What is Sphericity: Understanding Sphericity through an Example

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English In this lecture we will try to understand what is Sphericity? Sphericity is an important assumption in repeated measure ANOVA And let us try to understand what it is. Sphericity simply means a condition in which we presume that the differences between the variability of the scores of different groups are the same. So the idea of sphericity is very simple. If you got the different groups in case of repeated measure ANOVA and that you get when you take multiple observations on similar group over a period of time and find out the standard deviation for each group. Square those standard deviations and that will give you variance. And if these variances are very different then we say that the assumption of sphericity has been violated. for example, look at this data. This data is about the scores of subject who belong to different levels of noise conditions. So these scores are basically the performance scores of the subjects which have been measured in three different levels of noise conditions. So for example subject number one he got a score of 55 when the noise was low and then he got the score of 68 when the noise was medium and then his score was 44 when the noise was high. And similarly for subject 2 and 3 and so on. So by Sphericity we means we are going to look at the differences between these scores and once we have got these differences we are going to calculate the standard deviation of those differences and square those standard deviations to find out the variances of the groups and if these variances are very different from each other, then we say that the assumption of sphericity has been violated. So what I have done here for these 3 groups means for these 3 levels of noise for the same subjects, the difference has been calculated. So low-medium suggest the differences between low and medium scores. So first turns out to be minus 13 Then difference between medium and high score for first subject is 24 and then it is 11. So once you calculate all the differences then calculate the standard deviation for these differences that you can simply do by MS excel or I can even demonstrate you even right now. For example if you copy and paste this data in Excel, and write Use this function and if you are using Excel 2007 you can use this function STDEVA and then write your cell ranges from A2 till A6. so its 12.461 and I have written the exactly same 12.461 so 12.461is the standard deviation of the difference in scores between low and medium noise level. Similarly we have got the standard deviation between medium and high noise level as 14.94 and for low and high. Noise condition. The standard deviation of difference in the score is 5.02 And when you square it, you get these numbers and these numbers are fairly distinct You can see 25 is much lesser variance as compared to 223 and 155. So what we see that here in case of this small data, the assumption of sphericity has been violated and whenever the assumption of Sphericity is violated, there is a high chance of committing the type 1 error, that is detecting the significance when it is not or rejecting the null hypothesis when in fact it is true. So to avoid this problem, we must always check for the assumption of Sphericity while doing the repeated measure ANOVA.