Why Missing Lineage Outcomes Need a Model
A cell’s terminal lineage may be unknown because its fate label was not recorded, because too little of its trajectory was measured, or because follow-up ended before a branch-specific outcome became visible. These situations do not all contribute the same information. A missing label can still inform branch probabilities if the chance of recording depends on fate; a censored trajectory can be informative if the branches imply different probabilities of the observed partial history. Treating every incomplete record as though it were a complete, randomly sampled fate can therefore distort estimated branch probabilities.
Let \(Z_i\) denote the latent branch of cell \(i\), with \(Z_i\in\{A,B\}\), and let \(\pi_A=P(Z_i=A)\), \(\pi_B=P(Z_i=B)\), where \(\pi_A+\pi_B=1\). Conditional on branch \(k\), the cell’s trajectory or measured outcome \(Y_i\) has density or probability mass function \(f_k(y;\theta_k)\). The branch label is latent when it is not observed. The key operation is to sum over possible latent branches, weighting each branch-specific likelihood by its branch probability.
This extends the likelihood for branching trajectories by accounting for incomplete observation rather than assigning an unobserved branch label as if it were known. The model must describe both the lineage process and, when relevant, the process that determines what gets recorded.
Marginalizing an Unobserved Branch
Suppose a cell has an observed feature vector \(y_i\) but no branch label. If the feature model is \(f_k(y_i;\theta_k)\) on branch \(k\), and recording is independent of branch and outcome, its likelihood contribution is
The sum is required because either branch could have generated the observed data. It is not valid to use only the most likely branch unless the model explicitly treats that branch assignment as known. For a record containing only a subset \(y_{i,\mathrm{obs}}\) of the outcome, use the branch-specific likelihood of that subset, integrating the unobserved components out:
This expression assumes the observed-data model is correctly specified and that the mechanism causing components to be unobserved can be ignored for the parameters of interest. That assumption requires justification: in particular, observation availability must not selectively depend on the latent branch in an unmodeled way.
Worked Example: A Partially Observed Trajectory
Suppose the modeled branch probabilities are \(\pi_A=0.40\) and \(\pi_B=0.60\). A cell has a partially observed trajectory \(y_{\mathrm{obs}}\), for which the branch-specific likelihoods are \(f_A(y_{\mathrm{obs}})=0.40\) and \(f_B(y_{\mathrm{obs}})=0.15\). Assume the observation mechanism has the same probability under both branches, so its factor does not favor either branch.
The marginal likelihood contribution is
The two terms show how much each branch contributes to the observed-data likelihood. Normalizing them gives the model-based branch weights for this record: \(0.16/0.25=0.6400\) for A and \(0.09/0.25=0.3600\) for B. These weights use both the prior branch probabilities and the partial trajectory evidence.
If the trajectory were entirely absent and the probability of its being absent were the same under both branches, the contribution from that record would not distinguish the branches. For example, if each branch has probability \(0.20\) of an entirely missing trajectory, then
The relative branch contributions remain \(0.40\) and \(0.60\). The missing record supplies no branch-discriminating evidence under this assumption.
Fate-Dependent Recording Changes the Likelihood
Let \(R_i=1\) mean that a terminal fate label is recorded, and let \(q_k=P(R_i=1\mid Z_i=k)\) be the branch-specific recording probability. If a label is recorded, the probability of observing label \(k\) in the full cohort is \(\pi_kq_k\). The probability that a randomly selected cell has any recorded label is \(\pi_Aq_A+\pi_Bq_B\). Thus, among cells whose labels were recorded, the branch probabilities are
This is a selection effect: the observed labeled cells represent the underlying branch proportions only when recording is equally likely across branches, or when the unequal probabilities are otherwise accounted for. If only labeled cells are analyzed, their likelihood must be conditional on being labeled. For \(n_A\) recorded A labels and \(n_B\) recorded B labels, that likelihood is proportional to
The denominator is essential: it normalizes the branch probabilities among recorded labels. Omitting it and maximizing \((\pi_Aq_A)^{n_A}(\pi_Bq_B)^{n_B}\) would not correct the selection, because fixed factors \(q_A^{n_A}q_B^{n_B}\) do not change the maximizer of that incorrectly specified expression.
