One line per public function of dioptra-stats: what it computes, the reference it
implements, and the conventions that differ between Prism and SciPy. Conventions that hold
everywhere:
NaN is a missing value. Descriptive statistics and tests skip it; paired tests drop
incomplete pairs; outlier indices keep the caller’s positions. Infinities are rejected
with StatsError::InvalidArgument.- Two-sample differences are
x - y. Prism shows “B - A” (later column minus earlier); the
presentation layer flips the sign where it mimics Prism.
- Every test returns the two-sided p and both one-sided p-values (
p_less, p_greater)
plus p for the requested alternative. Prism’s “one-tailed” p is p_less or p_greater
for the predicted direction.
- Upper tails are computed directly, never as
1 - cdf; quantiles invert whichever tail
is smaller.
- The low-level
special kernels return NaN for invalid arguments; distribution
constructors and tests return StatsError.
| Function | Reference | Notes |
|---|
ln_gamma(x) | fdlibm lgamma_r (via libm) | ln abs Gamma(x), about 1 ulp. |
gamma(x) | musl tgamma (via libm) | |
digamma(x) | Abramowitz & Stegun 6.3.5, 6.3.7, 6.3.18 | Recurrence to x >= 10, asymptotic series; reflection for x < 0. |
ln_beta(a, b) | R lbeta (Ihaka) | Stirling-correction form when max(a,b) >= 8, no cancellation at a = b = 5000. |
beta(a, b) | exp(ln_beta) | |
ln_choose(n, k) | via ln_beta | Real n, k. |
erf(x), erfc(x) | fdlibm (via libm) | erfc keeps relative accuracy in the tail. |
log1p(x), expm1(x) | IEEE (std) | |
log1pmx(x) | R log1pmx (Welinder) | ln(1+x) - x without cancellation. |
stirling_error(z) | Loader (2000), R stirlerr; A&S 6.1.40 | ln Gamma(z) minus Stirling’s formula. |
binomial_deviance(x, np) | Loader (2000), R bd0 | |
binomial_pmf_raw(x, n, p, q) | Loader (2000), R dbinom_raw | Saddle-point binomial pmf, q supplied separately. |
ln_gamma_prefactor(a, x) | Temme (1987); DiDonato & Morris (1986) rcomp | ln(x^a e^-x / Gamma(a)), Stirling form for a >= 8. |
gammainc_pair(a, x) | Numerical Recipes 3rd ed. §6.2; DiDonato & Morris (1986), TOMS 654 | (P, Q): series for x < a+1, Lentz continued fraction for Q otherwise. |
gammainc_p, gammainc_q | as gammainc_pair | Regularised lower / upper incomplete gamma. |
gammainc_inv_pq(a, p, q), gammainc_inv, gammaincc_inv | DiDonato & Morris (1986) §5; Wilson & Hilferty (1931) start | Bracketed Halley iteration on the smaller tail. |
ln_beta_prefactor(a, b, x, y) | DiDonato & Morris (1992), TOMS 708 brcomp; Loader (2000) | ln(x^a y^b / B(a,b)), expanded around the mean when a, b >= 8. |
betainc_pair_xy(a, b, x, y) | Numerical Recipes 3rd ed. §6.4 (modified Lentz); TOMS 708 ideas (symmetry, bpser, brcomp) | (I, 1 - I) with y = 1 - x supplied; switch at x = (a+1)/(a+b+2). |
betainc_pair, betainc, betaincc | as above | Regularised incomplete beta and its complement. |
betainc_inv_pq(a, b, p, q), betainc_inv, betaincc_inv | Numerical Recipes 3rd ed. §6.14 start (A&S 26.5.22); bracketed Halley | Returns (x, 1 - x); solves for the smaller of x and 1 - x. |
norm_quantile(p), norm_quantile_upper(q), norm_quantile_pq | Wichura (1988), AS 241 PPND16 | About 1e-16; R’s qnorm algorithm. Upper tail by symmetry, so tiny q is exact. |
norm_quantile_ppnd7(p) | Wichura (1988), AS 241 PPND7 | Single-precision variant, kept for reference. |
erfc_inv(y), erf_inv(x) | via AS 241 | Central range uses the PPND16 rational directly (no 1 + x rounding). |
Each continuous distribution implements Continuous (pdf, cdf, sf, ppf, isf) and
has a cdf_sf or quantile_pq method taking both tails.
