def best_paired(y1, y2, name1 = 'Group1', name2 = 'Group2',
pum = 1000.0,
ROPE = [-0.1, 0.1],
ctd = ('mode', 'mode', 'mean'),
iter = 110000, burn = 10000,
printstats = False
):
y = y1 - y2 # get distribution of differences
mu_m = np.mean(y) # mean of distribution of group means
mu_p = 1/(pum*np.std(y))**2 # precision of distribution of group meanssigma_low = np.std(y)/1000 # lower bound is 1000 times less than std of data
sigma_low = np.std(y)/pum # lower bound is 1000 times less than std of data
sigma_high = np.std(y)*pum # upper bound is 1000 times more than std of data
group_mean = pymc.Normal('group_mean', mu_m, mu_p)
group_std = pymc.Uniform('group_std', sigma_low, sigma_high)
nu_minus_one = pymc.Exponential('nu_minus_one', 1/29)
# Normality nu = nu_minus_one + 1
@pymc.deterministic(plot = False)
def nu(n = nu_minus_one):
out = n+1
return out
# Scale lam = 1/std**2
@pymc.deterministic(plot=False)
def lam(s = group_std):
out = 1/s**2
return out
group = pymc.NoncentralT('group', group_mean, lam, nu, value = y, observed = True)
model = pymc.Model({group, group_mean, group_std, nu_minus_one})
# Define the MCMC object on the model
M = pymc.MCMC(model)
M.sample(iter = 110000, burn = 10000, thin = 1)
# Create a figure to hold 2 rows and 2 columns
f = plt.figure(figsize = (9,5), facecolor = 'white')
n_bins = 30
bins = np.linspace(min(y1.min(), y2.min()), max(y1.max(), y2.max()), n_bins)
# Left column: plot the original distributions of the data
ax1 = f.add_subplot(2, 2, 1, axisbg = 'none')
plot_posterior( y1,
ax = ax1,
bins = bins,
title = r'$\,%s$'%name1,
label = '',
ctd = ctd[0],
printstats = printstats
)
ax3 = f.add_subplot(2, 2, 3, axisbg = 'none')
plot_posterior( y2,
ax = ax3,
bins = bins,
title = r'$\,%s$'%name2,
label = '',
ctd = ctd[1],
printstats = printstats
)
# Right column, top panel: plot histogram of data differences
# and 50 of the t-distribution fits from the MCMC chain
orig_vals = M.get_node('group').value
bin_edges = np.linspace( np.min(orig_vals), np.max(orig_vals), n_bins )
ax2 = f.add_subplot(2, 2, 2, axisbg = 'none')
plot_data_and_prediction( M.get_node('group').value,
M.trace('group_mean')[:],
M.trace('group_std')[:],
M.trace('nu_minus_one')[:],
ax2,
bins = bin_edges,
group = '',
name = r'$\,%s - %s$'%(name1, name2),
colour = 'blue'
)
# Right column, bottom panel: plot the posterior distribution for the mean difference
ax4 = f.add_subplot(2, 2, 4, axisbg = 'none')
plot_posterior( M.trace('group_mean')[:],
ax = ax4,
bins = n_bins,
title = 'Difference of Means',
label = r'$\mu_{%s-%s}$'%(name1, name2),
ctd = ctd[2],
draw_zero = True,
compVal = 0.0,
ROPE = ROPE,
printstats = printstats
)
f.subplots_adjust(hspace = 0.5, top = 0.92, bottom = 0.09,
left = 0.09, right = 0.95, wspace = 0.25)
return f
def best(y1, y2, name1 = 'Group1', name2 = 'Group2',
pum = 1000.0,
ROPE = ([-0.1, 0.1], [-0.1, 0.1], [-0.1, 0.1]),
ctd = ('mean', 'mean', 'mode', 'mode', 'mode', 'mean', 'mode', 'mode'),
iter = 110000, burn = 10000,
printstats = False
):
y = np.concatenate((y1, y2)) # combine both groups
mu_m = np.mean(y) # mean of distribution of group means
mu_p = 1/(pum*np.std(y))**2 # precision of distribution of group means
sigma_low = np.std(y)/pum # lower bound is 1000 times less than std of data
sigma_high = np.std(y)*pum # upper bound is 1000 times more than std of data
group1_mean = pymc.Normal('group1_mean', mu_m, mu_p)
group2_mean = pymc.Normal('group2_mean', mu_m, mu_p)
group1_std = pymc.Uniform('group1_std', sigma_low, sigma_high)
group2_std = pymc.Uniform('group2_std', sigma_low, sigma_high)
nu_minus_one = pymc.Exponential('nu_minus_one', 1/29)
# Normality nu = nu_minus_one + 1
@pymc.deterministic(plot = False)
def nu(n = nu_minus_one):
out = n+1
return out
# Scale lam1 = 1/std**2
@pymc.deterministic(plot=False)
def lam1(s = group1_std):
out = 1/s**2
return out
# Scale lam2 = 1/std**2
@pymc.deterministic(plot=False)
def lam2(s = group2_std):
out = 1/s**2
return out
group1 = pymc.NoncentralT('group1', group1_mean, lam1, nu, value = y1, observed = True)
group2 = pymc.NoncentralT('group2', group2_mean, lam2, nu, value = y2, observed = True)
model = pymc.Model({group1, group2,
group1_mean, group2_mean,
group1_std, group2_std,
nu_minus_one})
