How to calculate error of percentage ratio?

Question

I have the following problem: I am counting a subset of cells from a tissue in Drosophila (fruit fly) at different days after "birth". At Day0 I obtain a population of flies, and dissect some of these flies at each time-point (I.e. I have a population of 30 flies at Day0. I dissect 3 of them right away, 3 more after 1 day, another 3 after 3 days, and so on up to 15 days after birth). To obtain these results I need to dissect (thus killing) the fly, so I cannot assay the differences in a single individual fly changing overtime, but I can assay the differences in the whole population. What I obtain is a n number of around 50 for each timepoint. I could just plot the average for each timepoint on a chart and show the variation overtime this way. I would use standard error of the mean to build up the error bars.

My problem is that I need to show these data not as raw data, but as a ratio over the first Day, in order to better show the decrease in the cell number overtime. To do this, I am showing my data as a percentage ratio over day0 [(Day1Mean/Day0Mean)*100].

To make it more clear:

Day0 mean: 5

Day1 mean: 4

Day3 mean: 3.5

Day6 mean: 3.5

Day9 mean: 3

Day12 mean: 2.5

Day15 mean: 2

Instead of making a dispersion chart with those data (that are the average of the number of cells I counted) I would like to show them this way:

Day0: (Day0/Day0)*100 = 100%

Day1: (Day1/Day0)*100 = 80%

Day3: (Day3/Day0)*100 = 70%

Day6: (Day6/Day0)*100 = 70%

Day9: (Day9/Day0)*100 = 60%

Day12: (Day12/Day0)*100 = 50%

Day15: (Day15/Day0)*100 = 40%

Now my problem is this: If I am using the "raw" data, I can just plot them in and use the standard error calculated on the mean. I cannot use the standard error with the second kind of setup since my number is completely different. If it was possible I would have liked to "convert" the standard error to the new visualization. I thought it could be possible with some kind of mathematical transformation, in a similar way as reported in this document: http://www.census.gov/acs/www/Downloads/data_documentation/Accuracy/PercChg.pdf

Many thanks for Your help

Answer 1

This reply illustrates a key issue and provides some (simple) guidance.

Consider an experiment in which the growth of biological subjects is tracked daily. Here, 24 subjects were measured on days 1 through 10:

Each set of 10 linked points represents a subject.

A collection of boxplots depicts the variation among subject sizes day by day: the box heights are directly proportional to this variation.

Using only this information, we would (naively) conclude there is a lot of uncertainty in the daily growth rates, expressed as ratios. But there's actually very little because the size of a subject is strongly correlated with its size the previous day and all subjects are growing at the same (albeit varying) rates day to day:

The extremely short heights of these boxes testify to an extraordinarily consistent growth rate among subjects each day--even though those rates varied from day to day (as shown by the variable vertical positions of the boxes).

(These are artificial data created to illustrate the point that temporal correlation matters.)

The delta method formula accounts for this correlation with a covariance term. The possible solutions, listed in descending order of accuracy and effectiveness, are

1. Compute subject-specific ratios from the experimental data and analyze those separately.

2. Compute covariances from the (raw) experimental data and use those in a delta-method calculation.

3. Do not report standard errors of the ratios.

Answer 2

The delta method is a way to approximate the variance of a function of random variable(s). So in your case you want the variance of (X/Y) given in a formula involving the things you know such as the variance of X and the variance of Y. It is just a Taylor series expansion of a function expressed in terms of random variables where expected values are taken and the higher order terms in the Taylor series are dropped. However what is important regarding the formula which you did not mention was whether or not the variables X for say Day1 and Y for Day0 are correlated. Common formulae based on the delta method assume independence.