Methods & assumptions

This tool implements ordinary one-way ANOVA for independent groups and one continuous outcome. Paired, repeated-measures and nested designs are unsupported. ANOVA for two groups is allowed. Product limits are 2–20 groups, ≥2 included values per group and ≤10,000 included observations.

Dose–response fitting

The dose-response workspace fits each series/biological-experiment combination independently. Technical repeats at a concentration are averaged; distinct dose means receive equal weight in ordinary nonlinear least squares. No pooled biological estimate, replicate weighting, automatic exclusion or model selection is performed. Use separate methods for correlated dose responses, heterogeneous errors, biphasic responses and nonmonotonic curves.

For x = log10(concentration in M), the model is Y = Bottom + (Top − Bottom) / [1 + 10^(−s × Hill × (x − midpoint))], where s = +1 for activation and −1 for inhibition. Top exceeds Bottom and Hill is positive. The 4PL fits both plateaus; the two 3PL options explicitly fix either Bottom or Top on the final response scale. EC50/IC50 is the relative midpoint between asymptotes, not an absolute 50% response. pEC50/pIC50 is minus log10 of the molar midpoint.

Optional normalization is 100 × response / mean vehicle, or 100 × (response − mean blank) / (mean vehicle − mean blank). Controls must match both series and experiment and have a positive denominator. Control rows are retained but not fitted. Normalization does not clip values; its shared-control uncertainty and induced correlations are not propagated into fit intervals.

Nine starting values feed a bounded Levenberg–Marquardt solver. Numerical bounds are Hill 0.05–10, midpoint within two log10 units beyond observed concentrations, and amplitude 0.0001–100 times the observed response span. The fitted bottom in a 4PL is bounded to −10 to +10 spans relative to the smallest dose mean. Fixed 3PL plateaus must lie within 100 spans of that origin. These guards are numerical limits, not biological priors. At least six distinct concentrations for 4PL or five for 3PL are required. Concentrations must fall between 10⁻¹⁵ and 10³ M; at most 24 curves, 100 doses per curve and 5,000 rows are supported.

Residual degrees of freedom equal distinct concentrations minus free parameters. RMSE = √(SSE/df). Covariance is inverse(JᵀJ) × SSE/df in scaled fitting coordinates. Intervals use a two-sided Student-t critical value, with exponentiation of log-midpoint and log-Hill intervals. Fixed parameters have no estimated interval. Intervals assume independent homoscedastic dose-mean errors and local linearity; they are not profile-likelihood intervals and do not estimate between-experiment uncertainty. Bounds or an infinity-norm condition estimate above 10¹⁰ suppress intervals. Missing plateau coverage and extrapolated midpoints are flagged. No global-optimum guarantee is made.

Figures show dose means ± technical sample SD, optional individual technical measurements, and model lines only within measured concentrations. A single technical measurement has no SD. Residuals use dose means. Figure selection never removes fits from results exports. References: GraphPad logistic equation and SciPy covariance limitations.

T-tests and control comparisons

The comparison workspace accepts one continuous outcome and one prespecified observation or summary per experimental unit. Technical repeats must be summarized according to the study design before import. Missing values and duplicate group/unit records are rejected. Independent and one-sample groups require distinct unit IDs across groups. Paired tests match exact unit IDs across every treatment and the selected control; incomplete pairs are rejected, not silently dropped. Limits: 20 groups, 5,000 observations, at least two units per group, and finite values between −10¹⁰⁰ and 10¹⁰⁰.

Differences are treatment minus reference. Independent tests offer Student’s equal-variance pooled standard error or Welch’s unequal-variance standard error and Satterthwaite degrees of freedom. Each control contrast uses its own two groups, not an ANOVA pooled error term. Paired tests are one-sample t-tests of within-unit differences against zero. One-sample tests compare each group mean with a declared known constant. An estimated or control-normalized reference is not automatically a known constant. Student/one-sample/paired inference relies on appropriate normal-error assumptions; Welch uses a degrees-of-freedom approximation. Observations or paired differences must be independent across units.

t = difference / SE. For Welch, SE² = s₁²/n₁ + s₂²/n₂ and df = (s₁²/n₁ + s₂²/n₂)² / [(s₁²/n₁)²/(n₁−1) + (s₂²/n₂)²/(n₂−1)]. Student uses the two-group pooled variance with df = n₁+n₂−2. Paired and one-sample tests use SE = sample SD/√n, df = n−1. Two-sided p-values use the complementary F-tail identity at t² with df₁ = 1. One-sided tails use the signed t statistic. Sample moments use centered compensated sums; binary64 input precision remains a limitation. Zero standard error yields unavailable inference, not an invented zero p-value.

