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	<header>
	<h1 class="title">Module <code>tinytroupe.experimentation.statistical_tests</code></h1>
	</header>
	<section id="section-intro">
	<details class="source">
	<summary>
	<span>Expand source code</span>
	</summary>
	<pre><code class="python">import numpy as np
	import scipy.stats as stats
	from typing import Dict, List, Union, Callable, Any, Optional

	from tinytroupe.experimentation import logger


	class StatisticalTester:
	"""
	A class to perform statistical tests on experiment results. To do so, a control is defined, and then one or
	more treatments are compared to the control. The class supports various statistical tests, including t-tests,
	Mann-Whitney U tests, and ANOVA. The user can specify the type of test to run, the significance level, and
	the specific metrics to analyze. The results of the tests are returned in a structured format.
	"""

	def __init__(self, control_experiment_data: Dict[str, list],
	treatments_experiment_data: Dict[str, Dict[str, list]],
	results_key:str = None):
	"""
	Initialize with experiment results.

	Args:
	control_experiment_data (dict): Dictionary containing control experiment results with keys
	as metric names and values as lists of values.
	e.g.,{"control_exp": {"metric1": [0.1, 0.2], "metric2": [0.3, 0.4], ...}}
	treatments_experiment_data (dict): Dictionary containing experiment results with keys
	as experiment IDs and values as dicts of metric names to lists of values.
	e.g., {"exp1": {"metric1": [0.1, 0.2], "metric2": [0.3, 0.4]},
	"exp2": {"metric1": [0.5, 0.6], "metric2": [0.7, 0.8]}, ...}
	"""

	# if results_key is provided, use it to extract the relevant data from the control and treatment data
	# e.g., {"exp1": {"results": {"metric1": [0.1, 0.2], "metric2": [0.3, 0.4]}}
	if results_key:
	control_experiment_data = {k: v[results_key] for k, v in control_experiment_data.items()}
	treatments_experiment_data = {k: v[results_key] for k, v in treatments_experiment_data.items()}

	self.control_experiment_data = control_experiment_data
	self.treatments_experiment_data = treatments_experiment_data

	# Validate input data
	self._validate_input_data()

	def _validate_input_data(self):
	"""Validate the input data formats and structure."""
	# Check that control and treatments are dictionaries
	if not isinstance(self.control_experiment_data, dict):
	raise TypeError("Control experiment data must be a dictionary")
	if not isinstance(self.treatments_experiment_data, dict):
	raise TypeError("Treatments experiment data must be a dictionary")

	# Check that control has at least one experiment
	if not self.control_experiment_data:
	raise ValueError("Control experiment data cannot be empty")

	# Check only one control
	if len(self.control_experiment_data) > 1:
	raise ValueError("Only one control experiment is allowed")

	# Validate control experiment structure
	for control_id, control_metrics in self.control_experiment_data.items():
	if not isinstance(control_metrics, dict):
	raise TypeError(f"Metrics for control experiment '{control_id}' must be a dictionary")

	# Check that the metrics dictionary is not empty
	if not control_metrics:
	raise ValueError(f"Control experiment '{control_id}' has no metrics")

	# Validate that metric values are lists
	for metric, values in control_metrics.items():
	if not isinstance(values, list):
	raise TypeError(f"Values for metric '{metric}' in control experiment '{control_id}' must be a list")

	# Check treatments have at least one experiment
	if not self.treatments_experiment_data:
	raise ValueError("Treatments experiment data cannot be empty")

	# Validate treatment experiment structure
	for treatment_id, treatment_data in self.treatments_experiment_data.items():
	if not isinstance(treatment_data, dict):
	raise TypeError(f"Data for treatment '{treatment_id}' must be a dictionary")

	# Check that the metrics dictionary is not empty
	if not treatment_data:
	raise ValueError(f"Treatment '{treatment_id}' has no metrics")

	# Get all control metrics for overlap checking
	all_control_metrics = set()
	for control_metrics in self.control_experiment_data.values():
	all_control_metrics.update(control_metrics.keys())

	# Check if there's any overlap between control and treatment metrics
	common_metrics = all_control_metrics.intersection(set(treatment_data.keys()))
	if not common_metrics:
	logger.warning(f"Treatment '{treatment_id}' has no metrics in common with any control experiment")

	# Check that treatment metrics are lists
	for metric, values in treatment_data.items():
	if not isinstance(values, list):
	raise TypeError(f"Values for metric '{metric}' in treatment '{treatment_id}' must be a list")

	def run_test(self,
	test_type: str="welch_t_test",
	alpha: float = 0.05,
	**kwargs) -> Dict[str, Dict[str, Any]]:
	"""
	Run the specified statistical test on the control and treatments data.

