In [1]:
# Monday 25 Sep 2023
#
# w03 Model comparison and hypothesis testing
#
import warnings
import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as stats
obs = [1,29,3,7]
exp = [3,27,1,9]
# df = degrees of freedom
# df = 4 - constraints
c = 2 # 3 constraints: N_v = 30, N_c = 10, f_H0 = 1/10
df = 4 - c # df = 1
# the chi2 val
# calculated directly
#
chi2val = 0
for i in range(len(obs)):
chi2val += (obs[i]-exp[i])*(obs[i]-exp[i])/exp[i]
print("The chi-square value calculated using the definition")
print("chi square ", chi2val, " at df = ", df)
The chi-square value calculated using the definition chi square 5.925925925925926 at df = 2
In [2]:
# The p-value calculated using the chi2 distribution
# + the definition of p-value
#
pval = 1-stats.chi2.cdf(chi2val, df, loc=0, scale=1)
print("\nThe p-value calculated using the chi2 distribution and the definition of p-value")
print("chi square ", chi2val, "p-value ", pval, " at df = ", df)
The p-value calculated using the chi2 distribution and the definition of p-value chi square 5.925925925925926 p-value 0.05166560687888455 at df = 2
In [3]:
# the chi2 val
# calculated using the chisquare function in stats
#
# why 2??
# The p-value is computed using a chi-squared distribution with
# k - 1 - ddof degrees of freedom, where k is the number of observed
# frequencies
chi2vala, pvala = stats.chisquare(obs, exp, ddof=2)
print("\nThe p-value calculated using the numpy function chisquare")
print("chi square ", chi2vala, "p-value ", pvala, " at df = ", df)
# why 2??
# from the documentation of chisquare:
#
# The p-value is computed using a chi-squared distribution with
# k - 1 - ddof degrees of freedom, where k is the number of observed
# frequencies
The p-value calculated using the numpy function chisquare chi square 5.925925925925926 p-value 0.014919696305821892 at df = 2
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df = 1
# The chisquare distribution
#
fig, ax = plt.subplots(2, 1)
x = np.linspace(stats.chi2.ppf(0.01, df), stats.chi2.ppf(0.99, df), 100)
# the PDF
ax[0].plot(x, stats.chi2.pdf(x, df), 'r-', lw=2, alpha=0.6, label='chi2 pdf')
ax[0].legend(loc='best', frameon=False)
ax[0].set_ylim([0,0.5])
# CDF
#
# 1-CDF = Survival = p-value
#
ax[1].plot(x, stats.chi2.cdf(x, df), 'r-', lw=2, alpha=0.6, label='chi2 cdf')
ax[1].plot(x, stats.chi2.sf(x, df), 'b-', lw=2, alpha=0.6, label='chi2 1 - cdf = survival')
ax[1].legend(loc='best', frameon=False)
plt.show()
In [5]:
import warnings
import numpy as np
import matplotlib.pyplot as plt
import math
import scipy.stats as stats
import scipy.special as special
import decimal
def binomial_posterior(f, N, n):
ff = 1.0 - f # probability of not contracting disease
# log[(N+1)!] - log[n!] - log[N-n)!]
logcoeff = np.real(special.loggamma(N+2))
logcoeff -= np.real(special.loggamma(N-n+1))
logcoeff -= np.real(special.loggamma(n+1))
# pdf = coeff * f^n * (1-f)^{N-n}
logpdf = logcoeff + n*np.log(f) + (N-n)*np.log(ff)
pdf = np.exp(logpdf)
return pdf, logpdf
def binomial_pdata(f, N, n):
