Statistics and Uncertainty

Statistics creates common sample analyses and presents native reports with their distribution counts, comparisons, test methods, intervals and assumptions. Grid calculates every result through the existing native formula runtime. The surface does not fit a model, compute a p-value, invent an interval, choose a significance conclusion or infer causality in the browser.

Start with the sample comparison model. Its two named numeric samples produce COUNT, AVERAGE, MEDIAN, STDEV.S, MIN, MAX and COUNTIF outputs, T.TEST results and CONFIDENCE.T mean intervals. All numerical results are native formulas. Assumptions and source labels are explicit authored declarations that the study owner must review.

The main view highlights each sample's mean, size and standard deviation, comparisons and intervals. Distribution bars display the report's existing native bin counts; bar length is a display scale, not a fitted distribution. The first authored assumption remains visible. Expand All assumptions and sources to review the rest, Full sample summaries for medians and ranges, and each test's hypotheses and threshold for its interpretation. Print expands these sections. Native errors stay visible in the primary results.

Are categories associated?

Choose New analysis → Are categories associated? and select a named observed count table or a rectangular range, for example Counts or Data!A1:C3. Use two–eight row categories and two–eight column categories, nonnegative integer counts and a positive total at most one billion. Keep headers and totals outside the source. Analysis options folds optional row/column names (one per line in source order), alpha and the report name. Blank names use row/column positions; authored names must match the table dimensions.

The native report shows each observed count alongside its expectation under independence, the counted total, Pearson chi-square, contingency degrees of freedom (rows-1)*(columns-1) and Cramér V. Cramér V describes association magnitude, without direction or a causal interpretation. Cell differences are inspection hints, not independent significance tests.

The Pearson upper-tail p-value requires positive row/column margins and every expected count at least five. For sparse 2×2 tables with positive margins and at most 1000 observations, the native report instead names its Fisher conditional two-sided exact test. It enumerates at most 501 supported tables with the same margins and sums probabilities no greater than the observed table's probability, with a relative tie tolerance of 1e-12. This probability-ordering convention follows the R Fisher test definition; it is neither twice a one-sided tail nor a mid-p test. Exact inference still uses floating-point arithmetic: nonpositive/nonfinite probabilities or normalization outside 1e-10 refuse the selected result, without an approximate fallback.

Larger sparse tables and larger exact budgets retain counts/descriptive estimates while refusing inference. A zero margin refuses chi-square and effect size too. There is no odds-ratio confidence interval, continuity correction or substitution of another study design. Counts must represent independent observations classified once into mutually exclusive categories; repeated, matched or clustered observations need an appropriate study-specific analysis.

The canonical model includes Category association and Sparse category association, with named Control/ Treatment and Success/Failure categories. Formula writes and separate report connections use the existing acknowledgement and draft-recovery workflow. Captures retain all native count-table cells; comparison matches recorded category positions and names, without calculating new differences or significance.

Start an analysis

Add a Statistics surface and choose New analysis. Pick a named numeric sample or cell range, such as SampleA or Data!A1:A100; named array samples appear as suggestions. One sample summary produces native summaries, counts below and at or above the mean, and a Student-t confidence interval for the mean. Compare two samples adds the mean difference and a two-sided Welch t-test. Analysis options holds alpha, paired or equal-variance study designs, and the report name. Paired design adds a summary of aligned A-minus-B differences and a confidence interval for their population mean. It requires matching sample shapes, complete numeric pairs and at least two pairs; missing or mismatched pairs produce visible native errors rather than being dropped or realigned. Pairs must be independent and differences approximately normal. The paired mean-difference interval replaces individual mean intervals in this workflow, so it answers the comparison directly. An interval containing zero is compatible with no mean difference under those assumptions; it is not proof of equivalence. Defaults use a fresh name; replacing an existing report requires an explicit checkbox.

Create analysis authors an ordinary Grid formula through the native source owner, then connects its saved result through a separate configuration write. The captured result appears while setup is open; it is labeled a saved preview until the connected live report is observed. If connection fails, the saved report remains available and is never recreated on retry. Keep analysis and stop waiting retains an uncertain operation; Observe saved analysis is a read-only recovery action. An ambiguous write is not repeated automatically. While the outcome is uncertain, controls keep the original pending report and options fixed. Observation must identify that exact report and setup before confirming it. Close setup without another write closes the form without claiming the analysis was saved or retaining the local form after closure. Use Report setup or Code to connect a retained report when recovery cannot prove the connection. Guided creation requires acknowledged source/configuration owners; otherwise the source editor remains available.

