Section 3.6 — Statistical design and error analysis#
This notebook contains the code examples from Section 3.6 Statistical design and error analysis of 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 ministats import plot_pdf
from ministats.utils 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/design"
<Figure size 640x480 with 0 Axes>
# set random seed for repeatability
np.random.seed(42)
#######################################################
Definitions#
# FIGURES ONLY
filename = os.path.join(DESTDIR, "H0_rejection_region.pdf")
from ministats import plot_alpha_beta_errors
with sns.axes_style("ticks"), plt.rc_context({"figure.figsize":(7,2.5)}):
ax = plot_alpha_beta_errors(cohend=0.1, ax=None, xlims=[-3,3], n=9, show_alt=False, show_concl=True, alpha_offset=(0.06,0.013))
savefigure(ax, filename)
Design params: n = 9 , alpha = 0.05 , beta = 0.910663745890349 , Delta = 0.2 , d = 0.1 , CV = 1.0965690846343148
Saved figure to figures/stats/design/H0_rejection_region.pdf
Saved figure to figures/stats/design/H0_rejection_region.png
# FIGURES ONLY
filename = os.path.join(DESTDIR, "H0_HA_distributions_cvalue.pdf")
from ministats import plot_alpha_beta_errors
with sns.axes_style("ticks"), plt.rc_context({"figure.figsize":(8,2.5)}):
ax = plot_alpha_beta_errors(cohend=0.8, show_dist_labels=True, show_concl=True,
alpha_offset=(0.06,0.013), beta_offset=(-0.06,0.02))
savefigure(ax, filename)
Design params: n = 9 , alpha = 0.05 , beta = 0.2250805806658559 , Delta = 1.6 , d = 0.8 , CV = 1.0965690846343148
Saved figure to figures/stats/design/H0_HA_distributions_cvalue.pdf
Saved figure to figures/stats/design/H0_HA_distributions_cvalue.png
Hypothesis decision rules#
Decision rule based on \(p\)-values#
# pre-data
alpha = ... # chosen in advance
# post-data
obst = ... # calculated from sample
rvT0 = ... # sampling distribution under H0
pvalue = ... # prob. of obst under rvT0
# make decision based on p-value
if pvalue <= alpha:
decision = "Reject H0"
else:
decision = "Fail to reject H0"
Simplified decision rule#
# pre-data
alpha = ... # chosen in advance
rvT0 = ... # sampling distribution under H0
CV_alpha = ... # calculated from alpha-quantile of rvT0
# post-data
obst = ... # calculated from sample
# make decision based on test statistic
if obst >= CV_alpha:
decision = "Reject H0"
else:
decision = "Fail to reject H0"
Statistical design#
# FIGURES ONLY
filename = os.path.join(DESTDIR, "panel_beta_for_different_effect_sizes.pdf")
d_small = 0.20
d_medium = 0.57 # chosen to avoid overlap between CV and Delta
d_large = 0.80
d_vlarge = 1.3
with sns.axes_style("ticks"):
fig, ((ax1, ax2), (ax3, ax4)) = plt.subplots(2,2, figsize=(6,3))
plot_alpha_beta_errors(cohend=d_small, xlims=[-1.7,2.8], n=15, ax=ax1, fontsize=10, show_es=True,
alpha_offset=(-0.03,0.005), beta_offset=(0.1,0.2))
ax1.set_title("(a) Small effect size", fontsize=11)
plot_alpha_beta_errors(cohend=d_medium, xlims=[-1.6,2.9], n=15, ax=ax2, fontsize=10, show_es=True,
alpha_offset=(-0.03,0.005), beta_offset=(0.05,0.08))
ax2.set_title("(b) Medium effect size", fontsize=11)
plot_alpha_beta_errors(cohend=d_large, xlims=[-1.5,3], n=15, ax=ax3, fontsize=10, show_es=True,
alpha_offset=(-0.03,0.005), beta_offset=(0,0.02))
ax3.set_title("(c) Large effect size", fontsize=11)
plot_alpha_beta_errors(cohend=d_vlarge, xlims=[-1.5,3], n=15, ax=ax4, fontsize=10, show_es=True,
alpha_offset=(-0.03,0.005), beta_offset=(-0.06,0.02))
ax4.set_title("(d) Very large effect size", fontsize=11)
savefigure(fig, filename)
