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Chapter 7 - Analysing the Data Part IV - Analysis of Variance Chapter 1 - Behavioural Science and research Chapter 2 - Research Design Chapter 3 - Collecting the Data Chapter 4 - Analysing the Data Part I - Descriptive Statistics Chapter 5 - Analysing the Data Part II - Inferential Statistics Chapter 6 - Analysing the Data Part III - Common Statistical Tests

 

Chapter 7: Analysing the Data
Part IV : Analysis of Variance

 

One-Way ANOVA
Matrix of ordered means

As a conventional way of displaying exactly which two means are significantly different and which are not, a 'Matrix of Ordered Means' sometimes needs to be constructed. The means are arranged in order from the smallest to the largest. These values with their group labels are put down the page and also across the page as shown.

 

Count

6.9

Rhyme

7.1

Adject

11.0

Intent

12.0

Imagery

13.4

Count 6.9

-

0.2

4.2*

5.1*

6.5*

Rhyme 7.1

 

-

3.9

4.9*

6.3*

Adject 11.0

   

-

1.0

2.4

Intent 12.0

     

-

1.4

Imagery 13.4

       

-

 

Notice, the means are put in an increasing order. Here Count and Rhyme are switched around from the original presentation and so are Imagery and Intent. The difference between the value on each row and column is then calculated and put into the corresponding cell. The value placed in each of the cells is the difference between those two means. We now compare our HSD value with each of these differences and determine which ones are greater than our "critical" HSD value. Traditionally we mark the significant ones (i.e., the ones greater than our critical HSD value) with an asterisk.

We can therefore conclude from this example, that the mean for the Count group is significantly different to the means for Adject, Intent, and Imagery groups. The mean for the Rhyme group is significantly different to the means for the Intent and Imagery groups. We can also conclude that the means for Rhyme and Count groups and the means for Rhyme and Adject are not significantly different from each other. Similarly, the means for the Adject group and the Intent and Imagery groups are not significantly different from each other and the Intent and Imagery groups are not significantly different.

Harmonic mean

Note that the above example assumed equal Ns in each of the four groups. If there are unequal numbers, which N you use in the formula becomes problematic. The recommendation is to use the harmonic mean of the sample sizes. (Note I tend to use big N and small n interchangeably!).

The harmonic mean is defined as:

where k = the number of means being compared, and the n1 to nk = the size of each of the k samples.

If four groups had sample sizes of 6, 8, 9, and 11, the harmonic mean would be

The normal arithmetic mean would be 34/4 = 8.5. Here there is not much difference (and there rarely is any appreciable difference). However, when the sample sizes are much more disparate, the difference between the harmonic N and the arithmetic N becomes greater.

Also note that the above description of how to calculate the harmonic N uses all the group sizes. When SPSS is used to calculate the harmonic N the default setting only considers the sample sizes of the two groups being compared. Because the error term we use is based on all the groups, it is a better idea to combine all the groups into one average than use a different value for each comparison. If the Ns are all the same or much the same there is very little problem.

 

 

 

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