Statistics stats

This section collects various statistical tests and tools. Some can be used independently of any models, some are intended as extension to the models and model results.

API Warning: The functions and objects in this category are spread out in various modules and might still be moved around. We expect that in future the statistical tests will return class instances with more informative reporting instead of only the raw numbers.

Residual Diagnostics and Specification Tests

durbin_watson(resids[, axis])

Calculates the Durbin-Watson statistic

jarque_bera(resids[, axis])

The Jarque-Bera test of normality.

omni_normtest(resids[, axis])

Omnibus test for normality

medcouple(y[, axis])

Calculate the medcouple robust measure of skew.

robust_skewness(y[, axis])

Calculates the four skewness measures in Kim & White

robust_kurtosis(y[, axis, ab, dg, excess])

Calculates the four kurtosis measures in Kim & White

expected_robust_kurtosis([ab, dg])

Calculates the expected value of the robust kurtosis measures in Kim and White assuming the data are normally distributed.

acorr_breusch_godfrey(res[, nlags, store])

Breusch-Godfrey Lagrange Multiplier tests for residual autocorrelation.

acorr_ljungbox(x[, lags, boxpierce, …])

Ljung-Box test of autocorrelation in residuals.

acorr_lm(resid[, nlags, autolag, store, …])

Lagrange Multiplier tests for autocorrelation.

breaks_cusumolsresid(resid[, ddof])

Cusum test for parameter stability based on ols residuals.


Test for model stability, breaks in parameters for ols, Hansen 1992

recursive_olsresiduals(res[, skip, lamda, …])

Calculate recursive ols with residuals and Cusum test statistic

compare_cox(results_x, results_z[, store])

Compute the Cox test for non-nested models

compare_encompassing(results_x, results_z[, …])

Davidson-MacKinnon encompassing test for comparing non-nested models

compare_j(results_x, results_z[, store])

Compute the J-test for non-nested models

het_arch(resid[, nlags, autolag, store, ddof])

Engle’s Test for Autoregressive Conditional Heteroscedasticity (ARCH).

het_breuschpagan(resid, exog_het)

Breusch-Pagan Lagrange Multiplier test for heteroscedasticity

het_goldfeldquandt(y, x[, idx, split, drop, …])

Goldfeld-Quandt homoskedasticity test.

het_white(resid, exog)

White’s Lagrange Multiplier Test for Heteroscedasticity.

spec_white(resid, exog)

White’s Two-Moment Specification Test

linear_harvey_collier(res[, order_by])

Harvey Collier test for linearity

linear_lm(resid, exog[, func])

Lagrange multiplier test for linearity against functional alternative

linear_rainbow(res[, frac, order_by, …])

Rainbow test for linearity

linear_reset(res[, power, test_type, use_f, …])

Ramsey’s RESET test for neglected nonlinearity

Outliers and influence measures


class to calculate outlier and influence measures for OLS result

GLMInfluence(results[, resid, endog, exog, …])

Influence and outlier measures (experimental)

MLEInfluence(results[, resid, endog, exog, …])

Local Influence and outlier measures (experimental)

variance_inflation_factor(exog, exog_idx)

variance inflation factor, VIF, for one exogenous variable

See also the notes on notes on regression diagnostics

Sandwich Robust Covariances

The following functions calculate covariance matrices and standard errors for the parameter estimates that are robust to heteroscedasticity and autocorrelation in the errors. Similar to the methods that are available for the LinearModelResults, these methods are designed for use with OLS.

sandwich_covariance.cov_hac(results[, …])

heteroscedasticity and autocorrelation robust covariance matrix (Newey-West)

sandwich_covariance.cov_nw_panel(results, …)

Panel HAC robust covariance matrix

sandwich_covariance.cov_nw_groupsum(results, …)

Driscoll and Kraay Panel robust covariance matrix

sandwich_covariance.cov_cluster(results, group)

cluster robust covariance matrix


cluster robust covariance matrix for two groups/clusters


heteroscedasticity robust covariance matrix (White)

The following are standalone versions of the heteroscedasticity robust standard errors attached to LinearModelResults


See statsmodels.RegressionResults


See statsmodels.RegressionResults


See statsmodels.RegressionResults


See statsmodels.RegressionResults


get standard deviation from covariance matrix

Goodness of Fit Tests and Measures

some tests for goodness of fit for univariate distributions

powerdiscrepancy(observed, expected[, …])

Calculates power discrepancy, a class of goodness-of-fit tests as a measure of discrepancy between observed and expected data.

