Collider Bias From Conditioning on Cell States

Why Conditioning on a Cell State Can Mislead

A cell-type QTL analysis asks whether genetic variation is associated with gene expression in a defined cellular context. That context may be a broad cell type, or it may be a more specific state, such as an activated, stressed, or cycling state within a cell type. Restricting an analysis to cells in a particular state can be scientifically useful, but it changes which cells contribute to the comparison. If genotype and other causes of expression both influence entry into that state, the restriction can create an association between genotype and those other causes.

This is collider bias. It is not evidence that the genotype affects expression. Rather, conditioning on a shared consequence of genotype and another expression-related factor can make the genotype and that factor statistically associated within the selected cells. The induced association can then appear as a genotype–expression association.

The Causal Diagram and Its Interpretation

Let \(G\) denote a genetic variant, \(S\) a cell state, \(U\) another cause of that state, and \(Y\) expression of a gene. The relevant diagram is:

The two arrows into \(S\) make state a collider. Before conditioning on \(S\), the path from \(G\) to \(Y\) through \(S\) is blocked at the collider. If \(G\) and \(U\) are otherwise independent after accounting for relevant population structure and other design factors, this diagram does not contain an open path associating \(G\) with \(Y\). Restricting the analysis to cells with \(S=1\), however, opens the path \(G \rightarrow S \leftarrow U \rightarrow Y\).

The mechanism can be understood informally: among cells selected to be in the state, a genotype that makes entry into the state more likely can be accompanied by a lower prevalence of another cause that also makes entry likely. If that other cause changes expression, the genotype will appear associated with expression in the selected group, even when it has no direct effect on expression.

The variables need not be measured at the same level. Genotype is typically donor-level, while state and expression may be cell-level. For example, a donor’s genotype may influence the probability that a cell enters an activation state; a donor-level exposure or a cell-level signaling input may also influence activation and expression. The causal question concerns how selection into the analyzed state changes the composition of the observations, not simply the level at which each variable was recorded.

A Numerical Demonstration

The following hypothetical calculation makes the induced association explicit. Let \(G\) and \(U\) each be binary, with values 0 and 1, and assume they are independent in the population: each of the four \((G,U)\) combinations has probability \(0.25\). Let \(S=1\) indicate that a cell is included in the analyzed state. Suppose the probabilities of being in that state are as follows.

Both \(G\) and \(U\) increase the probability of the state. Let expression follow \(Y=2+3U+\varepsilon\), where \(\varepsilon\) has conditional mean zero and is independent of \(G\), \(U\), and selection. In this model, changing \(G\) while holding \(U\) fixed does not change expected expression. The value of \(U\), however, changes expected expression by 3 units.

Among selected cells with \(G=1\), the relative selection weights for \(U=0\) and \(U=1\) are \(0.60\) and \(0.90\). Since the population probabilities of the two \(U\) values are equal, they cancel in the conditional probability:

For selected cells with \(G=0\), the corresponding weights are \(0.10\) and \(0.60\), so:

Thus, within the selected state, \(U=1\) is less common among cells with \(G=1\) than among cells with \(G=0\). Expected expression in each genotype group is obtained from the expression model:

The selected-sample contrast, defined as the mean for \(G=1\) minus the mean for \(G=0\), is \(3.80-4.571\approx -0.7714\) expression units. Yet the model contains no direct effect of \(G\) on \(Y\). The contrast arises because conditioning on \(S=1\) makes \(G\) and \(U\) associated, and \(U\) affects \(Y\).

Why the Induced Association Appears

The key calculation is a conditional probability. For binary \(U\), if \(P(U=1)=P(U=0)\) and \(G\) is independent of \(U\), then within genotype group \(g\):

The expression follows from Bayes’ rule: the numerator is proportional to the probability of \(U=1\) and selection, and the denominator sums that quantity over both possible values of \(U\). In the numerical example, the probabilities of \(U=0\) and \(U=1\) before selection are equal, so those common factors cancel. The resulting conditional prevalence depends on genotype whenever the state-selection probabilities differ across the \(G,U\) combinations.

More generally, suppose \(E[Y\mid G,U,S=1]=\mu+\tau G+\gamma U\), with a mean-zero residual in the selected group. The genotype contrast among selected observations is:

The first term, \(\tau\), is the model’s conditional genetic effect at a fixed value of \(U\). The second term is the contribution of the selection-induced difference in \(U\). If \(\tau=0\), the selected-group contrast can still be nonzero whenever \(\gamma\ne 0\) and the two conditional prevalences differ. In the numerical example, the prevalence difference is \(0.60-6/7=-9/35\), so the induced contrast is \(3(-9/35)=-27/35\approx -0.7714\) units.

Distinguishing Collider Bias From Mediation

Not every association involving a cell state is collider bias. If genotype affects state and state in turn affects expression, the diagram includes \(G \rightarrow S \rightarrow Y\). In that case, state lies on a causal pathway from genotype to expression, and restricting or adjusting for state may change the genetic effect being estimated. This is different from the collider path \(G \rightarrow S \leftarrow U \rightarrow Y\), although both structures can exist at once.

The scientific question matters. An analysis of expression among cells observed in a state estimates an association for that selected group. It does not automatically estimate the effect of genotype on expression in all cells, nor does it isolate a direct genetic effect. Interpret the result in relation to the selection rule and the causal paths that could influence both state membership and expression.

Practical Implications for Cell-Type QTLs

A cell-state restriction may be appropriate when the target is explicitly expression among cells in that state. The important point is to recognize that this target is conditional on state membership and may be vulnerable to collider bias. A state label is not a neutral filter if genotype, donor characteristics, environmental exposures, cell viability, or technical capture processes affect which cells receive that label or are observed.

A causal diagram helps make the assumptions visible. Before interpreting a state-specific association, ask whether genotype could affect state membership, whether another cause of state also affects expression, and whether the analysis selects or adjusts on state. Also consider whether population structure or other factors already associate genotype with those causes; such associations can create additional paths and are distinct from the collider mechanism described here.

If a plausible alternative cause is measured, modeling or stratifying on it may help assess whether it explains the induced association, as in the worked example. This is not a universal repair: adjustment choices depend on the causal structure and on the estimand. In particular, conditioning on additional variables can introduce other biases or change the question being answered. Reporting the state definition, the selection rule, and the causal interpretation of the estimated association makes these limitations easier to evaluate.

Common Mistake

Incorrect reasoning: “Genotype is fixed before the cell state, so selecting cells by state cannot bias a genotype–expression association.” The timing of genotype does not prevent selection bias. If genotype influences state membership and another cause of expression also influences state membership, conditioning on that shared consequence can associate genotype with the other cause.

Correction: Draw the relevant arrows and identify what the analysis conditions on. Under \(G \rightarrow S \leftarrow U \rightarrow Y\), conditioning on \(S\) opens a noncausal path from \(G\) to \(Y\), even when \(G\) and \(U\) were independent before selection and genotype has no direct effect on expression. The diagram establishes a possible mechanism, not proof that a particular result is biased.