Worked Example: Correcting Unequal Label Recording
In a hypothetical lineage dataset, 30 recorded labels are A and 36 are B. Suppose the label-recording probabilities are known to be \(q_A=0.50\) and \(q_B=0.90\). Estimate the underlying branch proportions using the conditional likelihood among recorded labels.
The observed proportions among recorded labels are \(30/66\) for A and \(36/66\) for B. Since the conditional A proportion is \(\pi_Aq_A/(\pi_Aq_A+\pi_Bq_B)\), the implied underlying branch weights are proportional to the observed counts divided by their recording probabilities:
Normalizing these weights gives
For a direct check, the fitted fraction of A labels among recorded cells is \((0.600)(0.50)/[(0.600)(0.50)+(0.400)(0.90)]=0.30/0.66=0.4545\), matching \(30/66=0.4545\). The estimated underlying A proportion, 0.600, is higher than the observed labeled proportion because A labels were less likely to be recorded.
This correction requires the recording probabilities to be known or estimable from additional information, and the conditional-likelihood model to be appropriate. If both the underlying branch proportions and the branch-specific recording probabilities are unknown, the labeled counts alone generally cannot separate them. The full cohort likelihood can use the number of cells with missing labels as well, but it still needs a model for their recording probabilities.
Censored Fate Data
Censoring differs from a missing label because it may provide a bound or partial history. For example, if a cell has not reached a terminal fate by follow-up time \(c\), its fate time \(T\) is known to exceed \(c\). Let \(S_k(c)=P(T>c\mid Z_i=k)\) be the branch-specific survival probability. If the censoring process is independent of the fate time conditional on branch and modeled covariates, and its distribution is common across branches, a cell observed without a terminal fate by \(c\) contributes
If a terminal event is observed at time \(t\) with branch \(k\) known, the corresponding fate-time contribution is \(\pi_k f_k(t)\), where \(f_k\) is the branch-specific event-time density. If the terminal branch is unknown, sum the compatible branch-specific event contributions instead. The censoring model can be omitted from these expressions only when it is independent as specified and its likelihood factor does not depend on the lineage parameters. If follow-up or recording depends on branch, that dependence must be included or justified away.
Worked Example: A Cell Censored Before Terminal Fate
Suppose \(\pi_A=0.60\) and \(\pi_B=0.40\). At a follow-up time of 10 days, the modeled probabilities of not yet observing a terminal fate are \(S_A(10)=0.70\) and \(S_B(10)=0.40\). Assume censoring is independent of the fate time conditional on branch, with a common censoring process across branches. A cell is followed for 10 days but has no terminal fate recorded.
Its marginal likelihood contribution is
The conditional branch weights for this censored record are \(0.42/0.58=0.7241\) for A and \(0.16/0.58=0.2759\) for B. They differ from the cohort proportions because remaining without a terminal fate by day 10 is more likely under branch A in this model. These are model-based weights for this record, not proof that the cell will ultimately take branch A.
Assumptions, Diagnostics, and Alternatives
Missingness assumptions describe how observation relates to observed and unobserved data. Under missing completely at random (MCAR), the chance of a value being missing does not depend on measured or unmeasured outcomes. Under missing at random (MAR), that chance may depend on observed information but not on the missing value after conditioning on the observed information in the model. Under missing not at random (MNAR), it still depends on unobserved outcomes, such as latent fate, after that conditioning. These labels are meaningful only relative to the variables included in the model.