Design params: n = 15 , alpha = 0.05 , beta = 0.8079200023112518 , Delta = 0.4 , d = 0.2 , CV = 0.8493987605509224
Design params: n = 15 , alpha = 0.05 , beta = 0.28680362819561334 , Delta = 1.14 , d = 0.57 , CV = 0.8493987605509224
Design params: n = 15 , alpha = 0.05 , beta = 0.07303790512845224 , Delta = 1.6 , d = 0.8 , CV = 0.8493987605509224
Design params: n = 15 , alpha = 0.05 , beta = 0.0003494316033385532 , Delta = 2.6 , d = 1.3 , CV = 0.8493987605509224
Saved figure to figures/stats/design/panel_beta_for_different_effect_sizes.pdf
Saved figure to figures/stats/design/panel_beta_for_different_effect_sizes.png
# FIGURES ONLY
filename = os.path.join(DESTDIR, "panel_beta_for_different_sample_sizes.pdf")
Delta = 0.6
with sns.axes_style("ticks"):
fig, (ax1, ax2) = plt.subplots(1,2, figsize=(7,1.6))
n1 = 15
plot_alpha_beta_errors(cohend=Delta, xlims=[-1.7,2.8], n=n1, ax=ax1, fontsize=10, show_es=True,
alpha_offset=(0.03,0.005), beta_offset=(0,0.05))
ax1.set_title(f"(a) Small sample size $n={n1}$", fontsize=12)
n2 = 30
plot_alpha_beta_errors(cohend=Delta, xlims=[-1.5,3], n=n2, ax=ax2, fontsize=10, show_es=True,
alpha_offset=(0.005,0), beta_offset=(0.04,0.01))
ax2.set_title(f"(b) Larger sample size $n={n2}$", fontsize=12)
savefigure(fig, filename)
Design params: n = 15 , alpha = 0.05 , beta = 0.24858908649659617 , Delta = 1.2 , d = 0.6 , CV = 0.8493987605509224
Design params: n = 30 , alpha = 0.05 , beta = 0.0503487295055579 , Delta = 1.2 , d = 0.6 , CV = 0.6006156235170058
Saved figure to figures/stats/design/panel_beta_for_different_sample_sizes.pdf
Saved figure to figures/stats/design/panel_beta_for_different_sample_sizes.png
# FIGURES ONLY
filename = os.path.join(DESTDIR, "panel_beta_for_different_alphas.pdf")
Delta = 0.6
with sns.axes_style("ticks"):
fig, (ax1, ax2) = plt.subplots(1,2, figsize=(7,1.6))
n1 = 15
alpha1 = 0.05
plot_alpha_beta_errors(alpha=alpha1, cohend=Delta, xlims=[-1.7,2.8], n=15, ax=ax1, fontsize=10, show_es=True,
alpha_offset=(-0.02,0.005), beta_offset=(0.1,0.1))
ax1.set_title(r"(a) Conventional alpha $\alpha=0.05$", fontsize=12)
alpha2 = 0.1
plot_alpha_beta_errors(alpha=alpha2, cohend=Delta, xlims=[-1.7,2.8], n=15, ax=ax2, fontsize=10, show_es=True,
alpha_offset=(0,0.005), beta_offset=(0.05,0.05))
ax2.set_title(r"(b) Larger alpha $\alpha=0.1$", fontsize=12)
savefigure(fig, filename)
Design params: n = 15 , alpha = 0.05 , beta = 0.24858908649659617 , Delta = 1.2 , d = 0.6 , CV = 0.8493987605509224
Design params: n = 15 , alpha = 0.1 , beta = 0.14865057213685162 , Delta = 1.2 , d = 0.6 , CV = 0.6617903827547039
Saved figure to figures/stats/design/panel_beta_for_different_alphas.pdf
Saved figure to figures/stats/design/panel_beta_for_different_alphas.png
Cohen’s d standardized effect size#
The effect size \(\Delta\) we use in statistical design is usually expressed as a standardized effect size like Cohen’s \(d\), which is defined as the observed “difference” divided by a standard deviation of the theoretical model under the null hypothesis.
One-sample case#
For a one-sample test of the mean relative to a null model with mean \(\mu_0\), Cohen’s \(d\) is calculated as:
\[
d = \frac{\overline{\mathbf{x}} - \mu_0}{ s_{\mathbf{x}} },
\]
where \(s_{\mathbf{x}}\) is the sample standard deviation.
When doing statistical design, we haven’t obtained the sample \(\mathbf{x}\) yet so we don’t know what its standard deviation \(s_{\mathbf{x}}\) will be, so we have to make an educated guess of this value, which usually means using the standard deviation of the theoretical model \(\sigma_{X_0}\).