Values use compact number formatting, with the exact native number available on hover and in report downloads. No browser calculation supplies statistical results. Review sampling assumptions before interpreting the analysis.

Compare several groups

Open examples/canonical/statistics-independent-groups.grid for a complete native teaching report.

Choose New analysis → Compare several groups for three–six independent numeric groups. Select one source per group; Add group includes another. Unequal sample sizes are allowed. Every group must be a complete finite numeric row/column vector with at least two values; the total is limited to 100000. Missing observations are refused rather than silently removed.

The default pooled report shows group summaries, pooled pointwise mean intervals, an omnibus one-way ANOVA F test, sums of squares/degrees of freedom and observed eta squared. The native F statistic divides between-group mean square by within-group mean square, with degrees of freedom k−1 and N−k. Group mean intervals also appear on a shared axis, with native estimates as dots and interval endpoints as bars. Exact values remain in the interval cards and captures. Overlap is not a pairwise test. An unavailable selected estimate or endpoint leaves the chart unavailable rather than displaying a partial comparison. See NIST's one-way ANOVA explanation. An omnibus result addresses whether all population means are equal; it does not identify particular differing pairs. Pairwise comparisons are an explicit optional analysis described below.

This analysis assumes independent observations and groups, normal within-group errors and equal population variances. It is unsuitable for paired, repeated or clustered measurements. Pooled remains the default; a variance-method choice never follows automatically from a sample variance or test result. Group mean intervals are individual intervals without multiplicity adjustment; overlap is not a pairwise test. Eta squared describes the observed fraction of variation between groups, not a causal or unbiased population effect. Zero pooled within-group variance or nonfinite sums refuse inference and pooled intervals explicitly. Captures, comparisons and CSV retain these native fields. Selected native work is linear in the bounded selected groups' observations; no browser statistics or additional ordinary runtime path is introduced.

Unequal variances: Welch

Choose Analysis options → Group variance method → Welch when the study calls for independent groups with unequal population variances. This selects native Welch one-way ANOVA. It retains the same complete-vector and 100000-total observation admission. Each group must have positive finite sample variance; zero variance, nonfinite moments/weights or a nonfinite corrected statistic refuses the omnibus result. No pooled or other fallback is substituted.

Native weights are n_i / s_i²; the weighted mean uses normalized weights. With q = Σ[(1 − w_i / Σw)² / (n_i − 1)] / (k² − 1), the corrected F statistic is Σ[w_i (mean_i − weighted_mean)²] / [(k − 1) (1 + 2(k − 2)q)] and degrees of freedom are k − 1 and 1 / (3q). The right-tail probability uses native F.DIST.RT, preserving fractional denominator degrees of freedom. This follows R Core's maintained Welch convention.

Welch group intervals use each group's own sample variance, standard error SQRT(s_i² / n_i) and native Student-t critical value with n_i − 1 degrees of freedom. They are individual pointwise intervals without multiplicity adjustment. Finite group intervals can remain available when only the omnibus variance weights overflow; every refused native value remains explicit. The report does not supply pooled eta squared, repeated-measures analysis, or verification of sampling assumptions. Pairwise comparisons remain optional.

Open examples/canonical/statistics-welch-groups.grid for the unequal-size, unequal-variance example. Its maintained native execution returns F 19.57531086181936, denominator df 5.4673073414025914 and right-tail p 0.0032183239277718545. Positive finite moments and finite uncertainty are required; missing observations are not removed or replaced.

Optional comparisons between every group pair

In Analysis options, enable Compare every group pair with Holm adjustment to request all pairs among the selected three–six groups (at most fifteen). It is off by default; both existing omnibus formulas remain unchanged when off. The signed difference is the left group mean minus the right group mean. The compact table retains every pair, raw two-sided probability and Holm-adjusted probability; its disclosure holds the complete ordered source family, method, alpha, hypotheses and exact native values. Omnibus significance does not select pairs or authorize a smaller family.

Pooled pairs use the omnibus common within-group variance and N−k degrees of freedom with native T.DIST.2T. Welch pairs use native T.TEST with two-sided, unequal-variance selection. They retain the respective independence and within-group distribution assumptions; Welch requires positive finite variance in every group. Missing observations are not dropped or substituted. Each valid probability participates in native Holm step-down adjustment: sort all m raw probabilities, take cumulative maxima of (m + 1 − rank) × p, clamp at one and restore the original pair order. This follows R Core's Holm convention. If any pair fails, every adjusted probability is unavailable. Native errors and other valid raw probabilities remain visible; no failed pair is removed.