	Args:
	test_type (str): Type of statistical test to run.
	Options: 't_test', 'welch_t_test', 'mann_whitney', 'anova', 'chi_square'
	alpha (float): Significance level, defaults to 0.05
	**kwargs: Additional arguments for specific test types.

	Returns:
	dict: Dictionary containing the results of the statistical tests for each treatment (vs the one control).
	Each key is the treatment ID and each value is a dictionary with test results.
	"""
	supported_tests = {
	't_test': self._run_t_test,
	'welch_t_test': self._run_welch_t_test,
	'mann_whitney': self._run_mann_whitney,
	'anova': self._run_anova,
	'chi_square': self._run_chi_square
	}

	if test_type not in supported_tests:
	raise ValueError(f"Unsupported test type: {test_type}. Supported types: {list(supported_tests.keys())}")

	results = {}
	for control_id, control_data in self.control_experiment_data.items():
	# get all metrics from control data
	metrics = set()
	metrics.update(control_data.keys())
	for treatment_id, treatment_data in self.treatments_experiment_data.items():
	results[treatment_id] = {}

	for metric in metrics:
	# Skip metrics not in treatment data
	if metric not in treatment_data:
	logger.warning(f"Metric '{metric}' not found in treatment '{treatment_id}'")
	continue

	control_values = control_data[metric]
	treatment_values = treatment_data[metric]

	# Skip if either control or treatment has no values
	if len(control_values) == 0 or len(treatment_values) == 0:
	logger.warning(f"Skipping metric '{metric}' for treatment '{treatment_id}' due to empty values")
	continue

	# Run the selected test and convert to JSON serializable types
	test_result = supported_tests[test_type](control_values, treatment_values, alpha, **kwargs)
	results[treatment_id][metric] = convert_to_serializable(test_result)

	return results

	def _run_t_test(self, control_values: list, treatment_values: list, alpha: float, **kwargs) -> Dict[str, Any]:
	"""Run Student's t-test (equal variance assumed)."""
	# Convert to numpy arrays for calculations
	control = np.array(control_values, dtype=float)
	treatment = np.array(treatment_values, dtype=float)

	# Calculate basic statistics
	control_mean = np.mean(control)
	treatment_mean = np.mean(treatment)
	mean_diff = treatment_mean - control_mean

	# Run the t-test
	t_stat, p_value = stats.ttest_ind(control, treatment, equal_var=True)

	# Calculate confidence interval
	control_std = np.std(control, ddof=1)
	treatment_std = np.std(treatment, ddof=1)
	pooled_std = np.sqrt(((len(control) - 1) * control_std**2 +
	(len(treatment) - 1) * treatment_std**2) /
	(len(control) + len(treatment) - 2))

	se = pooled_std * np.sqrt(1/len(control) + 1/len(treatment))
	critical_value = stats.t.ppf(1 - alpha/2, len(control) + len(treatment) - 2)
	margin_error = critical_value * se
	ci_lower = mean_diff - margin_error
	ci_upper = mean_diff + margin_error

	# Determine if the result is significant
	significant = p_value < alpha

	return {
	'test_type': 'Student t-test (equal variance)',
	'control_mean': control_mean,
	'treatment_mean': treatment_mean,
	'mean_difference': mean_diff,
	'percent_change': (mean_diff / control_mean * 100) if control_mean != 0 else float('inf'),
	't_statistic': t_stat,
	'p_value': p_value,
	'confidence_interval': (ci_lower, ci_upper),
	'confidence_level': 1 - alpha,
	'significant': significant,
	'control_sample_size': len(control),
	'treatment_sample_size': len(treatment),
	'control_std': control_std,
	'treatment_std': treatment_std,
	'effect_size': cohen_d(control, treatment)
	}

	def _run_welch_t_test(self, control_values: list, treatment_values: list, alpha: float, **kwargs) -> Dict[str, Any]:
	"""Run Welch's t-test (unequal variance)."""
	# Convert to numpy arrays for calculations
	control = np.array(control_values, dtype=float)
	treatment = np.array(treatment_values, dtype=float)

