If you’re here, chances are that you’ve probably used Julius to help you run descriptive statistics on your data to explore the relationships. You’ve likely tested your data for normality and homogeneity of variances. If you haven’t completed these steps, I recommend checking out my guides on running descriptive statistics with Julius and testing whether your data is suitable for parametric or non-parametric testing.
This guide will cover how to run the one-way ANOVA. For simplicity, let’s assume we have already conducted normality tests and descriptive statistics on this dataset (always do this!). I’ll focus on showing you how to set up and run the test in Julius.
Running a One-way ANOVA
Situation: A researcher wants to determine whether there is any difference in plant growth under three different fertilizers. They set up an experiment where they randomly assign 15 plants (Crimson clover; Trifolium incarnatum) to one of three groups: Group A receives Fertilizer 1, Group B receives Fertilizer 2, and Group C receives Fertilizer 3. After a month of treatment, plant height measurements are taken.
| fertilizer | plant_height |
|---|---|
| 1 | 20 |
| 2 | 22 |
| 3 | 24 |
| 1 | 26 |
| 2 | 28 |
| 3 | 21 |
| 1 | 23 |
| 2 | 25 |
Note: This is a small snippet of the dataset.
Question 1a: Is there a difference in plant height between the three fertilizer groups?
Prompt 1a: Can you see if there is a difference in average plant height between the three types of fertilizer groups (labelled 1, 2 and 3) by running a one-way ANOVA?
Be sure to specify the groups and their associated numbers in your prompt.
Great! We see that Julius gave us an F-value: 6.46 and a p-value: 0.003. We also need the degrees of freedom to properly report the results. Let’s prompt Julius for this:
Prompt 1b: What are the degrees of freedom for this dataset?
Now we can properly report our significant results as F(2,42) = 6.46, p ≤ 0.003. These results only indicate a statistically significant difference, but they do not specify where the difference lies.
So, what next? Our next step is to perform a post-hoc test. These tests are essential when you need to determine which groups differ from one another after obtaining a significant result. Let’s ask Julius to perform one!
Question 1c: Which groups are statistically significant from one another?
Prompt 1c: What post-hoc test can be used to determine which specific fertilizer groups differ in plant height?
Family wise error rate (FWER): A measure of the probability of making one or more false discoveries (Type I errors; false positive test results), when conducting multiple statistical tests.
Julius has provided Tukey’s HSD as an appropriate post-hoc test and explained why it is suitable for our dataset. There are multiple post-hoc tests to choose from, so knowing which one to perform on your dataset is crucial. Fortunately, Julius gave us some insight into why we use this one!
Reviewing the test results, we can see that Group 1 and 3 are significantly different from one another, with a p-value of 0.0024. Now let’s visualize these differences using a boxplot!
After prompting Julius to provide a visual representation of the differences in plant height, we can see that all three groups do differ in plant height. However, there is a noticeable (significant) difference between Groups 1 and 3, as confirmed by both our One-way ANOVA and post-hoc test.
You do not have to use a boxplot to show these visual differences – other types of figures can also be effective as well. It is best practice to highlight statistically significant differences between the groups, so providing a visualization that has an asterisks or letters to denote significance would be necessary after running these statistics.
Writing the Results
When you are write a results section for an article, you should always bring in your statistically significant results, post hoc results, and sometimes, your descriptive statistics. I’ll write an example using these results I have gotten from running this one-way ANOVA.
"A one-way ANOVA showed statistically significant differences between fertilizer treatment and plant height (F(42,2) = 60.000, p<0.05) (Table 1). Further post hoc tests revealed statistically significant differences between in plant height between treatment 1 (25.00±3.05) and treatment 3 (29.00±3.05)(p=0.0024), but no statistically significant difference was found between any other comparisons (Figure 1).
Table 1. A one-way ANOVA analysis for the effect of fertilizer on height of plants. MS (Mean Square), F-value and significant value are displayed. Asterisks denote significant effects (*p<0.05, **p<0.01).
| Measurement | MS | F | Sig |
|---|---|---|---|
| Between Groups | 60.000 | 6.462 | 0.004** |
| Within Groups | 9.286 |
Figure 1. Boxplot depicting the effect of three different fertilizer treatments on plant height. Asterisks denote statistically significant differences between groups (here there would be a double asterisk above boxplot 1 and 3).
This is just a nice formatted example of how you could create a small results section for these findings.
Keywords: AI statistics, AI statistical analysis, GPT, one-way ANOVA, ANOVA, normality test, parametric tests, descriptive statistics, post hoc, Tukey’s HSD
References & Further Reading
- Kim T. K. (2017). Understanding one-way ANOVA using conceptual figures. Korean journal of anesthesiology , 70 (1), 22–26. Understanding one-way ANOVA using conceptual figures
- Montgomery, D. C. (2019). Design and Analysis of Experiments (10th ed.). Wiley.