COMPSC — Compositional Synthetic Controls in the Aitchison Geometry =================================================================== When to use it -------------- Reach for COMPSC when the outcome is a composition — a vector of shares that sum to a whole — a single unit is treated, and what you care about is the *relative* structure of the mix rather than any one share on its own. An electricity generation mix split across gas, coal and renewables; a marketing budget split across channels; brand share within a product category; a modal split across car, transit and bicycle; vote share across parties. The defining feature is a sum constraint: if one share rises, others must fall. Two things go wrong if you ignore the geometry of shares. The first is multiplicity. Running a separate synthetic control per share gives each share its own donor mix, so the fitted shares need not sum to one and no single counterfactual unit exists. A common weight vector fixes this, and that is what :doc:`propsc` does. The second is scale dependence, and it is the reason COMPSC exists. A Euclidean objective on raw shares penalises absolute deviations, not proportional ones. Doubling a category that holds 2 percent of the total moves the objective by roughly :math:`(0.02)^2`, while the same proportional error in a category at 50 percent moves it by :math:`(0.5)^2` — some six hundred times more. The fit is therefore driven by the dominant categories, and a small emerging category is almost invisible to the optimiser even when its relative evolution is the whole question. Consider three regions splitting output across services, industry and agriculture: A at (82.5, 15.0, 2.5) percent, B at (72.5, 25.0, 2.5), and C at (92.5, 5.0, 2.5). A is the same Euclidean distance from B and from C. But the services-to-industry ratio is 5.5 in A, 2.9 in B and 18.5 in C — structurally, B is much the closer match, and only a log-ratio geometry sees that. COMPSC works in log-ratios, where those two problems dissolve together. If your composition has only large, comparably sized categories and you have a block of treated units adopting simultaneously, prefer :doc:`propsc`, whose synthetic-difference-in-differences machinery and jackknife inference are better suited to that design. COMPSC is for one treated unit and a relative-scale question. Notation -------- We observe :math:`J` units over :math:`T` periods. Unit :math:`j = 1` is treated from a known date :math:`T_0`, with :math:`1 < T_0 < T`; units :math:`j = 2, \dots, J` form the donor pool and are never treated. For each unit and period we observe shares across :math:`p` mutually exclusive and exhaustive categories, .. math:: \pi_{j,t} = (\pi_{1,j,t}, \dots, \pi_{p,j,t}) \in \mathcal{S}^{p-1}, \qquad \mathcal{S}^{p-1} = \Big\{ \pi \in \mathbb{R}^{p}_{+} : \textstyle\sum_{k} \pi_{k} = 1 \Big\}, where :math:`\mathcal{S}^{p-1}` is the probability simplex — the set of all valid compositions. Write :math:`\pi^{0}_{j,t}` and :math:`\pi^{1}_{j,t}` for the share vectors absent and under treatment. The object of interest is the treated unit's counterfactual composition :math:`\pi^{0}_{1,t}` for :math:`t > T_0`. The simplex is not a Euclidean space, and the transform that linearises it is the log-ratio. Two versions matter here. The additive log-ratio takes ratios against a nominated baseline category :math:`p`, .. math:: \operatorname{alr}(\pi)_{k} = \log \frac{\pi_{k}}{\pi_{p}}, \qquad k = 1, \dots, p - 1, and the centered log-ratio takes ratios against the geometric mean of all categories, .. math:: \operatorname{clr}(\pi)_{k} = \log \pi_{k} - \frac{1}{p} \sum_{m=1}^{p} \log \pi_{m}, \qquad k = 1, \dots, p. Distance on the simplex is the Aitchison distance, which compares every pair of categories by their log-ratio, .. math:: :label: aitchison \delta_{A}(\pi, \pi') = \Bigg[ \frac{1}{2p} \sum_{i=1}^{p} \sum_{j=1}^{p} \bigg( \log \frac{\pi_{i}}{\pi_{j}} - \log \frac{\pi'_{i}}{\pi'_{j}} \bigg)^{2} \Bigg]^{1/2} = \big\| \operatorname{clr}(\pi) - \operatorname{clr}(\pi') \big\|_{2}. The second equality is the useful one: the