Causal Mediation: When a x b Works and When It Does Not

อ่านฉบับภาษาไทย (Thai version)

Abstract

In the simplest mediation, the indirect effect really is ordinary multiplication: if exercise takes off 2 kg and each kilogram lost lowers blood pressure by 3 mmHg, about 6 mmHg of the fall runs through weight loss. The trouble starts when the second number has no single value. If each kilogram lowers pressure by 3 mmHg in one group and by 1 mmHg in another, the product is 6 or 2, and the multiplication is not wrong: we do not know which slope to multiply. That is an exposure-mediator interaction, and a yes-or-no outcome adds a second problem, because a logistic coefficient is not a change in risk. This article keeps the a x b intuition, shows where it stops, and builds causal mediation from the question of how a treatment works: total, direct and indirect effects, two imagined worlds, the effect scale, the order of thinking, a workflow and six common mistakes.

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How much of the fall runs through weight loss?

A primary care team runs a year-long exercise programme for adults with raised blood pressure. At the end of the year, the people who exercised have lower systolic blood pressure, and they have also lost weight. The team asks a natural question: how much of the fall in pressure happened because of the weight loss, and how much happened by other routes?

A colleague offers the familiar answer: multiply two numbers, the effect of exercise on weight and the effect of weight on blood pressure. That answer is better than its reputation. In the simple case it really is ordinary multiplication, and much of the confusion around mediation comes from jumping to the complicated cases too quickly. This article starts with the multiplication, shows exactly where it stops being enough, and then builds the causal way of thinking that takes over from there.

In the simple case, the indirect effect is ordinary multiplication

Write $A$ for the exposure, $M$ for the mediator and $Y$ for the outcome. A mediator is a variable on the causal path between them: the exposure changes the mediator, and the mediator in turn changes the outcome.

$$A \rightarrow M \rightarrow Y$$

In the exercise programme the chain is exercise → weight loss → lower blood pressure. Suppose exercise makes people lose 2 kg, and each kilogram lost lowers blood pressure by 3 mmHg. Call the first number $a$, the effect of the exposure on the mediator, and the second number $b$, the effect of the mediator on the outcome. The effect that passes through weight is the product of the two.

$$a \times b = 2 \times 3 = 6$$

In words: exercise lowers blood pressure by about 6 mmHg through weight loss. That is all there is to it in simple linear mediation, where both relations are straight lines and each has one slope that applies to everyone.

Hand example: 2 kg times 3 mmHg

The same arithmetic, one step at a time.

1. Exposure to mediator

   \[ a = 2 \ \text{kg lost} \]

   Exercise makes people lose 2 kg.

2. Mediator to outcome

   \[ b = 3 \ \text{mmHg lower per kg lost} \]

   Each kilogram lost lowers blood pressure by 3 mmHg.

3. Multiply

   \[ a \times b = 2 \times 3 = 6 \]

   The two effects multiply, because every kilogram that exercise removes carries the same 3 mmHg.

Result: About 6 mmHg of the fall in blood pressure runs through weight loss.

Signs carry direction. If the mediator is coded as weight change, a loss is negative (a = -2 kg), and if the outcome slope is per kilogram gained (b = +3 mmHg), the product is (-2) x (+3) = -6 mmHg: the same 6 mmHg fall, with the minus sign saying that it is a fall.

Why you cannot always multiply: b may have more than one value

The trouble starts when $b$ does not have one value. Suppose the effect of losing 1 kg on blood pressure looks like this:

• people not taking a blood-pressure drug: 3 mmHg lower;
• people already taking the drug: 1 mmHg lower.

Now $b = 3$ for one group and $b = 1$ for the other. With $a = 2$, the product is either

$$2 \times 3 = 6 \quad \text{or} \quad 2 \times 1 = 2$$

So what is the indirect effect: 6 or 2? That is the whole problem.

The multiplication itself is not wrong. What is missing is a single $b$ to put into it, because the effect of the mediator changes with another factor. When that factor is the exposure itself, so that the mediator's effect on the outcome depends on the level of the exposure, it is called an interaction, here an exposure-mediator interaction.

