1
$\begingroup$

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.

$\endgroup$
3
  • 1
    $\begingroup$ Build a hierarchical model where the $p_i$'s are distinguished from their noisy counterparts, the $q_i$'s. $\endgroup$
    – Xi'an
    Jan 18 at 15:01
  • $\begingroup$ 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? $\endgroup$
    – mef
    Jan 18 at 23:13
  • $\begingroup$ 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. $\endgroup$
    – user6384
    Jan 19 at 10:42
2
$\begingroup$

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.

$\endgroup$
4
  • $\begingroup$ Thank you for your answer Frank. I am a bit unclear on what you mean. Is the idea to set up the likelihood as: (p_1, p_2,1-p_1-p_2)~Dirichlet treating the p's as parameters and then have two measurement errors likelihoods for example: y_1~N(p_1,SD) and y_2~N(p_2,SD)? $\endgroup$
    – user6384
    Jan 19 at 10:33
  • $\begingroup$ Yes to the first part (I think) and not sure about the second part. $\endgroup$ Jan 19 at 13:15
  • $\begingroup$ Thank you. Do you think that by simulating values based on known parameters and retrieving those values would be good enough to be "confident" this approach is sensible? $\endgroup$
    – user6384
    Jan 20 at 5:53
  • $\begingroup$ This is more about math than about checking algorithm accuracy, but yes it's usually a good idea to simulate form a model and see what data you get. $\endgroup$ Jan 20 at 13:32

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.