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Rahul Mahajan <firstname.lastname@example.org>
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Fri, 14 Jun 2013 17:55:42 -0400
I have a question regarding significance testing for the difference in
the ratio of means.
The data consists of a control and a test group, each with and without
treatment. I am interested in testing if the treatment has a
significantly different effect (say, in terms of fold-activation) on
the test group compared to the control.
The form of the data with arbitrary n and not assuming equal variance:
m1 = mean of (control group) n = 7
m2 = mean of (control group w/ treatment) n= 10
m3 = mean of (test group) n = 8
m4 = mean of (test group w/ treatment) n = 9
H0: m2/m1 = m4/m3
H0: m2/m1 - m4/m3 = 0;
Method 1: Fieller's Intervals
Use fieller's theorum available in Stata as fieller package. This is
a promising way to compute standard error/confidence intervals for
each of the two ratios but will not yield p-values for significance
testing. Significance by non-overlap of confidence intervals is too
stringent a test and will lead to frequent type II errors.
Method 2: Bootstrap
Abandoning an analytical solution, we try a numerical solution. I can
repeatedly (1000 or 10,000 times) draw with replacement samples of
size 7,10,8,9 from m1,m2,m3,m4 respectively. Each iteration, I can
compute the ratio for m2/m1 and m4/m3 as well as the difference.
Standard deviations of the m2/m1 and the m4/m3 bootstrap distributions
can give me standard errors for these two ratios. Then, I can test to
see where "0" falls on the third distribution, the distribution of the
difference of the ratios. If 0 falls on one of the tails, beyond the
2.5th or 97.5th percentile, I can declare a significant difference in
the two ratios. My question here is if I can correctly report the
percentile location of "0" as the p-value?
Method 3: Permutation test
I understand the best way to obtain a p-value for the significance
test would be to resample under the null hypothesis. However, as I am
comparing the ratio of means, I do not have individual observations to
randomize between the groups. The best I can think to do is create an
exhaustive list of all (7x10) = 70 possible observations for m2/m1
from the data. Then create a similar list of all (8x9) = 72 possible
observations for m4/m3. Pool all (70+72) = 142 observations and
repeatedly randomly assign them to two groups of size 70 and 72 to
represent the two ratios and compute the difference in means. This
distribution could represent the distribution under the null
hypothesis and I could then measure where my observed value falls to
compute the p-value. This however, makes me uncomfortable as it seems
to treat the data as a "mean of ratios" rather than a "ratio of
Method 4: Combination of bootstrap and permutation test
Sample with replacement samples of size 7,10,8,9 from m1,m2,m3,m4
respectively as in method 2 above. Calculate the two ratios for these
4 samples (m2/m1 and m4/m3). Record these two ratios into a list.
Repeat this process an arbitrary (B) number of times and record the
two ratios into your growing list each time. Hence if B = 10, we will
have 20 observations of the ratios. Then proceed with permutation
testing with these 20 ratio observations by repeatedly randomizing
them into two equal groups of 10 and computing the difference in means
of the two groups as we did in method 3 above. This could potentially
yeild a distribution under the null hypothesis and p-values could be
obtained by localizing the observed value on this distribution. I am
unsure of appropriate values for B or if this method is valid at all.
Another complication would be the concern for multiple comparisons if
I wished to include additional test groups (m5 = testgroup2; m6 =
testgroup2 w/ treatment; m7 = testgroup3, m8 = testgoup3 w/
treatment...etc) and how that might be appropriately handled.
Method 2 seems the most intuitive to me. Bootstrapping this way will
likely yield appropriate Starndard Errors for the two ratios.
However, I am very much interested in appropriate p-values for the
comparison and I am not sure if localizing "0" on the bootstrap
distribution of the difference of means is appropriate.
Thank you in advance for your suggestions.
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