We combine WT counts into a vector

over types

*t* by adding contributions from

and

across all tissue samples. That is,

. Since we have combined multinomial counts governed by a common probability vector, the summary counts retain the multinomial form. Further, we let

*Y* record the MC MT data

from all tissue samples (these are subject to overdispersion) as well as any available SC-derived counts

. Here, we do not collapse by counting contributions over

*i*, since such summarized counts would entail undue information loss; instead

*Y* is a collection of vectors.

To emphasize notational distinctions,

*X* and

*Y* refer to random elements in our actual experiment, taking possible values

*x* and

*y*. As is sometimes done in

*p*-value discussions, we introduce a separate notation for data obtained from a hypothetical repeat of the experiment: let

denote a hypothetical repeated draw of the random vector

*X*. Having observed

and

, the proposed

*p*-value is

where

is a posterior predictive distribution for WT counts given observed MT counts:

Here,

*θ* is the vector of unknown probabilities over types. Although

*x*, the possible realization of WT counts, is used above, nowhere have we conditioned on the event

. Had we conditioned in (

4.1) instead on all the data

, then the

*p*-value would be a posterior predictive

*p*-value (

Gelman *and others*, 1996). This object can be unduly conservative, and so we decided to condition on part of the data only, as in the conditional predictive

*p*-value approach (

Bayarri and Berger, 1999,

2000;

Evans, 2000;

Robins *and others*, 2000). There, one conditions on part of the data,

*U* (in our case, the MT counts

*Y*), and generates a

*p*-value from the conditional distribution of a statistic

*T* that is as independent from

*U* as possible. Our particular choice to split data by MT and WT counts and to construct a test statistic from the conditional density is most similar to Evans’ cross-validatory surprise (

Evans, 2000), though examples and properties for count data seem not to have been developed previously.

A further justification of the proposed

*p*-value (

4.1) is its structural similarity to Fisher's exact test

*p*-value, which would be suitable in the absence of multinomial overdispersion. Following the notational conventions above, Fisher's

*p*-value is

where

holds the sufficient statistic vector for the null frequencies

*θ*, and where

is a generalized hypergeometric mass function. In the absence of overdispersion, this

*p*-value is exact in the frequentist sense of being dominated by the uniform distribution, but unaccounted sources of variation tend to deflate and invalidate

. Splitting the data into its natural components

*X* and

*Y* enables construction of a conditional

*p*-value that is similarly reliant on a conditional mass function as a test statistic. Further, in conditioning on one of the data components, it is more sensible to condition on

*Y*, as we propose, since

*Y* contains information on the frequency parameters

*θ* as well as the overdispersion parameter. The WT data

*X* informs only

*θ*, on the other hand conditioning on

*X* instead of

*Y* would make the computation more difficult and more sensitive to prior information.

The null sampling distribution of the conditional predictive

*p*-value proposed in (

4.1) is neither exactly uniformly distributed nor dominated by the uniform distribution. Since posterior simulation is used to average over unknown variables, this null distribution is difficult to determine. The proposed

*p*-value is a valid frequentist

*p*-value in the sense that it converges to a uniform distribution as sample sizes diverge (Section 6). A fully Bayesian test of homogeneity between MT and WT proportions provides an alternative approach to the inference problem. This could be pursued since Bayesian analysis is already used to generate conditional predictions. We take a different approach, partly because a full Bayesian development would require further non-null modeling assumptions.

Gelman and Shalizi (2012) discuss this and related factors supporting the use of predictive

*p*-values.