# Statistical method implemented in interSAE 0.1.0

Let `s` denote a probability sample with outcome `y`, design weight `d`, area
`a`, and categorical auxiliary vector `x`. Let `N` be the unknown vector of
counts in the complete area-by-auxiliary population table. Published census
margins are represented as

```text
A N = b,   N >= 0,
```

where each row of `A` aggregates complete cells to one published census cell.
Structural-zero declarations remove impossible complete cells before fitting.

## Population reconstruction

The weighted survey table, after adding a positive support prior, is used as a
seed. Generalized iterative proportional fitting computes the minimum-divergence
projection of the seed onto all supplied margin constraints. This fitted table
is not described as identified when the rank of `A` is smaller than the number
of feasible complete cells.

## Outcome estimator

A design-weighted generalized linear model estimates the cell prediction
`m_hat(x)`. For a reported domain `D`, the synthetic total is

```text
T_syn(D) = sum_{c in D} N_hat(c) m_hat(c).
```

Survey weights are calibrated within supported complete cells to their fitted
population counts. The model-assisted total adds a residual term:

```text
T_MA(D) = T_syn(D) + sum_{i in s and D} w_cal(i) [y(i) - m_hat(x(i))].
```

The domain mean or proportion is `T_MA(D) / N_hat(D)`. For a binary response,
the final proportion is restricted to the unit interval.

## Identification sensitivity

The fitted joint population table is one member of a generally non-singleton
feasible set. With cell prediction vector `p` and domain indicator `q`, a
synthetic domain mean is the linear-fractional target

```text
(p' N) / (q' N).
```

`identification_bounds()` applies the Charnes--Cooper transformation and linear
programming to minimize and maximize this target subject to the census
constraints. A finite sensitivity value `gamma` adds

```text
N_hat(c) / gamma <= N(c) <= gamma N_hat(c).
```

`gamma = Inf` removes these association restrictions. Bounds are conditional
on fitted cell predictions and concern latent-table non-identification; they do
not replace sampling inference.

## Sampling and refitting variation

The multiplier bootstrap draws one exponential multiplier per PSU, then
renormalizes weights within strata. Each replicate repeats population
reconstruction and outcome-model fitting. Supplied replicate-weight columns
can be used instead. The resulting standard error combines survey-weight and
model-refitting variation conditional on fixed published census margins.

## Required reporting

Applications should report the survey design, outcome formula, supplied
margins, structural zeros, reconstruction tolerance, nullity, bootstrap method,
and all sensitivity values. A narrow bootstrap interval does not resolve a wide
identification interval.
