Demystifying “drop-outs” in single-cell UMI data

Table 3 The alternative hypothesis H_A:p_g>e^−λ is robust to different model hypotheses

Underlying distribution	p under H₀	p under H_A
Mixture of Poisson	\(e^{-\lambda } = e^{-\sum _{k} \pi _{k} \lambda _{k}}\)	\(\sum _{k} \pi _{k} e^{-\lambda _{k}}\)
Negative binomial	e^−λ	\(\left (\frac {r}{r+\lambda }\right)^{r}\)
Zero-inflated negative binomial	e^−λ	\(\left (\frac {r}{r+\lambda }\right)^{r}+ \pi _{0}\)

In the first row, the right column is larger than the left column due to Jensen’s inequality. For negative binomial, the dispersion parameter r is constructed so that the variance is \(\frac {\lambda ^{2}}{r} + \lambda \), so that Poisson is a special case of negative binomial with r=∞. The zero-inflated negative binomial distribution is parameterized as π₀δ₀+(1−π₀) NB(λ,r)

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