	# Calculate basic statistics
	control_mean = np.mean(control)
	treatment_mean = np.mean(treatment)
	mean_diff = treatment_mean - control_mean

	# Run Welch's t-test
	t_stat, p_value = stats.ttest_ind(control, treatment, equal_var=False)

	# Calculate confidence interval (for Welch's t-test)
	control_var = np.var(control, ddof=1)
	treatment_var = np.var(treatment, ddof=1)

	# Calculate effective degrees of freedom (Welch-Satterthwaite equation)
	v_num = (control_var/len(control) + treatment_var/len(treatment))**2
	v_denom = (control_var/len(control))2/(len(control)-1) + (treatment_var/len(treatment))2/(len(treatment)-1)
	df = v_num / v_denom if v_denom > 0 else float('inf')

	se = np.sqrt(control_var/len(control) + treatment_var/len(treatment))
	critical_value = stats.t.ppf(1 - alpha/2, df)
	margin_error = critical_value * se
	ci_lower = mean_diff - margin_error
	ci_upper = mean_diff + margin_error

	control_std = np.std(control, ddof=1)
	treatment_std = np.std(treatment, ddof=1)

	# Determine if the result is significant
	significant = p_value < alpha

	return {
	'test_type': 'Welch t-test (unequal variance)',
	'control_mean': control_mean,
	'treatment_mean': treatment_mean,
	'mean_difference': mean_diff,
	'percent_change': (mean_diff / control_mean * 100) if control_mean != 0 else float('inf'),
	't_statistic': t_stat,
	'p_value': p_value,
	'confidence_interval': (ci_lower, ci_upper),
	'confidence_level': 1 - alpha,
	'significant': significant,
	'degrees_of_freedom': df,
	'control_sample_size': len(control),
	'treatment_sample_size': len(treatment),
	'control_std': control_std,
	'treatment_std': treatment_std,
	'effect_size': cohen_d(control, treatment)
	}

	def _run_mann_whitney(self, control_values: list, treatment_values: list, alpha: float, **kwargs) -> Dict[str, Any]:
	"""Run Mann-Whitney U test (non-parametric test)."""
	# Convert to numpy arrays
	control = np.array(control_values, dtype=float)
	treatment = np.array(treatment_values, dtype=float)

	# Calculate basic statistics
	control_median = np.median(control)
	treatment_median = np.median(treatment)
	median_diff = treatment_median - control_median

	# Run the Mann-Whitney U test
	u_stat, p_value = stats.mannwhitneyu(control, treatment, alternative='two-sided')

	# Calculate common language effect size
	# (probability that a randomly selected value from treatment is greater than control)
	count = 0
	for tc in treatment:
	for cc in control:
	if tc > cc:
	count += 1
	cles = count / (len(treatment) * len(control))

	# Calculate approximate confidence interval using bootstrap
	try:
	from scipy.stats import bootstrap

	def median_diff_func(x, y):
	return np.median(x) - np.median(y)

	res = bootstrap((control, treatment), median_diff_func,
	confidence_level=1-alpha,
	n_resamples=1000,
	random_state=42)
	ci_lower, ci_upper = res.confidence_interval
	except ImportError:
	# If bootstrap is not available, return None for confidence interval
	ci_lower, ci_upper = None, None
	logger.warning("SciPy bootstrap not available, skipping confidence interval calculation")

	# Determine if the result is significant
	significant = p_value < alpha

	return {
	'test_type': 'Mann-Whitney U test',
	'control_median': control_median,
	'treatment_median': treatment_median,
	'median_difference': median_diff,
	'percent_change': (median_diff / control_median * 100) if control_median != 0 else float('inf'),
	'u_statistic': u_stat,
	'p_value': p_value,
	'confidence_interval': (ci_lower, ci_upper) if ci_lower is not None else None,
	'confidence_level': 1 - alpha,
	'significant': significant,
	'control_sample_size': len(control),
	'treatment_sample_size': len(treatment),
	'effect_size': cles
	}

	def _run_anova(self, control_values: list, treatment_values: list, alpha: float, **kwargs) -> Dict[str, Any]:
	"""Run one-way ANOVA test."""
	# For ANOVA, we typically need multiple groups, but we can still run it with just two
	# Convert to numpy arrays
	control = np.array(control_values, dtype=float)
	treatment = np.array(treatment_values, dtype=float)

