Method

Mean model and estimand

With TMMwsp effective library size $L_j$, ComBat-refQL fits

$$\log\mu_{gj}=\log L_j+X_j\beta_g+\gamma_{g,b(j)}.$$

For reference batch $r$, the transport estimand is $\delta_{gb}=\gamma_{gb}-\gamma_{gr}$. The design must identify batch separately from supplied biological groups and covariates.

Hierarchical dispersion

The negative-binomial dispersion is

$$\phi_{gb}=\phi_{gr}\exp(d_b),\qquad d_r=0.$$

edgeR supplies the gene baseline; an information-weighted NB likelihood estimates one reference-centred multiplier per batch. QL dispersion quantifies coefficient uncertainty and never enters NB count probabilities.

Evidence-adaptive EB moderation

For raw QL contrast $\delta_g$, QL variance $s_g^2$, prior mean $m$, and prior variance $\tau^2$, the standard EB weight is

$$w_g=\frac{\tau^2}{\tau^2+s_g^2}.$$

ComBat-refQL then uses

$$z_g=\frac{|\delta_g-m|}{\sqrt{s_g^2+\tau^2}},\qquad a_g=\frac{z_g^2}{1+z_g^2},$$

$$w_g^*=w_g+(1-w_g)a_g, \qquad \delta_{g,\mathrm{final}}=w_g^*\delta_g+(1-w_g^*)m.$$

Weak or uncertain effects retain stronger EB stabilization; strongly separated effects approach raw QL continuously. There is no threshold or tuning parameter. This is evidence-adaptive EB moderation, not an exact conjugate posterior mean.

Counterfactual transport

For a source observation, the reference mean is $\mu_{gj}^{(r)}=\mu_{gj}\exp(-\delta_{g,\mathrm{final}})$. Given observed count $y$, the deterministic mid-P probability is

$$q=F_b(y-1;\mu_{gj},\phi_{gb})+ frac12P_b(Y=y;\mu_{gj},\phi_{gb}),$$

and $y^*=F_r^{-1}(q;\mu_{gj}^{(r)},\phi_{gr})$. Machine-safe clipping handles distribution tails. Reference observations bypass transport and remain exact.

TMMwsp identifies a relative compositional scale. An unanchored absolute global RNA-output shift is not identifiable without external information.