Holm controls family-wise error under valid individual tests. It does not verify sampling assumptions, measure effect size, or make the pointwise group mean intervals multiplicity-adjusted. Optional Bonferroni pairwise intervals are a separate selection described below. Captures and CSV preserve the admitted complete-family declaration and pair values. Capture comparison matches pairs only within the same declared ordered groups, variance method, alpha and adjustment; changed families appear unpaired.

Selected native work adds at most fifteen pair tests and a fifteen-element sort/scan; Welch may traverse each group for each participating pair. Only this option increases the generated formula cap from 16 to 32 KiB. There is no new runtime kernel, browser statistics, or ordinary-path cost. Report admission refuses incomplete families, duplicates, inconsistent indices, out-of-range probabilities and partial adjustment. The existing bounded CSV output and whole-file refusal apply unchanged; capture comparisons admit at most 964 scalar rows including two full fifteen-pair families.

Explore a relationship

Choose Relationship between samples in New analysis, then name a predictor X and outcome Y. Grid fits Y from X through native LINEST and reports Pearson correlation, slope, intercept and R². Slope describes the fitted change in Y per unit X; the intercept describes fitted Y at zero X and may be an inappropriate extrapolation. Association is not causation.

The native scatter plot shows up to the first 64 aligned observations and a line joining two native fitted endpoints. Larger samples identify the plotted subset explicitly; all admitted observations enter the fit. Expand Exact observed and fitted coordinates to inspect values without display rounding. Print includes the plot and the entire admitted coordinate collection.

For a response at one X, expand Analysis options and enter Predict at X (optional) before creating the report. Leaving it blank keeps a fit-only analysis. Grid computes a pointwise confidence interval for the population mean Y and a wider prediction interval for one independent new Y at that X, using the native residual standard error and Student-t degrees of freedom. Nominal level is the authored coverage level under the assumptions, not a probability that this particular observation or fixed mean lies in the reported interval. These are pointwise intervals, not a simultaneous band over the fitted curve. The report flags extrapolation when X is outside the observed range; the linear relationship may not extend there. New-observation coverage additionally requires an independent observation from the same population with approximately normal, constant-variance residuals. Captures/downloads retain the chosen X and exact interval values. To choose another X, create a new analysis or edit the saved native formula in Code; no browser-side interval is recomputed.

Open Residual diagnostics to inspect residuals against fitted Y with a zero reference. Curvature, changing spread or unusual points can prompt a closer review of the source and assumptions; the view does not certify those assumptions. This plot uses the same first 64 observations as the scatter plot. Native RMSE and residual standard error use the full admitted sample and remain in outcome units. RMSE uses residual sum of squares divided by sample size; residual standard error uses residual degrees of freedom (n − 2 for this fit). They summarize in-sample errors, not future prediction uncertainty. Native errors stay visible while diagnostics are folded. Captures/downloads retain exact summaries and both point collections; print unfolds diagnostics and includes exact residual summary values.

Relationship authoring requires matching, complete numeric row or column vectors of 3–100000 observations and positive variation in both X and Y. Missing rows are never dropped or realigned. The slope interval and two-sided slope test assume independent observations, a linear mean relationship, constant residual variance and approximately normal residuals. They are not prediction intervals or causal effects. Zero residual variation produces visible inference errors; a perfect fitted line does not manufacture statistical certainty. The default flow has one predictor. Additional predictors are optional as described below; nonlinear regression and simultaneous prediction bands remain outside guided authoring.

The canonical model includes a dose/response example with every result calculated natively.

Fit several predictors

In relationship setup, expand Additional predictors (optional) and choose up to three more numeric sample vectors. Grid fits the outcome from all two–four predictors through native LINEST pivoted QR. A coefficient describes the fitted change in outcome per unit of that predictor, holding the other authored predictors fixed. It is a conditional association, not a causal effect. The intercept is fitted outcome when every predictor is zero, which may be outside the observed data. Grid maps native coefficients back to the chosen predictor names and reports their native standard errors, individual Student-t intervals and two-sided coefficient tests, plus joint R² and residual summaries.

All predictor vectors must have the same complete row/column shape as the outcome, positive variation, at least k+2 observations for k predictors, and at most 100000 observations. Missing rows are never dropped or realigned. Collinear or numerically rank-deficient designs return native errors rather than guessed coefficients. Residual degrees of freedom are n−k−1. Coefficient uncertainty assumes independent observations, a linear conditional mean, constant variance and approximately normal residuals. Tests/intervals are individual comparisons; no multiplicity adjustment, automatic variable selection or causal conclusion is supplied. Native errors stay inspectable in all captured and reopened reports.

The main plot compares fitted Y with the observed outcome; it is a joint fit view, not a separate predictor effect or a multidimensional regression plane. It and the residual plot show the same first 64 aligned rows at most; all admitted rows contribute to the fit and error summaries. Exact native coefficients, standard errors, interval endpoints and plot points survive captures, workbook Files and comparisons. For a multivariable prediction, expand Analysis options → Predict at a profile (optional) and enter one numeric value per predictor. Leave the entire profile blank for coefficients only. An incomplete or nonfinite profile remains editable and never dispatches a model write. Clear all prediction values returns to a fit-only report. Changing predictor selection or profile values requires a new analysis or editing its native formula in Code.

The report supplies a pointwise mean-response confidence interval and an individual prediction interval for one independent new outcome. Native LINEST refits predictors shifted by the chosen profile: the new intercept is the fitted mean there, and its native standard error includes coefficient covariance. The individual interval additionally includes native residual variance. Both use native Student-t residual degrees of freedom; nominal coverage assumes the same independent, linear, constant-variance, approximately normal residual process. A perfect or degenerate fit refuses uncertainty rather than inventing precision. These are pointwise intervals, not a simultaneous band or a causal claim.

Native diagnostics identify each predictor value outside its observed range. Within all marginal ranges is not proof that the joint profile is supported by the observed combinations. Inspect the full predictor data and Prediction leverage, the native squared mean-SE/residual-SE ratio; larger leverage implies more estimation uncertainty, without an automatic cutoff or coverage guarantee. Rank/conditioning errors in the centered fit are retained as native errors. The chosen values remain in the report's authored assumptions in predictor-source order, and captures/export preserve exact estimates, standard errors and bounds. Generated reports retain descriptive source names and reuse native local aliases, so four predictors with 64-character names and a practical numeric profile fit within the 20 KiB guided relationship formula limit. Extremely long names or profile literals can still receive an editable authoring-budget refusal. Only a complete selected profile incurs the extra native fit. Nonlinear models and automatic variable selection remain outside guided authoring. The canonical model includes a two-predictor dose/baseline fit.

Connect an existing report

Choose Connect existing report or Set up Statistics, and name a report symbol. The report must be an object with schema grid.statistics-report/v1. Optional inputs.alpha and inputs.test_type, under Optional analysis controls, name already-declared numeric model inputs; they do not create formulas or model inputs. Configuration changes require a host with matching revision acknowledgements. A failed save retains the draft. Pending setup can stop waiting and observe the saved setup without repeating the write. Use Author report in Code to change the report formulas or sample sources.

Statistics!type = "statistics"
Statistics!config = <toml>
  version = 1
  kind = "statistics"
  title = "Sample comparison"
  report = "StatisticsReport"

  [inputs]
  alpha = "Alpha"
  test_type = "TestType"
</toml>

The complete example is the canonical schema illustration. Its report has:

Field	Content	Maximum
schema, title	Schema name and report title	512 characters per text
sources	Authored source names	8, at least one
assumptions	Explicit study assumptions	16, at least one
summaries	label, n, mean, median, stdev, minimum, maximum	8 rows
bins	label, count	32 rows
comparisons	label, method, estimate	8 rows
tests	label, method, null_hypothesis, alternative, p_value, alpha	8 rows
intervals	label, method, level, estimate, lower, upper, optional native standard_error	16 rows
diagnostics	label, detail	16 rows
Optional residuals	Native standard_error, rmse, degrees_of_freedom and plot (same coordinate shape as scatter)	Nonnegative summaries or native errors; 64 points, 0 or 2 zero reference endpoints
Optional scatter	Native x_label, y_label, total_count, observed points and fitted line coordinate records	64 observed points, 0 or 2 fitted endpoints

Each array field is required; unused sections use []. Numeric results must be finite native numbers or explicit native errors. Native errors remain visible; they do not become zero or a successful interval. Counts must be nonnegative integers; probabilities must be in range; interval endpoints must be ordered. An invalid or oversized report is refused as a whole, without silently dropping rows. The UI projection has an additional 4096-node, depth-20 bound. Existing symbol transport can materialize a larger authored value before this refusal; this is not a streaming large-dataset inspection API. Aggregate large samples in the native model and bind a bounded report.

