Compositional data: a rising share can hide unchanged activity
A share can increase while its underlying activity stays constant. A share can also fall while activity grows. The denominator is not an incidental display choice: dividing every component by their common total creates a constrained vector whose parts must sum to one. Research that treats those shares as unrelated measurements can mistake arithmetic dependence for an economic mechanism.
The deliverable is a linked ledger of original amounts, closed shares and explicitly chosen component ratios. Preserve the total whenever it is meaningful, document what each part measures, and classify zeros before calculating logarithms. The purpose is to answer a defined composition question without silently replacing it with an absolute-growth claim.
Define the whole before closing the parts
Assume a hypothetical research desk counts completed reviews in three mutually exclusive categories A, B and C during equal observation windows. A review has one category, all eligible reviews are captured, and definitions are stable. The amounts are counts in the same unit. The category universe, window and inclusion rule belong in the evidence record.
For nonnegative amounts x with a positive total S, define closure by p[i] = x[i]/S, where S is the sum of all included parts. Percentages are 100 times p. Multiplying every amount by the same positive constant leaves the shares unchanged. Consequently, a composition alone cannot tell whether the desk completed 100 reviews or 1,000 in the same proportions.
Van den Boogaart’s original compositions vignette, version 0.91 from May 2008, distinguishes analyses of amounts from analyses of compositions in section 2. Sections 4.3 and 6.1–6.3 discuss pairwise relations, retained totals and subcompositions. These concepts guide the worksheet; the review counts below are invented for transparent arithmetic.
Reconcile unchanged amounts with changing shares
Start with A=10, B=20 and C=70. Compare separate hypothetical later scenarios rather than treating them as consecutive months:
| Scenario | A amount | B amount | C amount | Total | A share | B share | C share | A/B |
|---|---|---|---|---|---|---|---|---|
| Baseline | 10 | 20 | 70 | 100 | 10% | 20% | 70% | 0.5 |
| C expands | 10 | 20 | 170 | 200 | 5% | 10% | 85% | 0.5 |
| C contracts | 10 | 20 | 20 | 50 | 20% | 40% | 40% | 0.5 |
| A genuinely expands | 20 | 20 | 70 | 110 | 18.18% | 18.18% | 63.64% | 1 |
In the expansion scenario, A’s share falls five percentage points, or 50% relative to its baseline share, while A’s count changes by zero. In the contraction scenario, A’s share doubles to 20%, again without any change in A’s count. Neither sentence about shares is arithmetically wrong. The unsupported step would be calling either one a change in A’s absolute activity.
In the final scenario, A actually doubles and its share rises by about 8.18 percentage points. B remains at 20 reviews but its share falls. A/B doubles because A changes relative to B; it did not change in either C-only scenario. This combination lets a reviewer distinguish an A-versus-B question from an A-versus-everything question.
Keep unrounded fractions for calculations. The displayed final shares are rounded, so their last decimals are presentation values. A rounding discrepancy should not be “fixed” by editing one category’s original count.
Use ratios for a stated relative question
For positive A and B, p[A]/p[B] = (A/S)/(B/S) = A/B. The common denominator cancels. If only A and B are retained and reclosed, their shares are A/(A+B) and B/(A+B); their ratio remains A/B. In all three baseline and C-only rows, the two-part composition is one-third A and two-thirds B.
That invariance is useful, but it has a boundary. The new one-third share is A’s share among A and B, not its original share among A, B and C. Dropping C changes the whole being described even though the A/B ratio is preserved. Record both the selected parts and the original category universe.
A natural log ratio ln(A/B) is approximately −0.6931 at baseline and zero when A and B are both 20. Its change is approximately +0.6931, the logarithm of a doubling in the ratio. Reversing the reference gives ln(B/A) and reverses the sign. Neither reference provides a universal “positive performance” score.
Choose the reference from the research question and retain it across comparisons. A stable ratio could reflect two constant amounts or two amounts growing proportionately. If A and B both double while C remains 70, A/B stays 0.5. Ratios remove the common scale information; they do not prove that the scale was constant.