The family is every imported non-control group versus the selected control, or every imported group versus the fixed reference. It is declared before analysis and never reduced by hiding plots or withholding an unavailable test. Bonferroni p-values are min(1, m × p). Holm uses sorted step-down factors m, m−1, …, 1 with monotonic cumulative maxima. If any raw p-value is unavailable, all Holm adjustments are withheld; Bonferroni can still adjust available p-values using the full m. Unadjusted tests are explicitly labelled and do not control family-wise error. These methods are not Dunnett’s test.

With either adjustment, intervals use Bonferroni α/m: two-sided difference ± t(1−α/(2m), df) × SE, or the corresponding one-sided bound with t(1−α/m, df). Unadjusted intervals use m = 1. Holm p-values and Bonferroni intervals can differ at the significance threshold because these intervals do not invert Holm’s decisions. One-sided intervals retain an unbounded endpoint; arrows in the difference plot indicate that endpoint. A raw-plot SD bar describes variability and is not a difference confidence interval.

These are prespecified endpoint comparisons, not longitudinal, factorial or hierarchical models. Statistical significance of a combination versus control does not establish interaction or synergy. Nonsignificance does not establish equivalence. Independence, distributional assumptions and selection of the family require researcher judgement; the confirmation checkbox cannot verify them. No automated test choice, normalization, data exclusion or outlier removal is performed.

Primary references: SciPy independent t-tests, SciPy paired t-tests, and R multiplicity adjustment definitions. Independent fixtures use SciPy 1.17.1; no R execution is claimed.

ANOVA calculations

Observations are centered on a common origin and accumulated with compensated sums. The grand mean is weighted by sample size. Between-group SS = Σ nᵢ(meanᵢ − grand mean)²; within-group SS = Σ(xᵢⱼ − meanᵢ)². Total SS is their sum, checked against a direct centered calculation. F = [SS between / (k−1)] / [SS within / (N−k)]. The upper F tail uses the complementary incomplete-beta identity.

Sample SD uses n−1; SEM = SD/√n. Individual mean confidence intervals are mean ± t(1−α/2, n−1) × SEM. Eta-squared = SS between / SS total. Raw omega-squared = [SS between − (k−1) MS within] / [SS total + MS within]; negative estimates are retained.

All-pairs inference

Tukey (balanced) or Tukey-Kramer (unequal sizes) always uses the complete family. Difference is first mean minus second. SE range = √[(MS within / 2)(1/nᵢ + 1/nⱼ)]. q = |difference| / SE range. Simultaneous intervals are difference ± q critical × SE range. Unequal sizes do not remove the equal-variance assumption. Omnibus and pairwise tests concern different hypotheses, so comparisons are not gated by the omnibus p-value.

Diagnostics and limitations

Residual-versus-fitted plots and Q-Q positions (i−0.5)/N use within-group residuals. Levene test, median-centered, applies ANOVA to absolute deviations from each group median; zero within-deviation variance makes it unavailable. Diagnostics cannot establish independence or certify normal errors and equal variances. No automatic outlier removal, imputation or test selection is performed.

Missing tokens are case-sensitive; blanks are missing. Duplicate numbers are retained. CSV record/column positions and original strings are saved. Only decimal points are supported; confirm delimiters in the mapping preview, since unquoted decimal commas in a comma-separated file can be inherently ambiguous.

See numerical limits before interpretation. Nonsignificance does not demonstrate equivalence and these analyses do not establish causality.

References: NIST ANOVA, R TukeyHSD, NIST Levene.