	# Run one-way ANOVA
	f_stat, p_value = stats.f_oneway(control, treatment)

	# Calculate effect size (eta-squared)
	total_values = np.concatenate([control, treatment])
	grand_mean = np.mean(total_values)

	ss_total = np.sum((total_values - grand_mean) ** 2)
	ss_between = (len(control) * (np.mean(control) - grand_mean) ** 2 +
	len(treatment) * (np.mean(treatment) - grand_mean) ** 2)

	eta_squared = ss_between / ss_total if ss_total > 0 else 0

	# Determine if the result is significant
	significant = p_value < alpha

	return {
	'test_type': 'One-way ANOVA',
	'f_statistic': f_stat,
	'p_value': p_value,
	'significant': significant,
	'control_sample_size': len(control),
	'treatment_sample_size': len(treatment),
	'effect_size': eta_squared,
	'effect_size_type': 'eta_squared'
	}

	def _run_chi_square(self, control_values: list, treatment_values: list, alpha: float, **kwargs) -> Dict[str, Any]:
	"""Run Chi-square test for categorical data."""
	# For chi-square, we assume the values represent counts in different categories
	# Convert to numpy arrays
	control = np.array(control_values, dtype=float)
	treatment = np.array(treatment_values, dtype=float)

	# Check if the arrays are the same length (same number of categories)
	if len(control) != len(treatment):
	raise ValueError("Control and treatment must have the same number of categories for chi-square test")

	# Run chi-square test
	contingency_table = np.vstack([control, treatment])
	chi2_stat, p_value, dof, expected = stats.chi2_contingency(contingency_table)

	# Calculate Cramer's V as effect size
	n = np.sum(contingency_table)
	min_dim = min(contingency_table.shape) - 1
	cramers_v = np.sqrt(chi2_stat / (n * min_dim)) if n * min_dim > 0 else 0

	# Determine if the result is significant
	significant = p_value < alpha

	return {
	'test_type': 'Chi-square test',
	'chi2_statistic': chi2_stat,
	'p_value': p_value,
	'degrees_of_freedom': dof,
	'significant': significant,
	'effect_size': cramers_v,
	'effect_size_type': 'cramers_v'
	}

	def check_assumptions(self, metric: str) -> Dict[str, Dict[str, Any]]:
	"""
	Check statistical assumptions for the given metric across all treatments.

	Args:
	metric (str): The metric to check assumptions for.

	Returns:
	dict: Dictionary with results of assumption checks for each treatment.
	"""
	if metric not in self.control_experiment_data:
	raise ValueError(f"Metric '{metric}' not found in control data")

	results = {}
	control_values = np.array(self.control_experiment_data[metric], dtype=float)

	# Check normality of control
	control_shapiro = stats.shapiro(control_values)
	control_normality = {
	'test': 'Shapiro-Wilk',
	'statistic': control_shapiro[0],
	'p_value': control_shapiro[1],
	'normal': control_shapiro[1] >= 0.05
	}

	for treatment_id, treatment_data in self.treatments_experiment_data.items():
	if metric not in treatment_data:
	logger.warning(f"Metric '{metric}' not found in treatment '{treatment_id}'")
	continue

	treatment_values = np.array(treatment_data[metric], dtype=float)

	# Check normality of treatment
	treatment_shapiro = stats.shapiro(treatment_values)
	treatment_normality = {
	'test': 'Shapiro-Wilk',
	'statistic': treatment_shapiro[0],
	'p_value': treatment_shapiro[1],
	'normal': treatment_shapiro[1] >= 0.05
	}

	# Check homogeneity of variance
	levene_test = stats.levene(control_values, treatment_values)
	variance_homogeneity = {
	'test': 'Levene',
	'statistic': levene_test[0],
	'p_value': levene_test[1],
	'equal_variance': levene_test[1] >= 0.05
	}