| Function | Reference | Notes |
|---|
std_normal_pdf, std_normal_ln_pdf | R dnorm argument splitting | Full relative accuracy for large z. |
std_normal_cdf, std_normal_sf | erfc (fdlibm) with argument-rounding correction; Laplace continued fraction for Mills’ ratio beyond 37 (A&S 26.2.14) | Relative accuracy to the underflow limit. |
std_normal_ln_cdf, std_normal_ln_sf | ln phi + ln Mills’ ratio for z < -5 | |
Normal | as above; AS 241 quantile | |
StudentT (cdf_sf, two_sided_p, quantile_pq) | A&S 26.7.1 via the incomplete beta | Non-integer df; df = inf is the normal. |
ChiSquared, Gamma | regularised incomplete gamma | |
FisherF | A&S 26.6.2 via the incomplete beta | Beats SciPy’s f.isf in the far upper tail. |
Beta | regularised incomplete beta | |
Binomial | Loader (2000) pmf; A&S 26.5.24 cdf/sf | |
Hypergeometric | R dhyper (Loader kernel), R pdhyper summation | SciPy’s hypergeom(M, n, N) parameter order: population, successes, draws. |
NoncentralT | Lenth (1989), AS 243; Guenther-style reflection for t < 0 | Tails below 1e-6 (and nc > 37 or df > 4e5) by adaptive Gauss–Kronrod quadrature of integral Phi(tu - delta) f(u) du, which keeps relative accuracy where AS 243 and SciPy’s nct fail (x = -3, df = 5, nc = 8: true 1.90e-20, SciPy 8.0e-20). |
StudentizedRange | Copenhaver & Holland (1988); Lund & Lund (1983), AS 190 | Inner composite Gauss–Legendre, outer adaptive Gauss–Kronrod over the chi scale; upper tail integrated directly. About 1.2-1.5 ms per cdf. |
Dunnett (new, balanced, from_group_sizes, cdf_sf, p_value, quantile) | Dunnett (1955); product-correlation reduction rho_ij = lambda_i lambda_j | Deterministic double integral (not SciPy’s randomized QMC, which is reproducible only to about 1e-3). Two-sided and one-sided; balanced and unbalanced. |
p_two_sided_t, p_less_t, p_greater_t | Student t tails | |
p_two_sided_z, p_less_z, p_greater_z | normal tails | |
p_upper_chi2, p_upper_f | upper tails, computed directly | |
t_critical_two_sided(alpha, df) | t quantile | |
| Function | Reference | Notes |
|---|
compensated_sum | Neumaier (1974) | |
clean, count | | Drop NaN; reject infinities. |
sum, mean | Higham (2002) two-pass corrected mean | |
variance, sd, sem, sd_population, cv | Chan, Golub & LeVeque (1983) corrected two-pass | n - 1 denominators (Prism, SciPy ddof=1); NumPy’s default is n. CV is a fraction (Prism shows %). |
min, max, range, median | | |
quantile, quantile_sorted, percentile, quartiles | Hyndman & Fan (1996) | QuantileMethod::Prism (type 6, (n+1)p, the default) vs Linear (type 7, SciPy/NumPy/Excel PERCENTILE.INC) vs TukeyHinges (R fivenum, for box plots). |
skewness_g1, kurtosis_g2 | moment estimators (SciPy bias=True) | |
skewness, kurtosis | Joanes & Gill (1998) G1, G2 | Bias-corrected, what Prism and Excel report; excess kurtosis. |
geometric_mean, geometric_sd_factor, ci_geometric_mean | Kirkwood (1979) | All values > 0. |
harmonic_mean | | All values > 0. |
ci_mean | t interval | Any level. |
lag1_autocorrelation | NIST StRD definition | Corrected for the rounding of the mean. |
Summary::compute | all of the above | Prism’s column statistics in one pass. |
| Function | Reference | Notes |
|---|
one_sample | Student (1908) | Estimate and CI are of mean - mu (SciPy’s CI is of the mean). |
unpaired | pooled-variance t | |
welch | Welch (1947); Satterthwaite (1946) df | Fractional df. Dioptra’s default unpaired test (DESIGN §2); Prism’s default is the pooled test. |
paired | one-sample t on x - y | Cohen’s d is d_z. |
ratio_paired | GraphPad “ratio paired t test” | Paired t on log10(x) - log10(y); geometric mean ratio and back-transformed CI. |
hedges_j(df) | Hedges & Olkin (1985) | Exact J = Gamma(df/2)/(sqrt(df/2) Gamma((df-1)/2)); hedges_g_approx uses 1 - 3/(4df - 1). Two-sample effect sizes use the pooled SD and n1 + n2 - 2 df, also for Welch. R^2 = t^2/(t^2 + df). |
| Function | Reference | Notes |
|---|
rank_with_ties, tie_term | midranks (SciPy rankdata('average')) | Tie sizes of groups >= 2; sum(t^3 - t). |
u_distribution(m, n) | Mann & Whitney (1947) recurrence (Löffler 1983) | Exact null distribution of U, additions only. |
mann_whitney | Mann & Whitney (1947); Hodges & Lehmann (1963); Moses CI (Hollander & Wolfe 1999 §4.3) | U1 is SciPy’s statistic, u = min(U1, U2) is Prism’s. Exact when there are no ties and n1 n2 <= 10,000; tie-corrected normal approximation with optional 0.5 continuity correction otherwise. Prism computes an exact p even with ties (up to 100 per group); we switch to the asymptotic method and say so in method. |
signed_rank_distribution(n) | generating function prod(1 + q^i) | |
signed_rank, paired, one_sample (wilcoxon) | Wilcoxon (1945); Pratt (1959) zeros; Cureton (1967) variance | ZeroMethod::Wilcoxon (drop zeros, Prism’s default) or Pratt. Exact for n <= 50 without ties or zeros; otherwise normal approximation with tie (and Pratt-zero) adjustment; continuity correction off by default (SciPy’s default). Hodges–Lehmann pseudo-median of the Walsh averages (zeros kept, and a zeros-dropped variant) with Tukey’s CI. Prism computes exact p with ties below 200 pairs; we use the asymptotic method there and say so. |
| Function | Reference | Notes |
|---|
shapiro_wilk | Royston (1992); Royston (1995), AS R94 | 3 <= n <= 5000. Normal scores from AS 241 PPND16 (as R’s swilk.c). |
dagostino_pearson | D’Agostino (1970); Anscombe & Glynn (1983); D’Agostino, Belanger & D’Agostino (1990) | Exactly SciPy’s normaltest; n >= 8. |
