# What is ANOVA (Analysis Of Variance) and what can I use it for?

One-Way Analysis of Variance (ANOVA) tells you if there are any statistical differences between the means of three or more independent groups.

## What is ANOVA?

Developed by Ronald Fisher, ANOVA stands for Analysis of Variance. One-Way Analysis of Variance tells you if there are any statistical differences between the means of three or more independent groups.

## When might you use ANOVA?

You might use Analysis of Variance (ANOVA) as a marketer, when you want to test a particular hypothesis. You would use ANOVA to help you understand how your different groups respond, with a null hypothesis for the test that the means of the different groups are equal. If there is a statistically significant result, then it means that the two populations are unequal (or different).

## How can ANOVA help?

The one-way ANOVA can help you know whether or not there are significant differences between the means of your independent variables (such as the first example: age, sex, income). When you understand how each independent variable’s mean is different from the others, you can begin to understand which of them has a connection to your dependent variable (landing page clicks), and begin to learn what is driving that behavior.

## Examples of using ANOVA

Do age, sex, or income have an effect on whether someone clicks on a landing page?

Do location, employment status, or education have an effect on NPS score?

One-way ANOVA can help you know whether or not there are significant differences between the groups of your independent variables (such as USA vs Canada vs Mexico when testing a Location variable). You may want to test multiple independent variables (such as Location, employment status or education). When you understand how the groups within the independent variable differ (such as USA vs Canada vs Mexico, not location, employment status, or education), you can begin to understand which of them has a connection to your dependent variable (NPS score).

“Do all your locations have the same average NPS score?”

Although, you should note that ANOVA will only tell you that the average NPS scores across all locations are the same or are not the same, it does not tell you which location has a significantly higher or lower average NPS score.

## What is the difference between one-way and two-way ANOVA tests?

This is defined by how many independent variables are included in the ANOVA test. One-way means the analysis of variance has one independent variable. Two-way means the test has two independent variables. An example of this may be the independent variable being a brand of drink (one-way), or independent variables of brand of drink and how many calories it has or whether it’s original or diet.

## How does ANOVA work?

Like other types of statistical tests, ANOVA compares the means of different groups and shows you if there are any statistical differences between the means. ANOVA is classified as an omnibus test statistic. This means that it can’t tell you which specific groups were statistically significantly different from each other, only that at least two of the groups were.

It’s important to remember that the main ANOVA research question is whether the sample means are from different populations. There are two assumptions upon which ANOVA rests:

First: Whatever the technique of data collection, the observations within each sampled population are normally distributed.

Second: The sampled population has a common variance of s2.

## How to conduct an ANOVA test

### Stats iQ and ANOVA

Stats iQ from Qualtrics can help you run an ANOVA test. When users select one categorical variable with three or more groups and one continuous or discrete variable, Stats iQ runs a one-way ANOVA (Welch’s F test) and a series of pairwise “post hoc” tests (Games-Howell tests).

The one-way ANOVA tests for an overall relationship between the two variables, and the pairwise tests test each possible pair of groups to see if one group tends to have higher values than the other.

### Users can run an ANOVA test through StatsiQ

The Overall Stat Test of Averages acts as an Analysis of Variance (ANOVA). An ANOVA tests the relationship between a categorical and a numeric variable by testing the differences between two or more means. This test produces a p-value to determine whether the relationship is significant or not.

In StatsiQ take the following steps:

• Click a variable with 3+ groups and one with numbers,
• Then click “Relate”,
• You’ll then get an ANOVA, a related “effect size”, and a simple, easy to understand summary.

### Qualtrics Crosstabs and ANOVA

You can run an ANOVA test through the Qualtrics Crosstabs feature too.

• Ensure your “banner” (column) variable has 3+ groups and your “stub” (rows) variable has numbers (like Age) or numeric recodes (like “Very Satisfied” = 7)
• Click “Overall stat test of averages”
• You’ll see a basic ANOVA p-value

## What does an ANOVA test reveal?

A one way ANOVA will allow you to distinguish that at least two groups were different from each other. Once you begin to understand the difference between the independent variables you will then be able to see how each behaves with your dependent variable. (See landing page example above)

## What are the limitations of ANOVA?

Whilst ANOVA will help you to analyse the difference in means between two independent variables, it won’t tell you which statistical groups were different from each other. If your test returns a significant f-statistic (this is the value you get when you run an ANOVA test), you may need to run an ad hoc test (like the Least Significant Difference test) to tell you exactly which groups had a difference in means.

## Welch’s F Test ANOVA

Stats iQ recommends an unranked Welch’s F test if several assumptions about the data hold:

• The sample size is greater than 10 times the number of groups in the calculation (groups with only one value are excluded), and therefore the Central Limit Theorem satisfies the requirement for normally distributed data.
• There are few or no outliers in the continuous/discrete data.

Unlike the slightly more common F test for equal variances, Welch’s F test does not assume that the variances of the groups being compared are equal. Assuming equal variances leads to less accurate results when variances are not in fact equal, and its results are very similar when variances are actually equal.

## Ranked ANOVA

When assumptions are violated, the unranked ANOVA may no longer be valid. In that case, Stats iQ recommends the ranked ANOVA (also called “ANOVA on ranks”); Stats iQ rank-transforms the data (replaces values with their rank ordering) and then runs the same ANOVA on that transformed data.

The ranked ANOVA is robust to outliers and non-normally distributed data. Rank transformation is a well-established method for protecting against assumption violation (a “nonparametric” method), and is most commonly seen in the difference between the Pearson and Spearman correlation. Rank transformation followed by Welch’s F test is similar in effect to the Kruskal-Wallis Test.

Note that Stats iQ’s ranked and unranked ANOVA effect sizes (Cohen’s f) are calculated using the F value from the F test for equal variances.

## Games-Howell Pairwise Test

Stats iQ runs Games-Howell tests regardless of the outcome of the ANOVA test (as per Zimmerman, 2010). Stats iQ shows unranked or ranked Games-Howell pairwise tests based on the same criteria as those used for ranked vs. unranked ANOVA, so if you see “Ranked ANOVA” in the advanced output, the pairwise tests will also be ranked.

The Games-Howell is essentially a t-test for unequal variances that accounts for the heightened likelihood of finding statistically significant results by chance when running many pairwise tests. Unlike the slightly more common Tukey’s b test, the Games-Howell test does not assume that the variances of the groups being compared are equal. Assuming equal variances leads to less accurate results when variances are not in fact equal, and its results are very similar when variances are actually equal (Howell, 2012).

Note that while the unranked pairwise test tests for the equality of the means of the two groups, the ranked pairwise test does not explicitly test for differences between the groups’ means or medians. Rather, it tests for a general tendency of one group to have larger values than the other.

Additionally, while Stats iQ does not show results of pairwise tests for any group with less than four values, those groups are included in calculating the degrees of freedom for the other pairwise tests.

With smaller sample sizes, data can still be visually inspected to determine if it is in fact normally distributed; if it is, unranked t-test results are still valid even for small samples. In practice, this assessment can be difficult to make, so Stats iQ recommends ranked t-tests by default for small samples.

With larger sample sizes, outliers are less likely to negatively affect results. Stats iQ uses Tukey’s “outside fence” to define outliers as points more than three times the intraquartile range above the 75th or below the 25th percentile point.

Data like “Highest level of education completed” or “Finishing order in marathon” are unambiguously ordinal. Though Likert scales (like a 1 to 7 scale where 1 is Very dissatisfied and 7 is Very satisfied) are technically ordinal, it is common practice in social sciences to treat them as though they are continuous (i.e., with an unranked t-test).