def generate_t_distribution_chart_data(alpha, t_stat, n_1, n_2, x_bar_1, x_bar_2, s_1, s_2):
welches_df = utils.welches_degrees_of_freedom(s_1, n_1, s_2, n_2)
d = utils.calculate_cohens_d(x_bar_1, s_1, n_1, x_bar_2, s_2, n_2)
alpha_upper = t.ppf(1 - alpha/2, df=welches_df)
alpha_lower = -alpha_upper
# Determine X axis range
x_min = -5
x_max = 5
x_axis = list(np.arange(x_min, x_max, (x_max - x_min) / 501))
x_axis_values = ["{:.3f}".format(x) for x in x_axis]
H0 = []
significant = []
for value in x_axis:
H0.append(t.pdf(value, df=welches_df))
if alpha_lower <= value <= alpha_upper:
significant.append(None)
else:
significant.append(t.pdf(value, df=welches_df))
return {
"title": "t-statistic distribution (effect size: {:.3f})".format(d),
"xAxisLabel": "t",
"yAxisLabel": "Density",
"labels": x_axis_values,
"verticalLine": {
"position": "{:.3f}".format(utils.find_closest_value(x_axis, t_stat)),
"label": "t statistic: {:.3f}".format(t_stat)
},
"dataset": [
{
"label": "H0",
"data": H0,
"pointBorderWidth": 0,
"pointRadius": 0.5,
"borderColor": colors.line_colors[0],
"backgroundColor": None
},
{
"label": "H0",
"data": significant,
"pointBorderWidth": 0,
"pointRadius": 0.5,
"borderColor": colors.line_colors[0],
"backgroundColor": colors.background_colors[0]
}
]
}
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_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 run_model(inputs):
sample_fields = inputs['sampleFields']
results = {"charts": {}}
d = None
# TARGET: SAMPLE SIZE
if inputs['target'] == "sample-size":
alpha = float(inputs['alpha'])
power = float(inputs['power'])
enrolment_ratio = float(inputs['enrolmentRatio'])
# Stats and Formulas
if utils.all_sample_info_provided(sample_fields):
mu_1 = float(sample_fields[0]['mean'])
mu_2 = float(sample_fields[1]['mean'])
sigma_1 = float(sample_fields[0]['stdDev'])
sigma_2 = float(sample_fields[1]['stdDev'])
results['statistics'] = calculate_sample_size_from_means(
mu_1=mu_1,
mu_2=mu_2,
sigma_1=sigma_1,
sigma_2=sigma_2,
alpha=alpha,
power=power,
enrolment_ratio=enrolment_ratio)
results['formulae'] = create_sample_size_from_means_formula(
mu_1=mu_1,
mu_2=mu_2,
sigma_1=sigma_1,
sigma_2=sigma_2,
alpha=alpha,
power=power,
enrolment_ratio=enrolment_ratio)
else:
d = float(inputs['effectSize'])
mu_1 = 0
mu_2 = mu_1 + d
sigma_1, sigma_2 = 1, 1
results['statistics'] = calculate_sample_size_from_cohens_d(
d=d, alpha=alpha, power=power, enrolment_ratio=enrolment_ratio)
results['formulae'] = create_sample_size_from_d_formula(
d=d, alpha=alpha, power=power, enrolment_ratio=enrolment_ratio)
# Calculate vars
n_1 = results['statistics'][0][1]
n_2 = results['statistics'][1][1]
if d is None:
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)
pooled_sd = utils.calculate_pooled_standard_deviation(
n_1, n_2, sigma_1, sigma_2)
# Notes
results['notes'] = generate_sample_size_notes(alpha, power)
results['chartText'] = generate_power_distributions_text(d=d,
mu_1=mu_1,
n_1=n_1,
mu_2=mu_2,
n_2=n_2,
alpha=alpha,
power=power)
# Charts
results['charts'][
'chartOne'] = generate_power_vs_sample_size_chart_data(
d=d, alpha=alpha, power=power, enrolment_ratio=enrolment_ratio)
results['charts'][
'chartTwo'] = generate_effect_size_vs_sample_size_chart_data(
d=d, alpha=alpha, power=power, enrolment_ratio=enrolment_ratio)
