Rigor MCP

Effect size for a one-way ANOVA, less biased than eta_squared for small samples since it subtracts out the variance explained by chance alone. Use alongside one_way_anova. Can be slightly negative when the true effect is near zero -- that's expected, not an error.

Test whether an observed proportion (successes out of n) differs from a hypothesized proportion p0 -- e.g. "is this coin fair (p0=0.5) given 55 heads in 100 flips?" Uses the normal approximation, which degrades for small n or p0 near 0 or 1; a warning is included when that assumption looks shaky. Returns the z-statistic, two-tailed p-value, a confidence interval for the true proportion, a citation, and warnings.

Test whether a sample's mean differs from a hypothesized value mu0. Returns the t-statistic, degrees of freedom, two-tailed p-value, a confidence interval for the mean, and any assumption warnings.

Test whether three or more independent groups have different means -- e.g. comparing average order value across three marketing channels. A significant result means at least one group differs from the others, not which one -- follow up with pairwise two_sample_t_test calls (correcting for multiple comparisons via bonferroni_correction or benjamini_hochberg_correction) to find which. Returns the F-statistic, between/within degrees of freedom, p-value, a citation, and a warning if within-group df is small.

Test whether the mean difference between paired observations (e.g. before/after measurements on the same subjects, or matched pairs) is zero. a[i] and b[i] must be the two measurements of the same pair -- use two_sample_t_test instead if the two samples are independent (different subjects in each group). Returns the t-statistic, degrees of freedom (n-1), two-tailed p-value, a confidence interval for the mean difference, a citation, and assumption warnings.

Run every pairwise comparison across 2+ groups and correct for multiple comparisons in one call, instead of orchestrating k*(k-1)/2 separate two_sample_t_test/mann_whitney_u calls plus a separate correction call by hand -- and forgetting the correction is one of the most common real mistakes this package exists to prevent. The natural follow-up after a significant one_way_anova/kruskal_wallis result: pass the same groups here to find *which* group(s) differ, not just whether any do. Returns every pair's statistic, raw p-value, whether it's still significant after correction, and an effect size…

Test for a *linear* association between two paired variables -- e.g. "does hours studied predict test score?" statistic is r itself (in [-1, 1]), not a t-statistic. Returns r, df (n-2), a two-tailed p-value (H0: r=0), a confidence interval for r via the Fisher z-transform, a citation, and warnings. Use spearman_correlation instead if the relationship may be monotonic but not linear, or if outliers shouldn't dominate the result. Use simple_linear_regression instead for the actual slope (units of y per unit of x), not just the strength of association.

Statistical power to detect a given Cohen's d with n observations, using a one-sample (or paired) t-test. Use for paired_t_test too -- it's a one-sample t-test on the differences, so the same power formula applies. Use sample_size_for_one_sample_t_test instead to solve for n given a target power. Returns a float in [alpha, 1].

Statistical power to detect a difference between two proportions (e.g. two conversion rates) with n_per_group observations in each group, using a two-proportion z-test. p1 and p2 are interchangeable (only their difference matters) -- e.g. current vs. new conversion rate. Use sample_size_for_two_proportion_test instead to solve for n given a target power. Returns a float in [alpha, 1].

Statistical power to detect a given Cohen's d with n_per_group observations per group, using a two-sample t-test. Power is the probability of correctly detecting a real effect of this size at the given alpha; a design with low power means a non-significant result would be inconclusive rather than good evidence the effect doesn't exist. Use sample_size_for_two_sample_t_test instead to solve for n given a target power. Returns a float in [alpha, 1].

Effect size for a Mann-Whitney U test. Call after mann_whitney_u, passing its statistic and the two sample sizes. Positive means sample 1's values tend to exceed sample 2's; negative means the reverse; 0 is no tendency either way. Returns a float in [-1, 1]; rough guidance mirrors Cohen's d: ~0.1 small, ~0.3 medium, ~0.5 large.

Not sure which rigor tool fits your question? Answer a few characteristics of the data and get back which tool to call, why, what to call instead if this test's assumptions look shaky, and what to run alongside it (an effect size, a power calculation, a natural follow-up). Every test in this package already documents this guidance in its own docstring for the sibling comparisons it knows about -- this tool exists so you don't have to have already read every other tool's docstring to find the one relevant cross-reference. Pure decision logic, no statistics computed here.

How many observations are needed to detect a given Cohen's d with a one-sample (or paired) t-test at the target power. Use for paired_t_test too -- a paired t-test is a one-sample t-test on the differences, so the same power formula applies. Returns a continuous value and a rounded-up integer to actually use.

How many observations per group are needed to detect a difference between two proportions (e.g. conversion rates) at the target power. p1 and p2 are interchangeable -- only their difference matters.

How many observations per group are needed to detect a given Cohen's d with a two-sample t-test at the target power. Returns a continuous value and a rounded-up integer to actually use.

Always-valid test of whether two proportions (e.g. two conversion rates in a live A/B test) differ, safe to call again after every new observation in either group -- the sequential-monitoring counterpart to two_proportion_z_test. Use this instead whenever the result will be checked more than once before the experiment ends, which is the normal case for a live dashboard rather than a one-shot analysis. p1 and p2 are interchangeable (only their difference matters). tau does not need to be exact -- reuse the minimum-detectable-effect you'd otherwise plug into sample_size_for_two_proportion_test.

Always-valid test of whether two groups' means differ, safe to call again after every new observation in either group -- unlike two_sample_t_test, which needs a sample size decided in advance and gives no such guarantee if checked repeatedly and stopped at the first significant look (that repeated-checking failure mode is exactly what inflates false positives; see naive_peeking_inflation for a demonstration). Use this instead of two_sample_t_test whenever a result will be (or already has been) checked more than once as data accumulates, e.g. monitoring a live experiment. Returns the current ef…

Fit y = intercept + slope * x by ordinary least squares -- single predictor only. Reports the slope (change in y per unit of x), the intercept, R^2 (proportion of y's variance explained by x), and a significance test + confidence interval for the slope (H0: slope=0). Use pearson_correlation instead if you only need the strength of a linear association, not its actual units/magnitude.

Test for a *monotonic* association between two paired variables, via the Pearson correlation of their ranks -- doesn't assume linearity and is far less sensitive to outliers' exact magnitude than pearson_correlation. Same return shape as pearson_correlation (statistic is rho itself, in [-1, 1]).

Test whether two independent proportions differ -- the standard test behind comparing conversion rates between two groups (e.g. an A/B test). Returns the z-statistic, two-tailed p-value, a confidence interval for the difference in proportions, a citation, and warnings.

Test whether two independent samples have different means. Defaults to Welch's t-test (does not assume equal variances); pass equal_var=true for the classic pooled-variance test.

The non-parametric alternative to paired_t_test -- use when that test's own small-n warning makes a normal-theory result suspect. Tests whether the median of the paired differences is zero, by ranking the absolute differences rather than assuming they're normally distributed. Pairs with a zero difference are dropped (and counted in a warning), the standard procedure. statistic is T = min(W+, W-).