centered log-ratio is an isometry, so ordinary Euclidean distance between clr vectors *is* Aitchison distance on the simplex. The additive log-ratio is not an isometry — a point we return to under Assumption 5. Where the log-ratios come from ------------------------------ The transform is not a convenience. Suppose the shares aggregate individual discrete choices: each person in unit :math:`j` at time :math:`t` picks one of the :math:`p` alternatives, with random utility :math:`U_{ik,j,t} = V_{k,j,t} + \varepsilon_{ik,j,t}` and type-I extreme value shocks. Then choice probabilities follow the multinomial logit, and the log-odds of category :math:`k` against the baseline equal the relative systematic utility exactly: .. math:: \log \frac{\pi^{0}_{k,j,t}}{\pi^{0}_{p,j,t}} = V^{0}_{k,j,t} - V^{0}_{p,j,t} =: \tilde{V}^{0}_{k,j,t}. The observable log-ratios *are* the preference parameters governing choice. That is why a factor model is imposed on them rather than on the shares, and why the donor weights read as preference similarity: a donor earns weight to the extent that its population's relative preferences track the treated unit's. Assumptions ----------- 1. Positivity. Every share is strictly positive: :math:`\pi^{0}_{k,j,t} > 0` for all :math:`k, j, t`. Remark. Log-ratios are undefined at zero. In practice a structural zero is repaired by adding a small constant and re-closing, the standard device in compositional data analysis; ``zero_replacement`` controls it and can be set to 0 to reject zeros instead. A category that is zero for a unit throughout the sample is better handled by dropping that unit from the donor pool, since the repaired value is arbitrary and its log-ratio is extreme. 2. Factor structure on relative utilities. For each category :math:`k`, .. math:: \tilde{V}^{0}_{k,j,t} = \delta_{k,t} + \theta'_{k,t} Z_{j} + \lambda'_{k,t} \mu_{j} + \varepsilon_{k,j,t}, where :math:`\mu_{j}` are unobserved unit loadings, common across all :math:`p - 1` relative-utility equations within a unit, and :math:`\varepsilon_{k,j,t}` is mean-zero given :math:`(Z_j, \mu_j)`. Remark. The loadings being shared across categories is what licenses fitting one weight vector to all of them, and what makes the stacking below an efficiency gain rather than an assumption of convenience. Note the factor model sits on the log-ratios, not on the shares: because the transform is nonlinear, a factor model on one does not imply a factor model on the other, and the utilities are the economically primitive object. 3. Convex hull. There exist weights :math:`w^{*}_{j} \ge 0` summing to one with :math:`Z_{1} = \sum_{j \ge 2} w^{*}_{j} Z_{j}` and :math:`\mu_{1} = \sum_{j \ge 2} w^{*}_{j} \mu_{j}`. Remark. This is the familiar synthetic-control requirement, transposed to the space of preference parameters: the treated unit's persistent preference structure must be expressible as a mixture of the donors'. It fails in the usual way — when the treated unit is extreme, here meaning its category ratios lie outside the range the donors span. Check it by reading ``pre_treatment_aitchison_rmspe``, not by eyeballing the share levels. 4. Uniqueness of the weights. The donor log-ratio matrix has full column rank over the pre-period, so :math:`w^{*}` is the unique minimiser. Remark. Consistency needs only :math:`(p-1) T_0 \to \infty`, so it can come from a long pre-period or from many categories. But a perfect pre-treatment fit is not evidence the weights are identified: if the donors lie in a low-dimensional factor space with no idiosyncratic variation, many weight vectors fit equally well and the reported one is arbitrary. Treat a near-zero pre-treatment distance with a short pre-period as a warning. 