Two-sample case#
For a two-sample test for the difference between means, Cohen’s \(d\) is defined as the difference between sample means divided by the pooled standard deviation:
\[
d = \frac{\overline{\mathbf{x}} - \overline{\mathbf{y}} }{ s_{p} }.
\]
where the formula for the pooled variance is \(s_{p}^2 = [(n -1)s_{\mathbf{x}}^2 + (m -1)s_{\mathbf{y}}^2]/(n + m -2)\).
When doing statistical design, we don’t know the standard deviations \(s_{\mathbf{x}}\) and \(s_{\mathbf{y}}\) so we replace them with the theoretical standard deviations \(\sigma_{X_0}\) and \(\sigma_{Y_0}\) under the null hypothesis.
Recall table of reference values for Cohen’s \(d\) standardized effect sizes, suggested by Cohen in (Cohen YYYY TODO).
Cohen’s d |
Effect size |
|---|---|
0.01 |
very small |
0.20 |
small |
0.50 |
medium |
0.80 |
large |
The formulas for sample size planning and power calculations we’ll present below are based on Cohen’s \(d\).
This means you’ll have to express your guess about the effect size \(\Delta\) as \(d = \frac{\Delta}{\sigma_0}\) where \(\sigma_0\) is your best about the standard deviation of the theoretical distribution.
Note Cohen’s \(d\) is based on the standard deviations of the “raw” theoretical population, and not standard error.
Example 1: detect kombuncha volume deviation from theory#
Design Type A: choose \(\alpha=0.05\), \(\beta=0.2\), \(\Delta_{\textrm{min}} = 4\;\text{ml}\) then calculate required sample size \(n\). We’ll assume \(\sigma=10\).
alpha = 0.05
beta = 0.2
Delta_min = 4
# assumption
sigma = 10
from scipy.stats import norm
rvZ = norm(loc=0, scale=1)
z_u = rvZ.ppf(1-alpha)
z_l = rvZ.ppf(beta)
n_approx = (z_u - z_l)**2 * sigma**2 / Delta_min**2
n_approx
38.64098270012354
Using statsmodels#
To use the statsmodels sample size estimation function,
we’ll need to express the effect size \(\Delta=4\) in terms of Cohen’s \(d\).
d = Delta_min / sigma
d
0.4
from statsmodels.stats.power import TTestPower
ttp = TTestPower()
n = ttp.solve_power(effect_size=d, alpha=0.05, power=0.8,
alternative="larger")
n
40.02907682056493
filename = os.path.join(DESTDIR, "plot_one_sample_t_power_vs_sample_size.pdf")
ds = np.array([0.3, 0.4, 0.5])
ns = np.arange(5, 101)
fig, ax = plt.subplots(figsize=(6,2.4))
ttp.plot_power(dep_var="nobs", ax=ax,
effect_size=ds, nobs=ns, alpha=0.05,
alternative="larger")
ax.set_xticks( np.arange(5,105,5) )
# ax.set_title("Power of t-test vs. sample size for different effect sizes")
ax.set_title(None)
savefigure(fig, filename)
Saved figure to figures/stats/design/plot_one_sample_t_power_vs_sample_size.pdf
Saved figure to figures/stats/design/plot_one_sample_t_power_vs_sample_size.png
filename = os.path.join(DESTDIR, "plot_one_sample_t_power_vs_effect_size.pdf")
ds = np.arange(0, 2, 0.01)
ns = np.array([5, 10, 50, 100])
fig, ax = plt.subplots(figsize=(6,2.4))
ttp.plot_power(dep_var="effect size", ax=ax,
effect_size=ds, nobs=ns, alpha=0.05,
alternative="larger")
ax.set_xticks( np.arange(0, 2+0.1, 0.1) )
# ax.set_title("Power of t-test vs. effect size for different sample sizes")
ax.set_title(None)
savefigure(fig, filename)
Saved figure to figures/stats/design/plot_one_sample_t_power_vs_effect_size.pdf
Saved figure to figures/stats/design/plot_one_sample_t_power_vs_effect_size.png
#######################################################
kombuchapop = pd.read_csv("../datasets/kombuchapop.csv")
batch55pop = kombuchapop[kombuchapop["batch"]==55]
kpop55 = batch55pop["volume"]
np.random.seed(42)
n = 41 # rounding up
ksample55 = kpop55.sample(n)
len(ksample55)
41
# ksample55.values
from ministats import ttest_mean
ttest_mean(ksample55, mu0=1000)
