# Weighting variables for an index

I have been tasked with trying to modify our current "index" which basically takes 4 observations per person and calculates a score based on what they achieve. Here is how the score is created (all variables are ordinal in nature, higher scores representing more desirable outcomes):

• Variable A - gets a score of 1 to 3
• Variable B - gets a score of 1 to 3
• Variable C - gets a score of 1 to 5
• Variable D - gets a score of 1 to 8

Basically what I am looking to do is revise our index but have little experience doing so using a statistical framework. What techniques exist? My "problem" is that I will most likely be required to keep the same max score (19, maybe 20) and use the same 4 variables. Lastly, I will need to be able to explain the score in a way that our MIS department can code the index back into our database.

Any help or insight you can provide to get me started will be very much appreciated.

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Apologies if this is too basic or I have missed the point. Perhaps it would be helpful to post the current method, and why it is regarded as needing revision.

I am guessing that currently the method is simply to add the four scores together. If so, Variable D will probably contribute a lot more of the variance in the final index than the other variables. Also, worth noting is that the index will range from 4 to 19 (so not really a score out of 19 as normally interpreted, as the minimum is not zero)

An important first question is what weight you want each variable to contribute to the final index. If the answer is that each variable should make an equal contribution, the best way to do this in a way that you have a chance of explaining is to scale each variable so they are on the same scale eg from zero to five. This would mean:

$A_{new} =(A-1)*\frac{5}{2}$

$B_{new} =(B-1)*\frac{5}{2}$

$C_{new} =(C-1)*\frac{5}{4}$

$D_{new} =(D-1)*\frac{5}{7}$

Then adding the four scores together. This gives you an index that ranges from zero to 20, in which each variable has been scaled so it potentially makes an equal contribution (only "potentially" because if in fact a variable has very low variance - maybe everyone gets 8/8 for D - it makes no contribution to the variance in the final index).

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Thanks for your response. Yes, everything above is correct. It is a simple sum, with scores ranging from 4-19. However, I do not necessarily need to give each score equal weight. Asked differently, is there a way to find the optimal weight for each variable given the possible points available across each of the 4 options? Bascially, the first index was created, pretty much, at random without regard to weight and the impact on the final score. If needed (which I believe it is) I have an outcome variable that can help define/optimize the weights for the 4 variables. Thanks in advance. –  Btibert3 May 9 '12 at 12:45
The method I have put above (first scaling each variable to a 0-5 scale) will be an improvement I think on your original approach. The next step in sophistication would be to fit a regression with your outcome as the response to determine the weights. Then future values of the index will be the predicted value of the outcome, given the four variables you have (I presume you won't have the outcome measure in the future, or you might as well just use that). If you go the regression route you don't need to rescale the variables first; but you will need a stats package. –  Peter Ellis May 9 '12 at 22:08