For reading up on some basics, see ANOVA - What Is It?

- One-Way ANOVA - Null Hypothesis
- ANOVA Assumptions
- SPSS ANOVA Flowchart
- SPSS One-Way ANOVA Dialog
- SPSS ANOVA Output
- ANOVA - APA Reporting Guidelines

## ANOVA Example - Effect of Fertilizers on Plants

A farmer wants to know which fertilizer is best for his parsley plants. So he tries different fertilizers on different plants and weighs these plants after 6 weeks. The data -partly shown below- are in parsley.sav.

## Quick Data Check - Split Histograms

After opening our data in SPSS, let's first see what they basically look like. A quick way for doing so is inspecting a histogram of weights for each fertilizer separately. The screenshot below guides you through.

After following these steps, clicking syntax below. Let's run it.

results in the*** Run split histograms.**

GRAPH

/HISTOGRAM=grams

/PANEL COLVAR=fertilizer COLOP=CROSS.

## Result

Importantly, these distributions look plausible and we don't see any outliers: our **data seem correct** to begin with -not always the case with real-world data!

Conclusion: the vast majority of weights are between some 40 and 65 grams and they seem reasonably normally distributed.

## Inspecting Sample Sizes and Means

Precisely how did the fertilizers affect the plants? Let's compare some descriptive statistics for fertilizers separately. The quickest way is using MEANS which we could paste from but just typing the syntax may be just as easy.

***Basic descriptives table for grams by fertilizer.**

means grams by fertilizer

/cells count mean stddev.

## Result

- We have sample sizes of
**n = 30 for each fertilizer**. - Second, the chemical fertilizer resulted in the highest mean weight of almost 57 grams. “None” performed worst at some 51 grams while “Biological” is in between.
- “Biological” has a slightly higher standard deviation than the other conditions but the difference is pretty small.

Now, this table tells us a lot about our *samples* of plants. But what do our *sample* means say about the *population* means? Can we say anything about the effects of fertilizers on *all* (future) plants? We'll try to do so by refuting the statement that all fertilizers perform equally: our null hypothesis.

## One-Way ANOVA - Null Hypothesis

The null hypothesis for ANOVA is that
all population means are equal.
If this is true, then our sample means will probably differ a bit anyway. However, *very* different sample means contradict the hypothesis that the population means are equal. In this case, we may conclude that this null hypothesis probably wasn't true after all.

ANOVA will basically tells us to what extent our null hypothesis is credible. However, it requires some assumptions regarding our data.

## ANOVA Assumptions

**independent observations**: each record in the data must be a distinct and independent entity.Precisely, the assumption is “independent and identically distributed variables” but a thorough explanation is way beyond the scope of this tutorial.**normality**: the dependent variable is normally distributed in the population. Normality is not needed for reasonable sample sizes, say each n ≥ 25.**homogeneity**: the variance of the dependent variable must be equal in each subpopulation. Homogeneity is only needed for (sharply) unequal sample sizes. In this case, Levene's test can be used to see if homogeneity is met.

So how to check if we meet these assumptions? And what to do if we violate them? The simple flowchart below guides us through.

## SPSS ANOVA Flowchart

So what about our data?

- Our plants seem to be
**independent observations**: each has a different id value (first variable). - Our means table shows that
**each n ≥ 25**so we don't need to meet normality. - Since our sample sizes are equal, we don't need the homogeneity assumption either.

So why do we inspect our sample sizes based on a **means table**? Why didn't we just look at the frequency distribution for fertilizer? Well, our ANOVA uses only cases without missing values on our dependent variable. And our means table shows precisely those.

A second reason is that we need to **report** the means and standard deviations per group. And the means table gives us precisely the statistics we want in the order we want them.

## SPSS One-Way ANOVA Dialog

We'll now run a basic ANOVA from the menu. The screenshot below guides you through.

The syntax below.

button creates the## One-Way ANOVA Syntax

***Basic one-way ANOVA syntax.**

ONEWAY grams BY fertilizer

/MISSING ANALYSIS.

## SPSS One-Way ANOVA Output

A general rule of thumb is that we
reject the null hypothesis if “Sig.” or p < 0.05
which is the case here. So we reject the null hypothesis that all population means are equal.

**Conclusion**: different fertilizers perform differently. The differences between our mean weights -ranging from 51 to 57 grams- are statistically significant.

## ANOVA - APA Reporting Guidelines

First and foremost, we'll report our means table. Regarding the significance test, the APA suggests we report

- the
**F**value; **df1**, the numerator**d**egrees of**f**reedom;**df2**, the denominator degrees of freedom;- the
**p value**

like so: *“our three fertilizer conditions resulted in different mean weights for the parsley plants, F(2,87) = 3.7, p = .028.”*

## One-Way ANOVA - Next Steps

For this example, there's 2 more things we *could* take a look at:

**Post hoc tests**: our ANOVA results tell us that not all population means are equal. But precisely which mean differs from which other mean? This is answered by running post hoc tests. For an outstanding tutorial, consult SPSS - One Way ANOVA with Post Hoc Tests Example.**Effect size**: we concluded that fertilizers affect mean weights but how strong is this effect? A common effect size measure for ANOVA is partial eta squared. Sadly, effect size is absent from the One-Way dialog.

Oddly, MEANS*does*include eta-squared but lacks other essential options such as Levene’s test. For complete output, you need to run your ANOVA twice from 2 different commands. This really is a**major stupidity in SPSS**. There. I said it.

## ANOVA with Eta-Squared from MEANS

***ANOVA from MEANS includes eta-squared but not Levene's test or power estimates.**

means grams by fertilizer

/statistics anova.

## Result

Right, so that's about the most basic SPSS ANOVA tutorial I could come up with. I hope you found it helpful. Let me know what you think by throwing me a comment below.

**Thanks for reading!**

## This tutorial has 40 comments

## By jacquey on July 10th, 2015

simple explanations easily understood

## By Ruben Geert van den Berg on April 29th, 2015

We'd like to add some but we've a huge, huge list of things that we need to work on. So, honestly, it's not on our priorities list and it won't happen anytime soon unless we get many more similar requests.

## By Akbar on April 29th, 2015

I wonder if this useful tutorial has exercises with downloadable data that I can practice.

## By Ruben Geert van den Berg on September 25th, 2014

Thank you. Our main focus right now is in explaining the statistical theory underlying these tests. The first example of this is assumption of infinity and we soon hope to add something on Gaussianity and independent observations as well.

## By Craig Marsden on September 24th, 2014

Great tutorial, really good to see the mention of assumptions.