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I would really appreciate any suggestions with the following data analysis issue. Please read till the end as the problem at first may appear trivial, but after much researching, I assure you it is not. The situation is a little complicated because I want to compare the ratios of means:

For example, in one experiment I have collected data (Electric current levels, a continuous variable) from 7-8 cells (replicates) which express a particular ion channel gene (Gene1). This gives me a mean current level for Gene1. I have then measured current levels from a different set of 7-8 replicate cells which also express Gene1 PLUS an activator gene (the "treatment"). Now I have a mean current value for Gene1+activator.

I repeat the two measurements for Gene2 (a different ion channel gene), again recording mean current values from 7-8 cell replicates for Gene2 alone and then for Gene2+activator.

The quantity I am interested in comparing is the %activation or fold activation caused by the activator for Gene1 versus Gene2. So, I would obtain a ratio by dividing the mean current for Gene1+activator by the mean current for Gene1 alone. This would give me the fold activation for Gene1. I would compare this to a similar ratio obtained for Gene2.

I have done some research on this and using Fieller's intervals to compute the error bars or confidence intervals seems promising. However, I don't know how to convert that to hypothesis test and get an appropriate p-value for the comparison of means being same/different. Furthermore, the best solution would also allow multiple comparisons and allow me to compare "fold-activations" for Gene1 and Gene2 and Gene3 and Gene4 at the same time.

Fieller's intervals seem like the perfect tool to compute 95% confidence intervals etc around each fold-activation but as we are submitting to a journal, they will insist on p-values for our comparisons. As of now, I am reduced to insisting the lack of CI overlap clearly signals a significant difference but I know that lack of 95%CI overlap is an overly stringent comparison test which represents an alpha<0.05. I would truly appreciate any suggestions to the appropriate comparison test (single or multiple comparison, either at this point will be helpful). Thanks in advance.

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    $\begingroup$ BUMP. Still hoping for an answer. Anyone? How about using bootstrapping/permutation testing to test the null hypothesis: H0: m1/m2 = m3/m4 restated, H0: m1/m2 - m3/m4 = 0; I can see how I may bootstrap the difference in raio of means by sampling with replacement from each of m1,2,3,4 many times and recomputing the difference. But I can't see how to set this up as a permutation test to test the above null hypothesis. What groups could I randomize across to assume the null hypothesis and obtain the distribution on which to test my observed value? $\endgroup$ – Electrophys Jun 2 '13 at 23:41
  • $\begingroup$ Would it make sense to use the logarithm of the currents? That way you could compare the differences of the log transformed values rather than the more awkward ratio. (Whoops, I now read EdM's answer. I agree with EdM.) $\endgroup$ – Michael Lew Oct 6 '15 at 20:26
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Although the general solution may be difficult, for this situation the solution is simple if analyses are performed in the log scale. I would argue that working in the log scale would make sense for this type of data even if ratios weren't being compared.

The experiments and desired comparisons described only make sense if all ion currents have the same sign (all inward or all outward with respect to the cells). Furthermore, errors in these types of measurements are often proportional to the values being measured rather than being independent of the values. So working in the log scale of (absolute value of) currents makes sense.

If $I$ represents current, 1 and 2 represent the genes, and $c$ and $a$ represent control and activator cases respectively, the desired comparison is $I_{1a}/I_{1c}$ versus $I_{2a}/I_{2c}$. In the log scale the null hypothesis becomes:

$$log(I_{1a})-log(I_{1c}) = log(I_{2a})-log(I_{2c})$$

which can be tested by examining an interaction between gene and activator in a standard linear model in this log scale. This approach has the advantage of easily being extended to multiple genes (and activators).

I suppose one could imagine situations where working in the log scale would be inappropriate, but for this type of comparison in biomedical experiments in practice the log transformation often makes sense in terms of what is being measured and avoids the hassles noted here for applying Fieller's intervals.

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