A complete guide to the 5x2cv paired t-test (Dietterich's test)-formula, hypotheses, assumptions, hand-worked numerical examples, why the ordinary paired t-test fails for comparing classifiers, and an MLxtend Python walkthrough for statistically comparing two machine learning models.
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A complete guide to the Anderson-Darling Test-formula, hypotheses, assumptions, hand-worked numerical examples, critical-value interpretation, tail sensitivity, and a scipy Python walkthrough for testing goodness-of-fit against a normal distribution.
A complete guide to the Augmented Dickey-Fuller (ADF) Test-formula, hypotheses, assumptions, hand-worked numerical examples, critical values, and a statsmodels.tsa.stattools.adfuller Python walkthrough for testing stationarity in time series data.
A complete guide to Bartlett's Test-formula, hypotheses, assumptions, hand-worked numerical examples, the correction factor, and a scipy.stats.bartlett Python walkthrough for testing equality of variances across normally distributed groups.
A complete guide to the Bayesian t-test-priors, posteriors, the Bayes factor, credible intervals, hand-worked numerical intuition, and a Python walkthrough using pingouin and PyMC.
A complete guide to the Breusch-Pagan Test-formula, hypotheses, assumptions, hand-worked numerical examples, the Koenker studentized version, and a statsmodels.stats.diagnostic.het_breuschpagan Python walkthrough for detecting heteroskedasticity in regression residuals.
A complete guide to the Brown-Forsythe Test-formula, hypotheses, assumptions, hand-worked numerical examples, its two related meanings (equal-variance test and unequal-variances ANOVA F* test), and a scipy.stats Python walkthrough.
A complete guide to the Chi-Square Test of Independence-formula, hypotheses, assumptions, hand-worked numerical examples, Cramer's V effect size, and a scipy.stats.chi2_contingency Python walkthrough for categorical data.
A complete guide to Cochran's Q Test-formula, hypotheses, assumptions, hand-worked numerical examples, post-hoc pairwise McNemar comparisons, effect size, and a statsmodels.stats.contingency_tables.cochrans_q Python walkthrough for repeated binary measures.
A complete guide to the Corrected Resampled t-test (Nadeau and Bengio's test)-formula, hypotheses, assumptions, hand-worked numerical examples, why the ordinary paired t-test underestimates variance across repeated random train/test splits, and a from-scratch Python walkthrough for statistically comparing two machine learning models.
A complete guide to the Cox Proportional Hazards Test-formula, hypotheses, assumptions, hand-worked numerical examples, hazard ratio interpretation, the proportional hazards assumption check, and a lifelines Python walkthrough for survival analysis.
A complete guide to the D'Agostino-Pearson Test-formula, hypotheses, assumptions, hand-worked numerical examples, skewness and kurtosis interpretation, and a scipy Python walkthrough for testing normality using an omnibus K-squared statistic.
A complete guide to DeLong's Test for comparing two correlated ROC AUCs-formula, hypotheses, assumptions, hand-worked numerical examples, the structural relationship to the Mann-Whitney U statistic, and a Python (roc_auc_score + delong-style covariance) walkthrough for paired classifier comparison.
A complete guide to Dunn's Post-Hoc Test-formula, hypotheses, assumptions, hand-worked numerical examples, p-value adjustment methods, and a scikit-posthocs Python walkthrough after a significant Kruskal-Wallis Test.
A complete guide to the Durbin-Watson Test-formula, hypotheses, assumptions, hand-worked numerical examples, the inconclusive zone, and a statsmodels.stats.stattools.durbin_watson Python walkthrough for detecting autocorrelation in regression residuals.
A complete guide to the F-Test for Equality of Variances-formula, hypotheses, assumptions, hand-worked numerical examples, one-tailed vs two-tailed usage, and a scipy.stats Python walkthrough for comparing the variances of two normally distributed groups.
A complete guide to Fisher's Exact Test-formula, hypotheses, assumptions, hand-worked numerical examples using the hypergeometric distribution, odds ratio effect size, one-sided vs two-sided tests, and a scipy.stats.fisher_exact Python walkthrough for small-sample categorical data.
A complete guide to the Friedman Test-formula, hypotheses, assumptions, hand-worked numerical examples, effect size, Nemenyi post-hoc comparisons, and a scipy.stats.friedmanchisquare Python walkthrough.
A complete guide to the Granger Causality Test-formula, hypotheses, assumptions, hand-worked numerical examples, the critical stationarity and lag-order steps, and a statsmodels.tsa.stattools.grangercausalitytests Python walkthrough for testing whether one time series helps predict another.
A complete guide to the Hosmer-Lemeshow Test-formula, deciles-of-risk table construction, hand-worked numerical examples, degrees-of-freedom rules, and a Python (statsmodels-compatible) walkthrough for checking goodness-of-fit in logistic regression models.
A complete guide to the independent samples t-test (two-sample t-test)-formula, hypotheses, assumptions, hand-worked numerical examples, effect size, Welch's correction, and a scipy.stats.ttest_ind Python walkthrough.
A complete guide to the Kolmogorov-Smirnov Test-formula, hypotheses, assumptions, hand-worked one-sample and two-sample numerical examples, ECDF comparison, and a scipy Python walkthrough for testing goodness-of-fit and comparing distributions.
