Which one is the most correct to use in statistical analysis? The mean (average) of ratios or the ratio of means (averages)?
Yes, and No.
Yes: either one, they are both used.
No: Neither is more correct than the other, but they can give you different results.
If I have a bunch of technical replicates I might be inclined to calculate the ratios first if that will get rid of the technical replicates.
A great deal depends on what question you are asking, and how you phrased the question.
So I want to know the male-to-female ratio. I take 200,000 samples in total by sampling 1000 individuals from various cities. Sadly, not everyone wanted to participate. Therefore, my sample sizes are somewhat different for the different cities. Furthermore, I happened to keep track of district within each city. So I can find:
1) The male:female ratio for each district
2) I can ignore district and find the ratio for each city. Alternatively, I can find the ratio for each district and take and average to get the ratio for the city.
3) I can ignore district and city, to get a global average. I could take the average from each district and ignore city. I could take the average for each city and ignore district. I could take the average for each district to get a city average and then take the average of those values to get the global answer.
With all those choices there is a different distribution of the variance in the data. Your statistical method is entirely dependent on the variance. So based on your question, what gives you the most relevant estimate of your variance. This is determined by correctly identifying the population of interest. In turn that will help you identify your experimental unit and avoid pseudoreplication.
That said, if you take two measures from the same experimental unit and you are interested in the ratio of X1 to X2, then it would not make sense to find the average of variable X1 across all experimental units and the average of variable X2 and then calculate the X1/X2 ratio.
An concatenation example based on: D J Hand Deconstructing statistical questions (with discussion) J. Royal Statistical Society, Series A, 159, 445 – 492, (1994)
2 cars each of Type 1 and Type 2. Type 1 cars: 10miles/gallon and 40miles/gallon
gives an average of 25miles/gallon; Type 2 cars: Both 20miles/gallon gives an average 20miles/gallon
Thus Type 1 cars are better on average? Travel more miles per unit of gas….
However, lets look at the data in terms of gallons used per mile traveled:
Type 1 cars: 0.1gallons/mile and 0.025 gallons/mile give an average of 0.0625 gallons/mile; Type 2 cars 0.05gallons/mile each give an average 0.05gallons/mile
Type 2 cars are better – use less gas/mile…
This apparently contradictory conclusion revolves around the fact that the mean of the recipricols isn’t the same as the recipricol of the means.....
BTW, I changed the liters and kilometers in the original to miles and gallons. Feel free to consult the original paper for the original units. The conclusion is the same.
Alan,
That is generally true for any transformation that changes the underlying distribution.
The mean of the reciprocal does not equal the reciprocal of the means.
The mean of the log transformed data does not equal the log of the means.
And so forth. Where this gets people in trouble is in plotting data. I have heard other people advising (and been advised) to take my graph of log transformed values and back transform the axes. Seems simple enough until you start playing with the numbers and realize that it isn't exactly clear how to interpret antilog(mean(log)) values. It is deceptive to simply label the axis mean value.
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