Tensions in Practice: The unit boundary moves the change between terms¶
The same two firms viewed at two resolutions
Each of two firms owns a low-score plant fixed at 1 and a high-score plant fixed at 3. Both firms retain half of the overall weight. Inside each firm, the plants’ shares move from one half each to one quarter low and three quarters high. The overall weighted mean rises from 2 to 2.5, without either plant’s score changing. If a unit is a plant, this is reweighting; if a unit is a whole firm, each unit’s average has changed.
Track movement among plants
Attribute the change at the level where plant scores and shares are observed.
Track firm-level performance
Summarize what happened to the average performance of each whole firm.
Why these aims pull against each other
Both exact accounts sum to the same change, but the unit boundary decides which term receives it. A firm-level “within” contribution does not prove that individual plants improved.
Choose an arrangement to see what changes and what remains difficult.
Reweight means the contribution from changing unit shares; Within means change inside the chosen units. Rows are contributions to the same overall mean change, and Reweight + Within = Total. The chosen unit is named by the arrangement.
What this choice protects
What it costs
When it fits
Compare the arrangements
Plants are the units
Keep the four plants separate. Each low plant’s global share falls from 1/4 to 1/8 and each high plant’s share rises from 1/4 to 3/8. Plant scores stay 1 and 3.
| Reweight | Within | Total | |
|---|---|---|---|
| Firm A | +0.25 | 0 | +0.25 |
| Firm B | +0.25 | 0 | +0.25 |
| Overall | +0.5 | 0 | +0.5 |
- What it protects
- The account exposes a pure change of weights among unchanged plants.
- What it costs
- It requires plant-level shares and comparable scores; it does not by itself answer a whole-firm change question in the firm-level vocabulary.
- When it fits
- Plausible when plant-level allocation is the relevant descriptive unit and the disaggregated measurements are available.
Illustration note: This is an editorial, deliberately bounded illustration. Its stated rules and any numbers are invented, not observations, recommended settings, or predictions.
Firms are the units
Collapse each firm’s plants into its own weighted average, which changes from 2 to 2.5. The two firm shares remain 1/2 each.
| Reweight | Within | Total | |
|---|---|---|---|
| Firm A | 0 | +0.25 | +0.25 |
| Firm B | 0 | +0.25 | +0.25 |
| Overall | 0 | +0.5 | +0.5 |
- What it protects
- The account reports a real rise in each firm’s weighted average using the coarser unit.
- What it costs
- It hides the internal reallocation responsible for that rise; reading “within firm” as improvement inside every plant would be wrong.
- When it fits
- Plausible when firm-level averages are the intended descriptive target, provided the unit boundary is disclosed.
Illustration note: This is an editorial, deliberately bounded illustration. Its stated rules and any numbers are invented, not observations, recommended settings, or predictions.
What this illustration does—and does not—establish
The source establishes the structural tension; the concrete alternatives and their conditional costs are editorial synthesis. No arrangement is a universal recommendation.
- All weights and scores are invented and comparable by stipulation. The identity is accounting, not evidence of why the shares changed or what intervention would reproduce it.
- The row contributions are to the overall mean, not each firm’s own mean; multiplying a firm’s +0.5 change by its 1/2 share gives +0.25.
- These fixed-support units have no entry, exit, score changes or measurement error. Other cases need those terms handled explicitly.
Source entries
Selection Vs Transmission Decomposition
Selection vs transmission decomposition Population Partition Choice as Substantive (Scopal) supplies the local tension. The setting, alternative arrangements, and stipulated consequences are editorial applications.
Population Partition Choice as Substantive (Scopal)
The cut presupposes a partition into units, but *which* partition is chosen determines what counts as selection versus transmission — change that is "within-unit" at the firm level becomes "between-unit selection" when firms are split into plants.