One-Way ANOVA

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Short for One-Way Analysis of Variance, a One-Way ANOVA is used to look at the differences between two or more groups (or levels) of  one independent variable. The independent variable is ideally nominalbut can be ordinal. The dependent variable is ideally ratiobut can be interval.


 * For example, if your independent variable is mood, the groups (or levels) could be mad, sad, ok, and happy. A possible dependent variable could be the score on a math test. Then you can see if there are differences in test scores between each level of mood.

This test works differently from a t test because it does not compare the means, it compares the variances between the groups (or levels). To do this, the variance in the data set is partitioned into two groups: Within and Between sample variance.

Why not just use a t test? Well if the independent variable only has two levels, we can use a t test! BUT when the independent variable has three or more levels, we have to use ANOVA. Using multiple t tests would increase our probability of making a Type I Error.

Independent and Repeated Measures
Since we are comparing samples with ANOVAs, we need to know who we're comparing.

Independent Measures
There are different participants in each group (or level).

To determine the presence of differences, the observed variability is compared to the expected variability (with individual differences). So you compare what you find in your samples to what would be expected by using similar but different samples (with no significant differences).


 * For example, say your independent variable is "Type of Animal" with the levels "Mammal", "Bird", and "Reptile". Then a dependent variable could be a rating of "Friendliness" on a scale of 1 (not friendly) to 10 (very friendly).


 * Since there are different animals in each level of the independent variable, you have to account for those differences. This makes the expected variability bigger which means your observed variability needs to be even bigger!

Independent Measures ANOVA need bigger differences than Repeated Measures ANVOA to be significant.

Repeated Measures
The same participants are in every group (or level).

To determine the presence of differences, the observed variability is compared to the expected variability (without individual differences). So you compare the measured performance of your sample in all of the groups (levels) to a situation where there are no differences.


 * For example, say your independent variable is "Study Setting" with levels "Underwater", "Snow", and "Indoors". Then a dependent variable could be a test score on a scale of 0 (no correct answers) to 100 (all correct answers).


 * Since you can have the same people try studying in each condition for each test, you don't have to account for individual differences. This makes your variability smaller.

Repeated Measures ANOVAs need smaller differences to find significance because they do not have to account for individual differences.

Significance
Testing for significance in an ANOVA only reveals that at least one group (or level) is significantly different from the others. Post Hoc tests reveal the location of the difference(s). So ANOVA only tells you if there is a difference.

To determine the significance of an ANOVA, we compare two types of variance, Within Sample Variance (variance due to differences within groups) and Between Sample Variance (variance due to the interaction between groups).

We want the Between Sample Variance to be bigger than the Within Sample Variance. This would mean our groups are different from one another  and  we are likely to find the results significant. So the higher the Between Sample Variance, the more different the groups.

We do not want the Between Sample Variance to be equal to or lower than the Within Sample Variance. This would mean there are no significant differences between the groups.

So to find significant results, we want larger differences between the groups and smaller differences within the groups.

If we only have two groups (or levels), then we know the difference is between those two. However, if we have three or more groups (or levels) then we only know that there is a difference but we don't know where. This is when we need to use a Post Hoc test.

Post Hoc Tests
To determine where the difference is between the groups (or levels), we need to use a Post Hoc test.


 * For example, say our levels were mad, sad, ok, and happy. If we have a significant ANOVA then we know there is a difference somewhere. It could be between mad and happy. It could be between mad, ok, and happy. It could be between all of the moods. So it could be any combination of differences!

Basically, an ANOVA determines if there is a difference. Then the Post Hoc test shows the location of the difference. Other than the Pairwise Comparisons, all Post Hoc tests can find the differences without inflating your alpha value (increasing your chance of having a Type I Error).

Pairwise Comparisons
This technique compares all of the means in an ANOVA, two at a time. In each comparison, we are looking for a significant difference between the means. It's like using multiple t tests (this is bad).

Each comparison increases the risk of having a Type I Error because it inflates your alpha value.

Tukey's Honestly Significant Difference (HSD)
This method uses one score and compares it to each of the groups (or levels). If the group (or level) exceeds the HSD score, there is a significant difference between those groups (levels).

This is a very common Post Hoc test.

Scheffé Test
This method uses one score to represent each comparison of the groups (or levels). If the groups are the same size, the same score can be used to compare all of the groups.

Only use this test if you expect there to be differences between clusters of groups (or levels).


 * For example, say we're comparing age groups (young, teen, middle-aged, and older) on how many hours they spend in school per week. We would expect young and teen age groups to be similar, middle-aged and older age groups to be similar, and the two younger groups to be different from the two older groups. Here we can use this test.

This test is not as common as Tukey's HSD but it is more conservative than Tukey's HSD. That means it is harder to find significant differences but if you do find a significant difference, it is more meaningful than if you found it with Tukey's HSD.

Fisher's Least Significant Difference (LSD)
This is the best test to use for three groups (or levels).

Student Newman-Kewls (SNK)
This method sorts the groups (or levels) into subsets by variance.

Ryan Procedure (REGW)
Requires equal sample sizes.

One-Way ANOVA in SPSS
The procedure for conducting an SPSS analysis depends on the type of One-Way ANOVA, independent or repeated.

Independent Measures in SPSS

 * 1) Click on 'Analyze' -> 'Compare Means' -> 'One-Way ANOVA'


 * 1) Move the DV into the box 'Dependent List'


 * 1) Move the IV into the box 'Factor'


 * 1) Click 'Post-Hoc' -> and click on the post hoc test you want to use


 * 1) Click 'Options' and select 'Descriptives'


 * 1) Click 'OK'


 * 1) Your output should appear

Repeated Measures in SPSS

 * 1) Click on 'Analyze' -> 'General Linear Model' -> 'Repeated Measures'


 * 1) Write the name of your IV (factor) inside the box labeled 'Within Subjects Factor Name'


 * 1) Write the number of levels of your factor in the box labeled 'Number of Levels'


 * 1) Click 'Add'


 * 1) Next you need to define the levels of your factor by clicking 'Define'


 * 1) Select your first level in the column on the left and click it over to the column on the right


 * 1) Repeat step 6 for each level of your factor


 * 1) Click on 'Options'


 * 1) Move your factor into the right hand box labeled 'Display Means For'


 * 1) Check the box for 'Compare Main Effects'


 * 1) Check the box for 'Descriptive Statistics'


 * 1) Click 'Continue'


 * 1) Click 'OK'


 * 1) Your output should appear