anderson_darling | Anderson & Darling (1954); Stephens (1974); D’Agostino & Stephens (1986) Table 4.9 | A*^2 = A^2 (1 + 0.75/n + 2.25/n^2); p from Table 4.9 (R nortest, Prism). Also SciPy’s critical values. SciPy >= 1.17 method='interpolated' p is a different method. |
ks_one_sample, kolmogorov_smirnov_normal | Kolmogorov (1933); exact p by kolmogn_sf | Parameters specified a priori. |
kolmogn_sf(n, d) | Simard & L’Ecuyer (2011) method selection; Durbin (1968) matrix as Marsaglia, Tsang & Wang (2003); Pelz & Good (1976) | SciPy kstwo uses Pomeranz (1974) for 0.754693 < n d^2 <= 4; Durbin’s matrix is exact there too. |
smirnov_sf(n, d) | Birnbaum & Tingey (1951) | One-sided exact tail. |
kolmogorov_sf(z) | Kolmogorov (1933) limiting series | SciPy kstwobign. |
lilliefors | Lilliefors (1967) | Estimated mean and SD (n - 1). Two p-values: p_table (statsmodels simulation table, 10^7 replications, reproduced in lilliefors_table.rs, BSD-3) and p_dallal_wilkinson (Dallal & Wilkinson 1986 when <= 0.1, else the table, as statsmodels pvalmethod='approx'). Prism uses Dallal–Wilkinson and reports “P > 0.10” above. |
lilliefors_p_table, lilliefors_p_dallal_wilkinson | as above | |
| Function | Reference | Notes |
|---|
grubbs(data, alpha) | Grubbs (1969); NIST/SEMATECH e-Handbook §1.3.5.17 | Two-sided, iterated while significant; G uses the SD of all remaining values; p is the Bonferroni bound 2n P(T > t_obs). |
rout_column(data, q) | Motulsky & Brown (2006), BMC Bioinformatics 7:123; GraphPad guide | Lorentzian fit of a constant with adaptive RSDR = P68 N/(N-K) (Prism’s guide; some renderings of the paper show sqrt(N/(N-K))); FDR flagging of the farthest 30% at Q on N-K df. rsdr_alt_mad (1.4826 MAD) is reported for comparison only. |
rout_fdr_flags(resid, rsdr, k, q) | Motulsky & Brown (2006) eqs. 17-18 | The FDR step shared by the column and curve versions: ranks abs(r), tests the farthest 30% on N-K df, flags the first significant point and every point farther out. |
rsdr(resid, k) | GraphPad Curve Fitting Guide | P68 of abs(residuals) (Hyndman-Fan type 7) times N/(N-K). |
rout_curve(eq, data, q, opts) | Motulsky & Brown (2006) | ROUT for a fitted curve; delegates to nonlinear::rout_fit. |
| Function | Reference | Notes |
|---|
gauss_legendre, gauss_legendre_cached, composite_nodes | Numerical Recipes 3rd ed. §4.6 gauleg | |
integrate | QUADPACK QAG with QK15 (Piessens et al. 1983) | Raw |
integrate_pair | QUADPACK QAG/QK15 with QUADPACK’s error scaling | Two integrands sharing nodes (a cdf and its complement). |
halley_bracketed | Numerical Recipes 3rd ed. §9.4 rtsafe with Halley correction | Bisection (geometric across decades) when a step leaves the bracket. |
brent, expand_bracket | Brent (1973), zeroin | |
Sums of squares are computed on data shifted by the first observation (exact by Sterbenz’s
lemma when values share leading digits) with two-pass means and compensated sums.
| Function | Reference | Notes |
|---|
one_way_anova(groups) | Fisher (1925); omega^2 Hays (1963) | SS/df/MS/F/p, R^2 = eta^2, omega^2, residual SD, group n/mean/SD/SEM, plus the five tests below as Options (None when a group has n < 2 or zero variance). |
welch_anova(groups) | Welch (1951), doi:10.2307/2332579 | w_i = n_i/s_i^2; df2 = (k^2-1)/(3 Lambda). statsmodels anova_oneway(use_var='unequal'), R oneway.test. |
brown_forsythe_anova(groups) | Brown & Forsythe (1974), doi:10.2307/1267501; Mehrotra (1997), doi:10.1080/03610919708813431 | Test of means: F* = SS_b / sum (1 - n_i/N) s_i^2, Satterthwaite df2. p uses k - 1 numerator df (the 1974 test, Prism’s “Brown-Forsythe ANOVA”); mehrotra_p uses Mehrotra’s df (statsmodels default). |
bartlett(groups) | Bartlett (1937); Snedecor & Cochran (1989) §13.9 | SciPy bartlett. |
levene(groups, center) | Levene (1960); Brown & Forsythe (1974), doi:10.1080/01621459.1974.10482955 | Median = Brown–Forsythe test of variances (Prism’s variance check), Mean = Levene. SciPy levene. |
cells_from_long(a, b, y) | | Long format (level indices) to the [row][col] -> values cell array. |
two_way_anova(cells, opts) | Searle (1987); Langsrud (2003), doi:10.1023/A:1023260610025 | Sum-to-zero coded GLM, refined Householder QR; type I/II/III SS as nested-model RSS differences = statsmodels anova_lm(typ=1/2/3) with C(., Sum). Type III (default) is Prism’s; treatment-coded type III is wrong and not offered. Empty cells allowed where estimable (df = rank added). No replicates: interaction dropped (additive model). Partial eta^2 per term. |
rm_anova(m) | statsmodels AnovaRM; Box (1954); Greenhouse & Geisser (1959), doi:10.1007/BF02289823; Huynh & Feldt (1976) with Lecoutre (1991), doi:10.3102/10769986016004371; Mauchly (1940), doi:10.1214/aoms/1177731915 | Subjects x conditions; listwise deletion (count reported). GG epsilon from orthonormal Helmert contrasts (and Box’s formula as a cross-check), HF capped at 1 (uncapped also reported), lower bound; corrected df and p; Mauchly W, chi-square, df, p; F for subjects. Prism reports GG only. |
kruskal_wallis(groups) | Kruskal & Wallis (1952), doi:10.1080/01621459.1952.10483441 | Tie-corrected H, chi-square p (SciPy kruskal); mean ranks; epsilon^2. Prism’s small-sample exact p is not implemented. |
friedman(m) | Friedman (1937), doi:10.1080/01621459.1937.10503522; Kendall & Babington Smith (1939) | Within-subject midranks, tie-corrected statistic (SciPy friedmanchisquare), Kendall’s W = chi2/(n(k-1)); listwise deletion. |
Comparison (i, j) estimates value_i - value_j; all-pairs lists i < j; many-to-one lists
(treatment, control). statsmodels’ Tukey reports mean_j - mean_i (opposite sign).