0.43686918358182525
# ALT.
# from ministats import simulation_test_mean
# simulation_test_mean(ksample55, mu0=1000, sigma0=10)
Example 2: comparison of East vs. West electricity prices#
#######################################################
eprices = pd.read_csv("../datasets/eprices.csv")
eprices.groupby("loc").describe()
| price | ||||||||
|---|---|---|---|---|---|---|---|---|
| count | mean | std | min | 25% | 50% | 75% | max | |
| loc | ||||||||
| East | 9.0 | 6.155556 | 0.877655 | 4.8 | 5.5 | 6.3 | 6.5 | 7.7 |
| West | 9.0 | 9.155556 | 1.562139 | 6.8 | 8.3 | 8.6 | 10.0 | 11.8 |
Need to calculate Cohen’s \(d\) that corresponds to the assumption \(\Delta_{\text{min}} = 1\) and assuming the standard deviation of the population is \(\sigma=1\).
Delta_min = 1
std_guess = 1
d_min = Delta_min / std_guess
Solving for a desired power#
The function solve_power takes as argument the chosen level of power,
and two of the three other design parameters alpha, nobs1, and effect_size,
and calculates the value of the third parameter required to achieve the chosen level of power \(=(1-\beta)\).
from statsmodels.stats.power import TTestIndPower
ttindp = TTestIndPower()
# power of two-sample t-test assuming d_min and n=m=9
ttindp.power(effect_size=d_min, nobs1=9,
alpha=0.05, alternative="larger")
0.6500703467709203
filename = os.path.join(DESTDIR, "plot_two_sample_t_power_vs_sample_size.pdf")
ds = np.array([0.5, 0.8, 1.2])
ns = np.arange(2, 71)
fig, ax = plt.subplots(figsize=(6,2.4))
ttindp.plot_power(dep_var="nobs", ax=ax,
effect_size=ds, nobs=ns, alpha=0.05,
alternative="larger")
ax.set_xticks( np.arange(5,75,5) )
# ax.set_title("Power of two-sample t-test vs. sample size $n=m$ for different effect sizes")
ax.set_title(None)
savefigure(fig, filename)
Saved figure to figures/stats/design/plot_two_sample_t_power_vs_sample_size.pdf
Saved figure to figures/stats/design/plot_two_sample_t_power_vs_sample_size.png
filename = os.path.join(DESTDIR, "plot_two_sample_t_power_vs_effect_size.pdf")
ds = np.arange(0, 2, 0.01)
ns = np.array([5, 10, 50, 100])
fig, ax = plt.subplots(figsize=(6,2.4))
ttindp.plot_power(dep_var="effect size", ax=ax,
effect_size=ds, nobs=ns, alpha=0.05,
alternative="larger")
ax.set_xticks( np.arange(0, 2+0.1, 0.1) )
# ax.set_title("Power of two-smaple t-test vs. effect size for different sample sizes")
ax.set_title(None)
savefigure(fig, filename)
Saved figure to figures/stats/design/plot_two_sample_t_power_vs_effect_size.pdf
Saved figure to figures/stats/design/plot_two_sample_t_power_vs_effect_size.png
Calculate minimum effect size#
# minimum effect size to achieve 80% power when n=m=9
ttindp.solve_power(alpha=0.05, power=0.8, nobs1=9, alternative="larger")
1.2250749306456195
(optional) Calculate sample size needed to achieve 0.8 power#
# minimum sample size required to achieve 80% power when effect size is d=1
ttindp.solve_power(effect_size=1.0, alpha=0.05, power=0.8, alternative="larger")
#######################################################
13.097761955067904
Alternative calculation methods#
Using pingouin#
import pingouin as pg
One-sample \(t\)-test#
Delta_min = 0.4
pg.power_ttest(d=Delta_min, alpha=0.05, power=0.8,
contrast="one-sample", alternative="greater")
40.02907615780519
# power of one-sample t-test assuming d=0.4 and n=41
pg.power_ttest(d=0.4, alpha=0.05, n=41,
contrast="one-sample", alternative="greater")
0.8085822366975572
Two-sample \(t\)-test#
# power of two-sample t-test assuming d_min and n=m=9
pg.power_ttest(d=d_min, alpha=0.05, n=9,
contrast="two-samples", alternative="greater")
0.65007034847832