Compare methods and coverage

Analysis settings contains Edit native controls, offering alpha and test design when the host identifies the declared writable inputs and supports native revision receipts. Paired mode uses T.TEST(..., 2, 1), equal-variance independent mode uses type 2, and Welch independent mode uses type 3 in the example. Alpha must be between zero and one. The native input owner applies the writes with expected revisions. No optimistic fallback is used. On a failed or missing acknowledgement, the draft remains visible. Keep inputs and stop waiting retains pending work; Observe native inputs requires advancing, resolved native revisions before reloading a draft. No uncertain write is repeated against the same revision. Blank or incomplete alpha text stays in the editor and never becomes zero. A changed configuration invalidates the input draft.

Choose the method from the study design. A paired test requires meaningful aligned pairs. Independence and approximate normality are assumptions, not facts established by displaying the report. Guided independent comparisons now show a Mean difference (A minus B) confidence interval alongside each population mean interval. Positive estimates indicate a larger mean in A. Welch uses unequal-variance standard error and Welch–Satterthwaite degrees of freedom; pooled uses a shared variance estimate and nA+nB−2 degrees of freedom, requiring equal population variances. Both require independent, approximately normal observations and at least two numeric observations per sample. Native numeric inclusion applies; inspect missing or mixed data. Zero or nonfinite standard error refuses the difference interval with native errors. The interval describes the population mean difference, not individual changes or predictions; containing zero is compatible with no mean difference under the assumptions. Existing authored reports retain their formulas. The canonical model includes separate native Welch/pooled mean-difference reports and a paired report whose interval describes the mean aligned difference. A p-value is not an effect size or the probability that the null hypothesis is true. Diagnostics are authored notes, not automatic checks for every statistical assumption. Corrected native T.TEST now distinguishes paired (difference variance, n−1 degrees of freedom), pooled equal-variance and Welch (Welch–Satterthwaite degrees of freedom). Older native results incorrectly ignored the design argument; previously saved snapshots can therefore differ from current results. Invalid flags, insufficient samples and zero standard error return native errors instead of invented probabilities. Constant paired differences also produce a native interval error; no interval is manufactured.

Capture evidence

Download report captures and downloads the current report in one action. For inspection before export, expand Report identity and snapshot; Capture report evidence freezes one bounded report with model identity, partition, model revision when supplied by the host, the report symbol's requested/resolved revisions, assumptions and capture time. It marks that capture stale after the observed model or report changes. Download captured evidence retains the JSON outside the session; closing or changing account, backend, model or surface clears the in-memory capture. Explicit Files saves described below preserve a named copy in the workbook. Download failure keeps the capture so export can be retried. Captures are at most 128 KiB.

Export captured results keeps Download captured CSV beside the complete JSON download in a folded disclosure. CSV exports the selected session snapshot or a verified reopened saved capture; it never refreshes or implicitly captures live results. A stale snapshot keeps its original values and identity.

The CSV is one tidy table: section, row, label/category, field, exact value, value kind and native method, with original capture/report identity repeated on every row. Sources and assumptions remain explicit declaration rows. Summaries, bins, comparisons (including fitted coefficients), tests, intervals, category observed and expected cells, residual metrics, fit review and bounded plot points/line endpoints are retained. No observations beyond the captured native report view are reconstructed. Native errors, blanks, missing optional fields and empty collections have explicit kinds. Numeric strings round-trip the admitted values, including negative zero; formula-like text receives an apostrophe prefix for spreadsheet safety. CSV adds no calculation, statistical inference or certificate.

Export admits the existing 128 KiB capture/report contract, then caps CSV at 2,000 rows and 4 MiB. An oversized output refuses the whole file without trimming; complete JSON remains available. Saved exports also retain the verified Files identity and original capture name. Values are the admitted Statistics projection, not a claim to preserve raw native fields that were never stored in the capture.

One-click report downloads retain the selected snapshot. A late completion or failure from a replaced bindings owner or an older session capture cannot change the current capture’s status.

A capture records what the native model returned at that observed revision. It is not a signed statistical certificate or a verified immutable dataset hash. Source names and assumptions are authored; unknown model revision is explicitly represented as unavailable. The session snapshot is not automatically stored in the workbook. Use the explicit Files save action to retain a named copy. Print keeps native values, errors, assumptions and provenance, with mutation controls omitted. Run mode hides configuration authoring but permits host-authorized native controls and evidence capture.

Save and reopen workbook captures

Expand Save and reopen report snapshots, give the capture a name and choose Save current report (or Save report snapshot when a session capture already exists). The latter saves that exact historical snapshot, even if the live analysis changed; its title and capture time remain visible. This action writes one named JSON file through the existing workbook Files owner and checks the native receipt and readback bytes. Save the Grid document to carry the file in its archive. Download remains available for keeping a separate copy.

