Guide: How to Run a One-way ANOVA (Parametric Test)

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 also probably tested your data to for normality and homogeneity of variances. If you haven’t done these steps, you can go check out my guides on how to run descriptive statistics on Julius and running test to determine if your data is suitable for parametric or non-parametric testing!

This guide is going to go over how to run a one-way ANOVA! For simplicity sake, we can assume that we have ran a normalcy test and descriptive statistics on this dataset (always do this!). I just want to show you how to set up for this test in Julius.

Running a One-way ANOVA

Situation: A researcher wants to see 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

To make this less lengthy, I only put a small snippet of the dataset.

Question 1a: is there a difference in plant height between the three different 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?

Make sure you in your prompt that you explain the groups and their associated numbers

Awesome, so we can see that Julius gave us an F-value: 6.46 and a p-value: 0.003. We should also get the proper degrees of freedom. I can prompt Julius for this:

Prompt 1b: What are the degrees of freedom for this dataset?

Okay, now we can properly report our significant results as: F(2,42) = 6.46, p ≤ 0.003. These results only tell us there is a statistically significant difference, but it does not tell us exactly where this difference is.

So, what next, you ask? Our next step here is to perform what is called a post-hoc test. These tests are essential when you want 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 together.

Julius has provided us with Tukey’s HSD, and as a bonus, has told us why this test was appropriate for our dataset. There are multiple post-hoc tests to choose from, so knowing which one to perform on your dataset is crucial. Good thing Julius gave us some insight into why we use this one!

Looking at 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 via a boxplot!

After prompting Julius to provide a visual representation of the differences in plant height, we can now see that all three groups do differ in plant height. But there is a noticeable (significant) difference between Groups 1 and 3, as both our One-way ANOVA and post-hoc test has confirmed. Keep in mind, you do not have to use a boxplot to show these visual differences, other figures can be used as well. Also, it is best practice to show the 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.

Keywords: AI statistics, AI statistical analysis, GPT, one-way ANOVA, ANOVA, normality test, parametric tests, descriptive statistics, post hoc, Tukey’s HSD

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Great guide! Really useful. I’m interested — how have you been taking the output from Julius and preparing written reports, or otherwise disseminating the results? What does that process look like for you?

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Hi Zach!

Great question, thank you for asking! Typically when you are writing a results section for an article, you want to 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).

Note: the way that I am portraying my setup is how I (an ecologist/biologist) was told to format it when writing publications. This may be different for other disciplines, so please double check how you are required to format it based on what your organization/advisor/mentor/journal/etc. tells you to.

So you can see that I provide a table with the overall results of the one-way ANOVA and then a figure that shows the statistically significant differences between Group 1 and 3. Your tables should always come before your figures, and remember captions on top of tables and below figures. For a results section you do not dive into why you may see this difference, you just display your results. The discussion section is where you go into the many reasons why you may see certain trends or findings.

If you are wondering where I got these numbers (25.00±3.05 and 29.00±3.05), this was from my descriptive statistics that I ran on this data (not shown here). As for Table 1, you can ask Julius to give you a nice little table giving you all of these numbers.

Hope this helps!

Excellent work! These bring back fond memories of my technical writing days. Additionally, for enhanced clarity, consider including units of variables within parentheses and incorporating legends to improve comprehension

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Thank you Philip!

Haha, I just finished defending my thesis last December, so a lot of my technical writing is still ingrained in my brain.

You are absolutely correct about adding a legend and also including your units of measurement. This was a quick example so I did not include them, but for technical writing purposes, such as a publication or theses, those suggestions are also crucial for better comprehension for your readers.

Thanks for the additional tip! :slight_smile: