[R] significance testing for the difference in the ratio of means

[Rahul Mahajan]

My apologies if my request is off topic and for my admittedly
half-baked understanding of the topic.  I'm afraid trying to talk with
the "local statistical help", and trying to post on several general
statistical forums to look for proper guidance has not yielded any
response much less any helpful ones.  I turned to this forum in
desperation because 1) I will be using R to implement the chosen
strategy and 2) looking through the archives of this forum seemed
promising especially because of past helpful posts as this:

https://stat.ethz.ch/pipermail/r-help/2009-April/194843.html

Perhaps you can suggest a resource which would cover the applicable
"standard methodology" and perhaps its implementation in R?  I would
truly appreciate any guidance.

My protocols/design = each observation within the 4 groups represents
a recording of a continuous variable (whole-cell current from 1 cell
in electrophysiology measurements).  The data for each group appears
roughly normal (albeit small n values from 7-10 per group).  The
variance is not equal among the groups because it seems to vary with
the mean, ie larger currents = larger absolute variance.  There is no
explicit randomization involved as these observations are merely the
measurements of wholecell currents for cells receiving an identical
experimental treatment.  I am interested comparing the
"fold-activation" effect of the treatment for control cells versus for
testgroup cells which have differing baseline pre-treatment current
values.

Best,
Rahul




On Fri, Jun 14, 2013 at 7:13 PM, Bert Gunter <gunter.berton at gene.com> wrote:
> Sigh...
>
> (Again!) These are primarily statistical, not R, issues.  I would urge
> that you seek local statistical help. You appear to be approaching
> this with a good deal of semi-informed adhoc-ery. Standard methodology
> should be applicable, but it would be presumptuous and ill-advised of
> me to offer specifics remotely  without understanding in detail the
> goals of your research, the nature of your design (e.g. protocols,
> randomization?), and the behavior of your data (what do appropriate
> plots tell you??)
>
> Others may be bolder. Proceed at your own risk.
>
> Cheers,
> Bert
>
> On Fri, Jun 14, 2013 at 2:07 PM, Rahul Mahajan <mahajanr at vcu.edu> wrote:
>> 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
>> restated,
>> H0: m2/m1 - m4/m3 = 0;
>>
>> Method 1: Fieller's Intervals
>> Use fieller's theorum available in R as part of the mratios 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 means".
>>
>> 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.
>>
>> -Rahul
>>
>>         [[alternative HTML version deleted]]
>>
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>> and provide commented, minimal, self-contained, reproducible code.
>
>
>
> --
>
> Bert Gunter
> Genentech Nonclinical Biostatistics
>
> Internal Contact Info:
> Phone: 467-7374
> Website:
> http://pharmadevelopment.roche.com/index/pdb/pdb-functional-groups/pdb-biostatistics/pdb-ncb-home.htm