Browse saved captures loads one native metadata page of at most 50 entries; Next capture page replaces that page. Select and Open saved capture to read one file, limited to 128 KiB. A UUID and SHA-256 in its path identify the exact JSON bytes. Opening checks size, content type, hash, name, provenance and report schema before displaying any saved result. Edited bytes with a mismatched identity are refused. Original model ID, partition, report/model revisions, capture time, native errors, results and authored assumptions stay historical, including after importing an archive into a model with a new ID. This is content identity, not a verified dataset identity or a signed certificate; source bytes, effective inputs, packages and external data are not frozen.

Compare with current report captures the currently displayed ready native report and places exact scalar values beside the saved values. It matches rows by section, label and occurrence, shows their methods and warns when report bindings, declared sources, authored assumptions, methods, test hypotheses, alpha or interval coverage differ. A numerical result changing under the same declared analysis does not trigger that warning. No browser delta, new statistic or significance conclusion is calculated. Plot coordinates remain inspectable in each report. If the live report changes, the comparison marks its current-side snapshot historical until you compare again. Print retains the opened report and every admitted comparison row with provenance. Everyday comparison tables show 20 rows per page. Folded Comparison options can show only changed exact values or row methods; errors and unpaired rows remain visible. Print includes the complete comparison regardless of that filter or page. The bounded scalar budget covers both maximum admitted reports, including complete simultaneous pairwise intervals.

An unknown save retains its exact planned path. Keep capture and stop waiting fences late responses; Observe pending capture reads and verifies that file without repeating an upload. Cancelling a later browse, open or observation also keeps the original pending file available for recovery. Failure or absence keeps the uncertain state. You can discard tracking explicitly, but a file may already exist in Files: browse before creating another copy. Opening this surface, print and collapsed management never start Files reads, uploads, polling or report recalculation.

Guided relationships now include Fit review across every admitted observation, including rows outside the first-64 plot preview. A plain summary reports how many observations cross screening thresholds; the folded details show the first eight flagged observations in aligned source order, exact residuals, residual/RSE and the largest absolute residual across the full source. Indices are sample positions, not worksheet row addresses. No rows are removed or refitted.

For single- and multivariable fits with an intercept, native formulas also supply leverage and Cook distance. Screening highlights absolute residual/RSE above 3, leverage above 2p/n (p includes the intercept), Cook distance above 4/n or unavailable Cook distance. These are conventional review hints, not an outlier test, automatic exclusion decision or certificate of a trustworthy fit. Residual/RSE is not a studentized residual. A zero/nonfinite residual standard error refuses scaled screening. An observation with leverage at least 1 keeps its unavailable Cook distance explicit.

Multivariable leverage uses one native thin SVD of the full centered/scaled design, including the intercept. Row sums of squared orthonormal basis entries give leverage; Cook distance uses native residual/RSE squared times leverage, divided by p times (1 minus leverage) squared. Shape, finite-value, orthogonality, range and trace checks refuse an unreliable basis. A refused influence calculation leaves positive-scale residual screening available and retains unavailable leverage/Cook values explicitly. Leverage within 1e-12 of one is retained as one and refuses Cook distance. No observation refits or n-by-n hat matrix are created. Inspect full sources and study design before interpreting conditional coefficients or predictions. Captures, comparisons and CSV export retain these native values without deriving new statistical conclusions. Guided relationship formulas admit at most 20 KiB; other guided analysis formulas retain 16 KiB. Full-source review selects a bounded additional native O(n·k) residual pass and linear screening at explicit relationship creation/resolution. Multivariable influence additionally selects one O(n·p²) thin decomposition with native bounded sweeps and O(n·p) storage, plus a p-by-p orthogonality check; p≤5. Disabled or unrelated surfaces select none of this work; there is no browser calculation or repeated regression fit per observation.

The pairwise group example uses the same guided native Welch formula with complete-family Holm adjustment for three independent groups. The omnibus test, pairwise tests and pointwise mean intervals retain their separate meanings.

Categorical analysis explicitly shapes native column marginals as a row before indexing each expected cell. Rectangular and asymmetric tables therefore use each column's own total; the same marginals govern sparse 2×2 Fisher admission.

Compare repeated conditions

Choose Compare repeated conditions for one numeric matrix: each row is one participant and each column is one condition, in the same order for everyone. Keep headers outside the range. Native admission requires at least two participants, three–six conditions and at most 100000 total measurements. Every measurement must be finite and numeric; incomplete participant rows are refused, never dropped or reordered. Optional condition names and alpha stay under Analysis options.

