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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 $\lambda$?

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

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  • $\begingroup$ Given that there is a prespecified number of tags per library, the distribution can't be negative binomial, which has no upper bound. The problem description makes it look like a standard binomial distribution; why do you believe it's overdispersed relative to the binomial? $\endgroup$
    – jbowman
    Feb 20, 2023 at 2:18
  • $\begingroup$ I edited the question to add the original literature. The counts of a tag in different libraries following an NB distribution is a common treatment. $\endgroup$
    – Dan Li
    Feb 20, 2023 at 17:58
  • $\begingroup$ What do you mean by "independent of $\lambda$"? If $\lambda = 0.1$, the MLE for $\phi$ will be different than if $\lambda = 0.9$, so... $\endgroup$
    – jbowman
    Feb 20, 2023 at 19:05
  • $\begingroup$ I read the paper and I see what they are referring to - the profile log-likelihood, where $\lambda$'s maximum likelihood estimate replaces $\lambda$. I'm not used to seeing that referred to as a "conditional maximum likelihood", but then that's just me. $\endgroup$
    – jbowman
    Feb 20, 2023 at 20:13
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    $\begingroup$ @jbowman Conditional likelihod is not profile likelihood. $\endgroup$
    – Gordon Smyth
    Mar 4, 2023 at 23:02

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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: $$l_{Y|Z=z}(\mathbf y; \phi)=\log f(\mathbf y;\mu,\phi)-\log f_{z}(z;n\mu,n^{-1}\phi)\\ =\sum\log \Gamma(y_i+\phi^{-1}) - n \log\Gamma(\phi^{-1}) + \sum y_i\log\left({\phi\mu \over 1 + \phi\mu}\right) + n\phi^{-1}\log\left({1 \over 1 + \phi\mu}\right)\\ -\log\Gamma(z+n\phi^{-1})+\log\Gamma(n\phi^{-1})-z\log\left({\phi\mu \over 1 + \phi\mu}\right) - n\phi^{-1}\log\left({1 \over 1 + \phi\mu}\right)\\ =\left[\sum_{i=1}^{n}\log\Gamma(y_i+\phi^{-1})\right]+\log\Gamma(n\phi^{-1})-\log\Gamma(z+n\phi^{-1})-n\log\Gamma(\phi^{-1})$$

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