| Function | Reference | Notes |
|---|
p_adjust(p, method) | Bonferroni; Šidák (1967); Holm (1979); Hochberg (1988); Hommel (1988); Benjamini & Hochberg (1995); Benjamini & Yekutieli (2001); Benjamini, Krieger & Yekutieli (2006) | statsmodels multipletests (holm-sidak is step-down Šidák; TwoStageBky{alpha} = fdr_tsbky, one iteration, depends on alpha). Šidák via expm1/log1p (exact for tiny p). NaNs pass through and do not count. |
Adjust::name() | | Display name. |
tukey_kramer(groups, alpha) | Tukey (1953); Kramer (1956), doi:10.2307/3001469 | q = |
dunnett(groups, control, alternative, alpha) | Dunnett (1955), doi:10.1080/01621459.1955.10501294 | Exact multivariate-t p and critical value (dist::Dunnett, deterministic double integral), two- or one-sided, simultaneous CIs. |
pairwise_t(groups, pairs, adjust, alpha) | Fisher (1935) LSD; Šidák (1967); Holm (1979) | Pooled-variance t tests on chosen pairs (all if None); CIs for none/Bonferroni/Šidák only (as Prism). |
dunn_kruskal(groups, control, adjust, alpha) | Dunn (1964), doi:10.1080/00401706.1964.10490181; Daniel, Applied Nonparametric Statistics | Pooled midranks, tie-corrected SE; all pairs or vs control. |
dunn_friedman(m, control, adjust, alpha) | Prism’s documented post test after Friedman | z = (Rbar_i - Rbar_j)/sqrt(k(k+1)/(6n)), no tie correction. |
compact_letters(k, comparisons) | Piepho (2004), J. Comput. Graph. Stat. 13:456 | Insert-absorb letter display from the significant flags. |
2 x 2 layout [[a, b], [c, d]]: rows = groups, columns = outcome (event, no event).
| Function | Reference | Notes |
|---|
chi_square_independence(table) | Pearson (1900); Yates (1934); Wilks (1938) G; Haberman (1973) residuals; Cochran (1954) rule | Pearson, Yates (2 x 2 only, |
chi_square_gof(observed, proportions, ddof) | Pearson (1900) | SciPy chisquare; proportions normalised; df = k - 1 - ddof. |
fisher_exact_2x2(table, ci_level) | Fisher (1935); Cornfield (1956) | Two-sided = sum of P <= P_obs (1 + 1e-7) (R, SciPy, Prism); one-sided tails; conditional MLE odds ratio and exact CI by Brent on ln(OR) (R fisher.test, SciPy odds_ratio(kind='conditional')). |
risk_measures(table, ci_level) | Woolf (1955), doi:10.1111/j.1469-1809.1955.tb01285.x; Katz et al. (1978), doi:10.2307/2530610 | OR with Woolf CI, RR with Katz log CI (None with zero cells), risk difference with Wald CI. Prism’s Koopman (RR) and Baptista–Pike (OR) intervals are not implemented. |
fisher_exact_rxc(table) | Freeman & Halton (1951); Mehta & Patel (1983), doi:10.1080/01621459.1983.10477989 | Network algorithm: column stages, nodes = multisets of remaining row totals, merged path values, closed-form node mass, relaxation bounds; exact p, observed-table P, table count. Limits FISHER_RXC_WORK_LIMIT, FISHER_RXC_PATH_LIMIT (NoConvergence beyond; see CORRECTNESS.md). |
mcnemar(table) | McNemar (1947), doi:10.1007/BF02295996; Edwards (1948) | Chi-square with and without continuity correction (Prism uses the corrected one), exact binomial p (statsmodels). |
cochran_armitage(successes, totals, scores) | Cochran (1954), doi:10.2307/3001616; Armitage (1955), doi:10.2307/3001775 | z without the N/(N-1) factor (chi2 = z^2 = R prop.trend.test, Prism’s “chi-square for trend”); the N/(N-1) variant reported too. |
wilson_ci(successes, n, level) | Wilson (1927) | Score interval, no continuity correction. |
diagnostic(table, ci_level) | Wilson (1927); Simel, Samsa & Matchar (1991) | [[TP, FP], [FN, TN]]: sensitivity, specificity, PPV, NPV with Wilson CIs; LR+/LR- with log CIs. |
| Function | Reference | Notes |
|---|
pearson(x, y, ci_level) | Pearson (1896); Fisher (1921) z | Two-pass centred sums; t test on n - 2 df; Fisher-z CI (Prism). |
spearman(x, y, ci_level) | Spearman (1904); Bonett & Wright (2000) CI | Pearson on midranks (correct with ties); t-approximation p (SciPy); exact permutation p for n <= 9 (SPEARMAN_EXACT_MAX_N), ties kept. R uses AS 89 for larger n without ties: not implemented. |
kendall(x, y) | Kendall (1945); Knight (1966), doi:10.1080/01621459.1966.10480879 | tau-b and tau-c in O(n log n); tie-corrected asymptotic p (SciPy); exact p without ties for n <= 170 from the inversion distribution; p_auto = SciPy method='auto'. |
kendall_exact_two_sided(n, d) | Kendall (1970) ch. 4 | 2 P(D <= min(d, n0 - d)), Mahonian recurrence in probabilities. |
correlation_matrix(columns, method, adjust) | | Pairwise deletion; p adjusted over the k(k-1)/2 pairs. |
partial_correlation(x, y, z, ci_level) | Fisher (1924) | Correlation of residuals on z (refined QR), t on n - 2 - q df. |
| Function | Reference | Notes |
|---|