The native one-way repeated-measures ANOVA removes each participant's mean before estimating condition and residual variation. It forms squared double-centered residuals directly, avoiding subtraction of nearly equal sums of squares. The report retains the omnibus F and right-tail probability, condition and residual degrees of freedom, sums of squares, and descriptive partial eta squared. Each condition also has its own summary and a marginal Student-t mean interval across participants, using that condition's variance and n−1 degrees of freedom. These intervals are pointwise and do not test paired differences or identify which conditions differ. The native report declares all condition means on the existing common-axis interval chart; any selected estimate/endpoint error refuses the entire chart, while exact values remain available.

The omnibus probability is uncorrected and assumes sphericity: equal variances of pairwise condition differences. Participants must be independent and within-person contrasts approximately normal. Marginal mean intervals also assume approximately normal participant values within each condition. Grid does not verify these assumptions, choose a correction automatically, or substitute an independent-groups test. An explicit Mauchly diagnostic is available in folded Analysis options. Explicit Greenhouse–Geisser and Huynh–Feldt options retain separately labeled corrected omnibus results. Zero/nonfinite residual variance or residual variation unresolved relative to condition variation (64 machine epsilons) refuses omnibus inference. A zero/nonfinite condition variance refuses that condition's mean interval. Complete reports, uncertainties, assumptions and refusals use the existing capture/JSON/CSV workflows.

The condition/subject/error decomposition and sphericity limitation follow the published repeated-measures statistical notes. Review the design with appropriate statistical guidance if sphericity is questionable or a mixed, clustered or incomplete design is needed.

The repeated-conditions example keeps four participant rows aligned across Before, During and After. Its native report illustrates the uncorrected omnibus test, distinct condition uncertainty and pointwise mean chart; the source records the teaching result for inspection.

Paired repeated-condition comparisons

Under Analysis options, enable Compare every condition pair with Holm adjustment and review one unique condition name per matrix column. Those three–six names declare the complete family, in column order; native admission requires the matrix dimensions to match. The option is off by default. Participant rows stay aligned, and no missing measurement is dropped.

Each pair uses native two-sided paired T.TEST on the corresponding participant columns and retains its signed mean difference, raw probability and Holm-adjusted probability. All declared pairs are tested independently of the omnibus result. Holm adjustment uses the same complete-family native sort/scan convention as independent group comparisons; one invalid raw result refuses every adjusted probability while preserving every raw outcome. Constant or zero-variance paired differences return native unavailable values, without inventing significance.

Individual paired tests assume independent participants and approximately normal within-participant differences. They do not require the omnibus ANOVA's sphericity assumption. These are mean-difference tests; optional Bonferroni intervals are a separate selection. Neither selection verifies assumptions or supplies a mixed-model correction. Matrix source, reviewed condition order, participant count and the complete family remain in captures, JSON, CSV and saved-report comparisons. Reports from different declared matrices or participant counts are not paired by position.

Greenhouse–Geisser omnibus correction

Under Analysis options, enable Also show Greenhouse–Geisser corrected omnibus test to retain both uncorrected and corrected probabilities. Grid estimates epsilon from the participant-centered condition contrast covariance and multiplies both the condition and residual degrees of freedom by epsilon, leaving F unchanged. The report includes epsilon and both fractional degrees of freedom. This convention follows SAS/STAT's repeated-measures method.

Native evaluation forms the sample covariance of the double-centered residual columns, with n−1 participant degrees of freedom. Residuals are scaled by a common positive finite maximum magnitude before the covariance product; scale cancels in trace(S)^2 / ((k−1) trace(S²)). Epsilon must be finite and in [1/(k−1), 1]; only roundoff within 64 machine epsilons of a bound may be clamped. Degenerate or invalid covariance refuses epsilon, while the corrected probability additionally requires the existing finite admitted omnibus F. The option never selects itself from the data, runs a sphericity test, fixes missing measurements or substitutes a mixed model. Independent participants and approximately normal within-person contrasts remain assumptions.

Paired Holm comparisons remain a separate optional family with their original native probabilities and assumptions. Correction does not change those tests, marginal mean intervals or the default uncorrected report. Captures, JSON and CSV retain both omnibus results and exact epsilon/degree-of-freedom values.