	# Store results and convert to JSON serializable types
	results[treatment_id] = convert_to_serializable({
	'control_normality': control_normality,
	'treatment_normality': treatment_normality,
	'variance_homogeneity': variance_homogeneity,
	'recommended_test': self._recommend_test(control_normality['normal'],
	treatment_normality['normal'],
	variance_homogeneity['equal_variance'])
	})

	return results

	def _recommend_test(self, control_normal: bool, treatment_normal: bool, equal_variance: bool) -> str:
	"""Recommend a statistical test based on assumption checks."""
	if control_normal and treatment_normal:
	if equal_variance:
	return 't_test'
	else:
	return 'welch_t_test'
	else:
	return 'mann_whitney'


	def cohen_d(x: Union[list, np.ndarray], y: Union[list, np.ndarray]) -> float:
	"""
	Calculate Cohen's d effect size for two samples.

	Args:
	x: First sample
	y: Second sample

	Returns:
	float: Cohen's d effect size
	"""
	nx = len(x)
	ny = len(y)

	# Convert to numpy arrays
	x = np.array(x, dtype=float)
	y = np.array(y, dtype=float)

	# Calculate means
	mx = np.mean(x)
	my = np.mean(y)

	# Calculate standard deviations
	sx = np.std(x, ddof=1)
	sy = np.std(y, ddof=1)

	# Pooled standard deviation
	pooled_sd = np.sqrt(((nx - 1) * sx*2 + (ny - 1) sy**2) / (nx + ny - 2))

	# Cohen's d
	return (my - mx) / pooled_sd if pooled_sd > 0 else 0


	def convert_to_serializable(obj):
	"""
	Convert NumPy types to native Python types recursively to ensure JSON serialization works.

	Args:
	obj: Any object that might contain NumPy types

	Returns:
	Object with NumPy types converted to Python native types
	"""
	if isinstance(obj, np.ndarray):
	return obj.tolist()
	elif isinstance(obj, (np.number, np.bool_)):
	return obj.item()
	elif isinstance(obj, dict):
	return {k: convert_to_serializable(v) for k, v in obj.items()}
	elif isinstance(obj, list):
	return [convert_to_serializable(i) for i in obj]
	elif isinstance(obj, tuple):
	return tuple(convert_to_serializable(i) for i in obj)
	else:
	return obj</code></pre>
	</details>
	</section>
	<section>
	</section>
	<section>
	</section>
	<section>
	<h2 class="section-title" id="header-functions">Functions</h2>
	<dl>
	<dt id="tinytroupe.experimentation.statistical_tests.cohen_d"><code class="name flex">
	<span>def <span class="ident">cohen_d</span></span>(<span>x: Union[list, numpy.ndarray], y: Union[list, numpy.ndarray]) ‑> float</span>
	</code></dt>
	<dd>
	<div class="desc"><p>Calculate Cohen's d effect size for two samples.</p>
	<h2 id="args">Args</h2>
	<dl>
	<dt><strong><code>x</code></strong></dt>
	<dd>First sample</dd>
	<dt><strong><code>y</code></strong></dt>
	<dd>Second sample</dd>
	</dl>
	<h2 id="returns">Returns</h2>
	<dl>
	<dt><code>float</code></dt>
	<dd>Cohen's d effect size</dd>
	</dl></div>
	<details class="source">
	<summary>
	<span>Expand source code</span>
	</summary>
	<pre><code class="python">def cohen_d(x: Union[list, np.ndarray], y: Union[list, np.ndarray]) -> float:
	"""
	Calculate Cohen's d effect size for two samples.

