# Is there a formula for calculating the zero-centered diffs?

Yesterday a client ask me about the meaning of zero-centered diffs. I was able to explain what they mean, but the client wanted to know exactly how these utilities are calculated. I told him, I will give him the formula in the upcoming days. I tried to calculate the ZC-Diffs from the part-worths of a HB/CBC estimation of a recent study, but was not able to get the right values for the ZC-Diffs. I already read the article from Sawtooth Solutions, 1999, Issue 10 and the explanation in die SSI Web Manual.

Can anybody help me with this issue, please? I would really appreciate it!

Frank

+1 vote
For each respondent...

1. Within each attribute, compute the mean utility.  Within each attribute, subtract the mean utility from each utility (this zero-centers the utilities within each attribute...which often doesn't have to be done since they are often already zero-centered in their raw form).

2. Then, for each attribute compute the difference between best and worst utilities.  Sum those across attributes.

3.  Take 100 x #attributes and divide it by the sum achieved in step 2.  This is a single multiplier that you use in step 4.

4.  Multiply all utilities from step 1 by the multiplier.  Now, the average difference between best and worst utilities per attribute is 100 utility points.
answered Jun 4, 2014 by Platinum (175,715 points)
Traditionally, one would use a utility and calculate the score as 100*(exp(utility)/(1+exp(utility)). Does Sawtooth/Lighthouse Studio provide that? All these are closely correlated, but is there a way to validate which one is the best approach: 100*(exp(utility)/(1+exp(utility)),  Zero-Anchored Interval Scale, Probability Scale, or raw utilities?
Is the number of attributes to calculate the multiplier also includes utility for the none option?

Thank you
The None utility is ignored when calculating the scaling multiplier for each respondent.  But, we do multiply the None utility by the scaling multiplier that was developed by referencing the other attributes.
Hi Bryan,

Could you provide some clarity around why we want the average difference between best and worst utilities per attribute to be 100 utility points? In other words, why do we want to multiply by the multiplier? I'm having trouble understanding why that is an important step and how the 100 point difference comes in as being important.

Thank you!
With MNL estimation, the magnitude of the utilities is inversely related to the response error.  So, in HB estimation, if one respondent answers with half the response error as another respondent, that respondent's utilities will all grow in magnitude by a factor of 2x compared to the second respondent.  For this reason, when we report average utilities, we are worried that different respondents will have different weight in the sample mean calculation depending on their response error.  So, to try to put respondents more-or-less on the same scale for reporting of average utilities, we have decided to choose a common scale and find the multiplier for each respondent that puts them on the same scale so each respondent gets essentially equal weight (pull) on the sample means.  We could have chosen 1, 100, 143, or whatever target value we wanted.  We chose 100 points to be the common "average" scaling of the utilities and find the multiplier for each respondent that makes the average difference between best and worst utilities for that respondent equal to 100.

When conducting market simulations, we use the raw utilities as they came out of the HB algorithm.  We don't use the normalized to 100-pt "zero-centered diffs" for computing shares of preference in market simulations.