Why zero margins change a chi-square test but not your table
Distinguish original table identity from active degrees of freedom, and interpret small expected counts and underflow without inventing observations.
Aggregate counts already support the table test
The public question asks how to avoid expanding millions of counts into individual rows. Its visible data is a complete 6×2 table totaling 333, and a reply recommends an asymptotic table test. Direct table calculation retains counts but cannot recover unprovided individual order, covariates or dependence structure.
Original identity and active degrees of freedom serve different purposes
A zero margin gives zero expected cells and cannot add a usable category to the test degrees of freedom. Its original labels still matter for audit. Retaining both shapes explains how an original 3×3 table can have active df 1 while remaining ineligible for original-2×2 Yates correction.
| Item | Meaning | Misreading to avoid |
|---|---|---|
| N | Sum of complete integer counts | Does not establish individual independence or reconstruct observations |
| Original shape | Identity of every source row and column | Excluding zero margins does not change original-2×2 eligibility |
| Active degrees of freedom | (Positive rows−1)×(positive columns−1) | Zero total or fewer than two positive margins is undefined |
| p and log p | Two numerical representations of one asymptotic upper tail | Floating-point underflow is not mathematical probability zero |
| Small expected counts | Warnings for positive expected cells below 5 | A smaller p does not remove approximation limitations |
Use cell diagnostics to explain the overall statistic
Chi-square contributions add across cells. Pearson residuals retain direction, while standardized residuals also account for margin proportions. Eligible Yates correction changes contributions without replacing signed residuals. Undefined residuals retain a state and blank value rather than becoming zero.
- Retain every cell, original index and source span so duplicate labels do not merge.
- Interpretation depends on the study design; this tool offers no simulation, Fisher or exact test.
- The public 333-count result does not reproduce the unprovided private million-count table.
References
- Original aggregate-table inference discussion
The complete public 6×2 table totals 333. A reply recommends an asymptotic table test; no final author confirmation or private million-count table is supplied.
Tools in this category
Expand a tool to see its steps, options and supported formats, then open its workspace.
Contingency table inferenceCalculate a chi-square independence test directly from aggregate counts, with complete expected cells, contributions and residuals.
Use a two-axis frequency table to inspect margins, asymptotic evidence and the cells behind it. Select the active input, alpha and original 2×2 correction explicitly.
Steps
- Select the active input and provide the complete labeled count table.
- Enter alpha explicitly and choose correction using the original shape.
- Check the total, active degrees of freedom, small expected counts and undefined states.
- Save all eight complete files and interpret the asymptotic result in the study design.
Available options
- Active input
- Paste complete CSV · One original CSV file
- Significance level α
- 0.05
Must be strictly between 0 and 1.
- Yates correction for an original 2×2 table
- Off by default
Capabilities and limits
- Select paste or file as the active input. Use strict UTF-8 comma-separated CSV with nonempty labels on both axes: 1–64 data rows, 1–64 data columns, at most 4,096 cells and 4 MiB. Duplicate labels retain their original indices.
- Counts are nonnegative integers with total N ≤ 10¹². Alpha is an explicit finite number between 0 and 1. Yates applies only to the original 2×2 shape; removing zero margins does not make a larger original table eligible.
- All original rows, columns and zero-margin cells remain. Positive-margin degrees of freedom must be ≤ 100. Zero total or insufficient positive margins produce explicit undefined states. Positive expected counts below 5 warn about the asymptotic approximation.
- The upper tail uses direct positive terms and log arithmetic. Representable small p values retain their value; genuine binary64 underflow retains finite log p. This is an asymptotic independence test and does not reconstruct individuals or perform exact, Fisher or simulation tests.
- One operation shares 10 million work units, 512 MiB memory and a complete 30-second deadline including source read, worker startup, computation, complete validation, cleanup and first publication. Files plus full text allow 32 MiB, compact typed data including native metadata 8 MiB, their aggregate 40 MiB, and binary plus transport metadata 64 MiB. A crossed budget fails the whole operation.
- Download five complete matrices, the full report, settings and the exact original CSV. Screen previews are bounded; full copy and downloads are complete. The original, its filename and settings are processed locally in the current browser.