simple_linear_regression(x, y, opts) | least squares; Cook (1977); Durbin & Watson (1950); Swed & Eisenhart (1943), doi:10.1214/aoms/1177731178; Draper & Smith (1998) §2.1 | Coefficients and SEs from the refined QR; R^2 (uncentred through the origin, as R/statsmodels/NIST; r_squared_centered is Prism’s), F, Sy.x, x-intercept, slope vs hypothesised value, leverage, studentised residuals, Cook’s D, Durbin–Watson (input order), runs test (x order, exact zeros dropped), replicates (lack-of-fit) F. |
SimpleRegression::bands(x0) | | Confidence and prediction bands at x0. |
runs_test_by_x(x, resid), runs_test(signs) | Wald & Wolfowitz (1940); Swed & Eisenhart (1943) | Runs, E[R], z (no continuity), exact P(R <= runs) (Prism’s one-sided p). |
runs_cdf(r, n1, n2) | Swed & Eisenhart (1943) | Exact cdf of the number of runs. |
multiple_regression(x_rows, y, opts) | Golub (1965); Björck (1967), BIT 7:257; Marquardt (1970) VIF | Equilibrated pivoted Householder QR with double-double augmented-system refinement; aliased columns reported (R lm convention, in column order); SEs from refined (X’X)^{-1}; ANOVA table; adj R^2; VIFs; covariance. statsmodels OLS. |
polynomial_regression(x, y, degree, opts) | as above | Powers x^k formed in double-double so the refinement sees the exact design of the double inputs (NIST Filip LRE 14.0). |
deming_regression(x, y, lambda, ci_level) | Deming (1943); Linnet (1993), doi:10.1093/clinchem/39.3.424 | lambda = sigma_y^2/sigma_x^2 (Prism input: (SD_y/SD_x)^2); jackknife SEs (NCSS, R mcr); t(n-2) CIs. Prism’s SE method is undocumented. |
Prism’s equation and parameter names (GraphPad Curve Fitting Guide, “Models built in to
Prism”). Each Equation holds ParamSpecs (name, InitialRule, default bounds, data-set
constant flag), DerivedSpecs and its XTransform. Gradients are hand-written for the
simple models and exact forward-mode automatic differentiation for the rest (5PL,
bell-shaped, two-site competition, fit Ki, [Agonist] variable slope, log-logistic,
growth, power series); equations::tests::gradients_match_finite_differences checks every
one against Richardson differences.
| Function | Reference | Notes |
|---|
Equation::custom(expr), Equation::custom_with(expr, vars, params) | Pratt (1973) top-down operator precedence; Griewank & Walther (2008) forward AD | Parses + - * / ^ (**), unary minus below ^, exp ln log log10 log2 sqrt abs sin cos tan asin acos atan sinh cosh tanh pow, pi, e; log is base 10 (Prism); identifiers case-insensitive; parameters are the free identifiers in order of appearance (or the given order). Exact derivatives. |
Equation::from_closure(name, params, f) | | Rust closure; central-difference gradient. |
Equation::eval, eval_grad, eval_grad_fd(.., richardson) | | Central differences with h = eps^(1/3) max(abs(p), 1), or Richardson from h and h/2 at h = eps^(1/5) max(abs(p), 1). |
Equation::initial_values(x, y), InitialRule::eval, DataSummary::new | GraphPad “Rules for initial values” | YMIN, YMAX, YMID, XMIN, XMAX, XMID, X at YMID (linear interpolation of the replicate means), Y at XMIN/XMAX, SIGN(trend), rate ln 2/half-time, LS slope/intercept, ranges, or a function. |
Equation::derived_values(p) | | EC50 = 10^LogEC50, IC50, Ki, half-life ln 2/K, tau 1/K, doubling time, spans, Intercept2, Gaussian area. |
straight_line, line_through_origin, polynomial(1..=6), segmental_linear | | Lines and polynomials are linear: fitted from the exact linear least-squares solution. |
one_phase_decay, two_phase_decay, plateau_then_one_phase_decay, one_phase_association, two_phase_association, plateau_then_one_phase_association, exponential_growth | Prism exponential family | Y0, Plateau, K (PercentFast, KFast, KSlow; X0). |
log_agonist_vs_response, log_agonist_variable_slope (4PL), log_agonist_normalized_variable_slope, log_inhibitor_vs_response, log_inhibitor_variable_slope, asymmetric_5pl, agonist_vs_response, agonist_variable_slope, bell_shaped, log_logistic | Prism dose-response; Gottschalk & Dunn (2005) 5PL | X = log10(concentration) where x_transform is Log10; logistic pieces evaluated without overflow. |
michaelis_menten, substrate_inhibition, allosteric_sigmoidal, competitive_inhibition | Prism enzyme kinetics | I (inhibitor) is a data-set constant. |
one_site_specific, specific_binding_hill, one_site_total, one_site_fit_logic50, one_site_fit_ki, two_sites_fit_logic50 | Prism binding; Cheng & Prusoff (1973) | HotNM, HotKdNM are data-set constants in Fit Ki. |
gaussian, lorentzian, sine_wave, sine_wave_baseline, power, power_series, boltzmann_sigmoid, logistic_growth, gompertz_growth | Prism Gaussian, sine, classic and growth equations | |
library(), by_name(name) | | Every built-in equation; lookup by Prism name or alias (4PL, 3PL, 5PL, line, MM). |
cheng_prusoff_ki(ic50, ligand, kd) | Cheng & Prusoff (1973), Biochem Pharmacol 22:3099 | IC50/(1 + [L]/Kd). |
