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Table 3 The alternative hypothesis HA:pg>eλ is robust to different model hypotheses

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

Underlying distribution p under H0 p under HA
Mixture of Poisson \(e^{-\lambda } = e^{-\sum _{k} \pi _{k} \lambda _{k}}\) \(\sum _{k} \pi _{k} e^{-\lambda _{k}}\)
Negative binomial eλ \(\left (\frac {r}{r+\lambda }\right)^{r}\)
Zero-inflated negative binomial eλ \(\left (\frac {r}{r+\lambda }\right)^{r}+ \pi _{0}\)
  1. 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)