A Theorem at the Core of Colliding Bias

1 Introduction

When two, marginally independent, variables affect a third variable, they might become dependent (associated) conditional on the latter [1]. If the causal structure is described by arrows (e.g., A→C←B), the shared effect (C) is called a collider on the path between its causes (A and B). In the context of causal inquiry, where effects are estimated by associations, a newly formed association after conditioning on a collider can add colliding bias – the bias that might arise from conditioning on every collider along a path between the cause and effect of interest.

For example, if A→C←B (Figure 1, Diagram A), conditioning on C might create bias when estimating the effect (here null) of A on B (e.g., Berkson’s bias). Or another example: under the M-structure (Figure 1, Diagram B) conditioning on C might create bias when estimating the effect of E on D [2, 3].

Colliding bias, known by at least half a dozen names [4], is the antithetical counterpart of confounding [5]. Both biases are well recognized in the literature on causal diagrams, and theorems based on d-separation allow for the removal of confounding without adding colliding bias [2]. Nonetheless, d-connection, which might arise after conditioning on a collider, does not necessarily result in bias, because d-connection – the opposite of d-separation – does not imply dependence.

To our knowledge no article was devoted to a basic underlying question: when does conditioning on a collider create an association between its causes? In fact, the literature contains various statements on the possible consequences of conditioning on a collider, some of which are non-specific and others sound like unproven theorems. We present here a general theorem at the core of colliding bias, the origin of which can be traced to the case-only design.

Figure 1

Two structures in which colliding bias might arise following conditioning on C.

2 Notation, definitions, and basic propositions

Throughout this paper let A, B, and C be discrete (non-degenerate) random variables. Let {ai}i=1n and {bk}k=1m be the values of A and B, respectively, where n and m may be either finite or infinite. The lower case letters a,b, and c will denote an arbitrary value of A,B, and C, respectively. For the time being, we fix a value c of C. We will consider the case where A and B are marginally independent causes of C, as may be depicted by the causal diagram A→C←B (Figure 1, Diagram A).

For completeness, we will provide definitions of effects on C and of effect modification between A and B. In particular, we will define both effects and effect modification in terms of the probability ratio, because our results depend on this measure of effect. We shall assume that none of the probabilities mentioned hereafter is zero – a standard assumption under indeterminism – so the probability ratio may always be defined. Specifically, we assume that P(A=a,B=b,C=c)≠0 for all a,b, and c.

First, we define the effect of A on C under the causal structure A→C←B.

The effect of A on C could depend on the value of B. When it does, we say that B modifies A’s effect on C as defined below.

Similarly, we can define the effect of B on C and modification of B’s effect on C by A:

Effect modification is a symmetric property as stated in the proposition below.

By proposition 1, we may speak of A and B modifying each other’s effects on C=c. If A and B do not modify each other’s effect on C=c, then rij(b) and skl(a) do not depend on b and a, respectively. That is, in the absence of effect modification, we may speak of A’s effect on C without noting the value of B and of B’s effect on C without noting the value of A (Definition 5).

Next, we prove a more convenient formula for the effects of A and B on C – in the absence of effect modification.

This result has been referred to as collapsibility of the probability ratio [6, 7] .

Now, we will define what it means for A and B to have a null effect on C.

Which naturally extends to the following definition:

Lastly, we will introduce the notion of an effect matrix.

Note that R(b) (or S(a)) will be an infinite matrix when A (or B) has an infinite number of values. With the notion of an effect matrix, we can more succinctly describe the properties of effect modification and null effects. Specifically, B does not modify A’s effect on C=c iff R(b) does not depend on b. And A has a null effect on C=c iff for every b, R(b) is the matrix whose entries are all 1.

3 Main theorem

In appendix A, we give an alternative proof of theorem 1.

In proving the theorem, we assumed that P(A=a,B=b,C=c)≠0 for all a,b, and c. It may be the case, though, that A and B are already already restricted to take certain values. (For example, when studying the effect of A on C we may fix A to only two values, often labeled “exposed” and “unexposed”.) Theorem 1 will still hold under such restrictions if we replace the phrase “modify each other’s effects on C=c” with the phrase “conditionally modify each other’s effects on C=c” where the latter phrase is defined as follows:

Note that when n (or m) is infinite the notation {1,2,...,n} (or {1,2,...,m}) denotes the positive integers.

4 Some special cases

Some versions of the following statement may be found in the literature:

“If the effects of A and B on C are not null, then A and B are dependent in at least one stratum of C.”

Using theorem 1, we will prove that the statement is true when C is binary, and we will provide a counterexample when C is not binary.

That the above statement holds when C is binary follows from the following proposition:

Proposition 3 and theorem 1 can be combined to give the following corollary.

Corollary 1 does not hold when C is not binary. In that case, it is possible that A and B have non-null effects on C and do not modify each each other’s effects on any value of C. It then follows from theorem 1 that A and B will be independent in every stratum of C. Our will be based on this reasoning.

Below we give an example of a trinary variable C (with values 1,2,3) and binary A and B (with values 0,1) such that A and B have non-null effects on C. Consider the probabilities below, which were found by guessing values that satisfy the conditions for the absence of modification (Example 1).

The effects of A (1 vs. 0) on C=1,2, and 3 when B=0 are 1.8,0.6, and 0.9, respectively.

The effects of A (1 vs. 0) on C=1,2, and 3 when B=1 are 1.8,0.6, and 0.9, respectively.

