Statistical validation against R

PivoGraph runs every calculation locally in your browser, and its statistical methods are checked against published formulas, worked textbook examples, and R. The tables on this page put PivoGraph’s results and R’s side by side for t-tests, Mann–Whitney and Wilcoxon tests, one-way ANOVA with Tukey, Dunnett and Dunn multiple comparisons, Kruskal–Wallis and Friedman tests, descriptive statistics, and more. The checks use data sets that ship with R, so anyone can reproduce them.

Results at a glance

Deviation is |PivoGraph − R| divided by the larger of 1 and |R|; for confidence limits it is divided by the half-width of R’s interval. “Identical” means agreement to machine precision.

Reproduce the checks

The R script and the reference values it produces are free to download. With R and the packages below installed, one command recreates every reference value on this page.

Reference software: R 4.6.1 with multcomp 1.4.32, PMCMRplus 1.9.12, coin 1.4.6, DescTools 0.99.60, effectsize 1.0.3, car 3.1.5, onewaytests 3.2, survival 3.8.6.

• R script (r-reference.R)
• Reference values (r-reference.json)

Rscript r-reference.R

Methods

Comparing two groups: t-tests, Mann–Whitney and Wilcoxon

On Column tables PivoGraph offers the unpaired Welch t-test, the unpaired Student t-test (equal variance), the paired t-test, the Mann–Whitney test, and the Wilcoxon matched-pairs signed rank test. The reported difference is B − A. The t-tests report the 95% CI of the difference, R² (η², or partial η² when paired), Cohen’s d with the pooled SD and Hedges’ g with the exact small-sample correction (Cohen’s d_z when paired), an F test comparing the two variances, and, for paired data, the correlation between the pairs. The rank tests give exact P values whenever each group (or the number of nonzero differences) is below 50, including data with tied values, using the permutation distribution of the observed midranks. Values are rounded to 12 significant digits before ranking, so equal values are always treated as ties. The Mann–Whitney test also reports the Hodges–Lehmann difference with its confidence interval, and exact intervals state their achieved confidence level.

One-way ANOVA and multiple comparisons

For three or more columns PivoGraph offers ordinary one-way ANOVA, Welch and Brown–Forsythe ANOVA for unequal SDs, repeated-measures ANOVA with the Geisser–Greenhouse correction (each row is one subject; rows with a blank cell are set aside and reported), and the Kruskal–Wallis and Friedman tests. Kruskal–Wallis and Friedman P values are exact, by complete enumeration conditional on ties, for up to 10⁶ arrangements, and use the χ² distribution beyond that. Ordinary ANOVA also reports the Brown–Forsythe and Bartlett tests for equal SDs. Multiple comparisons can test every pair, each column against a control, or selected pairs, with Tukey, Dunnett, Šídák, Bonferroni, Holm–Šídák, Games–Howell, Dunnett T3, or Dunn corrections, with false discovery rate control (Benjamini–Hochberg, or the two-stage Benjamini–Krieger–Yekutieli method with q values), or without correction (Fisher’s LSD, Welch t, paired t, uncorrected Dunn). Dunnett’s distribution is integrated numerically, so P values and simultaneous confidence intervals are exact for any group sizes. Differences are the first column minus the second, and a compact letter display marks columns that do not differ significantly with a shared letter.

Row-by-row t-tests on Grouped tables

On Grouped tables each row compares two selected data sets with the Welch, Student, or paired t-test. In a paired test, matching subcolumns within a row are treated as measurements from the same subject. Rows with too few observations are shown as untested rather than silently omitted.

Significance brackets

Brackets on Column and Grouped graphs take their labels from the table’s two-group test, from the one-way ANOVA multiple comparisons on Column tables (adjusted P values, or q values for false discovery rate), or from the default test settings when no analysis exists yet. The labels are * for P < 0.05, ** for P < 0.01, *** for P < 0.001, and **** for P < 0.0001, and “ns” for comparisons that are not significant at the chosen level.