gof_chisquare_discrete(distfn, arg, rvs, …)

perform chisquare test for random sample of a discrete distribution

gof_binning_discrete(rvs, distfn, arg[, nsupp])

get bins for chisquare type gof tests for a discrete distribution

chisquare_effectsize(probs0, probs1[, …])

effect size for a chisquare goodness-of-fit test

anderson_statistic(x[, dist, fit, params, axis])

Calculate the Anderson-Darling a2 statistic.

normal_ad(x[, axis])

Anderson-Darling test for normal distribution unknown mean and variance.

kstest_exponential(x, *[, dist, pvalmethod])

Test assumed normal or exponential distribution using Lilliefors’ test.

kstest_fit(x[, dist, pvalmethod])

Test assumed normal or exponential distribution using Lilliefors’ test.

kstest_normal(x[, dist, pvalmethod])

Test assumed normal or exponential distribution using Lilliefors’ test.

lilliefors(x[, dist, pvalmethod])

Test assumed normal or exponential distribution using Lilliefors’ test.

Non-Parametric Tests

mcnemar(x[, y, exact, correction])

McNemar test


Test for symmetry of a (k, k) square contingency table

median_test_ksample(x, groups)

chisquare test for equality of median/location

runstest_1samp(x[, cutoff, correction])

use runs test on binary discretized data above/below cutoff

runstest_2samp(x[, y, groups, correction])

Wald-Wolfowitz runstest for two samples


Cochran’s Q test for identical effect of k treatments


class for runs in a binary sequence

sign_test(samp[, mu0])

Signs test.

Interrater Reliability and Agreement

The main function that statsmodels has currently available for interrater agreement measures and tests is Cohen’s Kappa. Fleiss’ Kappa is currently only implemented as a measures but without associated results statistics.

cohens_kappa(table[, weights, …])

Compute Cohen’s kappa with variance and equal-zero test

fleiss_kappa(table[, method])

Fleiss’ and Randolph’s kappa multi-rater agreement measure

to_table(data[, bins])

convert raw data with shape (subject, rater) to (rater1, rater2)

aggregate_raters(data[, n_cat])

convert raw data with shape (subject, rater) to (subject, cat_counts)

Multiple Tests and Multiple Comparison Procedures

multipletests is a function for p-value correction, which also includes p-value correction based on fdr in fdrcorrection. tukeyhsd performs simultaneous testing for the comparison of (independent) means. These three functions are verified. GroupsStats and MultiComparison are convenience classes to multiple comparisons similar to one way ANOVA, but still in development

multipletests(pvals[, alpha, method, …])

Test results and p-value correction for multiple tests

fdrcorrection(pvals[, alpha, method, is_sorted])

pvalue correction for false discovery rate

GroupsStats(x[, useranks, uni, intlab])

statistics by groups (another version)

MultiComparison(data, groups[, group_order])

Tests for multiple comparisons

TukeyHSDResults(mc_object, results_table, q_crit)

Results from Tukey HSD test, with additional plot methods

pairwise_tukeyhsd(endog, groups[, alpha])

Calculate all pairwise comparisons with TukeyHSD confidence intervals

local_fdr(zscores[, null_proportion, …])

Calculate local FDR values for a list of Z-scores.

fdrcorrection_twostage(pvals[, alpha, …])

(iterated) two stage linear step-up procedure with estimation of number of true hypotheses

NullDistribution(zscores[, null_lb, …])

Estimate a Gaussian distribution for the null Z-scores.

RegressionFDR(endog, exog, regeffects[, method])

Control FDR in a regression procedure.


Marginal correlation effect sizes for FDR control.


OLS regression for knockoff analysis.


Forward selection effect sizes for FDR control.


OLS regression for knockoff analysis.

RegModelEffects(model_cls[, regularized, …])

Use any regression model for Regression FDR analysis.

The following functions are not (yet) public

varcorrection_pairs_unbalanced(nobs_all[, …])

correction factor for variance with unequal sample sizes for all pairs

varcorrection_pairs_unequal(var_all, …)

return joint variance from samples with unequal variances and unequal sample sizes for all pairs

varcorrection_unbalanced(nobs_all[, srange])

correction factor for variance with unequal sample sizes

varcorrection_unequal(var_all, nobs_all, df_all)

return joint variance from samples with unequal variances and unequal sample sizes