The simplest summary

A dead-or-alive outcome adds a second problem

Now suppose the outcome is not blood pressure but death, yes or no. That adds another problem, because logistic regression gives its coefficients on the log-odds scale, not as how much the risk goes up. So the hard cases carry two problems at the same time:

$$\boxed{\text{interaction} \rightarrow \text{no single } b}$$$$\boxed{\text{logistic} \rightarrow \text{coefficient} \neq \text{risk}}$$

Both problems return later in this article, each in its own section.

Why mediation analysis: how does the treatment work?

Causal research usually starts with a simple question: does an exposure or treatment affect an outcome?

$$\text{Treatment} \rightarrow \text{Outcome}$$

Suppose a treatment reduces mortality. Knowing that it works is useful, but researchers often want something deeper: how does the treatment produce that effect? Perhaps the treatment lowers inflammation, and lower inflammation in turn reduces mortality.

$$\text{Treatment} \rightarrow \text{Inflammation} \rightarrow \text{Mortality}$$

The intermediate variable, inflammation, is the mediator. A mediator lies on the causal pathway between an exposure and an outcome. Mediators, confounders, colliders (variables caused by two others on a path) and effect modifiers each play a different role in causal analysis, and the sections below keep them apart.

The basic causal structure: two pathways

Let $A$ be the exposure or treatment, $M$ the mediator and $Y$ the outcome. The simplest mediation structure is $A \rightarrow M \rightarrow Y$. Usually there is also a direct pathway, $A \rightarrow Y$, that does not go through the mediator. The complete picture therefore holds two different causal pathways: one through the mediator, and one that bypasses it.

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Total effect: every pathway from A to Y

The total effect answers the broadest causal question: what is the overall effect of changing the exposure? For a treatment, that is what happens to mortality if patients receive Treatment A rather than Treatment B. The total effect includes everything the treatment does, whatever the mechanism.

Total effect = all causal pathways from $A$ to $Y$.

That includes both the pathways through the mediator and the pathways that bypass it.

Direct effect: everything not through the mediator studied

The direct effect is the part of the exposure effect that does not operate through the mediator of interest. Suppose a drug lowers mortality through inflammation,

$$\text{Drug} \rightarrow \text{Inflammation} \rightarrow \text{Mortality}$$

but the drug also improves cardiac function independently of inflammation:

$$\text{Drug} \rightarrow \text{Cardiac function} \rightarrow \text{Mortality}$$

If inflammation is the mediator under study, the second pathway belongs to what is called the direct effect relative to that mediator.

A direct effect does not necessarily mean biologically direct.

There may be many unmeasured mechanisms between $A$ and $Y$. The direct effect simply means the part of the causal effect that is not operating through the mediator being studied.

Indirect effect: through the mediator

The indirect effect is the part of the exposure effect that operates through the mediator. If exercise lowers blood pressure partly because it causes weight loss, then

$$\text{Exercise} \rightarrow \text{Weight reduction} \rightarrow \text{Blood pressure}$$

is a mediated pathway. The indirect effect asks: how much of the exposure's effect on the outcome occurs because the exposure first changes the mediator?

$$\boxed{A \rightarrow M \rightarrow Y}$$

is the mediated pathway. In the simple case at the start of this article, it was the 6 mmHg that ran through weight loss.

A clinical example: how much runs through LDL?

Suppose researchers study whether a new treatment reduces cardiovascular (CV) events, and they believe part of its effect comes from lowering LDL cholesterol.

$$\text{Treatment} \rightarrow \text{LDL} \rightarrow \text{CV event}$$

The treatment may also act in other ways, through inflammation or through endothelial function:

$$\text{Treatment} \rightarrow \text{Inflammation} \rightarrow \text{CV event}$$$$\text{Treatment} \rightarrow \text{Endothelial function} \rightarrow \text{CV event}$$

If LDL is the mediator under study, mediation analysis asks: how much of the treatment effect is transmitted through changes in LDL? It does not claim that LDL is the only mechanism.