| Function | Reference | Notes |
|---|
fit(eq, data, opts), fit_global(eq, datasets, opts) | Moré (1978), doi:10.1007/BFb0067700; MINPACK lmder (Moré, Garbow & Hillstrom 1980) | Levenberg–Marquardt with scaled trust region (diag from Jacobian column norms, non-decreasing), pivoted QR, Moré lambda update; ftol = xtol = 1e-12, gtol 1e-15, 1000 iterations. Then Newton polishing (J’r compensated, Hessian J’J + sum r_i H_i with H_i from central differences of the exact Jacobian) accepted while the gradient norm falls. Box constraints: active-set projection (a parameter on a bound whose descent direction leaves the box is held for the iteration; the step is projected and its predicted reduction recomputed); at-bound parameters get no SE and flag “hit constraint”. Fixed, per-data-set constant and shared parameters (global fit); per-data-set local parameters. SEs sqrt(s^2 (J’J)^{-1}) from the refined QR; t CIs; covariance and correlation; dependency 1 - 1/(C_jj (J’J)_jj); “ambiguous” when a column is aliased or a dependency > 0.9999. SS, df, Sy.x, R^2 = 1 - SS/SS_tot (weighted with the final weights), adjusted R^2 = 1 - (1-R^2)(N-1)/(N-k), AICc with K = k + 1 (Prism). Runs test (x order) and replicates (lack-of-fit) test per data set. Residuals Y - f. |
FitOptions::weighting | Prism Method tab | None, 1/Y, 1/Y^2, 1/Y^K, 1/X, 1/X^2, 1/SD^2, Poisson (deviance residuals, unscaled covariance). WeightBase::Curve (Prism) weights by the fitted Y with the full derivative of (y - f)/abs(f)^{K/2}; Data uses fixed 1/abs(y)^K (SciPy sigma). |
FitMethod::Robust | Motulsky & Brown (2006) | Adaptive Lorentzian: start from the LS fit; alternate RSDR = P68 N/(N-K) of the current residuals with an LM minimisation of sum ln(1 + (r/RSDR)^2) (residuals transformed to RSDR sign(r) sqrt(ln(1 + (r/RSDR)^2)) so the LM objective is the merit) until parameters and RSDR change < 1e-10 (max 200 rounds). No SEs (as Prism). |
FitMethod::RobustFixedScale { loss, scale } | SciPy least_squares | Cauchy c^2 ln(1 + (r/c)^2) or soft-L1 2c^2 (sqrt(1 + (r/c)^2) - 1); robust_objective = sum. |
FitOptions::profile_ci | Venzon & Moolgavkar (1988), doi:10.2307/2347496; Bates & Watts (1988) ch. 6 | Prism’s “asymmetrical” CI: the two values of each parameter where the re-minimised SS = SS_min (1 + F(level; 1, df)/df); bracketed by steps of 1.6^k SE (at most 40 or the bound), then Brent. +/-inf if never reached; the bound if a constraint is reached first. Equals the t CI for linear models. |
FitOptions::jacobian | | Exact (analytic/AD; closures: central differences) or FiniteDifference { richardson } for any equation. |
FitOptions::outlier_q, rout_fit(eq, data, q, opts), rout_fit_global | Motulsky & Brown (2006) | Robust fit, RSDR, FDR test on the weighted residuals pooled over data sets with N - K df, least-squares refit without the outliers (their residuals still reported). |
bands(eq, fit, d, xs, opts), FitOptions::curve_points | Delta method; Seber & Wild (1989) §5.1 | Confidence band f +/- t sqrt(g’ Cov g), prediction band f +/- t sqrt(g’ Cov g + s^2/w(x)) (w from the curve for 1/Y^K; 1 for 1/SD^2). |
compare_summaries(simple, complex, alpha), compare_models(simple, complex), FitSummary::aicc | Motulsky & Christopoulos (2004) ch. 21-22; Burnham & Anderson (2002) | Extra-sum-of-squares F = ((SS1-SS2)/(df1-df2))/(SS2/df2); AICc with Prism’s K; delta = AICc(simple) - AICc(complex); Akaike weights; preferred model by each. |
fit_family(eq, datasets, opts) | Prism “one curve for all data sets?” | Separate fits (sharing removed) versus the global fit; F test and AICc with summed SS, df and parameters. |
CurveFit::values, params, se, summary | | Accessors. |
| Function | Reference | Notes |
|---|
kaplan_meier(time, event, opts) | Kaplan & Meier (1958); Greenwood (1926); Kalbfleisch & Prentice (1980) | Events before censorings at tied times; rows at distinct event times; Greenwood variance; CIs log-log (Prism “asymmetrical”, lifelines; default), linear clipped (Prism “symmetrical”), log (R default); undefined once S = 0. Median = first t with S <= 0.5 (+inf if never); Brookmeyer & Crowley (1982) CI by inverting the chosen band. Step curve and censor marks. |
KaplanMeier::survival_at, ci_at, at_risk_at | | Right-continuous lookups; at risk = subjects with time >= t. |
logrank(groups, weight, scores) | Mantel (1966); Gehan (1965)/Breslow (1970); Tarone & Ware (1977); Fleming & Harrington (1991) ch. 7 | k groups, hypergeometric variance, chi2 = U’V^{-1}U on k-1 df; weights 1, n, sqrt(n); O/E per group; test for trend with scores (default 1..k). lifelines logrank_test. |
