# Comparison tests for the One-Way ANOVA

## Contents

### Overview

The One Way ANOVA is a statistical procedure used to determine whether the mean of three or more samples are statistically different. In case the results of the analysis shows any evidence that the means of the population differ, a comparison test can be performed to determine which means among the ones analysed differ from the rest.

Different comparison methods exist and they differ for the distribution function the method are based on and for the way the overall significance level is calculated.

Some of the most commonly used comparison procedures are:

• Tukey Method
• Fisher Method
• Games-Howell Method

In case the different samples can be assumed to have equal variance, one of the first two methods can be used; the third one should be used instead in case the assumption of equal variance cannot be made.

Any Hypothesis test in statistics requires the definition of a null hypothesis and an alternate hypothesis. The goal of the test is to aid the analyst in deciding which one of the two hypothesis is true. The table below shows the null and the alternate hypothesis adopted for each couple of samples included in a One-Way ANOVA Comparison procedure:

Null hypothesis Alternate hypothesis
The two means are equal The two means are not equal

If the p-value obtained at the end of the test is less than the adopted significance level, the Null Hypothesis should be rejected in favour of the Alternate Hypothesis.

More details on the Comparison tests for the One Way ANOVA can be found here:

### Practical Example

Suppose a university Professor wants to compare the test scores of the end-semester exams, for the following groups of students attending his lectures:

• Students attending the Monday morning lecture
• Students attending the Tuesday afternoon lecture
• Students attending the online lecture

A One Way ANOVA test performed on the three samples produced a p-Value of 0.014. Being the p-value less than the adopted significance level of 0.05, the Professor should reject the null hypothesis and conclude that at least one student group performs different from the others.

The professor decides to perform a comparison test to determine how the three samples compare to each other. He chooses to use the Tukey method with a family error rate of 5%. The test produces following results:

Mean Difference p-Value
Morning – Afternoon 0.574
Morning – Online 0.002
Afternoon - Online 0.036

The test results show that there is sufficient evidence to indicate that the students attending the online lecture performs different from the one attending the morning and afternoon lectures.

### Within Grapheme

To perform a Comparison test for the One-Way ANOVA in Grapheme, click on Create New Analysis from the toolbar of the Statistical Analysis View. Assign a name to the panel and select “One-Way ANOVA Comparisons” from the list. Click on Next.

In the "Sources" tab add a column for each sample you want to include in the analysis and select, for each of them, the source table, the view and the view column.

Move to the "Configuration" tab and define the following analysis settings:

• select the method to be used for the analysis among the ones available
• select Save data to display Individual and Residual Plots if you want to have available, within the chart area, the individual plot for each sample and the related residual plots
• if you want Grapheme to evaluate the confidence interval range, check the setting Compute Confidence Interval and insert the family error rate to be used for its evaluation. Click on Finish.

Grapheme: One-Way ANOVA Comparisons Panel
##### Remarks
• All the data available in the panel are updated on the fly, so that any change in the table values, is immediately reflected by the panel tables and charts. Automatic update can be temporary suspended, by clicking on the lock button in the main panel toolbar.
• All the data contained in each table, can be copied to the clipboard for further reporting by clicking on the button available in the toolbar.
• Each chart can be exported as Image by clicking on the “Save as Image” button available in the toolbar