for each concentration (95%, 86%, 61%, 53%).
Just calculating the marginal means, however, isn't enough to determine if the different concentrations of drug result in statistically significant differences in tumor reduction. To do that, Jamal would have to take this data and perform an analysis of variance, commonly known as ANOVA. ANOVA is a statistical test that's used to determine if there are differences between groups when there are more than two treatment groups. There are different types of ANOVA that should be used depending on the situation, but in this case, Jamal should use a two-way ANOVA because he's testing the effects of two independent variables on the dependent variable.
While the main effects are caused autonomously by each independent variable, an interaction effect occurs if there is an interaction between the independent variables that affects the dependent variable.
In our example, it's possible that there is an interaction between the concentration of drug administered and the original tumor size. Perhaps larger tumors respond better to a higher concentration, while smaller tumors respond to a lower concentration. Testing for an interaction effect will help Jamal determine if this is happening in his study.
Luckily, a two-way ANOVA also tests for interaction effects, so there is only one statistical test that Jamal needs to perform.
Results of a Two-Way ANOVA
Analysis of the data using ANOVA will give Jamal three important numbers that he can use to determine if either of the main effects or the interaction effect are statistically significant. These important numbers are called F-ratios, and there will be one for each main effect and the interaction effect.
To determine if any of the effects are significant, the calculated F-ratio should be compared to a critical F-ratio that can be looked up in a table of F-ratios. If the calculated F-ratio for a certain effect is bigger than the critical F-ratio, then the effect is significant and there are differences in the dependent variable because of that effect.
Let's take a couple moments to review what we've learned about the main effect and interaction effect in analysis of variance. An independent variable is something that you can control and change about the experiment. The dependent variable is the outcome that you measure at the end of the experiment. It's dependent on the independent variables because it can change as a result of changes in the other variables.
In statistics, main effect is the effect of one of just one of the independent variables on the dependent variable. There will always be the same number of main effects as independent variables. An interaction effect occurs if there is an interaction between the independent variables that affect the dependent variable.
Analysis of variance (ANOVA) is a statistical test that's used to determine if there are differences between groups when there are more than two treatment groups. When there are two independent variables, you should use a two-way ANOVA to determine if the main effects or interaction effect are statistically significant.