def calculate_power_from_cohens_d(d, n_1, n_2, alpha):
df = n_1 + n_2 - 2
denominator = (1 / n_1 + 1 / n_2)**0.5
t_crit_os = t.ppf(q=1 - alpha, df=df)
t_crit_ts = t.ppf(q=1 - alpha / 2, df=df)
# Create Non-Centralized t-distribution
nc = abs(d) * (2 / (1 / n_1 + 1 / n_2) / 2)**0.5
nct_dist = utils.initialize_nct_distribution(df=df, nc=nc)
power_os = 1 - nct_dist.cdf(x=t_crit_os)
power_ts = 1 - nct_dist.cdf(x=t_crit_ts)
return [[power_os], [power_ts]]
def calculate_power_from_means(mu_1, sigma_1, n_1, mu_2, sigma_2, n_2, alpha):
diff = abs(mu_1 - mu_2)
df = utils.welches_degrees_of_freedom(sigma_1, n_1, sigma_2, n_2)
t_crit_os = t.ppf(q=1 - alpha, df=df)
t_crit_ts = t.ppf(q=1 - alpha / 2, df=df)
# Create Non-Centralized t-distribution
d = utils.calculate_cohens_d(mu_1, sigma_1, n_1, mu_2, sigma_2, n_2)
nc = d * (2 / (1 / n_1 + 1 / n_2) / 2)**0.5
nct_dist = utils.initialize_nct_distribution(df=df, nc=nc)
power_os = 1 - nct_dist.cdf(x=t_crit_os)
power_ts = 1 - nct_dist.cdf(x=t_crit_ts)
return [[power_os], [power_ts]]
def create_power_from_d_formula(d, n_1, n_2, alpha):
formulae = []
df = n_1 + n_2 - 2
sig = 1 - alpha/2
t_crit = t.ppf(q=sig, df=df)
step_1 = "t_{{crit}} = t_{{1-\\alpha/2, \ \\upsilon}} = t_{{{:.3f}, \ {}}} = {:.3f}"
formulae.append(step_1.format(sig, df, t_crit))
step_2 = "\\beta = P(T <= t_{{crit}})\ where\ T\ \\sim\ t_{{\\upsilon={},\ \\mu={:.3f}}}"
nc = abs(d) * (2 / (1/n_1 + 1/n_2) / 2)**0.5
formulae.append(step_2.format(df, nc))
nct_dist = utils.initialize_nct_distribution(df=df, nc=nc)
beta = nct_dist.cdf(x=t_crit)
step_3 = "\\beta = P(T <= {:.3f}) = {:.3f}"
formulae.append(step_3.format(t_crit, beta))
step_4 = "1 - \\beta = 1 - {:.3f} = {:.3f}"
formulae.append(step_4.format(beta, 1 - beta))
return formulae
def create_power_from_means_formula(mu_1, sigma_1, n_1, mu_2, sigma_2, n_2, alpha):
formulae = []
df = int(utils.welches_degrees_of_freedom(sigma_1, n_1, sigma_2, n_2))
sig = 1 - alpha/2
t_crit = t.ppf(q=sig, df=df)
step_1 = "t_{{crit}} = t_{{1-\\alpha/2, \ \\upsilon}} = t_{{{:.3f}, \ {}}} = {:.3f}"
formulae.append(step_1.format(sig, df, t_crit))
step_2 = "\\beta = P(T <= t_{{crit}})\ where\ T\ \\sim\ t_{{\\upsilon={},\ \\mu={:.3f}}}"
d = utils.calculate_cohens_d(mu_1, sigma_1, n_1, mu_2, sigma_2, n_2)
nc = abs(d) * (2 / (1/n_1 + 1/n_2) / 2)**0.5
formulae.append(step_2.format(df, nc))
nct_dist = utils.initialize_nct_distribution(df=df, nc=nc)
beta = nct_dist.cdf(x=t_crit)
step_3 = "\\beta = P(T <= {:.3f}) = {:.3f}"
formulae.append(step_3.format(t_crit, beta))
step_4 = "1 - \\beta = 1 - {:.3f} = {:.3f}"
formulae.append(step_4.format(beta, 1 - beta))
return formulae
def generate_sampling_distributions_chart_data(mu_1, mu_2, sigma_1, sigma_2, n_1, n_2, alpha):