hazard_ratio(g1, g2, ci_level) | GraphPad “The hazard ratio”; Berry, Kitchin & Mock (1991) | Log-rank method (O1/E1)/(O2/E2) with SE sqrt(1/E1 + 1/E2); Mantel–Haenszel exp((O1-E1)/V) with SE 1/sqrt(V). Group 1 relative to group 2. |
cox_ph(time, event, covariates, ties, ci_level) | Cox (1972); Efron (1977); Breslow (1974) | Newton–Raphson with step halving on the partial likelihood (covariates centred); SEs from the inverse information; Wald z, p, HR CI; likelihood-ratio test. Matches the 50-digit maximum where lifelines stops early. |
cox_log_likelihood(time, event, covariates, beta, ties) | | Partial log-likelihood at a given beta. |
| Function | Reference | Notes |
|---|
roc(positives, negatives, opts), roc_from_labels(scores, labels, opts) | DeLong, DeLong & Clarke-Pearson (1988), doi:10.2307/2531595; Sun & Xu (2014), doi:10.1109/LSP.2014.2337313; Hanley & McNeil (1982); Youden (1950) | Positive if score >= threshold; thresholds +inf then every distinct score descending (sklearn drop_intermediate=False). Sensitivity, specificity with Wilson (default) or Clopper–Pearson CIs, LR+ and LR-, Youden J (first maximum), closest to (0,1). AUC by Mann–Whitney (ties 1/2); DeLong SE by midranks; Hanley–McNeil SE; CIs clipped to [0, 1]; z test of AUC = 0.5 with the DeLong SE. |
proportion_ci(x, n, level, method) | Wilson (1927); Clopper & Pearson (1934); Brown, Cai & DasGupta (2001) | Prism’s “Wilson/Brown” hybrid is not reproduced; plain Wilson is the oracle’s. |
delong_paired(a, b, labels, ci_level) | DeLong et al. (1988) | Two correlated AUCs; covariance of the structural components. No oracle fixture (identity tests only). |
| Function | Reference | Notes |
|---|
bland_altman(a, b, kind, ci_level) | Bland & Altman (1986), Lancet 327:307; (1999), Stat Methods Med Res 8:135 | Difference A - B, ratio A/B or 100 (A-B)/mean; bias, SD (n-1), limits bias +/- 1.96 SD (Prism’s constant at 95%; z quantile otherwise; exact-z limits also reported); bias CI with t(n-1); SE of a limit SD sqrt(1/n + 1.96^2/(2(n-1))) with t CIs; proportional bias by OLS of difference on mean. |
icc(rows, ci_level) | Shrout & Fleiss (1979), Psychol Bull 86:420; McGraw & Wong (1996), Psychol Methods 1:30 | ICC(1,1), (2,1), (3,1), (1,k), (2,k), (3,k) with F tests and F-based CIs (Satterthwaite df for ICC(2,.)), as R psych::ICC. Hand-tested on the Shrout–Fleiss example. |
cohen_kappa(table, weights, ci_level), cohen_kappa_ratings(r1, r2, weights, ci_level) | Cohen (1960); Fleiss, Cohen & Everitt (1969), Psychol Bull 72:323; Cicchetti & Allison (1971); Fleiss & Cohen (1973) | Unweighted, linear or quadratic weights; asymptotic SE (CI) and null SE (z test). Hand-tested. |
| Function | Reference | Notes |
|---|
pca(rows, scaling) | Jolliffe (2002) ch. 1-3 | SVD of the centred (standardized by default, n-1 SD) matrix; eigenvalues S^2/(n-1), explained and cumulative ratios, eigenvectors (rows = components) flipped so the largest-magnitude entry is positive (first on ties), loadings eigenvector x sqrt(eigenvalue), scores U S, Kaiser count. sklearn PCA on the standardized matrix. |
parallel_analysis(fit, replicates, percentile, seed) | Horn (1965), Psychometrika 30:179; Glorfeld (1995) | Eigenvalues of seeded standard-normal data of the same shape and scaling; mean and percentile per component; components retained while above the percentile. |
covariance_matrix(rows) | | n-1 denominator, compensated two-pass sums, complete rows. |
| Function | Reference | Notes |
|---|
Xoshiro256::new(seed), next_u64, next_f64, below(n), normal, shuffle | Blackman & Vigna (2021), ACM TOMS 47:36; Steele, Lea & Flood (2014) SplitMix64; Lemire (2019) | xoshiro256** seeded by SplitMix64; 53-bit uniforms; unbiased bounded integers; normals by inversion; Fisher–Yates. Deterministic on every platform. |
bootstrap_ci(samples, stat, replicates, ci_level, method, seed) | Efron (1979); Efron (1987), JASA 82:171; Efron & Tibshirani (1993) ch. 14 | Each sample resampled within itself; percentile or BCa (z0 with ties counted 1/2, acceleration from the leave-one-out jackknife of every observation); quantiles of the replicates by type 7 interpolation; non-finite replicates dropped. |
bootstrap_mean_difference(a, b, replicates, ci_level, method, seed) | | mean(a) - mean(b). |
permutation_test(a, b, exact_limit, monte_carlo, seed) | Fisher (1935); Pitman (1937); Phipson & Smyth (2010) | Difference in means; exact enumeration of all C(n, n_a) splits when within exact_limit, else Monte Carlo with p = (count + 1)/(B + 1); two-sided and one-sided p. |
dioptra-core::compute turns each analysis specification into a results sheet on top of
the functions above. Every sheet has the same parts, in the order the UI shows them:
assumption checks, summary sentences, a methods line, typed sections, a checklist.