5. Choice of geometry. The fitting objective is the pre-treatment distance in log-ratio coordinates, and which coordinates you choose is a modelling decision exposed as ``geometry``. Remark. The source paper's Algorithm 1 uses the additive log-ratio and describes the result as minimising Aitchison distance. Those are different objectives: alr is not an isometry for :eq:`aitchison`, so the alr fit depends on which category is nominated as the baseline — an arbitrary labelling choice. ``geometry="clr"`` is the default here and does minimise pre-treatment Aitchison distance, invariantly to relabelling. ``geometry="alr"`` with an explicit ``baseline`` reproduces the paper. The result carries ``baseline_sensitivity``, the largest change in the fitted weights across the possible baselines; when it is large the two routes give materially different synthetic units and the clr fit is the defensible one. Identification and estimation ----------------------------- Under Assumptions 1–3 the treated unit's counterfactual log-ratios are a convex combination of the donors', and the counterfactual composition follows by inverting the transform. The weights solve a convex quadratic program on the simplex, .. math:: \hat{w} = \operatorname*{arg\,min}_{w \ge 0,\ \sum_{j} w_{j} = 1} \sum_{t=1}^{T_0} \sum_{k} \Big( \ell_{k}(\pi_{1,t}) - \sum_{j \ge 2} w_{j}\, \ell_{k}(\pi_{j,t}) \Big)^{2}, where :math:`\ell` is the configured log-ratio map. Because the coordinate equations share the unit loadings :math:`\mu_j`, they stack into a single regression with :math:`(p-1) T_0` scalar observations rather than :math:`T_0` vector-valued ones — the multi-outcome stacking of Sun, Ben-Michael and Feller (2025), which :doc:`scmo` implements for general multiple outcomes. Given the weights, the counterfactual is the Fréchet barycenter of the donor compositions under the Aitchison metric, which works out to the closure of their weighted geometric mean: .. math:: \hat{\pi}^{0}_{1,t} = \mathcal{C}\Big( \textstyle\prod_{j \ge 2} \pi_{k,j,t}^{\hat{w}_{j}} \Big)_{k=1}^{p}, with :math:`\mathcal{C}` the closure (renormalisation). Standard synthetic control takes the arithmetic mean of the donors; COMPSC takes the geometric mean. The result is a valid composition automatically — strictly positive, summing to one — so the share effects sum to zero by construction rather than by correction. Estimands --------- Three read-outs, all on the result object. ``share_effects`` is the per-period effect on each share in share units, :math:`\pi^{1}_{k,1,t} - \hat{\pi}^{0}_{k,1,t}`, summing to zero across categories. This is the compositional analogue of the usual ATT. ``log_ratio_effects`` is the treatment-induced change in the odds of one category against another, .. math:: \mathrm{LRTE}_{k\ell,t} = \log \frac{\pi^{1}_{k,1,t}}{\pi^{1}_{\ell,1,t}} - \log \frac{\hat{\pi}^{0}_{k,1,t}}{\hat{\pi}^{0}_{\ell,1,t}}, reported for every ordered pair. A value of 1 means the odds of :math:`k` against :math:`\ell` are :math:`e \approx 2.7` times their counterfactual. This is the natural estimand under the simplex geometry, and it is the one that makes a small category legible: a category can gain ground relative to another while losing share in absolute terms, and vice versa. ``aitchison_distance`` is the scalar summary :math:`\delta_A` of total compositional displacement per period — how far the whole observed mix has moved from its counterfactual. Inference and diagnostics ------------------------- Inference is the Abadie permutation test, with the Aitchison distance in place of the usual root mean squared prediction error so the test statistic matches the estimation objective. Each donor is treated in turn as if it had been intervened on at :math:`T_0`, refitting against the others; the statistic is the post-to-pre Aitchison RMSPE ratio. Donors whose pre-treatment fit is worse than ``placebo_screen`` times the treated unit's are dropped before the comparison (the paper uses 5), and the p-value is the share of the retained set whose ratio is at least the treated unit's. Set ``inference="none"`` to skip it. The power of any such permutation test is bounded by the size of the retained set: with 36 units retained the smallest attainable p-value is :math:`1/36 \approx 0.028`, so a donor pool of 20 cannot produce evidence at the 1 percent level however large the effect. ``placebo.n_retained`` reports the number actually used. Two further diagnostics. ``pre_treatment_aitchison_rmspe`` is