ff = 1.0 - f # probability of not contracting disease
# log[(N)!] - log[n!] - log[N-n)!]
logcoeff = np.real(special.loggamma(N+1))
logcoeff -= np.real(special.loggamma(N-n+1))
logcoeff -= np.real(special.loggamma(n+1))
# pdf = coeff * f^n * (1-f)^{N-n}
logpdf = logcoeff + n*np.log(f) + (N-n)*np.log(ff)
pdf = np.exp(logpdf)
return pdf, logpdf
# Class example
#
Nv = 30 # volunteers that have received the vaccine
Nc = 10 # volunteers in control group (no vaccine)
nv = 1 # received vaccine and contracted plague
nc = 3 # no vaccine and contracted plague
f_max = 0.9999999999
f_min = 0.0000000001
f_bin = 1/(100*np.sqrt(max(Nv,Nc)))
f_nbin = int((f_max-f_min)/f_bin)
f = np.linspace(f_min, f_max, f_nbin) # the probability of contracting disease
# the posteriors: P(f_v) and P(f_c)
#
pdf_v = binomial_posterior(f, Nv, nv)
pdf_c = binomial_posterior(f, Nc, nc)
# max value
#
f_v_max = nv/Nv
f_c_max = nc/Nc
plt.plot(f, pdf_v[0], label = "Posterior(p of disease | vaccine)")
plt.plot(f, pdf_c[0], label = "Posterior(p of disease | no vaccine)")
plt.axvline(x=f_v_max, color='b', ls=':', label='nv/Nv')
plt.axvline(x=f_c_max, color='b', ls=':', label='nc/Nc')
plt.legend(loc="upper right")
plt.show()
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# when there is very little data
#
Nv = 3 # volunteers that have received the vaccine
Nc = 2 # volunteers in control group (no vaccine)
nv = 0 # received vaccine and contracted disease
nc = 1 # no vaccine and contracted disease
f_bin = 1/(100*np.sqrt(max(Nv,Nc)))
f_nbin = int((f_max-f_min)/f_bin)
# the probability of contracting disease
f = np.linspace(f_min, f_max, f_nbin)
# posterior probs as a function of data
pdf_v = binomial_posterior(f, Nv, nv)
pdf_c = binomial_posterior(f, Nc, nc)
plt.plot(f, pdf_v[0], label = "Posterior(f | vaccine)")
plt.plot(f, pdf_c[0], label = "Posterior(f | no vaccine)")
plt.legend(loc="upper right")
plt.show()
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# H1 = fv < fc
#
# P(H1| data) = P(fv < fc| data)
#
Nv = 30 # volunteers that have received the vaccine
Nc = 10 # volunteers in control group (no vaccine)
nv = 1 # received vaccine and contracted plague
nc = 3 # no vaccine and contracted plague
f_bin = 1/(100*np.sqrt(max(Nv,Nc)))
f_nbin = int((f_max-f_min)/f_bin)
# the probability of contracting disease
f = np.linspace(f_min, f_max, f_nbin)
# posterior probs as a function of data
pdf_v = binomial_posterior(f, Nv, nv)
pdf_c = binomial_posterior(f, Nc, nc)
prob_H1 = 0
for bc in range(f_nbin):
fc = f_min + bc*f_bin
for bv in range(f_nbin):
fv = f_min + bv*f_bin
if fv < fc:
prob_H1 += pdf_v[0][bv]*pdf_c[0][bc]
prob_H1 *= f_bin * f_bin
print("P(H1 = [fv < fc] | data) = ", prob_H1)
P(H1 = [fv < fc] | data) = 0.9806714796684223
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# H0 = [fc = fv]
# P(H0 | data)
#
prob_H0 = 0
for b in range(f_nbin):
f = f_min + b*f_bin
prob_H0 += pdf_v[0][b]*pdf_c[0][b]
prob_H0 *= f_bin
print("P(H0 = [fc = fv] |data) =", prob_H0)
# ratio
print ("P(H1|data)/P(H0|data) =", prob_H1/prob_H0)
P(H0 = [fc = fv] |data) = 0.3265926157714178 P(H1|data)/P(H0|data) = 3.0027362295135123
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# posterior of the efficacy = 1-fv/fc
#
# 0 <= a = 1-fv/fc <= 1
#
# or fv = (1-a)fc
#
a_min = 0.00001
a_max = 1.0
a_nbin = 100
a = np.linspace(a_min, a_max, a_nbin)
Nv = 30 # volunteers that received the vaccine
Nc = 10 # volunteers in control group (no vaccine)
nv = 1 # received vaccine and contracted plague
nc = 3 # no vaccine and contracted plague
#Nv = 15000 # volunteers that received the vaccine
#Nc = 15000 # volunteers in control group (no vaccine)
#nv = 5 # received vaccine and contracted Covid-19
#nc = 90 # no vaccine and contracted Covid-19
f_min = 0.0000001
f_max = 0.9999999
f_bsize = 1/(100*np.sqrt(max(Nc,Nv)))
f_nbin = int((f_max-f_min)/f_bsize)
prior = 1/(f_max-f_min)
