Statistical methods

After active filters, each retained record contributes 1 to an ordinary workspace cell. A supported saved survey design may instead execute its declared weight and variance contract. An untyped weight selection returns no table. Missing row or column values are excluded by default; the Include missing setting retains them as an explicit (Missing) category.

• For observed cell O_ij, row total R_i, column total C_j, and grand total N, row percentage is 100O_ij/R_i, column percentage is 100O_ij/C_j, and total percentage is 100O_ij/N.
• Expected frequency under independence is E_ij = R_iC_j/N.
• A valid inferential table needs a positive grand total and positive totals for every displayed row and column.

Saved projects use Research Execution Contract v1. Row sets, multibanners, table and banner filters, weights, category order, missing-value rules, wave mappings, and test profiles are compiled into one versioned input. A value is applied exactly or the table is rejected; saved choices are not silently ignored.

• Column-proportion tests compare each pair of testable banner columns with a two-sided pooled z-test. A saved profile controls 90%, 95%, or 99% confidence tiers, Bonferroni or no correction, minimum base, unweighted or weighted effective bases, and display mode.
• Ad-hoc workspace defaults are 95%/90% confidence, Bonferroni correction across testable column pairs, minimum unweighted respondent base 30, and letter display.
• An untyped numeric weight cannot produce a table or column tests. A saved project may execute ordinary tests for an explicit non-negative integer frequency weight, and a supported saved survey design executes design-aware inference. Calibration and analytic weights remain unsupported for ordinary inference.
• Every contract-bearing result includes engine and method versions, per-setting diagnostics, and canonical SHA-256 input/result digests. The digests prove deterministic equality under this contract version; they are not signatures or evidence that a sample is representative.

Pearson's statistic is χ² = Σ(Oij − Eij)²/Eij, with df = (r − 1)(c − 1). Its p-value is the upper tail of the chi-square distribution. The likelihood-ratio alternative is G = 2Σ Oij log(Oij/Eij) over nonzero observed cells, using the same degrees of freedom and an asymptotic reference distribution.

Adjusted residuals are (Oij − Eij) / √[Eij(1 − Ri/N)(1 − Cj/N)]. Cells beyond the saved profile's strict two-sided threshold receive a directional flag (90%: ±1.645; 95%: ±1.960; 99%: ±2.576). Crosstabs does not adjust these cellwise residual flags for multiple comparisons.

Yates' correction uses Σ max(|O − E| − 0.5, 0)²/E. It is displayed for 2×2 tables and is intended for frequency counts. Fisher's exact test is a two-sided Fisher–Irwin, fixed-margin hypergeometric test: it sums feasible tables whose probability is no greater than the observed table. It is available only when all four cells are non-negative safe integers and the fixed-margin support contains at most 100,000 tables; it is hidden for weighted workspace tables.

For [[a,b],[c,d]], the odds ratio is ad/bc. The displayed 95% confidence interval is the large-sample Woolf interval exp(log(OR) ± 1.96√(1/a + 1/b + 1/c + 1/d)), not an exact interval. A zero cell produces the uninformative interval [0, ∞].

• Cramér's V is the uncorrected V = √[χ²/(N min(r−1,c−1))]. Labels such as small, medium, and large use heuristic Cohen-style thresholds; they are not intrinsic classifications.
• Phi is the signed 2×2 coefficient. The contingency coefficient is √[χ²/(χ²+N)] and its attainable maximum depends on table dimensions.
• Goodman–Kruskal lambda is proportional reduction in prediction error. Gamma is (concordant−discordant)/(concordant+discordant).
• Kendall tau-b and tau-c account for ties and table shape. Somers' d is reported symmetrically and directionally as d(row|column) and d(column|row).
• Theil's U reports uncertainty reduction in each direction and symmetrically.

Gamma, Kendall tau, and Somers' d are meaningful only when the displayed category sequence represents the intended ordinal order. Manual count calculators preserve entered order. Saved row and banner category order executes exactly; unlisted observed levels follow the deterministic numeric-aware label sort. Check the recorded order before interpreting ordinal measures.

Outcome Studio fits a local binary logistic model for one selected success value versus all other observed values. Numeric drivers are standardized; categorical drivers use reference-level indicator terms. The fit applies an L2 ridge penalty of 0.5 for stability and is bounded to eight selected drivers and 40 encoded terms.

• Rows missing the outcome or any selected driver are excluded listwise. Successful results show used and excluded counts and the exclusion diagnostic.
• Validation uses one deterministic, outcome-stratified 80/20 split. AUC, Brier score, accuracy at a 0.50 threshold, and calibration gap are calculated on held-out rows.
• Outcome Studio is currently unweighted. Any selected workspace weight—including one attached to a supported crosstab survey design—blocks Outcome modeling until a typed, design-aware regression policy exists; no rows are evaluated and no run is persisted.
• At least 60 complete rows, 25 successes, and 25 other outcomes are required. Predictors with no variance are omitted; categorical predictors above 20 levels and designs above 40 encoded terms are blocked before model allocation.
• Driver rank is based on the largest absolute standardized coefficient per variable. Odds ratios and p-values are model summaries, not causal effects; regularization and single-holdout selection must be considered when interpreting them.
• Observed segments are descriptive. Three-group k-means diagnostics run only with at least two numeric drivers and use at most 1,500 deterministic observed rows. Scenario and synthetic-cohort values are model outputs and never become observed evidence.
• The model is not survival analysis, multilevel modeling, causal inference, true multiclass classification, or complex-survey variance estimation. Repeated holdout, cross-validation, fairness audits, and external validation remain analyst responsibilities.

The public capacity and workload envelope lists the deterministic domain fixtures exercised by the release suite.

• Inference assumes independent observations, mutually exclusive categories, meaningful sampling, positive margins, and adequate expected counts.
• Numeric weight candidates are preserved, but an ad-hoc or legacy untyped selection fails closed before results or evidence are returned.
• Ordinary weighted chi-square, p-values, residuals, effect sizes, and column tests are unavailable until an explicit supported semantic policy authorizes them. A declared frequency weight runs ordinary tests on its whole-number counts.
• With a declared survey (sampling) weight, column letters and column-mean letters use the effective-base test (approximation, as in SPSS Custom Tables/Q): each banner column's Kish effective base n_eff = (Σw)²/Σw² replaces its sample size in pooled t-tests, with the same correction and minimum-base rules. Chi-square, residuals and effect sizes are not reported; full design-based tests stay in Complex Samples.
• Saved tab-book tables can bind a validated Taylor-linearized or replicate-weight survey design. Supported runs disclose strata, PSU, FPC or replicate policy, design degrees of freedom, t intervals, design effects, effective bases, and Rao–Scott or adjusted-Wald association. Invalid designs fail closed; independent statistical review is still pending.
• Warnings identify small samples, expected counts below 5, when more than 20% of expected counts are below 5, blocked or invalid weight contracts, and very wide tables. They are diagnostics, not a guarantee that inference is valid.

The answer-first survey weighting and raking guide distinguishes design, calibration, and frequency weights and explains the quick-workspace fail-closed boundary and supported saved-design path.