From: David Novo (novod@fhs.csu.McMaster.CA)
Date: Tue Feb 10 1998 - 21:55:03 EST
On Tue, 10 Feb 1998, Darren Hickerson wrote: > > Several have written in response to the latest inquiries conerning how to properly ratio data. I still have unanswered questions, though. I don't have all the letters in one place, so I can't quote exactly who said what (sorry). I do greatly appreciate the feedback, so please pardon my humble ignorance on this topic. > One comment stated that resolution was too poor in the upper decades of the log axis scale since numbers were so far apart, so using these values to calculate ratios was less accurate. > Here's my question from actual data. I run a sample and get mean 2000 on FL3, and mean 50 on FL1. After treatment, I get mean 200 on FL3 and mean 1000 on FL1. This is all visualized on the "log" axes. If I divide the numerical values given for the means, I get ratios of 0.025 pre and 5.0 post, for a fold change in ratio of 200. > First question: do I invoke the mathematics of dividing log values when what I'm really dividing are "linear" values presented on a log scale? Is the above division appropriate? > > Another writer stated the mathematical problems involved in dividing true logarithmic values. > Darren - to get a true ratio you should probably be doing more that dividing the means of your two populations by each other. A ratio should be calculated for every point and then a mean (or more likely median) of the calculated population taken. We could probably come up with a case where the mean of the ratios does not equal the ratios of the means. Second, as Howard Shapiro mentioned earlier, if you want to take the ratio of data acquired on a log scale you should subtract one population from the other - this calculated population will be log(a/b). Third, while it is true that log amps do not give 100% accurate log conversion, problems usually arise only in the lower decade (which is usually noise anyway) and in the upper half of the top decade. If you stick to the second and third decades you should get a pretty good response. If you are paranoid (or I guess another word would be meticulous) you could run a simple calibration to see how good your log amps are. Fourth - on most machines the log scale is NOT simply linear data graphed on a logarithmic histogram/dot plot. While the data in the FCSfile will be stored from 0-1023 (assuming a 10 bit A->D converter) on both linear and log they will mean different things. On Linear: channel 10 is 10x channel 1 channel 100 is 10x channel 10 channel 1000 is 10x channel 100 Notice that you only get 3 decades of range (and the first decade only has 10 channels). On Log channel 256 is 10x channel 1 channel 512 is 10x channel 256 channel 768 is 10x channel 512 channel 1023 is 10x channel 768. The magic of log amps - you now have 4 decades of dynamic range. As you can see then, when you are looking at a histogram of log amplified data it is not simply the linear data on a log graph. In fact on a 10 bit machine you would not be able to convert back to the linear value because as you see anything in the bottom decade of the log scale is off the charts on the linear scale, and anything in the second decade of the log scale corresponds to a mere 10 channels on the linear scale (the first linear decade). So the results you would get when converting from log to linear would be very ratty. Hope some of this helped, Dave p.s. for everything you never wanted to know about logarithmic amplification there is a good section in Practical Flow Cytometry by Howard Shapiro (pp166). Any little wisdom (and it may be little indeed) comes from here.
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