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I have a dataset with many ratios gathered from a large group of individuals. Each ratio is between repeated measurements from the same individual. Ratios normalize the concentrations, which vary between individuals. The total variation of each sample is a mixed distribution of a biological component (unknown distribution) and an analytical noise (Gaussian). I am trying to work out the biological component, expressed an expected relative change.

I have characterized a BoxCox normal distribution (by σ, µ & λ) of the ratios. I believe that the total variation of a single sample should also be Gaussian on the BoxCox scale, with σ / √2 . I reason this given that the ratio distribution is the product of two samples with the same variation/distribution. I have tested this by generating ratio distributions with Monte Carlo.

Now I need to find a way to remove the analytical noise, which is where I got stuck. I know the total variation on the normal scale (by inverting the BoxCox) and the Gaussian distribution of the analytical noise (centered at mean = 1). I have tried to solve this deconvolution problem using the package Decon in R but have yet to be successful. Any suggestions would be appreciated.

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I'd suggest a different approach, rather than using ratios, to account for varying concentration levels among individuals. Although ratios can seem to be a good choice for this at first, the complications that you are encountering, potential problems when the denominator of a ratio is close to 0, and the (sometimes hidden) assumptions that you need to make can pose problems.

If this is a type of time-course study, one particularly useful approach is to use the baseline concentration as a predictor in the model, and then model the concentration over later times as a function of an individual's baseline concentration and other predictors of interest related to your "biological component." Repeated measurements that aren't primarily a time course (e.g., responses of the same individuals to different treatments) can typically be re-formulated into a structure that's equivalent to a time-course study.

Chapter 7 of Frank Harrell's online notes discusses several approaches to this situation, with an emphasis on generalized least squares. Generalized least squares doesn't directly model the variance among individuals, which you also don't seem to be doing. This situation is also frequently addressed with a linear mixed model, which explicitly models a distribution of responses among individuals.

The residual variance in such a model would include both analytical measurement error and other un-modeled sources of variance. (That's also true of your modeling of the ratios.) If you have some estimate of the analytical error and analytical error is independent of other sources of variance, it's fairly simple to estimate the magnitude of the other sources of variance.

If you then want to re-express the results in terms of ratios you could then do that, applying standard error-propagation methods to the error estimates from the linear model.

You also might consider modeling concentrations in a logarithmic scale in this way. That can be useful if measurement error is proportional to the value instead of independent of it. In that case, differences in the log scale are the logs of corresponding ratios.

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