Statistical Methods

Crosstabs reports descriptive tables and common association statistics. This page states the calculation rules, assumptions, warnings, and current editorial-review status.

Ownership and review status

This methodology is published and maintained by crosstabs.com. A named statistical owner and independent reviewer have not yet been published, so the site does not claim that an anonymous “methods team” has reviewed it. For high-stakes work, verify the result with a qualified statistician or established statistical package.

Report a suspected error or unclear assumption to support@crosstabs.com. Confirmed corrections will update this page, affected outputs, and the changelog; published review metadata changes only after a real review of the affected revision.

Content updated .

Table construction and notation

After active filters, each retained record contributes 1, or its positive finite weight, to one cell. Missing row or column values are excluded by default; the Include missing setting retains them as an explicit (Missing) category.

  • For observed cell Oᵢⱼ, row total Rᵢ, column total Cⱼ, and grand total N, row percentage is 100Oᵢⱼ/Rᵢ, column percentage is 100Oᵢⱼ/Cⱼ, and total percentage is 100Oᵢⱼ/N.
  • Expected frequency under independence is Eᵢⱼ = RᵢCⱼ/N.
  • A valid inferential table needs a positive grand total and positive totals for every displayed row and column.

Pearson, G-test, and residuals

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 ±1.96 receive a nominal two-sided 5% flag. Crosstabs does not adjust these cellwise flags for multiple comparisons.

2×2 calculations

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, ∞].

Association measures

  • 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; the uploaded-data workspace currently sorts category labels lexicographically. Check that order before interpreting ordinal measures.

Outcome Studio predictive model

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, plus rows with missing, non-numeric, zero, or negative selected weights, 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; survey weights are honored in every displayed holdout metric.
  • 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.

Assumptions, weights, and warnings

  • Inference assumes independent observations, mutually exclusive categories, meaningful sampling, positive margins, and adequate expected counts.
  • Numeric weights can be applied to descriptive counts and percentages; missing, non-numeric, zero, or negative weights are excluded and reported.
  • Weighted chi-square, p-values, residuals, and effect sizes are descriptive or approximate unless the weights are genuine frequency weights.
  • Crosstabs does not apply clustering, stratification, finite-population corrections, replicate weights, complex-survey variance estimation, or Rao–Scott corrections.
  • Warnings identify small samples, expected counts below 5, when more than 20% of expected counts are below 5, invalid weights, 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 workspace's descriptive weighting boundary.

Validation evidence

Selected Pearson chi-square, Fisher exact, Cramér's V, and directional Somers' d cases are compared with SciPy through a public reference manifest and automated tests. This evidence is intentionally narrower than a blanket software-equivalence claim and does not replace named review.

Inspect the values, tolerance, reproduction contract, and limits on the statistical validation page.

Primary references

These are intellectual sources for the implemented methods. Listing them does not mean their authors reviewed or endorsed crosstabs.com.