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I would like to fit a regression model with continuous response and predictors. A fraction of the response is a non-negative linear combination of several predictors. What is not covered by this predictors (the residuals) has to be non-negative as well. To a small degree, the residuals could be negative due to measurement errors, but not from their physical nature. These measurement errors should be comparably small.

I am currently working with the package brms in R. The model families are similar to those in the glm function from the base package, but more exhaustive. So far, I tried Gamma, weibull and the inverse Gaussian family. Because of the non-negativity of the distributions, I had expected the residuals to be non-negative as well, but this did not work out. Seemingly, I have a misunderstanding there.

Maybe this is a very simple question and I am just lacking the right direction, but so far, I could not find solution that works.

Just some background information: it is about fluorescence of certain chemical compounds and the weight of total carbon. As there is neither negative fluorescence nor negative weight, I need to restrict the model.

Edit: It is true, that the predicted values are non-negative. Also, there are parts of the measured carbon compounds, that are not fluorescing. I understand it, that besides the actual residuum (positive or negative), there is an unknown "predictor", that cannot be negative. I assume the influence of this predictor larger than the residuum, therefore, hardly any negative residuals. My first idea was, to cover this unknown predictor in the residuum, but it would be even better, if there exists a method to properly cover such a case. I just have not found anything and I assume, it will not be easy to separate two unknowns without any information.

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    $\begingroup$ I think you mean that the predicted value cannot be negative. The residual still could be negative: sometimes you’ll undershoot the true value, and sometimes you’ll overshoot. Where, if anywhere, does this break down in your particular case? $\endgroup$
    – Dave
    Commented Feb 9, 2020 at 1:14

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First, even after your edit, you seem to have a misunderstanding. Residuals can be negative or positive. Indeed, they have a mean of 0 (I don't know of any models that are exceptions to this).

Second, you wrote

there is an unknown "predictor", that cannot be negative. I assume the influence of this predictor larger than the residuum, therefore, hardly any negative residuals.

I'm not sure what you mean by an "unknown predictor". Do you mean an omitted variable? If it's unknown, how do you know it can't be negative? It's nice when the influence of the (known) predictors is larger than the residuals, but it's dangerous to assume this -- that's part of the reason you do tests! Is signal greater than noise? And, regardless, if the distribution of the residuals is symmetric then roughly half will be negative and half positive.

Finally, the problem of a response that is always positive has been discussed here several times. See e.g. here and here.

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