n = n_1 + n_2 - 2
df = utils.welches_degrees_of_freedom(sigma_1, n_1, sigma_2, n_2)
H0_mean = 0
HA_mean = mu_2 - mu_1
sd_pooled = utils.calculate_pooled_standard_deviation(n_1, n_2, sigma_1, sigma_2)
se = sd_pooled * (1/n_1 + 1/n_2)**0.5
d = utils.calculate_cohens_d(mu_1=mu_1, sigma_1=sigma_1, n_1=n_1, mu_2=mu_2, sigma_2=sigma_2, n_2=n_2)
nc = d * (2 / (1/n_1 + 1/n_2) / 2)**0.5
if mu_1 > mu_2:
nc *= -1
# Determine X axis range
x_min = min(H0_mean, nc) - 4
x_max = max(H0_mean, nc) + 4
x_axis_values = list(np.linspace(start=x_min, stop=x_max, num=1000, endpoint=True))
alpha_lower = t.ppf(q=alpha/2, df=df)
alpha_upper = -1 * alpha_lower
# Insert key values
for val in [H0_mean, nc, alpha_lower, alpha_upper]:
if val not in x_axis_values:
bisect.insort(x_axis_values, val)
H0_significant = []
H0_not_significant = []
HA_powered = []
HA_unpowered = []
threshold = alpha_upper if HA_mean >= H0_mean else alpha_lower
nct_dist = utils.initialize_nct_distribution(df=df, nc=nc)
for value in x_axis_values:
# Null Hypothesis
H0_not_significant.append(t.pdf(x=value, df=df))
if value < alpha_lower or value > alpha_upper:
H0_significant.append(t.pdf(x=value, df=df))
else:
H0_significant.append(None)
# Alternative Hypothesis
HA_powered.append(nct_dist.pdf(x=value))
if HA_mean < H0_mean and value > alpha_lower:
HA_unpowered.append(nct_dist.pdf(x=value))
elif HA_mean >= H0_mean and value < alpha_upper:
HA_unpowered.append(nct_dist.pdf(x=value))
else:
HA_unpowered.append(None)
if HA_mean < H0_mean:
power = nct_dist.cdf(x=alpha_lower)
threshold = alpha_lower
else:
power = 1 - nct_dist.cdf(x=alpha_upper)
threshold = alpha_upper
decimal_points = utils.determine_decimal_points(x_max)
format_string = "{:." + str(decimal_points) + "f}"
return {
"title": "Central and Noncentral Distributions (effect size: {:0.3f}, α: {:0.3f}, power (1 - β): {:.1%})".format(d, alpha, power),
"xAxisLabel": "t statistic",
"yAxisLabel": "Density",
"labels": [format_string.format(x) for x in x_axis_values],
"verticalLine": {
"position": format_string.format(utils.find_closest_value(x_axis_values, threshold)),
"label": "t crit: " + format_string.format(threshold)
},
"hidePoints": True,
"dataset": [
{
"label": "H0 - Significant",
"data": H0_significant,
"borderColor": colors.line_colors[0],
"backgroundColor": colors.background_colors[0]
},
{
"label": "H0",
"data": H0_not_significant,
"borderColor": colors.line_colors[0],
"backgroundColor": None
},
{
"label": "HA - Powered",
"data": HA_powered,
"borderColor": colors.line_colors[1],
"backgroundColor": None
},
{
"label": "HA",
"data": HA_unpowered,
"borderColor": colors.line_colors[1],
"backgroundColor": colors.background_colors[1]
}
]
}