StepDown(vals, nobs_all, var_all[, df])

a class for step down methods



compare_ordered(vals, alpha)

simple ordered sequential comparison of means

distance_st_range(mean_all, nobs_all, var_all)

pairwise distance matrix, outsourced from tukeyhsd


no frills empirical cdf used in fdrcorrection

get_tukeyQcrit(k, df[, alpha])

return critical values for Tukey’s HSD (Q)

homogeneous_subsets(vals, dcrit)

recursively check all pairs of vals for minimum distance


find all up zero crossings and return the index of the highest


find all up zero crossings and return the index of the highest

mcfdr([nrepl, nobs, ntests, ntrue, mu, …])

MonteCarlo to test fdrcorrection


str(object=’’) -> str str(bytes_or_buffer[, encoding[, errors]]) -> str

randmvn(rho[, size, standardize])

create random draws from equi-correlated multivariate normal distribution


rankdata, equivalent to scipy.stats.rankdata

rejectionline(n[, alpha])

reference line for rejection in multiple tests


extract a partition from a list of tuples


remove sets that are subsets of another set from a list of tuples


should be equivalent of scipy.stats.tiecorrect

Basic Statistics and t-Tests with frequency weights

Besides basic statistics, like mean, variance, covariance and correlation for data with case weights, the classes here provide one and two sample tests for means. The t-tests have more options than those in scipy.stats, but are more restrictive in the shape of the arrays. Confidence intervals for means are provided based on the same assumptions as the t-tests.

Additionally, tests for equivalence of means are available for one sample and for two, either paired or independent, samples. These tests are based on TOST, two one-sided tests, which have as null hypothesis that the means are not “close” to each other.

DescrStatsW(data[, weights, ddof])

descriptive statistics and tests with weights for case weights

CompareMeans(d1, d2)

class for two sample comparison

ttest_ind(x1, x2[, alternative, usevar, …])

ttest independent sample

ttost_ind(x1, x2, low, upp[, usevar, …])

test of (non-)equivalence for two independent samples

ttost_paired(x1, x2, low, upp[, transform, …])

test of (non-)equivalence for two dependent, paired sample

ztest(x1[, x2, value, alternative, usevar, ddof])

test for mean based on normal distribution, one or two samples

ztost(x1, low, upp[, x2, usevar, ddof])

Equivalence test based on normal distribution

zconfint(x1[, x2, value, alpha, …])

confidence interval based on normal distribution z-test

weightstats also contains tests and confidence intervals based on summary data

_tconfint_generic(mean, std_mean, dof, …)

generic t-confint to save typing

_tstat_generic(value1, value2, std_diff, …)

generic ttest to save typing

_zconfint_generic(mean, std_mean, alpha, …)

generic normal-confint to save typing

_zstat_generic(value1, value2, std_diff, …)

generic (normal) z-test to save typing

_zstat_generic2(value, std_diff, alternative)

generic (normal) z-test to save typing

Power and Sample Size Calculations

The power module currently implements power and sample size calculations for the t-tests, normal based test, F-tests and Chisquare goodness of fit test. The implementation is class based, but the module also provides three shortcut functions, tt_solve_power, tt_ind_solve_power and zt_ind_solve_power to solve for any one of the parameters of the power equations.


Statistical Power calculations for t-test for two independent sample


Statistical Power calculations for one sample or paired sample t-test


Statistical Power calculations for one sample chisquare test


Statistical Power calculations for z-test for two independent samples.


Statistical Power calculations F-test for one factor balanced ANOVA


Statistical Power calculations for generic F-test

tt_solve_power([effect_size, nobs, alpha, …])

solve for any one parameter of the power of a one sample t-test

tt_ind_solve_power([effect_size, nobs1, …])

solve for any one parameter of the power of a two sample t-test

zt_ind_solve_power([effect_size, nobs1, …])

solve for any one parameter of the power of a two sample z-test


Also available are hypothesis test, confidence intervals and effect size for proportions that can be used with NormalIndPower.

proportion_confint(count, nobs[, alpha, method])

confidence interval for a binomial proportion

proportion_effectsize(prop1, prop2[, method])

Effect size for a test comparing two proportions

binom_test(count, nobs[, prop, alternative])

Perform a test that the probability of success is p.

binom_test_reject_interval(value, nobs[, …])

rejection region for binomial test for one sample proportion

binom_tost(count, nobs, low, upp)

exact TOST test for one proportion using binomial distribution

binom_tost_reject_interval(low, upp, nobs[, …])

rejection region for binomial TOST

multinomial_proportions_confint(counts[, …])

Confidence intervals for multinomial proportions.

proportions_ztest(count, nobs[, value, …])

Test for proportions based on normal (z) test

proportions_ztost(count, nobs, low, upp[, …])