Mediation is different from confounding

This distinction is critical. A confounder comes before both the exposure and the outcome in the causal structure. Age, for example, affects who receives a treatment and also affects mortality:

$$\text{Age} \rightarrow \text{Treatment}, \quad \text{Age} \rightarrow \text{Mortality}$$

Age creates an alternative explanation for the observed treatment-outcome association, so confounding usually needs to be controlled. A mediator is different:

$$\text{Treatment} \rightarrow \text{Mediator} \rightarrow \text{Outcome}$$

The mediator is produced by the exposure. If the research question concerns the total treatment effect, automatically adjusting for the mediator can remove part of the very treatment effect being estimated.

$$\boxed{\text{Confounder} \neq \text{Mediator}}$$

Confounders, colliders, mediators and effect modifiers should not be handled identically in an analysis.

Why simply adjusting for the mediator can mislead

A common traditional approach runs two models [1]:

• Model 1 estimates the effect of the treatment on the outcome, $\text{Treatment} \rightarrow \text{Outcome}$.
• Model 2 adds the mediator, $\text{Treatment} + \text{Mediator} \rightarrow \text{Outcome}$.

Researchers then look at how much the treatment coefficient shrinks from Model 1 to Model 2 and call that shrinkage "mediation". This can sometimes give intuition, but it is not a universal causal solution, because regression coefficients do not automatically correspond to causal pathways. Several complications can occur:

• the outcome may be binary;
• the relationship may be nonlinear;
• the exposure and the mediator may interact;
• mediator-outcome confounding may exist;
• variables affected by the treatment may themselves confound the mediator-outcome relationship.

Causal mediation analysis therefore asks a more explicit question:

What causal contrast are we actually trying to estimate?

The counterfactual idea: two imagined worlds

The easiest way into modern causal mediation is an imaginary experiment. Let $A$ be a treatment and $M$ inflammation, and suppose the treatment reduces inflammation. Instead of looking straight at regression coefficients, imagine two hypothetical worlds for the same patient [2, 3].

World 1. The patient receives the treatment, and the patient's inflammation responds naturally to it.

World 2. The patient still receives the treatment, but somehow their inflammation is set to the level it would have had without treatment.

Now ask: would the outcome differ between these two worlds? The treatment is the same in both. The main thing that has changed is the mediator. The difference between the two worlds is therefore the part of the treatment effect that operates through the mediator.

This is the intuition behind counterfactual mediation analysis. A counterfactual is what would have happened under a condition that did not actually occur, such as this patient's inflammation without the treatment they did receive.

Why this is useful

The counterfactual framework forces an exact definition of the direct effect, the indirect effect and the total effect. Rather than saying "this regression coefficient is the mediated effect", the analyst asks: which hypothetical situations are being compared? That is a much clearer causal question.

Natural indirect effect: the intuition

The natural indirect effect asks, roughly: what would happen if the exposure stayed fixed, but the mediator moved from the value it would naturally have under one exposure condition to the value it would naturally have under the other?

In the inflammation example, the treatment is kept present, and inflammation is moved between the level expected under treatment and the level expected without treatment. If the outcome changes, that difference reflects the pathway through inflammation.

There is no need to memorise the mathematical notation first. The core idea is:

$$\boxed{\begin{array}{c}\text{keep the exposure fixed}\\ \downarrow\\ \text{change the mediator}\\ \downarrow\\ \text{observe the outcome difference}\end{array}}$$

That isolates the mediated pathway conceptually.

Natural direct effect: the intuition

For the natural direct effect the logic is reversed. Hold the mediator at the level it would naturally have under one exposure condition, then change the exposure itself.

$$\boxed{\begin{array}{c}\text{keep the mediator fixed}\\ \downarrow\\ \text{change the exposure}\\ \downarrow\\ \text{observe the outcome difference}\end{array}}$$

This is the part of the exposure effect that does not operate through the mediator, under that particular definition.

Why interaction makes mediation more complicated

Return to $\text{Treatment} \rightarrow \text{Inflammation} \rightarrow \text{Mortality}$, but now suppose the effect of inflammation on mortality differs depending on whether the patient received the treatment.

The effect of the mediator depends on the exposure.

This is called an exposure-mediator interaction. Among untreated patients, high inflammation may strongly increase mortality; among treated patients, another mechanism of the treatment may partly protect them from the harm of inflammation. The effect of the mediator is no longer a single fixed quantity.