	Args:
	x: First sample
	y: Second sample

	Returns:
	float: Cohen's d effect size
	"""
	nx = len(x)
	ny = len(y)

	# Convert to numpy arrays
	x = np.array(x, dtype=float)
	y = np.array(y, dtype=float)

	# Calculate means
	mx = np.mean(x)
	my = np.mean(y)

	# Calculate standard deviations
	sx = np.std(x, ddof=1)
	sy = np.std(y, ddof=1)

	# Pooled standard deviation
	pooled_sd = np.sqrt(((nx - 1) * sx*2 + (ny - 1) sy**2) / (nx + ny - 2))

	# Cohen's d
	return (my - mx) / pooled_sd if pooled_sd > 0 else 0</code></pre>
	</details>
	</dd>
	<dt id="tinytroupe.experimentation.statistical_tests.convert_to_serializable"><code class="name flex">
	<span>def <span class="ident">convert_to_serializable</span></span>(<span>obj)</span>
	</code></dt>
	<dd>
	<div class="desc"><p>Convert NumPy types to native Python types recursively to ensure JSON serialization works.</p>
	<h2 id="args">Args</h2>
	<dl>
	<dt><strong><code>obj</code></strong></dt>
	<dd>Any object that might contain NumPy types</dd>
	</dl>
	<h2 id="returns">Returns</h2>
	<p>Object with NumPy types converted to Python native types</p></div>
	<details class="source">
	<summary>
	<span>Expand source code</span>
	</summary>
	<pre><code class="python">def convert_to_serializable(obj):
	"""
	Convert NumPy types to native Python types recursively to ensure JSON serialization works.

	Args:
	obj: Any object that might contain NumPy types

	Returns:
	Object with NumPy types converted to Python native types
	"""
	if isinstance(obj, np.ndarray):
	return obj.tolist()
	elif isinstance(obj, (np.number, np.bool_)):
	return obj.item()
	elif isinstance(obj, dict):
	return {k: convert_to_serializable(v) for k, v in obj.items()}
	elif isinstance(obj, list):
	return [convert_to_serializable(i) for i in obj]
	elif isinstance(obj, tuple):
	return tuple(convert_to_serializable(i) for i in obj)
	else:
	return obj</code></pre>
	</details>
	</dd>
	</dl>
	</section>
	<section>
	<h2 class="section-title" id="header-classes">Classes</h2>
	<dl>
	<dt id="tinytroupe.experimentation.statistical_tests.StatisticalTester"><code class="flex name class">
	<span>class <span class="ident">StatisticalTester</span></span>
	<span>(</span><span>control_experiment_data: Dict[str, list], treatments_experiment_data: Dict[str, Dict[str, list]], results_key: str = None)</span>
	</code></dt>
	<dd>
	<div class="desc"><p>A class to perform statistical tests on experiment results. To do so, a control is defined, and then one or
	more treatments are compared to the control. The class supports various statistical tests, including t-tests,
	Mann-Whitney U tests, and ANOVA. The user can specify the type of test to run, the significance level, and
	the specific metrics to analyze. The results of the tests are returned in a structured format.</p>
	<p>Initialize with experiment results.</p>
	<h2 id="args">Args</h2>
	<dl>
	<dt>control_experiment_data (dict): Dictionary containing control experiment results with keys</dt>
	<dt>as metric names and values as lists of values.</dt>
	<dt>e.g.,{"control_exp": {"metric1": [0.1, 0.2], "metric2": [0.3, 0.4], …}}</dt>
	<dt><strong><code>treatments_experiment_data</code></strong> :&ensp;<code>dict</code></dt>
	<dd>Dictionary containing experiment results with keys
	as experiment IDs and values as dicts of metric names to lists of values.
	e.g., {"exp1": {"metric1": [0.1, 0.2], "metric2": [0.3, 0.4]},
	"exp2": {"metric1": [0.5, 0.6], "metric2": [0.7, 0.8]}, …}</dd>
	</dl></div>
	<details class="source">
	<summary>
	<span>Expand source code</span>
	</summary>
	<pre><code class="python">class StatisticalTester:
	"""
	A class to perform statistical tests on experiment results. To do so, a control is defined, and then one or
	more treatments are compared to the control. The class supports various statistical tests, including t-tests,
	Mann-Whitney U tests, and ANOVA. The user can specify the type of test to run, the significance level, and
	the specific metrics to analyze. The results of the tests are returned in a structured format.
	"""

	def __init__(self, control_experiment_data: Dict[str, list],
	treatments_experiment_data: Dict[str, Dict[str, list]],
	results_key:str = None):
	"""
	Initialize with experiment results.