results['charts'][
'chartThree'] = generate_sampling_distributions_chart_data(
mu_1=mu_1,
mu_2=mu_2,
sigma_1=sigma_1,
sigma_2=sigma_2,
n_1=n_1,
n_2=n_2,
alpha=alpha)
# Labels
results['labels'] = {
"columns": ["", "One-sided test", "Two-sided test"],
"rows": [
"Sample 1 (n<sub>1</sub>)", "Sample 2 (n<sub>2</sub>)",
"All Samples (n<sub>1</sub> + n<sub>2</sub>)"
],
}
# TARGET: POWER
elif inputs['target'] == "power":
n_1 = int(sample_fields[0]['n'])
n_2 = int(sample_fields[1]['n'])
alpha = float(inputs['alpha'])
# Statistics and Formulas
if utils.all_sample_info_provided(sample_fields):
mu_1 = float(sample_fields[0]['mean'])
mu_2 = float(sample_fields[1]['mean'])
sigma_1 = float(sample_fields[0]['stdDev'])
sigma_2 = float(sample_fields[1]['stdDev'])
d = utils.calculate_cohens_d(mu_1, sigma_1, n_1, mu_2, sigma_2,
n_2)
results['statistics'] = calculate_power_from_means(mu_1=mu_1,
sigma_1=sigma_1,
n_1=n_1,
mu_2=mu_2,
sigma_2=sigma_2,
n_2=n_2,
alpha=alpha)
results['formulae'] = create_power_from_means_formula(
mu_1=mu_1,
sigma_1=sigma_1,
n_1=n_1,
mu_2=mu_2,
sigma_2=sigma_2,
n_2=n_2,
alpha=alpha)
else:
d = float(inputs['effectSize'])
mu_1 = 0
mu_2 = mu_1 + d
sigma_1, sigma_2 = 1, 1
results['statistics'] = calculate_power_from_cohens_d(d=d,
n_1=n_1,
n_2=n_2,
alpha=alpha)
results['formulae'] = create_power_from_d_formula(d=d,
n_1=n_1,
n_2=n_2,
alpha=alpha)
# Calculate vars
power = results['statistics'][1][0]
if d is None:
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)
pooled_sd = utils.calculate_pooled_standard_deviation(
n_1, n_2, sigma_1, sigma_2)
# Notes
welches_df = utils.welches_degrees_of_freedom(sigma_1, n_1, sigma_2,
n_2)
results['notes'] = generate_power_notes(alpha=alpha, df=welches_df)
results['chartText'] = generate_power_distributions_text(d=d,
mu_1=mu_1,
n_1=n_1,
mu_2=mu_2,
n_2=n_2,
alpha=alpha,
power=power)
# Charts
results['charts'][
'chartOne'] = generate_sample_size_vs_power_chart_data(d=d,
alpha=alpha,
power=power,
n_1=n_1,
n_2=n_2)
results['charts'][
'chartTwo'] = generate_effect_size_vs_power_chart_data(d=d,
alpha=alpha,
n_1=n_1,
n_2=n_2)
results['charts'][
'chartThree'] = generate_sampling_distributions_chart_data(
mu_1=mu_1,
mu_2=mu_2,
sigma_1=sigma_1,
sigma_2=sigma_2,
n_1=n_1,
n_2=n_2,
alpha=alpha)
# Labels
results['labels'] = {
"columns": ["Test type", "Statistical Power (1 - β)"],
"rows": ["One-sided test", "Two-sided test"],
}
# TARGET: MIN EFFECT SIZE
elif inputs['target'] == "min-effect":
n_1 = int(sample_fields[0]['n'])
n_2 = int(sample_fields[1]['n'])
alpha = float(inputs['alpha'])
power = float(inputs['power'])
results['statistics'] = calculate_min_effect_size(n_1=n_1,
n_2=n_2,
alpha=alpha,
power=power)
results['formulae'] = create_min_effect_size_formula(n_1=n_1,
n_2=n_2,
alpha=alpha,
power=power)
# Calculate Vars
d = results['statistics'][1][0]
mu_1 = 0
mu_2 = mu_1 + d
sigma_1, sigma_2 = 1, 1
pooled_sd = utils.calculate_pooled_standard_deviation(
n_1, n_2, sigma_1, sigma_2)
# Notes
welches_df = utils.welches_degrees_of_freedom(sigma_1, n_1, sigma_2,
n_2)
results['notes'] = generate_min_effect_size_notes(alpha=alpha,
power=power,
df=welches_df)
results['chartText'] = generate_power_distributions_text(d=d,
mu_1=mu_1,
n_1=n_1,
mu_2=mu_2,
n_2=n_2,
alpha=alpha,
