I recently came across this paper "http://host.uniroma3.it/progetti/capabitaly/db/docs/Mazziotta-Pareto_14_aprile_2014.pdf " , which is based on Mazziotta Pareto Index. My query is regarding the normalization technique. The formula used here is :
xij=100±((Xij - Mxj )*10)/Sxj
What is this 10 here ? Is there some generalized form for this normalization technique? Also, is there any upper bound in this index ? My purpose is to see how close my variables are to the "benchmark" of a perfect balance? Shall I take penalty to be 0 to construct this benchmark?
The formula converts individual indicators to a common scale with a mean of 100 and standard deviation of 10. So, the transformed values will fall approximately in the range (70; 130). The z-scores are adjusted to avoid negative values and obtain more visually manageable scores.
For more details, see: M. Mazziotta & A. Pareto (2016). On a Generalized Non-compensatory Composite Index for Measuring Socio-economic Phenomena, Soc. Indic. Res. 127: 983-1003. doi:10.1007/s11205-015-0998-2).
Thank you so much for your response.
If I need to compare my scores to a benchmark, say the most balanced scores , can I take that benchmark to be the average where penalty is 0 ?
Also , could you please share some source which talks about the mathematical construct of this index? Doee this paper elucidates the same ?
Thanks again !
1) Yes, you can.
2) For the mathematical construct of the index, you can see: http://link.springer.com/article/10.1007/s11205-015-0998-2
Since the index values will be unique for every item, can I take the benchmark to be 130 on each indicator (say for 3 indicators), which gives the average value of 130 (without a penalty) ?
Just a clarification, in the following formula of MPI
MPIi=Mxij - Sxij*CVi
If my data is as follows (before normalization):
Indicator A Indicator B Indicator C
Technology 1 -1 3 5
Technology 2 4 2 1
and so on
Then MPI for every technology (after normalization, between 70 & 130) will be average across three indicators A, B and C , minus [(standard deviation across A, B and C)* (coefficient of variation across A, B and C)] . Am I correct ?
Yes, you are.
However, I see that you have integer values.
Please note that methodological and statistical texts are clear that mean and standard deviation are inappropriate for ordinal data. So you can use MPI only for data expressed in an interval (or ratio) scale.
My data is rating data, -5 to 5. Actually its a survey data, and I have taken mean responses of the respondents. So there are decimal values as well, like 1.5 or -0.5. However with MPI, I am getting quite good results. Suggestions?
Ok, so you can use "Integer scoring", an equal-interval strategy of assigning rank-order numbers as category values, and then compute MPI.
For more details, see: Powers, D. A., Xie, Y. (2000). Statistical methods for categorical data analysis. San Diego: Academic Press (page 202).
I am not able to get free access to this paper. Is it there on researchgate?
Basically if I have an integer scoring, I can apply MPI. Correct?
I will try other methods as well, and see how the results differ.
Yes, it is on researchgate.
https://www.google.it/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&cad=rja&uact=8&ved=0ahUKEwiTlr_tot3SAhVERhQKHeOLBZQQFgg2MAI&url=https%3A%2F%2Fwww.researchgate.net%2Ffile.PostFileLoader.html%3Fid%3D5703e7e6dc332d4e1c2f7e20%26assetKey%3DAS%253A347549920710656%25401459873766230&usg=AFQjCNEGXqq4v1tp06rloApMPfCy7HBosg
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