The effects of B (1 vs. 0) on C=1,2, and 3 when A=0 are 1.3ˉ,2, and 0.6ˉ, respectively.

The effects of B (1 vs. 0) on C=1,2, and 3 when A=1 are 1.3ˉ,2, and 0.6ˉ, respectively.

Therefore, A and B do not modify each other’s effects on any value of C nor do they have null effects on C. Appendix B includes tables with counts, illustrating that A and B are independent in every stratum of C.

5 Related work and possible extensions

Heuristic arguments for binary variables are often used to explain why conditioning on a common effect sometimes creates an association between its causes [8]. For instance, if each of two drugs, A and B, is a deterministic cause of bleeding (C=1), and we are told that John did not take drug B (B=0), we are inclined to guess that John took drug A (A=1) – once we are informed that John suffered bleeding (C=1). That rational guess intuitively points to a conditional association between A and B. Notice, however, that the deterministic story above hides extreme effect modification between A and B:

Previous work has extended the heuristic explanation to a formal deterministic model when effects are monotonic (e.g., taking drug A cannot cause the death of John and prevent the death of Jane) and all variables are binary [9]. Under these constraints and others, it is possible to infer the sign of the conditional covariance between two causes of C in the strata of C [9]. Although a zero covariance is equivalent to independence for binary variables, most of the paper’s results show that the covariance will be non-negative or non-positive, guaranteeing neither dependence nor independence. Result 2 part 6 in that paper, however, is a (weaker) formulation of the following statement: For statement: For binary A,B, and C, if A and B don’t modify each other’s effects on C=c (on the probability ratio scale), then A and B will be independent conditional on C=c. A recent article [10] implicitly proved our theorem 1 in the restricted case of binary variables.

The case of binary variables was also explored, in retrospect, in the context of the case-only design [2]. It was shown that it is possible to estimate the magnitude of a multiplicative interaction between A and B (in our notation) in the stratum of cases (C=1), assuming that A and B are independent in the population from which the cases arose [11, 12, 13, 14]. Stated generically, two marginally independent binary causes do not modify each other’s effect on one value of a binary outcome (on the probability ratio scale), if and only if they are independent conditional on that value. That statement, though not articulated [12, 13, 14], is a specific case of our general theorem.

Notwithstanding the case-only design, the link between effect modification on the multiplicative scale and the consequences of conditioning on a shared effect did not receive much attention, perhaps because the very concept of effect modification is still debated [15, 16, 17, 18]. Moreover, methodological literature tends to favor a deterministic model, which downplays the concept of effect modification and is tightly connected to measuring effects on the additive scale. Whether the preference for difference measures of effect is justified is an open question, but the connection between effect modification and colliding bias may be one reason to prefer the multiplicative scale [19]. As shown here, if we are worried about colliding bias, we should look for heterogeneity of the probability ratio – not the probability difference – even if we eventually choose to estimate probability differences.

Some authors have suggested that A and B (in our notation) will “almost certainly” be associated in at least one stratum of C. Our results provide a formal justification of that phrase and a deeper insight into its meaning. The exceptions are instances in which there is precisely no effect modification on any value of C (e.g., Appendix B). Since no effect modification is just one point (the null) within an infinite range of possibilities, the phrase “almost certainly” is appropriate. Nonetheless, it may be improved. In general, we may say that A and B will “almost “almost certainly” be associated in every stratum of C. And when C is binary, A and B will certainly be associated in at least one stratum of C (Corollary 1). Still, the phrase “almost certainly” is relevant only under a Bayesian framework, conveying a belief that modification is almost always present. (The same reasoning may be used to claim that effects are “almost certainly” never null.) Under a non-Bayesian framework, the state of affairs is fixed and independent of our beliefs: either effect modification is present or it is not.

The consequences of conditioning on a collider through regression do not immediately follow from our theorem. In linear regression, for instance, effect modification is defined by the presence of an interaction term, which is not comparable to our definition of effect modification except under a log probability model. Furthermore, it is unclear how conditioning on a continuous variable through regression corresponds to other methods of conditioning (e.g. restriction and stratification).

A couple of extensions of our work are possible: (1) The main result applies to discrete variables but should hold for continuous variables by replacing probabilities with probability densities (assuming a joint density function) and converting sums to integrals. It may be more complicated to prove the theorem when some variables are continuous and others are discrete. (2) The results are limited to the independent-dependent dichotomy. We conjecture that if A and B are marginally dependent (due to a common cause, for instance), the association between them will be altered upon conditioning if effect modification is present.

A Alternative proof of theorem 1

The proof below formalizes and extends the results for case-only studies [12, 13], utilizing a measure of effect modification that equals a measure of a conditional association between the colliding variables. The connection between two such measures was recently alluded to elsewhere [20].

B Tables illustrating counterexample

We show below tables illustrating that A and B, two causes of C, can be independent in every stratum of C. Consider the next table, in which A and B are marginally independent.

The following tables depict the effects of A on C, stratified on B, and the effects of B on C, stratified on A. The counts and effects correspond to the probabilities mentioned in the text.

As can be seen, there is no effect modification between A and B on any value of C. Using the tables above, we can depict the relation (counts, column percentages) between A and B in each stratum of C:

Evidently, A and B, which were marginally independent, are conditionally independent as well. No association was created between A and B in any stratum of C.