A complete guide to the KPSS Test (Kwiatkowski-Phillips-Schmidt-Shin Test)-formula, hypotheses, assumptions, hand-worked numerical examples, critical values, and a statsmodels.tsa.stattools.kpss Python walkthrough for testing stationarity in time series data.
A complete guide to the Kruskal-Wallis Test-formula, hypotheses, assumptions, hand-worked numerical examples, effect size, Dunn's post-hoc test, and a scipy.stats.kruskal Python walkthrough.
A complete guide to Levene's Test-formula, hypotheses, assumptions, hand-worked numerical examples, the Brown-Forsythe modification, and a scipy.stats.levene Python walkthrough for testing equality of variances across groups.
A complete guide to the Likelihood Ratio Test for comparing nested statistical models-formula, hypotheses, assumptions, hand-worked numerical examples, Wilks' theorem, and a Python (statsmodels / scipy) walkthrough for comparing nested regression and generalized linear models.
A complete guide to the Lilliefors Test-formula, hypotheses, the corrected critical value table, hand-worked numerical examples, and a statsmodels Python walkthrough for testing normality when the mean and variance are estimated from the sample.
A complete guide to the Ljung-Box Test-formula, hypotheses, assumptions, hand-worked numerical examples, chi-square critical values, and a statsmodels.stats.diagnostic.acorr_ljungbox Python walkthrough for detecting autocorrelation across multiple lags in time series and residuals.
A complete guide to the Log-Rank Test-formula, hypotheses, assumptions, hand-worked numerical examples, Kaplan-Meier curve comparison, stratified and trend variants, and a lifelines Python walkthrough for comparing survival between groups.
A complete guide to the Mann-Whitney U Test (Wilcoxon Rank-Sum Test)-formula, hypotheses, assumptions, hand-worked numerical examples, effect size, and a scipy.stats.mannwhitneyu Python walkthrough.
A complete guide to McNemar's Test-formula, hypotheses, assumptions, hand-worked numerical examples, continuity correction, exact binomial version, odds ratio effect size, and a statsmodels.stats.contingency_tables.mcnemar Python walkthrough for paired binary data.
A clear, practical guide to the Morris Elementary Effects Method-what it is, the mu-star and sigma statistics, a hand-worked numerical example you can check yourself, and a SALib Python walkthrough for quickly screening which model inputs actually matter.
A complete guide to the Nemenyi Post-Hoc Test-formula, hypotheses, assumptions, hand-worked numerical examples, critical difference diagrams, and a scikit-posthocs Python walkthrough after a significant Friedman Test.
A complete guide to the paired t-test (dependent t-test)-formula, hypotheses, assumptions, hand-worked numerical examples, effect size, and a scipy.stats.ttest_rel Python walkthrough.
A complete guide to the Permutation Test (randomization test)-core idea, when to use it, hand-worked numerical examples, relationship to the Mann-Whitney U Test and t-test, and a Python walkthrough using scipy.stats.permutation_test.
A complete guide to the Ramsey RESET Test-formula, hypotheses, assumptions, hand-worked numerical examples, and a statsmodels.stats.diagnostic.linear_reset Python walkthrough for detecting omitted nonlinearity and functional-form misspecification in regression models.
A complete guide to the Score Test (Lagrange Multiplier Test)-formula, intuition, relationship to the Wald and Likelihood Ratio tests, hand-worked numerical examples, and a Python (statsmodels/scipy) walkthrough.
A complete guide to the Shapiro-Wilk Test-formula, hypotheses, assumptions, hand-worked numerical examples, Q-Q plot interpretation, and a scipy Python walkthrough for testing whether sample data comes from a normal distribution.
A complete guide to Sobol Sensitivity Analysis-first-order and total-order indices, variance decomposition, hand-worked numerical intuition, and a Python walkthrough using SALib applied to an SIR epidemic model.
A complete guide to Tukey's Honest Significant Difference (HSD) Test-formula, hypotheses, assumptions, hand-worked numerical examples, confidence intervals, and a statsmodels pairwise_tukeyhsd Python walkthrough.
A complete guide to the Wald Test for testing a single parameter (or a set of parameters) in a maximum likelihood model-formula, hypotheses, assumptions, hand-worked numerical examples, the relationship to the t-test and z-test, and a Python (statsmodels) walkthrough for regression coefficient testing.
A complete guide to White's Test for Heteroskedasticity-formula, hypotheses, assumptions, hand-worked numerical examples, and a statsmodels.stats.diagnostic.het_white Python walkthrough for detecting heteroskedasticity in regression residuals without assuming a linear variance pattern.
A complete guide to the Wilcoxon Signed-Rank Test-formula, hypotheses, assumptions, hand-worked numerical examples, effect size, and a scipy.stats.wilcoxon Python walkthrough.
A complete guide to using the Wilcoxon Signed-Rank Test to compare two machine learning models-formula, hypotheses, assumptions, hand-worked numerical examples across multiple datasets, effect size, and a scipy.stats.wilcoxon Python walkthrough for statistically comparing model performance.