	Args:
	control_experiment_data (dict): Dictionary containing control experiment results with keys
	as metric names and values as lists of values.
	e.g.,{"control_exp": {"metric1": [0.1, 0.2], "metric2": [0.3, 0.4], ...}}
	treatments_experiment_data (dict): Dictionary containing experiment results with keys
	as experiment IDs and values as dicts of metric names to lists of values.
	e.g., {"exp1": {"metric1": [0.1, 0.2], "metric2": [0.3, 0.4]},
	"exp2": {"metric1": [0.5, 0.6], "metric2": [0.7, 0.8]}, ...}
	"""

	# if results_key is provided, use it to extract the relevant data from the control and treatment data
	# e.g., {"exp1": {"results": {"metric1": [0.1, 0.2], "metric2": [0.3, 0.4]}}
	if results_key:
	control_experiment_data = {k: v[results_key] for k, v in control_experiment_data.items()}
	treatments_experiment_data = {k: v[results_key] for k, v in treatments_experiment_data.items()}

	self.control_experiment_data = control_experiment_data
	self.treatments_experiment_data = treatments_experiment_data

	# Validate input data
	self._validate_input_data()

	def _validate_input_data(self):
	"""Validate the input data formats and structure."""
	# Check that control and treatments are dictionaries
	if not isinstance(self.control_experiment_data, dict):
	raise TypeError("Control experiment data must be a dictionary")
	if not isinstance(self.treatments_experiment_data, dict):
	raise TypeError("Treatments experiment data must be a dictionary")

	# Check that control has at least one experiment
	if not self.control_experiment_data:
	raise ValueError("Control experiment data cannot be empty")

	# Check only one control
	if len(self.control_experiment_data) > 1:
	raise ValueError("Only one control experiment is allowed")

	# Validate control experiment structure
	for control_id, control_metrics in self.control_experiment_data.items():
	if not isinstance(control_metrics, dict):
	raise TypeError(f"Metrics for control experiment '{control_id}' must be a dictionary")

	# Check that the metrics dictionary is not empty
	if not control_metrics:
	raise ValueError(f"Control experiment '{control_id}' has no metrics")

	# Validate that metric values are lists
	for metric, values in control_metrics.items():
	if not isinstance(values, list):
	raise TypeError(f"Values for metric '{metric}' in control experiment '{control_id}' must be a list")

	# Check treatments have at least one experiment
	if not self.treatments_experiment_data:
	raise ValueError("Treatments experiment data cannot be empty")

	# Validate treatment experiment structure
	for treatment_id, treatment_data in self.treatments_experiment_data.items():
	if not isinstance(treatment_data, dict):
	raise TypeError(f"Data for treatment '{treatment_id}' must be a dictionary")

	# Check that the metrics dictionary is not empty
	if not treatment_data:
	raise ValueError(f"Treatment '{treatment_id}' has no metrics")

	# Get all control metrics for overlap checking
	all_control_metrics = set()
	for control_metrics in self.control_experiment_data.values():
	all_control_metrics.update(control_metrics.keys())

	# Check if there's any overlap between control and treatment metrics
	common_metrics = all_control_metrics.intersection(set(treatment_data.keys()))
	if not common_metrics:
	logger.warning(f"Treatment '{treatment_id}' has no metrics in common with any control experiment")

	# Check that treatment metrics are lists
	for metric, values in treatment_data.items():
	if not isinstance(values, list):
	raise TypeError(f"Values for metric '{metric}' in treatment '{treatment_id}' must be a list")

	def run_test(self,
	test_type: str="welch_t_test",
	alpha: float = 0.05,
	**kwargs) -> Dict[str, Dict[str, Any]]:
	"""
	Run the specified statistical test on the control and treatments data.

	Args:
	test_type (str): Type of statistical test to run.
	Options: 't_test', 'welch_t_test', 'mann_whitney', 'anova', 'chi_square'
	alpha (float): Significance level, defaults to 0.05
	**kwargs: Additional arguments for specific test types.

