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SPSS One-Way ANOVA with Post Hoc Tests Tutorial

Post Hoc ANOVA - Data View

A hospital wants to know how a homeopathic medicine for depression performs in comparison to alternatives. They adminstered 4 treatments to 100 patients for 2 weeks and then measured their depression levels. The data, part of which are shown above, are in depression.sav.

Data Inspection - Split Histogram

Before running any statistical tests, let's first just take a look at our data. In this case, a split histogram basically tells the whole story in a single chart. We don't see many SPSS users run such charts but you'll see in a minute how incredibly useful it is. The screenshots below show how to create it.

Post Hoc ANOVA - Histogram Menu 1

In step below, you can add a nice title to your chart. We settled for “Distribution BDI per Medicine”.

Post Hoc ANOVA - Histogram Menu 2

Syntax for Split Histogram

Clicking Paste results in the syntax below. Running it generates our chart.

*Run histogram of BDI scores for the four treatments separately.

GRAPH
/HISTOGRAM=bdi
/PANEL ROWVAR=medicine ROWOP=CROSS
/TITLE='Distribution BDI per Medicine'.

Result

Post Hoc ANOVA - Split Histogram

Means Table

We'll now take a more precise look at our data by running a means table with the syntax below.

*Run basic means table.

means bdi by medicine/cells count min max mean variance.

Result

Post Hoc ANOVA - Data View

Unsurprisingly, our table mostly confirms what we already saw in our histogram. Note (under “N”) that each medicine has 25 observations so these two variables don't contain any missing values.
So can we conclude that “Pharmaceutical” performs best and “None” performs worst? Well, for our sample we can. For our population (all people suffering from depression) we can't.
The basic problem is that samples differ from the populations from which they are drawn. If our four medicines perform equally well in our population, then we may still see some differences between our sample means. However, large sample differences are unlikely if all medicines perform equally in our population. This basic reasoning is explained further in ANOVA - What Is It?.
The question we'll now answer is: are the sample means different enough to reject the hypothesis that the mean BDI scores in our population are equal?

ANOVA Basics

We'll try to demonstrate that some medicines perform better than others by rejecting the null hypothesis that the mean BDI scores for our four medicines are all equal in our population. In short, our ANOVA tests whether all 4 means are equal. If they aren't then we'd like to know exactly which means are unequal with post hoc (Latin for “after that”) tests.
Our ANOVA will run fine in SPSS but in order to have confidence in its results, we need to satisfy some assumptions.

ANOVA - Main Assumptions

Running our ANOVA in SPSS

Post Hoc ANOVA - GLM Menu

There's many ways to run the exact same ANOVA in SPSS. Today, we'll go for General Linear Model because it'll provide us with partial eta squared as an estimate for the effect size of our model.

Post Hoc ANOVA - Univariate Menu

We'll briefly jump into Post Hoc and Options before pasting our syntax.

Post Hoc ANOVA - Post Hoc Menu

The post hoc test we'll run is Tukey’s HSD (Honestly Significant Difference), denoted as “Tukey”. We'll explain how it works when we'll discuss the output.

Post Hoc ANOVA - Options Menu

“Estimates of effect size” refers to partial eta squared. “Homogeneity tests” includes Levene’s test for equal variances in our output.

Post Hoc ANOVA Syntax

Following the previous screenshots results in the syntax below. We'll run it and explain the output.

*ANOVA syntax with Post Hoc (Tukey) test, Homoscedasticity (Levene's test) and effect size (partial eta squared).

UNIANOVA bdi BY medicine
/METHOD=SSTYPE(3)
/INTERCEPT=INCLUDE
/POSTHOC=medicine(TUKEY)
/PRINT=ETASQ HOMOGENEITY
/CRITERIA=ALPHA(.05)
/DESIGN=medicine.

SPSS ANOVA Output - Levene’s Test

Post Hoc ANOVA - Data View

Levene’s Test checks whether the population variances of BDI for the four medicine groups are all equal, which is a requirement for ANOVA. “Sig.” = 0.949 so there's a 94.9% probability of finding the slightly different variances that we see in our sample. This sample outcome is very likely under the null hypothesis of homoscedasticity; we satisfy this assumption for our ANOVA.

SPSS ANOVA Output - Between Subjects Effects

Post Hoc ANOVA - Data View

If our population means are really equal, there's a 0% chance of finding the sample differences we observed. We reject the null hypothesis of equal population means.
The different medicines administered account for some 39% of the variance in the BDI scores. This is the effect size as indicated by partial eta squared.
Partial Eta Squared is the Sums of Squares for medicine divided by the corrected total sums of squares (2780 / 7071 = 0.39).
Sums of Squares Error represents the variance in BDI scores not accounted for by medicine. Note that + = .

SPSS ANOVA Output - Multiple Comparisons

So far, we only concluded that all four means being equal is very unlikely. So exactly which mean differs from which mean? Well, the histograms and means tables we ran before our ANOVA point us in the right direction but we'll try and back that up with a more formal test: Tukey’s HSD as shown in the multiple comparisons table.

Post Hoc ANOVA - Data View

Right, now comparing 4 means results in (4 - 1) x 4 x 0.5 = 6 distinct comparisons, each of which is listed twice in this table. There's three ways for telling which means are likely to be different:
Statistically significant mean differences are flagged with an asterisk (*). For instance, the very first line tells us that “None” has a mean BDI score of 6.7 points higher than the placebo -which is quite a lot actually since BDI scores can range from 0 through 63.
As a rule of thumb, “Sig.” < 0.05 indicates a statistically significant difference between two means.
A confidence interval not including zero means that a zero difference between these means in the population is unlikely.
Obviously, , and result in the same conclusions.

So that's it for now. I hope this tutorial helps you to run ANOVA with post hoc tests confidently. If you have any suggestions, please let me know by leaving a comment below.

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