the quantity the clr objective minimises, and is what to read when judging Assumption 3; ``pre_treatment_share_rmspe`` is the same fit measured on the level scale, for comparison against a conventional synthetic control. And ``baseline_sensitivity`` is the geometry check described under Assumption 5. Example ------- A three-category compositional panel from a latent factor model, with one treated unit and a shift toward the first category after period 8. .. code-block:: python import numpy as np import pandas as pd from mlsynth import COMPSC rng = np.random.default_rng(0) J, T, K, T0 = 12, 16, 3, 8 factors = rng.standard_normal((T, 2, K - 1)) loadings = rng.standard_normal((J, 2)) V = np.einsum("tfk,jf->jtk", factors, loadings) V += 0.3 * rng.standard_normal(V.shape) V[0] = 0.5 * V[1] + 0.3 * V[2] + 0.2 * V[3] # treated in the hull V[0] += 0.15 * rng.standard_normal(V[0].shape) # its own transitory shocks V[0, T0:, 0] += 1.2 # the treatment logits = np.concatenate([V, np.zeros((J, T, 1))], axis=-1) P = np.exp(logits) P /= P.sum(axis=-1, keepdims=True) rows = [] for j in range(J): for t in range(T): rows.append({"unit": f"u{j:02d}", "time": t, "treat": int(j == 0 and t >= T0), "gas": P[j, t, 0], "coal": P[j, t, 1], "renew": P[j, t, 2]}) df = pd.DataFrame(rows) res = COMPSC({"df": df, "outcomes": ["gas", "coal", "renew"], "treat": "treat", "unitid": "unit", "time": "time", "target": "gas", "display_graphs": False}).fit() print(res.att_vector) # per-category effects, summing to zero print(res.sum_constraint) # ~0 to machine precision print(res.log_ratio_effects[-1, 0, 1]) # final-period gas-vs-coal log-odds effect print(res.placebo.p_value) Verification ------------ COMPSC reproduces Boussim (2026) exactly on the paper's Pennsylvania application. Under ``geometry="alr"`` with renewables as the baseline it returns all nine donor weights of Table 1 to three decimals (Illinois 0.233, Wyoming 0.208, Iowa 0.168, Nebraska 0.149, Massachusetts 0.102, Connecticut 0.048, New Jersey 0.043, Indiana 0.032, Montana 0.016), an Aitchison pre-treatment RMSPE of 0.187 and share RMSPE of 1.34 percentage points, every cell of Table 2 across all eleven reported years, and the placebo result of section 6.4 — a treated ratio of 9.18 against 36 retained units for a p-value of 0.111, with Michigan, Louisiana and Florida the three exceeding placebos. The correction of Assumption 5 is not cosmetic on that panel: switching to the default clr geometry moves the weights by 0.85 in :math:`L_1`, changes the donor set, improves the Aitchison pre-treatment fit from 0.187 to 0.177, and reduces the 2022 natural-gas effect from 60.4 to 51.1 percentage points. The benchmark case that pins all of this is `benchmarks/cases/compsc_pennsylvania.py `_, running on ``basedata/pa_aeps_generation.csv``. See :doc:`replications/compsc` for the full validation, including the category recipe recovered from the paper's balance table and the geometry comparison. Core API -------- .. autoclass:: mlsynth.COMPSC :members: :undoc-members: :show-inheritance: .. autoclass:: mlsynth.config_models.COMPSCConfig :members: :undoc-members: :show-inheritance: .. autoclass:: mlsynth.utils.compsc_helpers.structures.COMPSCResults :members: :undoc-members: :show-inheritance: References ---------- Boussim, O. (2026). "Compositional Synthetic Controls." arXiv:2607.16991. Aitchison, J. (1982). "The Statistical Analysis of Compositional Data." Journal of the Royal Statistical Society: Series B 44(2), 139–160. Egozcue, J. J., V. Pawlowsky-Glahn, G. Mateu-Figueras, and C. Barceló-Vidal (2003). "Isometric Logratio Transformations for Compositional Data Analysis." Mathematical Geology 35(3), 279–300. Sun, L., E. Ben-Michael, and A. Feller (2025). "Using Multiple Outcomes to Improve the Synthetic Control Method." Review of Economics and Statistics. Abadie, A., A. Diamond, and J. Hainmueller (2010). "Synthetic Control Methods for Comparative Case Studies: Estimating the Effect of California's Tobacco Control Program." Journal of the American Statistical Association 105(490), 493–505.