Descriptive statistics, box plots and violin plots

Descriptive statistics report N, mean, sample SD (n − 1), SEM, the 90%, 95%, or 99% CI of the mean, median, quartiles, minimum, maximum, range, a percentile of your choice, the CI of the median with its achieved confidence level, the coefficient of variation, the geometric mean with its SD factor and CI (positive values), skewness (G1), and excess kurtosis (G2). Quartiles and percentiles use the (n + 1)p rule (Hyndman & Fan type 6) throughout, so descriptive statistics, box plots, violin plots, and interquartile-range summaries always show the same numbers. The CI of the median uses order statistics with exact binomial coverage. Box-and-whisker plots can draw whiskers from minimum to maximum, by Tukey’s 1.5 × IQR rule (whisker ends on observed values, more extreme points drawn individually), or at the 10th–90th, 5th–95th, 2.5th–97.5th, or 1st–99th percentiles. Violin plots use Gaussian kernel density estimation with Silverman’s bandwidth, truncated at the sample minimum and maximum.

Linear and nonlinear regression

Linear regression on XY tables fits each data set by ordinary least squares, with every replicate as an individual point, and reports the slope and intercept with standard errors, R², Sy.x, degrees of freedom, the P value for a zero slope, coefficient confidence intervals, and a confidence band at the chosen level (95% by default); these are checked against jStat reference values for a textbook data set. Nonlinear regression fits four-parameter logistic (variable-slope dose-response) models to log₁₀ or untransformed concentrations with the Levenberg–Marquardt algorithm and reports Bottom, Top, HillSlope, LogEC50 / EC50, R², Sy.x, and n. Bottom is reported as no greater than Top, so the sign of HillSlope shows the direction of the curve, and a fit that does not converge is clearly reported instead of drawn. Parameter-recovery tests cover curves with and without replicate noise.

Area under the curve and correlation

The area under the curve on XY tables uses the trapezoid rule on the mean Y at each X, relative to a chosen baseline (0 by default), and reports the area above, the area below, and the net area; segments that cross the baseline are split at the crossing point, and the standard error of the net area is reported when every X has at least two replicates. Correlation of X and Y reports Pearson r with its Fisher z 95% CI, or Spearman r_s, for each data set, with two-sided P values from the t distribution.

Survival analysis

Kaplan–Meier curves use the product-limit method with Greenwood standard errors and log-log 95% confidence limits. Survival curves are compared with the log-rank (Mantel–Cox) and Gehan–Breslow–Wilcoxon tests (k − 1 degrees of freedom); with three or more groups, the log-rank test for trend scores the groups 1, 2, 3, … in table order. Each group’s median survival comes with a 95% CI read from the log-log limits (Brookmeyer and Crowley); an upper limit the curve never falls to is reported as not reached. Two-group comparisons add Mantel–Haenszel and log-rank hazard ratios and the ratio of median survival, each with a 95% CI (the CI of the median ratio assumes exponential survival). Event status is coded 1 for an event and 0 for a censored observation, with times greater than zero. The Gehan–Breslow–Wilcoxon statistic for the Freireich 6-MP remission data equals Gehan’s published W = 271, and the tables below compare every estimate with R’s survival package.

Frequency distributions and correlation matrices

A frequency distribution turns a Column table into a Grouped results table and histogram, with automatic bins (Sturges bin count, widths rounded to 1, 2, or 5 × 10^k) or bins you choose, as counts, fractions, or percentages, optionally cumulative; each bin includes its lower bound. A correlation matrix turns a Multiple variables table into a heat map of Pearson or Spearman coefficients computed by jStat from pairwise complete observations.

PivoGraph and R side by side

Open any data set to see each value PivoGraph reports next to the value from R. These tables are generated from the same comparison that PivoGraph’s test suite runs, each time the site is built.

How to cite

Cite the PivoGraph version you used, the project URL, and your access date.

PivoGraph, version X.Y. https://pivograph.com/ (accessed YYYY-MM-DD).

For reproducibility, record the PivoGraph version, test design, variance assumption, and other analysis settings used for the figure.

Choosing graphing software for your lab? See how PivoGraph compares with desktop alternatives.