Friday, September 02, 2005

A naive question

Question:

I have 3 samples which were each measured for protein content of IgG, and several other proteins (in mgs/ml). As you can see, all three of my samples for IgG do not fall within the acceptance limits, everything else looks fine. For my purposes, I am less interested in whether the exact mg/ml range specifications have been met.
I am more interested in knowing if the ratio of IgG to Total Protein is consistent. Since measurements of total protein, and each individual protein, have limits (unequal ranges), is it possible to derive the ranges for IgG/total protein ratios?

IgG: sample1=0.9213, sample2=0.8769, sample3=1.5984
Acceptance limit (+/- 4SD): low=0.15, high=0.79

Total Protein: sample1=31.635, sample2=32.856, sample3= 34.299
Acceptance limit (+/- 4SD): low= 29.2, high= 36.4

As before if the Acceptance limits are (for IgG): 0.15 and 0.79, the mean would be 0.47 (s= 0.08) (and for total): 29.2 and 36.4, the mean would be 32.8 (s=0.9) if I make a ratio of the means 0.47/32.8 = 0.014 , what is the new standard deviation? (I can assume the number of measurements made for each mean are the same).

Response:

I understand you are trying to derive the standard deviation, therefore the acceptance limit for ratio based on the standard deviation of the numerator and the denominator. But this approach does not work mathematically or statistically. You can calculate the mean/standard deviation for numerator is 0.47 (0.08) and for the denominator is 32.8 (0.9). But there is no statistical formula for deriving the standard deviation for the ratio. If the numerator and denominator follows the normal distribution, the ratio will NOT follow the normal distribution.
To calculate the standard deviation for IgG/total Protein ratio, you would need to calculate the ratio for each sample first.
ratio1, ratio2, ..., ration
SD of ratio = square root of (sum of (each individual ratio - mean of ratio)/n)

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