Recognize the dependence introduced by closure
With only two positive parts, p[B]=1−p[A]. Whenever A’s share varies, ordinary Pearson correlation between the two shares is exactly −1. This algebra holds even without evidence that one category displaces the other through a competitive process. A negative association can therefore be generated by the representation itself.
With three or more parts, every pair need not have negative correlation. In the C-only scenarios, A and B shares move together because both use the changing common total. The fixed sum constrains the full covariance structure, but it does not dictate the sign of every pair. Use this distinction when reviewing a claim that all compositional associations are automatically negative.
If shares enter a model with an intercept, including every share also creates exact linear dependence because their sum equals the intercept column. Dropping a part or choosing appropriate log-ratio coordinates requires a meaningful interpretation; it is not proof that the remaining slopes identify substitution or causation. A descriptive composition question may be answered well by shares without making either claim.
Classify zero and missing entries before a log ratio
The compositions 2.0-9 reference manual, missing.compositions on printed pages 130–133 and print.acomp on page 190, distinguishes structural zeros, below-detection values and missingness types. A blank and a genuine zero should not become interchangeable merely because both interrupt a logarithm.
For an original boundary example, suppose category A is not available under one desk’s operating mandate. A=0 is structural there. ln(A/B) has no finite value, and adding one invented review changes the support and the question. Consider a separately defined comparable positive subcomposition or an analysis that explicitly represents presence and amounts; describe the changed target population.
Alternatively, a data provider may round a small positive amount to zero or suppress values below a stated threshold. Preserve the threshold and reporting rule. If B=20 and the only information is 0<A<1, then ln(A/B)<ln(1/20), approximately −2.9957, with no finite lower bound from that information alone. Replacing A with 0.5 would select one assumed value, not recover an observed one.
Adding a small constant is not neutral. For the same displayed A=0 and B=20, adding 0.1 to both produces ratio 0.1/20.1, whereas adding one produces 1/21. Their logs are about −5.3033 and −3.0445. The materially different answers are driven by invented amounts. Any justified replacement model needs its own assumptions and sensitivity record.
An unreported category amount is missing, not zero. Closing only reported parts silently changes the universe. A total of zero makes closure undefined; it does not produce an all-zero vector summing to one. Keep these cases flagged until measurement evidence supports a treatment.
Build a reproducible review package
Retrieve original category files and any independently reported total. Reconcile their inclusion boundaries before calculating shares. If one source excludes unfinished reviews and another includes them, their totals need not agree; explain the scope difference rather than averaging incompatible totals.
For example, if A, B and C sum to 100 but a matching complete count is 120, twenty reviews remain unreconciled. Investigate omitted categories, duplicate identities and date boundaries. Closing the known three parts to 100% describes the classified subset. Label it accordingly and retain the unclassified amount and possible selection mechanism; adding a category later also requires a comparable historical universe.
| Review step | Evidence to retain | Action for an unresolved problem |
|---|---|---|
| Define parts | category codebook, units, window and exclusivity | separate overlaps and changed definitions |
| Preserve scale | original amounts and independently measured total | label absolute growth unidentified if unavailable |
| Close | sum, denominator and unrounded shares | flag incomplete or zero totals |
| Choose comparison | reference part, pair ratio and selected universe | explain a changed subcomposition |
| Classify entries | genuine zero, detection threshold, missing code | avoid automatic pseudocounts |
| Interpret | amount change, share change and ratio change | keep mechanism claims tied to other evidence |
If only percentages are available, report the composition and its limitations. An independently verified total for the same universe can restore amounts by multiplication, subject to rounding and measurement error. A total for another date or population cannot do that job.
Update the ledger when coverage expands, a category is reclassified, a suppressed amount is released or a matching total becomes available. Recompute historical comparisons under a consistent codebook where possible and preserve the original version. The revised narrative should identify whether the new conclusion comes from activity, the denominator, the reference component or improved measurement.
FAQ
Does a larger share prove that a category grew?
No. The C-contraction scenario doubles A’s share while A stays at ten. Obtain a compatible total or original amount to assess absolute growth.
Can adding a tiny constant always fix a zero?
No. First determine why the zero appears. A structural absence, a detection limit and a missing report require different treatment and may support different targets.