How to calculate the conditional maximum likelihood of independent negative binomial variables conditioned on the likelihood of the sum?

Question

In the paper Small-sample estimation of negative binomial dispersion, with applications to SAGE data, section 4.1 Conditional maximum likelihood:

For RNA sequencing data, assume the counts for a single tag across $n$ libraries is a negative binomial random variable. Consider $Y_1, \cdots , Y_n$ as independent and $NB(\mu_i = m_i \lambda, \phi)$, where $m_i$ is the library size (i.e. total number of tags sequenced for library $i$) and $\lambda$ represents the proportion of the library that is a particular tag. The probability mass function is:

$$f(y_i;\mu,\phi)=P(Y=y_i)=\frac{\Gamma(\phi^{-1}+y_i)}{\Gamma(\phi^{-1})\Gamma(y_i+1)}\left(\frac{1}{\phi^{-1}\mu^{-1}+1}\right)^{y_i}\left(\frac{1}{\phi\mu+1}\right)^{\phi^{-1}} \tag 1$$

$$\text{E}(Y)=\mu$$ and $$\text{Var}(Y)=\mu+\phi\mu^2$$

If all libraries are the same size (i.e. $m_i ≡ m$), the sum $$Z = Y_1 + \cdots + Y_n \sim NB(nm\lambda, \phi n^{−1})$$

How to derive conditional maximum likelihood for $\phi$ that is independent of $\lambda$?

$$l_{Y|Z=z}(\phi)=\left[\sum_{i=1}^{n}\log\Gamma(y_i+\phi^{-1})\right]+\log\Gamma(n\phi^{-1})-\log\Gamma(z+n\phi^{-1})-n\log\Gamma(\phi^{-1})$$

Answer 1

Based on the Conditional Likelihood defined in https://rss.onlinelibrary.wiley.com/doi/epdf/10.1111/j.2517-6161.1996.tb02101.x, the conditional log-likelihood for $\phi$ of $Y$ conditioned on $Z$, dropping terms that don't involve $\phi$, is: $$l_{Y|Z=z}(\mathbf y; \phi)=\log f(\mathbf y;\mu,\phi)-\log f_{z}(z;n\mu,n^{-1}\phi)\\ =\sum\log \Gamma(y_i+\phi^{-1}) - n \log\Gamma(\phi^{-1}) + \sum y_i\log\left({\phi\mu \over 1 + \phi\mu}\right) + n\phi^{-1}\log\left({1 \over 1 + \phi\mu}\right)\\ -\log\Gamma(z+n\phi^{-1})+\log\Gamma(n\phi^{-1})-z\log\left({\phi\mu \over 1 + \phi\mu}\right) - n\phi^{-1}\log\left({1 \over 1 + \phi\mu}\right)\\ =\left[\sum_{i=1}^{n}\log\Gamma(y_i+\phi^{-1})\right]+\log\Gamma(n\phi^{-1})-\log\Gamma(z+n\phi^{-1})-n\log\Gamma(\phi^{-1})$$