power=power)
# Charts
results['charts'][
'chartOne'] = generate_sample_size_vs_effect_size_data(d=d,
alpha=alpha,
power=power,
n_1=n_1,
n_2=n_2)
results['charts']['chartTwo'] = generate_power_vs_effect_size_data(
d=d, alpha=alpha, power=power, n_1=n_1, n_2=n_2)
results['charts'][
'chartThree'] = generate_sampling_distributions_chart_data(
mu_1=mu_1,
mu_2=mu_2,
sigma_1=sigma_1,
sigma_2=sigma_2,
n_1=n_1,
n_2=n_2,
alpha=alpha)
# Labels
results['labels'] = {
"columns": ["Test type", "Minimum effect size"],
"rows": ["One-sided test", "Two-sided test"],
}
# TARGET: T-STATISTIC
elif inputs['target'] == "t-stat":
n_1 = int(sample_fields[0]['n'])
n_2 = int(sample_fields[1]['n'])
alpha = float(inputs['alpha'])
# Statistics
if utils.all_sample_info_provided(sample_fields):
x_bar_1 = float(sample_fields[0]['mean'])
s_1 = float(sample_fields[0]['stdDev'])
x_bar_2 = float(sample_fields[1]['mean'])
s_2 = float(sample_fields[1]['stdDev'])
d = utils.calculate_cohens_d(x_bar_1, s_1, n_1, x_bar_2, s_2, n_2)
t_stat = calculate_t_stat_from_means(x_bar_1=x_bar_1,
s_1=s_1,
n_1=n_1,
x_bar_2=x_bar_2,
s_2=s_2,
n_2=n_2)
results['formulae'] = create_t_stat_from_means_formula(
x_bar_1=x_bar_1,
s_1=s_1,
n_1=n_1,
x_bar_2=x_bar_2,
s_2=s_2,
n_2=n_2)
else:
d = float(inputs['effectSize'])
x_bar_1, x_bar_2 = 1, 1 + d
s_1, s_2 = 1, 1
t_stat = calculate_t_stat_from_cohens_d(d=d, n_1=n_1, n_2=n_2)
results['formulae'] = create_t_stat_from_d_formula(d=d,
n_1=n_1,
n_2=n_2)
# Format Stats
welches_df = utils.welches_degrees_of_freedom(s_1, n_1, s_2, n_2)
t_critical_os = t.ppf(1 - alpha, df=welches_df)
t_critical_ts = t.ppf(1 - alpha / 2, df=welches_df)
results['statistics'] = [
[t_stat, t_critical_os],
[t_stat, t_critical_ts],
]
# Notes
results['notes'] = generate_t_stat_notes(n_1, n_2, d, t_stat)
results['chartText'] = generate_test_distribution_text(
alpha=alpha, n_1=n_1, n_2=n_2, df=int(welches_df))
# Charts
t_stat = results['statistics'][0][0]
results['charts'][
'chartOne'] = generate_t_statistic_vs_effect_size_chart_data(
n_1=n_1,
n_2=n_2,
x_bar_1=x_bar_1,
x_bar_2=x_bar_2,
s_1=s_1,
s_2=s_2,
alpha=alpha)
results['charts'][
'chartTwo'] = generate_t_statistic_vs_sample_size_chart_data(
n_1=n_1,
n_2=n_2,
x_bar_1=x_bar_1,
x_bar_2=x_bar_2,
s_1=s_1,
s_2=s_2,
alpha=alpha)
results['charts']['chartThree'] = generate_t_distribution_chart_data(
alpha=alpha,
t_stat=t_stat,
n_1=n_1,
n_2=n_2,
x_bar_1=x_bar_1,
x_bar_2=x_bar_2,
s_1=s_1,
s_2=s_2)
# Labels
results['labels'] = {
"columns": ["Test Type", "t-statistic", 't-critical'],
"rows": ["One-sided test", "Two-sided test"],
}
# TARGET: P-VALUE
elif inputs['target'] == "p-value":
n_1 = int(sample_fields[0]['n'])
n_2 = int(sample_fields[1]['n'])
alpha = float(inputs['alpha'])
# Statistics
if utils.all_sample_info_provided(sample_fields):
x_bar_1 = float(sample_fields[0]['mean'])
s_1 = float(sample_fields[0]['stdDev'])
x_bar_2 = float(sample_fields[1]['mean'])
s_2 = float(sample_fields[1]['stdDev'])
d = utils.calculate_cohens_d(x_bar_1, s_1, n_1, x_bar_2, s_2, n_2)
welches_df = utils.welches_degrees_of_freedom(s_1, n_1, s_2, n_2)
t_stat = calculate_t_stat_from_means(x_bar_1=x_bar_1,
s_1=s_1,
n_1=n_1,
x_bar_2=x_bar_2,
s_2=s_2,
n_2=n_2)
results['statistics'] = calculate_p_value(t_stat=t_stat,
df=welches_df)
