Section 3.7 — Inventory of statistical tests#

This notebook contains the code examples from Section 3.7 Inventory of statistical tests from the No Bullshit Guide to Statistics.

Notebook setup#

# load Python modules
import os
import numpy as np
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
# Plot helper functions
from plot_helpers import plot_pdf
from plot_helpers import savefigure
# Figures setup
plt.clf()  # needed otherwise `sns.set_theme` doesn't work
from plot_helpers import RCPARAMS
RCPARAMS.update({'figure.figsize': (10, 3)})   # good for screen
# RCPARAMS.update({'figure.figsize': (5, 1.6)})  # good for print
sns.set_theme(
    context="paper",
    style="whitegrid",
    palette="colorblind",
    rc=RCPARAMS,
)

# Useful colors
snspal = sns.color_palette()
blue, orange, purple = snspal[0], snspal[1], snspal[4]

# High-resolution please
%config InlineBackend.figure_format = 'retina'

# Where to store figures
DESTDIR = "figures/stats/inventory"
<Figure size 640x480 with 0 Axes>
# set random seed for repeatability
np.random.seed(42)
#######################################################

Definitions#

Assumptions#

NHST procedure#

Categorization of statistical test recipes#

Z-Tests#

One-sample \(z\)-test#

See the examples/one_sample_z-test.ipynb notebook.

Proportion tests#

One-sample \(z\)-test for proportions#

Binomial test#

Two-sample \(z\)-test for proportions#

from statsmodels.stats.proportion import proportions_ztest

T-tests#

One sample \(t\)-test#

See the examples/one_sample_t-test.ipynb notebook.

Welch’s two-sample \(t\)-test#

(explain pooled variance as special case “Two sample t-test”, but inferior)

Paired \(t\)-test#

Chi-square tests#

Chi-square test for goodness of fit#

Example: are digits of \(\pi\) random?#

pidigits = [99959,  99757, 100026, 100230, 100230, 100359,  99548,  99800, 99985, 100106]
# obtained using   np.bincount(list(str(sympy.N(sympy.pi, 1_000_000)).replace('.','')))

os = pidigits           # observed
es = [1_000_000/10]*10  # expected (uniform)

from scipy.stats import chisquare
chisquare(f_obs=os, f_exp=es)
Power_divergenceResult(statistic=5.51852, pvalue=0.7869706202650393)

See original blog post for useful historical context about this https://probabilityandstats.wordpress.com/2017/03/14/are-digits-of-pi-random/

Chi-square test of independence#

Chi-square test for homogeneity#

Chi-square test for the population variance#

Analysis of variance (ANOVA) tests#

One-way analysis of variance (ANOVA)#

Two-way ANOVA#

Nonparametric tests#

Use when assumptions for other tests not valid

Sign test for the population median#

via https://vitalflux.com/sign-test-hypothesis-python-examples/

from scipy.stats import binomtest

n_pos = 6
n_neg = 9
n_min = min(n_pos, n_neg)
n_tot = n_pos + n_neg

# Calculate p-value (two-tailed) using the binomial test
binomtest(k=n_min, n=n_tot, p=0.5, alternative='two-sided')
BinomTestResult(k=6, n=15, alternative='two-sided', statistic=0.4, pvalue=0.6072387695312499)
n_max = max(n_pos, n_neg)
binomtest(k=n_max, n=n_tot, p=0.5, alternative='two-sided')
BinomTestResult(k=9, n=15, alternative='two-sided', statistic=0.6, pvalue=0.6072387695312499)

One-sample Wilcoxon signed-rank test#

Mann-Whitney U-test#

example via https://www.reneshbedre.com/blog/mann-whitney-u-test.html

dfw = pd.read_csv("https://reneshbedre.github.io/assets/posts/mann_whitney/genotype.csv")
dfw.shape
# dfw
(23, 2)
from scipy.stats import mannwhitneyu

mannwhitneyu(x=dfw["A"], y=dfw["B"], alternative="two-sided")
MannwhitneyuResult(statistic=489.5, pvalue=7.004695394561307e-07)

Kruskal-Wallis analysis of variance by ranks#

Resampling methods#

Simulation tests#

from stats_helpers import simulation_test

%psource simulation_test

Two-sample permutation test#

from stats_helpers import permutation_test

%psource permutation_test

Permutation ANOVA#

from stats_helpers import permutation_anova

%psource permutation_anova
# test on three samples
from scipy.stats import norm

# Random samples
np.random.seed(43)
sample1 = norm(loc=0).rvs(size=30)
sample2 = norm(loc=0).rvs(size=30)
sample3 = norm(loc=0.7).rvs(size=30)

np.random.seed(45)
permutation_anova([sample1, sample2, sample3])
0.0297
# compare with analytical formula
from scipy.stats import f_oneway

f_oneway(sample1, sample2, sample3)
F_onewayResult(statistic=3.6808227678358856, pvalue=0.029206733498721497)

Equivalence tests#

See examples/two_sample_equivalence_test.ipynb for an example.

Distribution checks#

Kolmogorov-Smirnov test#

Shapiro-Wilk normality test#

Discussion#

Discussion#

Exercises#