Simple mediation approaches picture one path $A \rightarrow M$ followed by one fixed effect $M \rightarrow Y$. With interaction, the effect of $M$ on $Y$ depends on $A$, so there is no longer one universal "mediator coefficient" that represents the whole mediated effect. This is the 6-or-2 problem from the start of the article in general form, and counterfactual mediation methods that keep the interaction were built for it [4]. Interaction itself always needs a scale, which Part 1 of this series works through.

A simple example: kidneys and blood pressure

Imagine a treatment that improves kidney function partly by lowering blood pressure. Normally, higher blood pressure leads to a worse kidney outcome. But suppose the treatment also protects kidney tissue directly. Then the harm from high blood pressure may be weaker among treated patients than among untreated ones:

• Untreated: higher blood pressure → substantial kidney damage;
• Treated: higher blood pressure → a smaller increase in kidney damage.

The relation between the mediator and the outcome now depends on treatment status. That is why a more flexible causal mediation framework is needed.

Why binary outcomes add another layer

Suppose the outcome is death, yes or no, or disease, yes or no. Binary outcomes like these are commonly analysed with logistic regression. Logistic regression is useful, but its coefficients are not direct changes in probability: a coefficient operates on the log-odds scale.

A regression coefficient is not automatically a change in absolute risk.

This matters in mediation analysis because the aim is to split an effect into causal quantities that a clinician can interpret.

Why the effect scale matters

Consider mortality falling from 20% to 10%. The same change can be described with several effect measures, and they are different numerical descriptions of the same underlying comparison. The measures themselves are set out in GLM families and effect measures.

1. Risk difference

   \[ 10\% - 20\% = -10\% \]

   The absolute risk fell by 10 percentage points.

2. Risk ratio

   \[ 10\% / 20\% = 0.50 \]

   The risk was cut in half.

3. Odds ratio

   The same data can also be expressed as an odds ratio, a third number for the same comparison.

Result: A mediation analysis must make clear on what effect scale the direct and indirect effects are reported: risk difference, risk ratio, odds ratio or another clinically appropriate scale.

Without the scale, saying that an indirect effect equals 0.5 is difficult to interpret.

Why multiplying two coefficients is not always enough

In simple linear mediation, the traditional idea is

$$\text{indirect effect} \approx a \times b$$

where $a$ is the effect of the exposure on the mediator and $b$ is the association between the mediator and the outcome [1]. This is useful in simple settings, as the 2 kg and 3 mmHg example showed, but it should not become a universal rule. Problems arise when:

• the outcome model is nonlinear;
• the exposure and the mediator interact;
• the mediator is nonlinear;
• there are complex confounding structures;
• the causal effect wanted is on a different scale.

In these situations:

A regression coefficient is not automatically a causal indirect effect.

The causal effect has to be defined first. The statistical model is then used to estimate that causal quantity.

The order of thinking

A common mistake runs in this order:

1. Run the regression.
2. Look at the coefficients.
3. Decide what they mean.

Causal mediation should work in the opposite direction:

1. Define the causal question.
2. Draw the causal structure.
3. Define the direct and indirect effects.
4. Choose the effect scale.
5. Choose appropriate statistical models.
6. Estimate the effects.

This is the same broad principle that runs through causal inference in general.

Mediation and effect modification are different

Another common source of confusion is the difference between mediation and effect modification.

Mediation. $A \rightarrow M \rightarrow Y$: the variable lies on the causal pathway.

Effect modification. The effect of $A$ differs according to another variable. A drug may work strongly in severe disease and weakly in mild disease; severity modifies the treatment effect.

A variable may sometimes take part in complicated causal structures, but conceptually the two answer different questions:

• Mediation = how does the effect happen?
• Effect modification = for whom, or under what condition, is the effect different?

That is why mediator and effect modifier are treated as separate variable roles.

What to consider before claiming causal mediation

Finding that $A$ is related to $M$, and that $M$ is related to $Y$, is not enough to prove mediation. A causal reading requires thinking about confounding. At a minimum, consider potential confounders of three relations:

• the exposure-outcome relation, $A$ and $Y$;
• the exposure-mediator relation, $A$ and $M$;
• the mediator-outcome relation, $M$ and $Y$.

This is why DAG-based thinking matters [5]. Before running any mediation model, ask: could another variable explain the relation between the mediator and the outcome?

