# How to deal with negative proportions in Dirichlet Regression

I am trying to work on a Dirichlet regression problem where I look at three true proportions the first two are actually measured while the third is computed as the complement:

$$p_3=1-p_1-p_2$$

The issue that I have is that $$p_1$$ and $$p_2$$ are measured with error and hence $$p_3$$ will sometimes be negative i.e. when $$p_1+p_2>1$$.

Is there any way to “correct” for this problem? I know there might be a way to do this with a Bayesian approach where you model the measurement error. But I would rather look at a simpler approach such as a proper way to normalise the data.

• Build a hierarchical model where the $p_i$'s are distinguished from their noisy counterparts, the $q_i$'s. Jan 18 at 15:01
• Since $p_1$ and $p_2$ are measured with error, how do you guarantee that each of them is in the unit interval? If that happens "naturally" then why not compute $p_3$ the "same" way and then normalize the three of them to sum to one?
– mef
Jan 18 at 23:13
• Thanks mef, the issue is that I don't have p3. It would be equivalent to measuring a leaf area initially all with healthy tissue and then you have a leaf disease where at a later time you measure (in proportion of the initial area) the proportion of leaf diseased (p1), healthy(p2) and then you have some leaf that disintegrated that you cannot measure. I was trying to avoid having to use a hierarchical model as Xi'an suggested. Jan 19 at 10:42

Think of $$p_3$$ as a derived parameter that is not part of the model. Once you get posterior draws of $$(p_{1}, p_{2})$$ compute $$p_3$$ from these draws by subtraction. Then everything remains consistent.