# Rates affecting rates

Stats noob here. I've been trying to find out if there is a technique to perform a certain type of analysis but so far a lot of searching hasn't yielded anything.

My question is, if a large peak or trough is identified in an organisation's overall rate, is there a way of identifying which of the organisation's production sites has contributed the most to this rate change?

The data describes an organisation and it's performance against a rate based performance measure (let's say % wastage on a washer production line). The dataset contains 4 fields. These fields are:

1. Site (There are 100+ of these)
2. Month
3. Numerator (Number of washers lost to wastage)
4. Denominator (Total number of washers produced)

The organisation monitors it's performance based on its overall rate (sum of all numerators / sum of all denominators) by month. This gives a trend line. If there is a sudden large increase/decrease in rate then the organisation wants to target the sites which have most contributed to this rate change.

I'm not certain if there is an answer to this question, as one site could see a decrease in denominator and another could see an increase in its numerator value, meaning these sites both show a small change, whilst the organisation as a whole may see a larger scale change when the figures are combined and the overall rate is calculated.

I was originally thinking about something like a logistic regression with weighting but wasn't sure if this was quite the way to go...

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The overall rate can be expressed as the average rate of each of the sites, weighted by their denominators... So there are basically two orthogonal things to look for: an above- or below-average site changing its production with no rate change, and a rate change for an individual site. –  Jonathan Oct 17 '12 at 16:45
You seem to be concerned that the overall trend will be more extreme than the trends that are averaged together, but I wouldn't worry about that, it's usually it's the reverse. With a large number of sites, trends in opposite directions will tend to cancel out (assuming the rates are independent - if it's caused by parts supplier used by multiple sites, they may be correlated). –  Jonathan Oct 17 '12 at 16:49
Thanks for your help, I've penned a quick answer but if you want to put your comments in as an answer then will select as accepted answer. –  Tumbledown Oct 19 '12 at 9:15

Full credit to Jonathan for pointing me in the right direction on this.

I weighted each Site's rate by its denominator value and then looked at the differences between these values across months. Embarrassingly easy now I think about it!

So: Site A in May has a wastage rate of 50% and a denominator of 40. This denomination value equates to 10% of the organisation's total denominator for that month. This means that if the organisation had a wastage rate of 30% then 5% (50% x 10%) of this is directly contributed to by Site A.

What I've done is calculate this by month and by site and then calculate differences across months. The sites ranked highest are the ones which may require investigating.

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I think there might be a little more to do here... I created a random dataset to test, and one of the sites tied for the highest contribution actually had below-average waste rate, because it was a larger site. I'll see if I can think of another approach... –  Jonathan Oct 19 '12 at 16:58
I think that's bound to happen as you'll always have high volume vs high rate as factors in the overall rate, but if each sites level of contribution to the overall rate remains fairly static over time, this shouldn't be a problem. It's only where that rate changes across months that it will be scrutinised. One issue is that one site could have a huge numerical change upwards (but no rate change) which will obviously cause other site's contribution to the overall rate to reduce. –  Tumbledown Oct 22 '12 at 7:51
Wait, this is Simpson's paradox isn't it? –  Tumbledown Oct 22 '12 at 10:17