docs/ANALYSES.md describes each analysis kind.
Data. Excluded cells are never used; empty and text cells are missing. Paired and
repeated-measures analyses align values by row and drop a row with a missing member,
saying how many. Nested tables are averaged within each subgroup first (n counts the
experimental units). Data sets entered as mean/SD/N (or mean/SEM/N, mean/CI/N) are used
where the result depends only on the means, SDs and sizes: t tests (one-sample, Student,
Welch), one-way ANOVA with its post tests, and two-way ANOVA cells. Dioptra builds a
synthetic sample with exactly that mean, SD and N, so these results are exact; rank
tests, paired designs, normality and outlier tests say they are not possible from summary
data. Contingency counts must be non-negative whole numbers; the first bad cell is named
and the table is left as it is. Survival codes must be 1 (event) or 0 (censored).
Assumption checks. Each check has a status (pass, caution, fail, not_checked),
a one-sentence meaning, and an alternative when the status is not pass. Every check
uses α = 0.05.
| Check | Rule | Used by |
|---|
| Normality | Shapiro–Wilk (3 ≤ n ≤ 5000), D’Agostino–Pearson as a second opinion (n ≥ 8). Pass when every p ≥ .05; caution when the smallest p is in [.01, .05) or only D’Agostino–Pearson objects; fail below .01; not checked below n = 3. Applied to each group (t tests, correlation), the paired differences or log ratios (paired tests), the residuals (ANOVA, regression, curve fits), the differences (Bland–Altman). | t tests, ANOVA, correlation (Pearson), regressions, Bland–Altman, outliers |
| Equal variances | Brown–Forsythe (Levene on medians) decides; Bartlett is shown. Caution below .05, fail below .01. Informational (pass) when the test in use does not assume them (Welch, Welch’s ANOVA). From summary data: the SD ratio (t test) or Bartlett (ANOVA). | t tests, one-way and two-way ANOVA, Mann–Whitney (as a shape check) |
| Sphericity | Mauchly’s W; pass when the Greenhouse–Geisser correction is on. Not checked for the two-way repeated design. | Repeated-measures ANOVA |
| Expected counts | Cochran’s rule: caution when more than 20% are below 5, fail when any is below 1; Fisher’s exact test offered. | Chi-square |
| Sample size | Fail below 2 per group; caution below a minimum per analysis (5 for t tests and ANOVA, 6 pairs for Wilcoxon, 10 for correlation and ROC, 30 pairs for Bland–Altman). | most |
| Linearity | Runs test (too few runs, exact Swed–Eisenhart p); replicates (lack-of-fit) test when X repeats. | Linear and nonlinear regression |
| Outliers | Grubbs at α = 0.05 on each group, as a note only (caution, never fail). Nonlinear fits use ROUT when asked. | t tests, ANOVA, correlation |
| Convergence | Caution when a parameter hit a constraint or is ambiguous; fail when the fit did not converge. | Nonlinear regression |
| Stated, not tested | Independence (every analysis), proportional hazards and uninformative censoring (survival), X without error (regression), representative samples (ROC), acceptable agreement (Bland–Altman). | — |
Summary sentences. APA style, estimate first: the effect with its CI, then the test
statistic with its degrees of freedom, then p in the project’s style (p_phrase), then an
effect size. Examples: “The mean difference (A − B) was 3.2 (95% CI 1.1 to 5.3),
t(17.4) = 3.21, p = .005, Hedges’ g = 1.40.”; “One-way ANOVA: F(2, 27) = 4.85, p = .016,
η² = .26 (ω² = .20).” plus one sentence per multiple-comparison family naming the
comparisons that differ. Estimates and CI limits share their decimals: two significant
digits of the CI width (at least two of the estimate), scientific notation below 0.001 or
from 10⁶. Test statistics have two decimals; fractional df one decimal (two below 10);
quantities bounded by 1 (r, η², R², AUC) drop the leading zero in APA style.
Methods line. Test, sidedness, correction or adjustment, CI level, data form (for
example “computed from the entered mean, SD and N”) and the software version, ready to
paste: “Welch’s unpaired t test, two-sided, 95% CI. Dioptra 0.2.0.”
Graphs. A sheet names a default graph family (graph_family) and carries runtime
graph material: the analysed data sets, significance brackets (adjusted p of each
comparison; all of them up to six, otherwise the significant ones), fit curves with
confidence or prediction bands evaluated at 200 X values (linear and, when X > 0, log
spaced), survival curves with bands and numbers at risk, ROC points, replacement series
(correlation pairs, PCA scores and scree, Bland–Altman ratio or percent differences).