A clinical DAG example

Suppose we study $\text{Exercise} \rightarrow \text{Weight} \rightarrow \text{Diabetes}$, but age affects both weight and diabetes. Then age confounds the mediator-outcome relation, and ignoring this structure may make the apparent mediated pathway misleading.

The purpose of a causal diagram is not to make a paper look sophisticated. It is to force an answer to one question: why do we believe the pathway represents causation?

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Mediators should not automatically be adjusted away

Suppose the main research question is: what is the total effect of Treatment A on mortality? If a biological variable lies on the pathway,

$$\text{Treatment} \rightarrow \text{Mediator} \rightarrow \text{Mortality}$$

and you simply adjust for that mediator, you may remove part of the treatment effect you actually want to estimate.

Mediation is about mechanisms, not just prediction

Prediction asks: can this variable help predict what happens next? Mediation asks something fundamentally different: does this variable take part in the causal mechanism through which the exposure changes the outcome?

A highly predictive biomarker is not necessarily a mediator. Likewise, a true mediator does not have to be the strongest predictor.

$$\boxed{\text{Prediction importance} \neq \text{causal mediation}}$$

Predictive analysis and causal explanation are different tasks.

A practical workflow in eight steps

1. Define the causal question. Identify the exposure $A$, the mediator $M$ and the outcome $Y$. Ask: why do we believe $M$ lies on the pathway between $A$ and $Y$?
2. Specify the temporal order. The sequence $A \rightarrow M \rightarrow Y$ should make clinical sense: in the proposed mechanism, the mediator occurs after the exposure and before the outcome.
3. Draw the DAG. Mark the confounders, the mediator, the outcome, possible competing pathways and a possible exposure-mediator interaction.
4. Decide what effect you want: the total, the direct or the indirect effect. Do not begin by looking at regression coefficients; begin by defining the causal estimand, the exact quantity to be estimated.
5. Choose the effect scale. For a binary outcome, decide whether the risk difference, the risk ratio, the odds ratio or another measure is most useful, and state the scale explicitly.
6. Examine interaction. Ask: does the effect of the mediator on the outcome differ by exposure? If it does, the mediation method must allow an exposure-mediator interaction. Do not simply delete the interaction because it makes interpretation harder.
7. Estimate the mediation effects with a method able to estimate the causal effects defined above. In complex situations, counterfactual mediation methods help because they define effects through explicit hypothetical comparisons rather than treating single regression coefficients as causal effects [4].
8. Interpret clinically. Do not stop at "the indirect effect was statistically significant". Ask what biological or clinical mechanism the result supports, and how large the mediated effect is on a clinically meaningful scale.

Six common mistakes

• Calling every intermediate variable a mediator

  A variable measured between the exposure and the outcome is not automatically a causal mediator. There must be a plausible causal pathway A → M → Y.

  Fix: Before calling a variable a mediator, state why it lies on the causal pathway and check that it occurs after the exposure and before the outcome.

• Adjusting for the mediator and calling the remaining coefficient the direct effect

  This can fail in nonlinear or confounded settings. Regression adjustment alone does not automatically identify a causal direct effect.

  Fix: Define the direct effect first as a comparison of hypothetical worlds, then estimate it with a method built for that definition.

• Assuming the mediator coefficient is the indirect effect

  The coefficient for M describes a relation inside a model. The indirect effect describes a causal pathway. They are not automatically the same thing.

  Fix: Treat the indirect effect as a defined causal quantity and estimate it as such, rather than reading it off one coefficient.

• Ignoring exposure-mediator interaction

  If the effect of M on Y depends on A, that interaction is scientifically meaningful. Removing it purely for convenience can misrepresent the underlying causal structure.

  Fix: Keep the interaction and use a mediation method that allows it.

• Dichotomising a continuous mediator unnecessarily

  Turning a continuous mediator into low versus high throws information away and may distort the mediator-outcome relationship.

  Fix: Keep continuous information whenever reasonable.

• Reporting an effect without its scale

  An indirect effect must be interpretable, and a value such as 0.5 means little until its scale is known.

  Fix: Always name the scale: risk difference, risk ratio, odds ratio or another stated scale.

