Factors that Impact a Weighted Average the Most

Question

I am trying to determine how to calculate the factors that have the most significant impact on a weighted average. For example, let's say I am reviewing the number of patients that responded to a survey saying that they had a positive experience. The overall percentage of patients that had this response was 80%. When reviewing the data by doctor, it is broken down as follows:

• Doctor 1: 30 / 40 = 75%
• Doctor 2: 20 / 25 = 80%
• Doctor 3: 5 / 5 = 100%
• Doctor 4: 5 / 8 = 62.5%
• Doctor 5: 20 / 22 = 91%

How could I rank the 5 doctors in terms of having the most significant positive impact on the overall percentage? Since this is a weighted average, there are 2 factors at play: each doctor's percentage, and the size of their denominator ("weight"). For example, Doctor 3 has the highest percentage, but Doctor 5 probably had the most significant positive impact on the overall percentage because he has a weight that is over 4x larger than doctor 3. Is there a formula to calculate overall impact while taking both factors into account?

I found the following post on this site that has a (somewhat) similar problem: Contribution (weighted average) of change in rate over time

but I don't see a solution. And, this post is wanting to calculate the impact to a change in rate/percentage over time; I'm taking that complexity out and just want to know the impact on a rate/percentage at one point in time. As a side note, to solve this other post's problem, I think you could use a price index approximation technique, such as the Marshall Edgeworth price index. However, I don't think that is a solution to my problem.

Thanks for any help/insight you can provide!

Answer 1

The best way I have found is to find the percentage impact to the weighted average.

Find the difference from the individual rate to the total average rate, then multiply it by the individual denominator over the total denominator, then divide by the total average rate. They should sum to 0 and show the % above / below the weighted average relative to individual weight.

It would look like this for you example:

• Doctor 1: (75% - 80%)*(40/100)/80% = -2.5%
• Doctor 2: (80% - 80%)*(25/100)/80% = 0.0%
• Doctor 3: (100% - 80%)*(5/100)/80% = 1.1%
• Doctor 4: (63% - 80%)*(8/100)/80% = -1.8%
• Doctor 5: (91% - 80%)*(22/100)/80% = 3.0%

You could then conclude that the order of impact (from highest to lowest) is Doctor 5, Doctor 3, Doctor 2 (no net impact), Doctor 4 (negative impact), and Doctor 1 (most negative impact).