Simultaneous pairwise intervals

After selecting every group or condition pair, enable Also show simultaneous pairwise intervals in Analysis options. It is off by default. For the declared family of m pairs and alpha, native evaluation constructs each signed mean-difference interval using T.INV.2T(alpha / m, df) and its standard error. Bonferroni provides family confidence at least 1−alpha when the individual interval assumptions hold; each interval has individual level 1−alpha/m. This follows NIST's Bonferroni method. These intervals are separate from the Holm-adjusted probabilities and do not invert Holm tests.

Pooled group intervals use the ANOVA common within-group variance and N−k degrees of freedom. Welch intervals use each pair's separate sample variances and Satterthwaite fractional degrees of freedom; their coverage is approximate. Repeated-condition intervals use aligned participant differences and n−1 degrees of freedom, without requiring sphericity. Assumptions remain authored, and no missing participant or failed pair is removed.

The table adds interval endpoints only when selected; folded details retain exact native standard errors and degrees of freedom. If any declared interval is unavailable, the report refuses complete-family interval availability while retaining every endpoint/error and the separate original Holm outcomes. Captures, JSON, CSV and saved-report comparisons retain the selected interval method and complete family. Pointwise group/condition mean intervals remain unchanged.

Huynh–Feldt omnibus correction

In folded Analysis options, enable Also show Huynh–Feldt corrected omnibus test to retain an additional native corrected result. The option is independent of Greenhouse–Geisser display and off by default; correction is never selected from the observed data. The existing scaled contrast covariance is calculated once when either correction is selected.

For this complete, intercept-only repeated design, let d=k−1 and n be the participant count. Native evaluation uses (n*d*epsilon_GG−2) / (d*((n−1)−d*epsilon_GG)), retaining the raw estimate and applying min(1, estimate) before multiplying both original degrees of freedom. The same F is retained. This follows the published Huynh–Feldt approximation with design rank one. It supplies no between-subject model or automatic choice of correction.

At least three complete participants, admitted covariance, a finite positive estimator denominator and a finite raw estimate within the theoretical lower bound are required. Only roundoff within 64 machine epsilons below that bound may be clamped; estimates above one are capped by the method itself. Small-sample or degenerate estimator failures remain unavailable without substituting another correction. Uncorrected, selected Greenhouse–Geisser, paired tests, Bonferroni intervals and pointwise mean intervals retain their separate outcomes. Captures, JSON, CSV and comparisons preserve the raw and applied epsilon, corrected degrees of freedom and probability.

Optional sphericity diagnostic

In folded Analysis options, select Also show Mauchly sphericity test to retain a separate native diagnostic. It does not enable either correction. The first-order Bartlett-corrected chi-square approximation assumes independent, multivariate-normal participant contrasts; small-sample probabilities can be inaccurate, and nonrejection does not establish sphericity.

Native orthonormal Helmert contrasts remove the constant-condition direction. For d=k−1, contrast covariance A and participant count n, the calculation uses W=det(A)/(trace(A)/d)^d, rho=1−(2*d^2+d+2)/(6*d*(n−1)), chi_square=−(n−1)*rho*ln(W) and df=d*(d+1)/2−1. The trace normalization follows R's maintained implementation; the first-order approximation follows the published Mauchly contract. Grid does not add R's second-order tail correction.

At least as many complete participants as conditions, positive-definite native contrast covariance and a finite positive Bartlett factor are required. Singular, degenerate or invalid matrices retain an unavailable diagnostic. Only upper roundoff within 64 machine epsilons may clamp W to one. Scaled covariance is shared once with any selected Greenhouse–Geisser or Huynh–Feldt calculation. The original omnibus and pairwise results remain separately labeled.

Scientific JSON downloads and named Files captures preserve negative zero as an actual JSON number, including through verified reopening. Existing byte limits apply to the complete serialized record.

Probability and confidence displays keep their compact usual formatting. When rounding would turn an interior probability into 1 or a confidence level into 0%/100%, the display retains enough of the native value to avoid inventing an endpoint. Exact native values remain available in tooltips, downloads and print.

Opening a saved Statistics capture shows its primary results immediately, including retained native failures. Saved capture provenance and the report’s advanced assumptions remain folded; reopening makes no calculation or write request.

Paged saved/current comparisons offer Find a recorded value across retained labels, methods and exact native outcomes. Search combines with changed-only filtering; short comparisons stay simple. Printing includes every admitted comparison row regardless of search or display filters.

Complete named saved-capture JSON downloads and exact-value text preserve native negative zero, including chart titles and tooltips. This uses the same signed-number persistence boundary as retained Files captures; scientific calculations and configuration serialization are unchanged.