# minimum effect size to achieve 80% when n=m=9
pg.power_ttest(alpha=0.05, power=0.8, n=9,
contrast="two-samples", alternative="greater")
1.2250749193350323
# (optional) sample size required to achieve d_min at 80% power
pg.power_ttest(d=d_min, alpha=0.05, power=0.8,
contrast="two-samples", alternative="greater")
13.097761619826874
Using G*Power#
TODO: add sceenshots
Explanations#
Unique value proposition of hypothesis testing#
One-sided and two-sided rejection regions#
from scipy.stats import t as tdist
rvT0 = tdist(df=9)
alpha = 0.05
# right-tailed rejection region
rvT0.ppf(alpha)
-1.8331129326536337
# left-tailed rejection region
rvT0.ppf(1-alpha)
1.8331129326536335
# two-sided rejection region
rvT0.ppf(alpha/2), rvT0.ppf(1 - alpha/2)
(-2.262157162740992, 2.2621571627409915)
filename = os.path.join(DESTDIR, "panel_rejection_regions_left_twotailed_right.pdf")
from scipy.stats import t as tdist
rvT = tdist(df=9)
xs = np.linspace(-4, 4, 1000)
ys = rvT.pdf(xs)
# design choices
transp = 0.3
alpha_color = "#4A25FF"
beta_color = "#0CB0D6"
axis_color = "#808080"
def plot_rejection_region(ax, xs, ys, alt, title):
ax.set_title(title, fontsize=11)
# manually add arrowhead to x-axis + label t at the end
ax.plot(1, 0, ">", color=axis_color, transform=ax.get_yaxis_transform(), clip_on=False)
ax.set_xlabel("t")
ax.xaxis.set_label_coords(1, 0.2)
sns.lineplot(x=xs, y=ys, ax=ax, color="k")
ax.set_xlim(-4, 4)
ax.set_ylim(0, 0.42)
ax.set_xticklabels([])
ax.set_yticks([])
ax.spines[['left', 'right', 'top']].set_visible(False)
ax.spines['bottom'].set_color(axis_color)
ax.tick_params(axis='x', colors=axis_color)
ax.xaxis.label.set_color(axis_color)
if alt == "greater":
ax.set_xticks([2])
# highlight the right tail
mask = (xs > 2)
ax.fill_between(xs[mask], y1=ys[mask], alpha=transp, facecolor=alpha_color)
ax.vlines([2], ymin=0, ymax=rvT.pdf(2), linestyle="-", alpha=transp+0.2, color=alpha_color)
ax.text(2, -0.03, r"$\mathrm{CV}_{\alpha}^+$", verticalalignment="top", horizontalalignment="center")
elif alt == "less":
ax.set_xticks([-2])
# highlight the left tail
mask = (xs < -2)
ax.fill_between(xs[mask], y1=ys[mask], alpha=transp, facecolor=alpha_color)
ax.vlines([-2], ymin=0, ymax=rvT.pdf(-2), linestyle="-", alpha=transp+0.2, color=alpha_color)
ax.text(-2, -0.03, r"$\mathrm{CV}_{\alpha}^-$", verticalalignment="top", horizontalalignment="center")
elif alt == "two-sided":
ax.set_xticks([-2,2])
# highlight the left and right tails
mask = (xs < -2)
ax.fill_between(xs[mask], y1=ys[mask], alpha=transp, facecolor=alpha_color)
ax.vlines([-2], ymin=0, ymax=rvT.pdf(-2), linestyle="-", alpha=transp+0.2, color=alpha_color)
ax.text(-2, -0.03, r"$\mathrm{CV}_{\alpha/2}^-$", verticalalignment="top", horizontalalignment="center")
mask = (xs > 2)
ax.fill_between(xs[mask], y1=ys[mask], alpha=transp, facecolor=alpha_color)
ax.vlines([2], ymin=0, ymax=rvT.pdf(2), linestyle="-", alpha=transp+0.2, color=alpha_color)
ax.text(2, -0.03, r"$\mathrm{CV}_{\alpha/2}^+$", verticalalignment="top", horizontalalignment="center")
with sns.axes_style("ticks"), plt.rc_context({"figure.figsize":(7,1.6)}):
fig, (ax3, ax1, ax2) = plt.subplots(1,3)
# RIGHT
title = '(a) right-tailed rejection region'
plot_rejection_region(ax3, xs, ys, "greater", title)
# LEFT
title = '(b) left-tailed rejection region'
plot_rejection_region(ax1, xs, ys, "less", title)
# TWO-TAILED
title = '(c) two-tailed rejection region'
plot_rejection_region(ax2, xs, ys, "two-sided", title)
savefigure(fig, filename)
Saved figure to figures/stats/design/panel_rejection_regions_left_twotailed_right.pdf
Saved figure to figures/stats/design/panel_rejection_regions_left_twotailed_right.png