The deepest lesson

Causal mediation analysis is not mainly about finding a special regression formula. It is about understanding how an exposure produces an outcome.

$$\begin{array}{c}\text{Exposure}\\ \downarrow\\ \text{Mediator}\\ \downarrow\\ \text{Outcome}\end{array}$$

Then ask: what would happen if we could alter one part of this pathway while holding another part conceptually fixed? That question leads naturally to the counterfactual framework.

Final memory box

Glossary

mediator (ตัวแปรสื่อกลาง)

A variable on the causal pathway between an exposure and an outcome: the exposure changes it, and it in turn changes the outcome.

total effect

The overall effect of changing the exposure, through all causal pathways from the exposure to the outcome.

direct effect

The part of the exposure effect that does not operate through the mediator being studied; not necessarily biologically direct.

indirect effect

The part of the exposure effect that operates through the mediator.

natural indirect effect (ผลทางอ้อมตามธรรมชาติ)

The outcome difference when the exposure is kept fixed and the mediator moves from its natural value under one exposure condition to its natural value under the other.

natural direct effect (ผลทางตรงตามธรรมชาติ)

The outcome difference when the mediator is held at its natural value under one exposure condition and the exposure itself is changed.

counterfactual

What would have happened under a condition that did not actually occur, such as a treated patient's mediator set to its untreated level.

exposure-mediator interaction

The effect of the mediator on the outcome depends on the level of the exposure, so there is no single mediator effect to multiply.

effect modification (การปรับผล)

The effect of the exposure differs according to another variable; it asks for whom or under what condition the effect is different, not how it happens.

confounder

A variable that comes before both the exposure and the outcome and offers an alternative explanation for their association; unlike a mediator, it is not produced by the exposure.

risk difference (ผลต่างความเสี่ยง)

One risk minus another, an absolute change counted in percentage points.

estimand

The exact causal quantity an analysis sets out to estimate, defined before any model is fitted.

References

1. Baron RM, Kenny DA. The moderator-mediator variable distinction in social psychological research: conceptual, strategic, and statistical considerations. J Pers Soc Psychol. 1986;51(6):1173-1182. doi:10.1037/0022-3514.51.6.1173 https://doi.org/10.1037/0022-3514.51.6.1173
2. Robins JM, Greenland S. Identifiability and exchangeability for direct and indirect effects. Epidemiology. 1992;3(2):143-155. doi:10.1097/00001648-199203000-00013 https://doi.org/10.1097/00001648-199203000-00013
3. Pearl J. Direct and indirect effects. In: Proceedings of the 17th Conference on Uncertainty in Artificial Intelligence (UAI 2001). Morgan Kaufmann; 2001. p. 411-420. https://arxiv.org/abs/1301.2300
4. Valeri L, VanderWeele TJ. Mediation analysis allowing for exposure-mediator interactions and causal interpretation: theoretical assumptions and implementation with SAS and SPSS macros. Psychol Methods. 2013;18(2):137-150. doi:10.1037/a0031034 https://doi.org/10.1037/a0031034
5. VanderWeele TJ. Explanation in causal inference: methods for mediation and interaction. Oxford University Press; 2015. https://hsph.harvard.edu/research/vanderweele-group/books/

Key takeaways

• In simple linear mediation the indirect effect is ordinary multiplication: 2 kg lost times 3 mmHg per kg gives about 6 mmHg of the fall through weight loss.
• When the mediator's effect changes with the exposure, b has no single value (3 or 1, so 6 or 2): the multiplication is not wrong, but nobody knows which b to multiply; a logistic outcome adds a second problem, because its coefficients are log-odds, not risks.
• Mediation asks how an exposure causes an outcome: the total effect covers every pathway, the indirect effect runs through the mediator, and the direct effect is the rest, not necessarily biologically direct.
• A mediator is not a confounder: adjusting for it when the total effect is wanted removes part of that effect, and the shrinkage of a coefficient between two models is not a causal indirect effect.
• Define the causal effect and its scale first, then choose the model: a regression coefficient is not automatically a causal direct or indirect effect.

Related in the wiki: [[causal-interaction-in-clinical-epidemiology-concepts-measurement-and-interpretation]]