Equivalence test based on normal distribution

proportions_chisquare(count, nobs[, value])

test for proportions based on chisquare test

proportions_chisquare_allpairs(count, nobs)

chisquare test of proportions for all pairs of k samples

proportions_chisquare_pairscontrol(count, nobs)

chisquare test of proportions for pairs of k samples compared to control

proportion_effectsize(prop1, prop2[, method])

Effect size for a test comparing two proportions

power_binom_tost(low, upp, nobs[, p_alt, alpha])

power_ztost_prop(low, upp, nobs, p_alt[, …])

Power of proportions equivalence test based on normal distribution

samplesize_confint_proportion(proportion, …)

find sample size to get desired confidence interval length

Moment Helpers

When there are missing values, then it is possible that a correlation or covariance matrix is not positive semi-definite. The following three functions can be used to find a correlation or covariance matrix that is positive definite and close to the original matrix.

corr_clipped(corr[, threshold])

Find a near correlation matrix that is positive semi-definite

corr_nearest(corr[, threshold, n_fact])

Find the nearest correlation matrix that is positive semi-definite.

corr_nearest_factor(corr, rank[, ctol, …])

Find the nearest correlation matrix with factor structure to a given square matrix.

corr_thresholded(data[, minabs, max_elt])

Construct a sparse matrix containing the thresholded row-wise correlation matrix from a data array.

cov_nearest(cov[, method, threshold, …])

Find the nearest covariance matrix that is positive (semi-) definite

cov_nearest_factor_homog(cov, rank)

Approximate an arbitrary square matrix with a factor-structured matrix of the form k*I + XX’.

FactoredPSDMatrix(diag, root)

Representation of a positive semidefinite matrix in factored form.

kernel_covariance(exog, loc, groups[, …])

Use kernel averaging to estimate a multivariate covariance function.

These are utility functions to convert between central and non-central moments, skew, kurtosis and cummulants.


convert non-central moments to cumulants recursive formula produces as many cumulants as moments


convert central to non-central moments, uses recursive formula optionally adjusts first moment to return mean


convert central moments to mean, variance, skew, kurtosis


convert non-central moments to cumulants recursive formula produces as many cumulants as moments

mnc2mc(mnc[, wmean])

convert non-central to central moments, uses recursive formula optionally adjusts first moment to return mean


convert central moments to mean, variance, skew, kurtosis


convert mean, variance, skew, kurtosis to central moments


convert mean, variance, skew, kurtosis to non-central moments

cov2corr(cov[, return_std])

convert covariance matrix to correlation matrix

corr2cov(corr, std)

convert correlation matrix to covariance matrix given standard deviation


get standard deviation from covariance matrix

Mediation Analysis

Mediation analysis focuses on the relationships among three key variables: an ‘outcome’, a ‘treatment’, and a ‘mediator’. Since mediation analysis is a form of causal inference, there are several assumptions involved that are difficult or impossible to verify. Ideally, mediation analysis is conducted in the context of an experiment such as this one in which the treatment is randomly assigned. It is also common for people to conduct mediation analyses using observational data in which the treatment may be thought of as an ‘exposure’. The assumptions behind mediation analysis are even more difficult to verify in an observational setting.

Mediation(outcome_model, mediator_model, …)

Conduct a mediation analysis.

MediationResults(indirect_effects, …)

A class for holding the results of a mediation analysis.

Oaxaca-Blinder Decomposition

The Oaxaca-Blinder, or Blinder-Oaxaca as some call it, decomposition attempts to explain gaps in means of groups. It uses the linear models of two given regression equations to show what is explained by regression coefficients and known data and what is unexplained using the same data. There are two types of Oaxaca-Blinder decompositions, the two-fold and the three-fold, both of which can and are used in Economics Literature to discuss differences in groups. This method helps classify discrimination or unobserved effects. This function attempts to port the functionality of the oaxaca command in STATA to Python.

OaxacaBlinder(endog, exog, bifurcate[, …])

Class to perform Oaxaca-Blinder Decomposition.

OaxacaResults(results, model_type)

This class summarizes the fit of the OaxacaBlinder model.

Distance Dependence Measures

Distance dependence measures and the Distance Covariance (dCov) test.

distance_covariance_test(x, y[, B, method])

The Distance Covariance (dCov) test

distance_statistics(x, y[, x_dist, y_dist])

Calculate various distance dependence statistics.

distance_correlation(x, y)

Distance correlation.

distance_covariance(x, y)

Distance covariance.


Distance variance.