Tensions in Practice: Whole-system response in tension with an isolated direct path¶
An invented machine with a controllable relay
A machine has mode A equal to 0 or 1. Normally its relay follows M = A, and output is Y = 2A + 3M. Switching A from 0 to 1 changes output from 0 to 5. If an independent clamp holds M at 0 in both tests, the same mode switch changes output from 0 to 2. The first contrast measures the total response; the second deliberately excludes the relay path.
Measure the total response
Let the machine’s intermediate relay respond normally.
Isolate the direct contribution
Hold the relay fixed while changing the mode.
Why these aims pull against each other
Controlling the intermediate variable changes the causal question. A direct-path test cannot be reported as the full operating effect.
Choose an arrangement to see what changes and what remains difficult.
Arrows represent the declared causal equations. Clamping the intermediate relay removes A→M while retaining the direct A→Y contribution; computed outputs are given beside each arrangement.
What this choice protects
What it costs
When it fits
Compare the arrangements
Let the relay respond
Compare A = 0 with A = 1 under the native equation M = A. Outputs are 0 and 5.
- What it protects
- The contrast includes both the direct contribution 2 and the relay contribution 3.
- What it costs
- It does not isolate the direct contribution from the relay-mediated one.
- When it fits
- Fits measuring the full effect of the mode change during normal operation.
Illustration note: The total difference is 5 only under the declared equations and absence of other changes.
Hold the relay at zero
Use an external control to set M = 0 in both runs. Outputs are 0 and 2 as A changes.
- What it protects
- The contrast isolates the direct A-to-Y path in this system.
- What it costs
- It requires an independent relay clamp and no longer represents normal operation’s total response.
- When it fits
- Fits diagnosing the direct path when clamping M is feasible and introduces no other output effects.
Illustration note: This is a controlled intervention on M, not a claim that statistical matching on a post-treatment variable generally identifies a direct effect.
What this illustration does—and does not—establish
The source supplies the structural tension; the invented example makes one relation inspectable. Costs and conditions are part of each arrangement, not exceptions to a universal recommendation.
- The equations are a complete invented causal model, not findings from observational data.
- The source warns about matching away a mediator when the goal is the total effect. Here the changed direct-effect target is explicitly intentional.
- Real mediator adjustment can introduce confounding or selection bias; no identification recipe for such data is asserted.
Source entries
Control Sample
The canonical tension motivates this comparison. The setting, finite values and arrangements are declared editorial illustrations, not measured findings.
Scopal: Matching On Everything Can Match Away the Effect
The prime says the match-list and the confound-list are the same list, but over-matching is a real failure: if the control is matched on a variable that lies on the causal pathway from treatment to outcome, the difference operator subtracts out part of the very effect under study.
The source operation
A control sample is a deliberately matched comparator held alongside the case of interest so that the *difference* between them isolates the effect of the factor being tested from the mass of factors present in both. The structural pattern is *paired observation under contrast*: one group, batch, or specimen receives the manipulation and another, otherwise indistinguishable, does not, and the inferential weight rests on what changes between them rather than on what happens to either alone.