MAR can make a missingness mechanism ignorable for some outcome-model parameters when the model conditions on the relevant observed information and the missingness and outcome parameters are distinct. It does not mean that unequal branch representation among recorded cells can be ignored. If availability depends on branch, estimate or model that dependence, use an appropriate conditional likelihood, or use a design with known recording probabilities. For censored event times, independent censoring is a related but specific condition: conditional on modeled covariates, censoring must not carry additional information about the event time. Branch-dependent censoring may require a branch-specific censoring model.
A short derivation clarifies why the conditional normalization matters. Write \(r_A=P(Z=A\mid R=1)\) and \(r_B=1-r_A\). The conditional label likelihood is proportional to \(r_A^{n_A}r_B^{n_B}\), which is maximized at \(r_A=n_A/(n_A+n_B)\). Since \(r_A=\pi_Aq_A/[\pi_Aq_A+(1-\pi_A)q_B]\), solving for the implied branch odds gives
Consequently, \(\pi_A=(n_A/q_A)/(n_A/q_A+n_B/q_B)\). This derivation also shows the limitation: when \(q_A\) and \(q_B\) are unknown, many combinations of recording rates and true branch proportions can produce the same observed fraction.
Useful checks include comparing recording rates across known branches in a validation subset, tabulating missingness by observed covariates, and checking whether the fitted model reproduces the observed proportions of recorded labels and censored records. Sensitivity analysis is important when MNAR mechanisms are plausible: vary branch-specific recording probabilities over defensible ranges and examine how the estimated \(\pi_k\) changes. A model that fits recorded trajectories well can still give biased branch proportions if its selection assumptions are wrong.
Complete-case analysis is simple but can target the recorded-cell composition rather than the underlying composition. Inverse-probability weighting can correct known selection probabilities by weighting recorded observations by their inverse recording probability, but it can be unstable when some probabilities are very small and does not itself solve unknown MNAR selection. A joint likelihood explicitly models both lineage outcomes and observation or censoring; it uses more assumptions but can use all observed information coherently. These approaches are not interchangeable: the right choice depends on what is known about the recording and follow-up process.
Common pitfalls include treating a censored cell as though it had no information, assigning every incomplete cell to its most likely branch before fitting, and assuming that missingness is harmless because it is unrelated to the recorded expression values. The correction is to specify what is observed, what is latent, and how the observation process could depend on branch or outcome before writing the likelihood. A marginal likelihood retains uncertainty about the latent branch, while an explicit missingness model addresses selection when that selection affects the estimand.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A fraction \(r_A\) of recorded labels are A. If \(q_A\) and \(q_B\) are the branch-specific recording probabilities, write the relationship between \(r_A\) and \(\pi_A\).
- In a hypothetical dataset, 20 A labels and 30 B labels are recorded, with \(q_A=0.50\) and \(q_B=0.75\). What are the corrected branch-proportion estimates?
- A cell’s outcome is wholly missing, and its missingness probability is identical under A and B. Does that record change the relative branch weights? Explain.
- For a cell censored at time \(c\), write its likelihood contribution if the branch-specific survival probabilities are \(S_A(c)\) and \(S_B(c)\).
- Why can labeled-cell counts alone fail to identify both the true branch proportions and branch-specific recording probabilities?
Solutions. (1) \(r_A=\pi_Aq_A/[\pi_Aq_A+(1-\pi_A)q_B]\). (2) The adjusted weights are \(20/0.50=40\) and \(30/0.75=40\), so \(\hat{\pi}_A=40/80=0.500\) and \(\hat{\pi}_B=0.500\). (3) No. A common missingness factor multiplies both branch contributions equally, so their normalized relative weights remain \(\pi_A\) and \(\pi_B\). (4) The contribution is \(\pi_AS_A(c)+\pi_BS_B(c)\), under the stated independent-censoring assumptions. (5) The observed A fraction depends on the products \(\pi_Aq_A\) and \(\pi_Bq_B\), so different underlying proportions and recording probabilities can yield the same conditional recorded-label proportions.