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So, the intuition is a t test which accommodates more than two groups to compare. For example, when you want to compare user performance of three interaction techniques, ANOVA is one of the methods you want to consider. I don't think I fully understand it yet. It is not like it is impossible, but it is very complicated, and you may have to have some knowledge about linear models and manipulations for these in R.

But if you have time, I recommend you to read a statistics book to understand ANOVA and to double-check what I am saying here as well as this page. There is one clear reason why we cannot use multiple t test instead of ANOVA for saying whether a factor which has more than two groups e.

Let's say, you have Technique A, Technique B, and Technique C, and you have measured performance time of these three techniques.

What you want to know here is whether these techniques have a significant effect on performance time, that is whether there exists at least one significant difference in performance time against the techniques. In other words, your null hypothesis is that there is no significant difference in the means among three techniques. If you test the t test null hypothesis i. Remember that your null hypothesis you really need to test is there is no significant difference in the means among three techniques.

You need to do two more tests for B and C, and A and C. If all three tests reject the null hypothesis, you can say that there is no significant difference in the means among three techniques. Obviously, a test with Thus, you cannot simply do multiple t tests if you have more than two groups to compare. Repeated measure is a similar concept of paired in t test. If you gain the values for the multiple conditions from the same subject i. One common case in which you need to use repeated measure ANOVA is that you did a within-participant design experiment.

For the normality assumption, you can see another page. If you think you can assume the normality, you need to make sure that the second assumption holds. We are going to look into how you can test the homogeneity of variances or sphericity.

Another thing you need to be careful about is whether the data are balanced or unbalanced. Balanced means that the sample sizes are equal across all the groups to compare. There are a couple of ways to check the homogeneity of variances. Here, I show three kinds of tests with R codes. You can do it in R too, but need to include car package. So, in this example, p is over 0. One solution is to do a data transformation. There are several ways to do so, and which transformation to use depends on the relationship between the means and variances.

Some details are available in a separate page. Another thing you can do is to switch to non-parametric methods, such as Kruskal-Wallis. This is a better way to take if you are not sure what kind of data transformation you should perform.

A standard test for Sphericity is Mauchly's test. Unfortunately, doing a test for checking the sphericity is not easy in R This is automatically done in SPSS including the corrections for the cases where the sphericity assumption is violated.

There are two ways to do a sphericity test. The following procedure using car package is based on the blog entry here. You need to make a matrix such that the rows are the within-subject factor Participant and the columns are the groups to compare Group.

Now, you need to define the design of the study, which basically means that you need to make a list of the independent variable. Another way to do a sphericity test is to use ez package. We use the same data in this example too. In this example, we specify the dependent variable dv , the indices indicating different cases wid , and the within-subject factor within.

After executing the command, you get the results. The both procedures explained above give us the same results. Here, I take the results of the procedure using car package to explain how we can interpret the results. The test statistics of Mauchly's test is W , which is 0. For instance, the degree of freedom of Group is 2.

So, we can assume the sphericity and don't need for making any correction on the data. What if we cannot assume the sphericity? They have a value called epsilon. The epsilon means the departure from the sphericity, in other words, how far the data is from the ideal sphericity.

The epsilon is a number between 0 and 1, if the epsilon is equal to 1, the data have the sphericity. I omit the difference between Greenhouse-Geisser and Huynh-Feldt you actually see another value called lower-bound in SPSS, which means the possible lowest value for the epsilon , but generally the epsilon of Greenhouse-Geisser is used.

If the epsilon of Greenhouse-Geisser is greater than 0. This is because Greenhouse-Geisse tends to make the analysis too strict when the epsilon is large. It is also considered that you should use Huynh-Feldt when you have a small sample size like You can make a correction by multiplying the degree of freedom by the epsilon. Let's assume that we need to make a correction. The degree of freedom of design is 2. The epsilon of Greenhouse-Geisser is 0.

Thus, the new degree of freedom after the correction is 1. You also do the same thing for the second degree of freedom i. So now you say you have F 1. The p value with the adjusted F value is available right next to the epsilon.

In this example, the p value is 0. You don't have to do anything for the effect size. If the degree of freedom of your factor is 1, you don't need any correction. Thus, R won't show any information about the correction for that factor. So, if your factor's degree of freedom is 5, the theoretical lower bound of epsilon is 0. Thus, the adjusted degree of freedom cannot be below 1 in any case. In some cases, the two methods described above cause some errors.

Particularly you may see an error like this when you use Anova. In this case, you can run a Mauchly's test in a more manual way. Another thing we need to talk about is the effect size.

There are a few standard metrics to present the effect size for ANOVA, but the most common metrics are eta squared and partial eta squared.

The eta squared is a square of the correlation ratio, and represents how much the factor we are talking about affects the results of the total sum of square. We can calculate eta squared as follows:. However, when you have many factors or independent variables , the eta squared often becomes too small.

To address this, the partial eta squared is used. You can use either of them SPSS uses partial eta squared by default , but because there is no way to convert eta squared to partial eta squared or vice versa, you need to clarify which you used in your report. And here are the values which are considered as a small, medium, and large effect size. Another disappointment of the partial eta-squared is that there is no straightforward way to calculate its confidence interval in R if you know, please let me know!

We can do it for the eta squared by using the MBESS package, which is explained in the example codes. One-way ANOVA means you have one independent variable to use for the comparison of the dependent variables. First, prepare the data. Here, we have three groups. These values are measured in a between-subject manner. Thus, each of Value came from different participants. Before running a one-way ANOVA test, you need to make sure that the homogeneity of variances assumption holds.

There are a couple of ways to do that, but we use Barlett's test here. Because the p value is over 0. Your independent variable is Group, and dependent variable is Value. Because the F value for Group is large enough, the p value is below 0. Thus, you have found a significant effect of Group in the means of Value. For the effect size, both eta squared and partial eta squared become the same in this case.

It is If you need to calculate the confidence interval of the effect size, it is safer to use the eta squared at least there is an easy way to calculate its confidence interval in R. Then, you just need to call the ci. You need the F value, first and second degree of freedom, and sample size. This procedure assumes the sphericity implicitly.

Analysis of Variance (ANOVA) for comparing the means

So, the intuition is a t test which accommodates more than two groups to compare. For example, when you want to compare user performance of three interaction techniques, ANOVA is one of the methods you want to consider. I don't think I fully understand it yet. It is not like it is impossible, but it is very complicated, and you may have to have some knowledge about linear models and manipulations for these in R. But if you have time, I recommend you to read a statistics book to understand ANOVA and to double-check what I am saying here as well as this page.

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