results['formulae'] = create_p_value_from_means_formula(
x_bar_1=x_bar_1,
s_1=s_1,
n_1=n_1,
x_bar_2=x_bar_2,
s_2=s_2,
n_2=n_2)
else:
d = float(inputs['effectSize'])
x_bar_1, x_bar_2 = 1, 1 + d
s_1, s_2 = 1, 1
t_stat = calculate_t_stat_from_cohens_d(d=d, n_1=n_1, n_2=n_2)
results['statistics'] = calculate_p_value(t_stat=t_stat,
df=(n_1 + n_2 - 2))
results['formulae'] = create_p_value_from_d_formula(d=d,
n_1=n_1,
n_2=n_2)
# Notes
results['notes'] = generate_p_value_notes(
n_1=n_1,
n_2=n_2,
d=d,
p_one_sided=results['statistics'][0][0],
p_two_sided=results['statistics'][1][0],
t_stat=t_stat)
welches_df = utils.welches_degrees_of_freedom(s_1, n_1, s_2, n_2)
results['chartText'] = generate_test_distribution_text(
alpha=alpha, n_1=n_1, n_2=n_2, df=int(welches_df))
# Charts
results['charts'][
'chartOne'] = generate_p_value_vs_effect_size_chart_data(
n_1=n_1,
n_2=n_2,
x_bar_1=x_bar_1,
x_bar_2=x_bar_2,
s_1=s_1,
s_2=s_2,
alpha=alpha)
results['charts'][
'chartTwo'] = generate_p_value_vs_sample_size_chart_data(
n_1=n_1,
n_2=n_2,
x_bar_1=x_bar_1,
x_bar_2=x_bar_2,
s_1=s_1,
s_2=s_2,
alpha=alpha)
results['charts']['chartThree'] = generate_t_distribution_chart_data(
alpha=alpha,
t_stat=t_stat,
n_1=n_1,
n_2=n_2,
x_bar_1=x_bar_1,
x_bar_2=x_bar_2,
s_1=s_1,
s_2=s_2)
# Labels
results['labels'] = {
"columns": ["Test Type", "p-value"],
"rows": ["One-sided test", "Two-sided test"],
}
return results
def generate_p_value_vs_effect_size_chart_data(n_1, n_2, x_bar_1, x_bar_2, s_1, s_2, alpha):
d_raw = utils.calculate_cohens_d(x_bar_1, s_1, n_1, x_bar_2, s_2, n_2)
d_adjustment = utils.calculate_d_adjustment(s_1, n_1, s_2, n_2)
d_results = tt.calculate_min_effect_size(n_1=n_1, n_2=n_2, alpha=alpha, power=0.5)
d_target = d_results[1][0] / d_adjustment
ff = 0.5
if d_raw < 0:
x_min = min(-d_target, d_raw) * (1 + ff)
x_max = max(-d_target, d_raw) * (1 - ff)
else:
x_min = min(d_target, d_raw) * (1 - ff)
x_max = max(d_target, d_raw) * (1 + ff)
# Rounding to ensure matches
step = (x_max - x_min) / 500
dps = utils.determine_decimal_points(x_max)
effect_sizes = [round(x, dps) for x in np.arange(x_min, x_max, step)]
d_actual = utils.find_closest_value(effect_sizes, d_raw)
d_target = utils.find_closest_value(effect_sizes, d_target)
os_lower = []
ts_lower = []
os_higher = []
ts_higher = []
for d in effect_sizes:
welches_df = utils.welches_degrees_of_freedom(s_1, n_1, s_2, n_2)
t_stat = tt.calculate_t_stat_from_cohens_d(d, n_1, n_2) * d_adjustment
results = tt.calculate_p_value(t_stat, welches_df)
if d <= d_target:
os_lower.append(results[0][0])
ts_lower.append(results[1][0])
else:
os_lower.append(None)
ts_lower.append(None)
if d >= d_target:
os_higher.append(results[0][0])
ts_higher.append(results[1][0])
else:
os_higher.append(None)
ts_higher.append(None)
# Determine X axis range
format_string = "{:." + str(dps) + "f}"
x_axis_values = [format_string.format(x) for x in list(effect_sizes)]
return {
"title": "p-value vs Effect Size (sample size: {}, enrolment ratio: {:.3f})".format(n_1 + n_2, n_1/n_2),
"xAxisLabel": "Effect Size (d)",
"yAxisLabel": "p-value",
"labels": x_axis_values,
"verticalLine": {
"position": format_string.format(d_actual),
"label": "Effect Size: " + format_string.format(d_raw)
},
"dataset": [
{
"label": "One Sided Test",
"data": os_lower,
"pointBorderWidth": 0,
