Placebo Inference

GeoSC uses SparseSC in-space placebo inference because geo experiments often have too few treated units for conventional large-sample approximations.

For each treated-sized subset of eligible controls, SparseSC constructs an honest placebo effect. The top-level p-value compares the absolute average treated effect with the absolute average placebo effects and applies an add-one correction. It is two-sided.

When feasible assignments exceed sparse_sc_max_n_pl, SparseSC samples assignments using sparse_sc_placebo_seed; otherwise it enumerates them. The result reports the possible and effective counts and labels the mode exact, sampled, or unknown.

Finite support matters. With $M^*$ effective placebos the nominal resolution is $1/(M^*+1)$ . The decision rule is p_value < alpha. If the minimum p-value is greater than or equal to alpha, that rule cannot be met. Sampling adds reference-set variability even when the optimiser is stable; preserve the seed.

The p-value is calibrated only relative to the placebo assignment logic and its exchangeability assumptions. Non-comparable donors, spillover, differential measurement changes, or poor pre-fit can make a numerically small p-value misleading. SparseSC placebo intervals are likewise finite-reference-set quantities, not model-free guarantees.