post_a = []
post_a_norm = 0
post_max = 0
a_at_max = 0
for x in range(len(a)):
post = 0
for b in range(f_nbin):
fc = f_min + b*f_bsize
fv = fc * (1-a[x])
if fv > 0:
pdf_c, logpdf_c = binomial_posterior(fc, Nc, nc)
pdf_v, logpdf_v = binomial_posterior(fv, Nv, nv)
#post += pdf_c * pdf_v
post += np.exp(logpdf_c + logpdf_v)
post_a.append(post)
post_a_norm += post
# find max prob_a and efficiency (a) at which max happens
if post > post_max:
post_max = post
a_at_max = a[x]
# median
post_a_mode = stats.mode(post_a)
print("mode a=", a_at_max)
plt.plot(a, post_a/post_a_norm, label = "Posterior of 1-fv/fc")
plt.axvline(x=a_at_max, color='b', ls=':', label='mode')
plt.legend()
plt.xlabel ('1-fv/fc')
plt.ylabel ('Posterior')
plt.show()
mode a= 0.8989909090909091
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# Evidence for the covid-19 Moderna data, 2020
#
Nv = 15000 # volunteers that received the vaccine
Nc = 15000 # volunteers in control group (no vaccine)
nv = 5 # received vaccine and contracted Covid-19
nc = 90 # no vaccine and contracted Covid-19
f_min = 0.0000001
f_max = 0.9999999
f_bsize = 1/(100*np.sqrt(max(Nc,Nv)))
f_nbin = int((f_max-f_min)/f_bsize)
prior = 1/(f_max-f_min)
a_min = 0.00001
a_max = 1.0
a_nbin = 100
a = np.linspace(a_min, a_max, a_nbin)
post_a = []
post_a_norm = 0
post_max = 0
a_at_max = 0
for x in range(len(a)):
post = 0
for b in range(f_nbin):
fc = f_min + b*f_bsize
fv = fc * (1-a[x])
if fv > 0:
pdf_c, logpdf_c = binomial_posterior(fc, Nc, nc)
pdf_v, logpdf_v = binomial_posterior(fv, Nv, nv)
#post += pdf_c * pdf_v
post += np.exp(logpdf_c + logpdf_v)
post_a.append(post)
post_a_norm += post
# find max prob_a and efficiency (a) at which max happens
if post > post_max:
post_max = post
a_at_max = a[x]
# median
post_a_mode = stats.mode(post_a)
print("mode a=", a_at_max)
plt.plot(a, post_a/post_a_norm, label = "Posterior of 1-fv/fc")
plt.axvline(x=a_at_max, color='b', ls=':', label='mode')
plt.legend()
plt.xlabel ('1-fv/fc')
plt.ylabel ('Posterior')
plt.show()
mode a= 0.9494954545454545
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# Wednesday 9/27
#
# Using the data evidence P(data|H) to compare hypotheses
#--------------------------------------------------------
# let's test that:
#
# P(H1|data)/P(H0|data) = P(data|H1)/P(data|H0)
#
# Calculate the evidence
#
# P(data|H1 [f_v < f_c])
#
Nv = 30 # volunteers that have received the vaccine
Nc = 10 # volunteers in control group (no vaccine)
nv = 1 # received vaccine and contracted plague
nc = 3 # no vaccine and contracted plague
#integration range
#
f_bin = 1/(100*np.sqrt(max(Nv,Nc)))
f_nbin = int((f_max-f_min)/f_bin)
# the probability of contracting disease
f = np.linspace(f_min, f_max, f_nbin)
# the probability of the data given f
pdata_v = binomial_pdata(f, Nv, nv)
pdata_c = binomial_pdata(f, Nc, nc)
plt.plot(f, pdata_c[0], label = "P(data_c|f_c)")
plt.plot(f, pdata_v[0], label = "P(data_v|f_c)")
plt.axvline(x=nv/Nv, color='b', ls=':', label='nv/Nv')
plt.axvline(x=nc/Nc, color='b', ls=':', label='nc/Nc')
plt.legend(loc="upper right")
plt.show()
area_H1 = 1
prior = 1/area_H1
pdata_H1 = 0
for bc in range(f_nbin):
fc = f_min + bc*f_bin
for bv in range(f_nbin):
fv = f_min + bv*f_bin
if fv < fc:
pdata_H1 += pdata_v[0][bv]*pdata_c[0][bc] * prior
pdata_H1 *= f_bin * f_bin
print("P(D|H1 [fv < fc]) = ", pdata_H1)
P(D|H1 [fv < fc]) = 0.0028758708438290826
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# Calculate the evidence for
# P(data | H0 [fc = fv])
#
#
area_H0 = 1
prior = 1/area_H0
pdata_H0 = 0
for b in range(f_nbin):
pdata_H0 += pdata_v[0][b]*pdata_c[0][b] * prior
pdata_H0 *= f_bin
print("P(data|H0, fv=fc) = ", pdata_H0)
# P(H1|data) / P(H0|data) = P(data|H1) / P(data|H0)
#
print("H1/H0 = ", pdata_H1/pdata_H0)
P(data|H0, fv=fc) = 0.0009577497977263333 H1/H0 = 3.002737093399869
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