Hi all,
It is easy to show that the reliability of the MEAN score is the same as that of SUM score but I have not found any article/source of that. If you know such an article/source, please, send a note.
When forming the score of, for example, an attitude scale, we can form it by using either the SUM or MEAN operation. The alpha coefficient of reliability of the scale/score (KR20 or "Cronbach's alpha") uses the variance of the SUMMED scale/score in the formula/calculations and, hence, we cannot use the basic formula when estimating the reliability of a MEAN type of "sum". This is because the variance of the SUM is greater than the variance of MEAN and, if using the variance from MEAN type of "sum", we get faulty (out of range) result by using the classical formula. It's easy to show the reliability estimate is the same in both cases but I have not found a reference for that. Too obvious?
Unless I missed something, it is just too obvious. Mean value is simply the sum value divided by the number of items - for each individual. When computing the reliability, the divider simply cancels out.
That's the point. but this knowledge cannot be utilized if using the classical coefficient which is using strictly the variance of the SUM in the calculation (and not the variance of the MEAN). It is easy to show that if using the Mean typé of sum, we should ament the original variance of score By k x VAR(mean) but I have never seen that kind of formula. Another version of Alpha should/could be used when using the mean operation. Which brings the thinking to the use of othe formats of alpha - such as Lord & Novick formula (which uses the item total correlation instead of the score variance) or the form used is SPSS (which uses the covarainces and correlations).
My original point was that I could have been willing to say in my Methodology book that "the reliability of the MEAN type of sum is the same as the one made By SUM operation." and have a clear reference there. But I have not found.
While doing some exploration of this I found out that the way SPSS estimates the reliability is very unstable and it actually does not tell the reliability of the score at all if there are missing values in the dataset. It is not because of the formulae but because of the LISTwise deletion procedure which gives us a total different "SUM" than what is acquired By using the MEAN operation. In SUM and MEAN operations the sum is formed for EACH case but the alpha reliability in SPSS is calculated for a score where all those cases are omitted if there is even one missing value. So, these cases are included in the sum with MEAN operation but the SPSS alpha does not recognize these cases. If using the procedure based on correlation with CASEwise deletion (as is default if calculating the correlation in SPSS), the same formula would produce a proper reliability (for the formed SUM/MEAN). The current procedure does not support this. The SPSS aplha is then, more or less, grap if we use mean operation and there are missing values in te dataset. I wrote something of this. Willing to see?
I would add another risk of error (or inaccuracy) when comparing the results of different reliability tests.
Depending on how we test item reliability / test plan / - if the reliability information is collected while all items are be damaged - the result will be one.
If the reliability information is collected until then, when a certain number of items are be damaged, the result will be different.
If the reliability information is collected for a certain duration of work - the result will be third. Etc.
This makes it difficult to compare the reliability data and makes the results debatable
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