"pointRadius": 0.5,
"borderColor": colors.line_colors[0],
"backgroundColor": colors.background_colors[0]
},
{
"label": "One Sided Test",
"data": os_higher,
"pointBorderWidth": 0,
"pointRadius": 0.5,
"borderColor": colors.line_colors[1],
"backgroundColor": colors.background_colors[1]
},
{
"label": "Two Sided Test",
"data": ts_lower,
"pointBorderWidth": 0,
"pointRadius": 0.5,
"borderColor": colors.line_colors[0],
"backgroundColor": colors.background_colors[0]
},
{
"label": "Two Sided Test",
"data": ts_higher,
"pointBorderWidth": 0,
"pointRadius": 0.5,
"borderColor": colors.line_colors[1],
"backgroundColor": colors.background_colors[1]
}
]
}
def generate_p_value_vs_sample_size_chart_data(n_1, n_2, x_bar_1, x_bar_2, s_1, s_2, alpha):
d_actual = utils.calculate_cohens_d(x_bar_1, s_1, n_1, x_bar_2, s_2, n_2)
d_adjustment = utils.calculate_d_adjustment(s_1, n_1, s_2, n_2)
n_raw = n_1 + n_2
r_e = n_1 / n_2
n_results = tt.calculate_sample_size_from_means(mu_1=x_bar_1, mu_2=x_bar_2, sigma_1=s_1, sigma_2=s_2, alpha=alpha, power=0.5, enrolment_ratio=r_e)
n_target = n_results[0][1] + n_results[1][1]
ff = 0.1
x_min = int(max(4, min(n_raw * (1 - ff), n_target * (1 - ff))))
x_max = int(max(n_raw * (1 + ff), n_target * (1 + ff)))
step = int(max(1, (x_max - x_min) / 500))
sample_sizes = np.arange(x_min, x_max, step)
n_actual = utils.find_closest_value(sample_sizes, n_raw)
n_target = utils.find_closest_value(sample_sizes, n_target)
os_lower = []
ts_lower = []
os_higher = []
ts_higher = []
for n in sample_sizes:
cn_1 = math.ceil(n * r_e / (1 + r_e))
cn_2 = math.ceil(n - cn_1)
welches_df = utils.welches_degrees_of_freedom(s_1, cn_1, s_2, cn_2)
d = utils.calculate_cohens_d(x_bar_1, s_1, cn_1, x_bar_2, s_2, cn_2)
t_stat = tt.calculate_t_stat_from_cohens_d(d, cn_1, cn_2) * d_adjustment
results = tt.calculate_p_value(t_stat, welches_df)
if n <= n_target:
os_lower.append(results[0][0])
ts_lower.append(results[1][0])
else:
os_lower.append(None)
ts_lower.append(None)
if n >= n_target:
os_higher.append(results[0][0])
ts_higher.append(results[1][0])
else:
os_higher.append(None)
ts_higher.append(None)
# Determine X axis range
x_axis_values = [str(x) for x in list(sample_sizes)]
return {
"title": "Sample Size vs p-value (effect size: {:.3f}, enrolment ratio: {:.3f})".format(d_actual, n_1/n_2),
"xAxisLabel": "Total Samples",
"yAxisLabel": "p-value",
"labels": x_axis_values,
"verticalLine": {
"position": str(n_actual),
"label": "Sample Size: {}".format(n_raw)
},
"dataset": [
{
"label": "One Sided Test",
"data": os_lower,
"pointBorderWidth": 0,
"pointRadius": 0.5,
"borderColor": colors.line_colors[0],
"backgroundColor": colors.background_colors[0]
},
{
"label": "One Sided Test",
"data": os_higher,
"pointBorderWidth": 0,
"pointRadius": 0.5,
"borderColor": colors.line_colors[1],
"backgroundColor": colors.background_colors[1]
},
{
"label": "Two Sided Test",
"data": ts_lower,
"pointBorderWidth": 0,
"pointRadius": 0.5,
"borderColor": colors.line_colors[0],
"backgroundColor": colors.background_colors[0]
},
{
"label": "Two Sided Test",
"data": ts_higher,
"pointBorderWidth": 0,
"pointRadius": 0.5,
"borderColor": colors.line_colors[1],
"backgroundColor": colors.background_colors[1]
}
]
}
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]
}
]
}