# Define the MCMC object on the model
M = pymc.MCMC(model)
M.sample(iter = iter, burn = burn, thin = 1)
n_bins = 30
# Get the posterior distributions of the means and their difference
posterior_mean1 = M.trace('group1_mean')[:]
posterior_mean2 = M.trace('group2_mean')[:]
diff_means = posterior_mean1 - posterior_mean2
# Calculate common bin edges for both posteriors
posterior_means = np.concatenate((posterior_mean1, posterior_mean2))
bin_edges_means = np.linspace( np.min(posterior_means), np.max(posterior_means), n_bins )
# Get the posteriaor distributions of the std and their difference
posterior_std1 = M.trace('group1_std')[:]
posterior_std2 = M.trace('group2_std')[:]
diff_stds = posterior_std1 - posterior_std2
# Calculate common bin edges for both posteriors
posterior_stds = np.concatenate((posterior_std1, posterior_std2))
bin_edges_stds = np.linspace(np.min(posterior_stds), np.max(posterior_stds), n_bins)
# Calculate effect size
effect_size = diff_means/np.sqrt((posterior_std1**2 + posterior_std2**2)/2)
# Get the distribution of the normality parameter
post_nu_minus_one = M.trace('nu_minus_one')[:]
# and convert to log10 scale
lognu = np.log10(post_nu_minus_one + 1)
# Create a figure to hold 5 rows and 2 columns
f = plt.figure(figsize = (9,12), facecolor = 'white')
# Left column, top two: plot the distribution of the means
ax1 = f.add_subplot(5, 2, 1, axisbg = 'none')
plot_posterior( posterior_mean1,
ax = ax1,
bins = bin_edges_means,
title = name1 + ' Mean',
label = r'$\mu_1$',
ctd = ctd[0],
printstats = printstats
)
ax3 = f.add_subplot(5, 2, 3, axisbg = 'none')
plot_posterior( posterior_mean2,
ax = ax3,
bins = bin_edges_means,
title = name2 + ' Mean',
label = r'$\mu_2$',
ctd = ctd[1],
printstats = printstats
)
# Left column, next two: plot the distribution of the stds
ax5 = f.add_subplot(5, 2, 5, axisbg = 'none')
plot_posterior( posterior_std1,
ax = ax5,
bins = bin_edges_stds,
title = name1 + ' Std. Dev.',
label = r'$\sigma_1$',
ctd = ctd[2],
printstats = printstats
)
ax7 = f.add_subplot(5, 2, 7, axisbg = 'none')
plot_posterior( posterior_std2,
ax = ax7,
bins = bin_edges_stds,
title = name2 + ' Std. Dev.',
label = r'$\sigma_2$',
ctd = ctd[3],
printstats = printstats
)
# Left column, bottom row: Plot log10(nu)
ax9 = f.add_subplot(5, 2, 9, axisbg = 'none')
plot_posterior( lognu,
ax = ax9,
bins = n_bins,
title = 'Normality',
label = r'$\mathrm{log10}(\nu)$',
ctd = ctd[4],
printstats = printstats
)
#Right column, top two: plot histogram of data and 50 of the t-distribution fits from the MCMC chain
orig_vals = np.concatenate( (M.get_node('group1').value, M.get_node('group1').value) )
bin_edges = np.linspace( np.min(orig_vals), np.max(orig_vals), n_bins )
ax2 = f.add_subplot(5, 2, 2, axisbg = 'none')
plot_data_and_prediction( M.get_node('group1').value,
posterior_mean1,
posterior_std1,
post_nu_minus_one,
ax2,
bins = bin_edges,
group = 1,
name = name1,
)
ax4 = f.add_subplot(5, 2, 4, axisbg = 'none', sharex = ax2, sharey = ax2)
plot_data_and_prediction( M.get_node('group2').value,
posterior_mean2,
posterior_std2,
post_nu_minus_one,
ax4,
bins = bin_edges,
group = 2,
name = name2,
)
# Right column, third panel: plot the distribution of the differences of the means.
ax6 = f.add_subplot(5, 2, 6, axisbg = 'none')
plot_posterior( diff_means,
ax = ax6,
bins = n_bins,
title = 'Difference of Means',
label = r'$\mu_1 - \mu_2$',
ctd = ctd[5],
draw_zero = True,
compVal = 0.0,
ROPE = ROPE[0],
printstats = printstats
)
# Right column, fourth panel: plot the distribution of the differences of the stds.
ax8 = f.add_subplot(5, 2, 8, axisbg = 'none')
plot_posterior( diff_stds,
ax = ax8,
bins = n_bins,
title = 'Difference of Std. Dev.s',
label = r'$\sigma_1 - \sigma_2$',
ctd = ctd[6],
draw_zero = True,
compVal = 0.0,
ROPE = ROPE[1],
printstats = printstats
)
# Right column, bottom panel: plot the effect size
ax10 = f.add_subplot(5, 2, 10, axisbg = 'none')
plot_posterior( effect_size,
ax = ax10,
bins = n_bins,
title = 'Effect Size',
label = r'$(\mu_1 - \mu_2)/\sqrt{(\sigma_1^2 + \sigma_2^2)/2}$',
ctd = ctd[7],
draw_zero = True,
compVal = 0.0,
ROPE = ROPE[2],
printstats = printstats
)
f.subplots_adjust(hspace = 0.5, top = 0.92, bottom = 0.09,
left = 0.09, right = 0.95, wspace = 0.25)
return f
