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Lets say for two samples, treatment and control, there are three constituent molecules each and their corresponding amounts are as follows:

Tr. Ct.  
2   1
5   2
6   3

I wish to find out the individual ratios for these three constituent molecules. But however if for some reason the amount of treatment sample was more due to experimental error it might increase the ratio Tr:Ct artificially and vice versa. Is there any way to compensate for this effect for ratio calculation by normalizing or any other techniques?

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    $\begingroup$ Compensate how? What kind of statistical analysis are you proposing to do with these data? $\endgroup$
    – whuber
    Commented Sep 19, 2012 at 19:16
  • $\begingroup$ @whuber is there any way calculate the ratio that can mitigate the effect of unequal sample amount? I've come across methods like dividing all Tr. and Ct. values by mean(ct) but that doesn't change the ratiosarticle $\endgroup$
    – The August
    Commented Sep 19, 2012 at 19:34
  • $\begingroup$ What do you mean by the "effect of unequal sample amount"? What effect? So far, you mention computing ratios. So go ahead and compute them. But then what? What will you be doing with them? Another bit of helpful information you could provide would concern the nature of the measurement. Some measurement errors (especially in chemometrics) are multiplicative, which might resolve the entire issue. Also, if there is any possibility of correlation among measurements, that would have a bearing on this problem. $\endgroup$
    – whuber
    Commented Sep 19, 2012 at 19:42
  • $\begingroup$ By effect I mean, If the Tr. sample is added more compared to the Ct. the ratio would be artificially high.Can there be some proportion based method to bypass this problem? Or you need to have some other third standard to get an amount corrected ratio?So far I just need the ratios, and not to do any statistical calculations on them. The ratios are to be measured by corresponding intensity values. $\endgroup$
    – The August
    Commented Sep 19, 2012 at 20:13
  • $\begingroup$ If you just need the ratios, compute the ratios! But I think I begin to understand your concern. Would I be correct in saying you are viewing your data as ordered pairs $(x,y)$ where $x$ and $y$ are independent realizations of random variables with means $\mu$ and $\nu$, respectively, and you are concerned that the average of the $x_i/y_i$ may be a biased estimate of $\mu/\nu$? Even if this is the case, we need more information to respond well: information about the nature of these distributions and more specifics about the actual data you have (the example isn't too helpful). $\endgroup$
    – whuber
    Commented Sep 19, 2012 at 20:19

1 Answer 1

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I think I understand your question. When you take ratios of random variables the ratios can be hihgly variable. In fact as discussed in other recent questions if the denominator can be 0 or very close to it the ratio may not even have a finite mean let alone a variance. The Cauchy distribution is the canonical example.

Let R=U/V be the random variable you are interested in. As long as U and V have variances and V has a distribution bounded away from 0 R will have a finite variance that can be approximated by the delta method See this recent post.

Using the formula given by the delta method and plugging in sample estimates to get an estimate of this variance you have a good idea as to how variable your estimate is. If you can derive the density of R you should be able to construct confidence intervals for these ratios that account for this high variability.

You can't directly compensate for the small errors that propagate large error in the ratio because you don't know what they are or when they occur. But the variance estimate provides a measure of how these error affect the ratio.

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