	Returns:
	dict: Dictionary containing the results of the statistical tests for each treatment (vs the one control).
	Each key is the treatment ID and each value is a dictionary with test results.
	"""
	supported_tests = {
	't_test': self._run_t_test,
	'welch_t_test': self._run_welch_t_test,
	'mann_whitney': self._run_mann_whitney,
	'anova': self._run_anova,
	'chi_square': self._run_chi_square
	}

	if test_type not in supported_tests:
	raise ValueError(f"Unsupported test type: {test_type}. Supported types: {list(supported_tests.keys())}")

	results = {}
	for control_id, control_data in self.control_experiment_data.items():
	# get all metrics from control data
	metrics = set()
	metrics.update(control_data.keys())
	for treatment_id, treatment_data in self.treatments_experiment_data.items():
	results[treatment_id] = {}

	for metric in metrics:
	# Skip metrics not in treatment data
	if metric not in treatment_data:
	logger.warning(f"Metric '{metric}' not found in treatment '{treatment_id}'")
	continue

	control_values = control_data[metric]
	treatment_values = treatment_data[metric]

	# Skip if either control or treatment has no values
	if len(control_values) == 0 or len(treatment_values) == 0:
	logger.warning(f"Skipping metric '{metric}' for treatment '{treatment_id}' due to empty values")
	continue

	# Run the selected test and convert to JSON serializable types
	test_result = supported_tests[test_type](control_values, treatment_values, alpha, **kwargs)
	results[treatment_id][metric] = convert_to_serializable(test_result)

	return results

	def _run_t_test(self, control_values: list, treatment_values: list, alpha: float, **kwargs) -> Dict[str, Any]:
	"""Run Student's t-test (equal variance assumed)."""
	# Convert to numpy arrays for calculations
	control = np.array(control_values, dtype=float)
	treatment = np.array(treatment_values, dtype=float)

	# Calculate basic statistics
	control_mean = np.mean(control)
	treatment_mean = np.mean(treatment)
	mean_diff = treatment_mean - control_mean

	# Run the t-test
	t_stat, p_value = stats.ttest_ind(control, treatment, equal_var=True)

	# Calculate confidence interval
	control_std = np.std(control, ddof=1)
	treatment_std = np.std(treatment, ddof=1)
	pooled_std = np.sqrt(((len(control) - 1) * control_std**2 +
	(len(treatment) - 1) * treatment_std**2) /
	(len(control) + len(treatment) - 2))

	se = pooled_std * np.sqrt(1/len(control) + 1/len(treatment))
	critical_value = stats.t.ppf(1 - alpha/2, len(control) + len(treatment) - 2)
	margin_error = critical_value * se
	ci_lower = mean_diff - margin_error
	ci_upper = mean_diff + margin_error

	# Determine if the result is significant
	significant = p_value < alpha

	return {
	'test_type': 'Student t-test (equal variance)',
	'control_mean': control_mean,
	'treatment_mean': treatment_mean,
	'mean_difference': mean_diff,
	'percent_change': (mean_diff / control_mean * 100) if control_mean != 0 else float('inf'),
	't_statistic': t_stat,
	'p_value': p_value,
	'confidence_interval': (ci_lower, ci_upper),
	'confidence_level': 1 - alpha,
	'significant': significant,
	'control_sample_size': len(control),
	'treatment_sample_size': len(treatment),
	'control_std': control_std,
	'treatment_std': treatment_std,
	'effect_size': cohen_d(control, treatment)
	}

	def _run_welch_t_test(self, control_values: list, treatment_values: list, alpha: float, **kwargs) -> Dict[str, Any]:
	"""Run Welch's t-test (unequal variance)."""
	# Convert to numpy arrays for calculations
	control = np.array(control_values, dtype=float)
	treatment = np.array(treatment_values, dtype=float)

	# Calculate basic statistics
	control_mean = np.mean(control)
	treatment_mean = np.mean(treatment)
	mean_diff = treatment_mean - control_mean

	# Run Welch's t-test
	t_stat, p_value = stats.ttest_ind(control, treatment, equal_var=False)

	# Calculate confidence interval (for Welch's t-test)
	control_var = np.var(control, ddof=1)
	treatment_var = np.var(treatment, ddof=1)

	# Calculate effective degrees of freedom (Welch-Satterthwaite equation)
	v_num = (control_var/len(control) + treatment_var/len(treatment))**2
	v_denom = (control_var/len(control))2/(len(control)-1) + (treatment_var/len(treatment))2/(len(treatment)-1)
	df = v_num / v_denom if v_denom > 0 else float('inf')

	se = np.sqrt(control_var/len(control) + treatment_var/len(treatment))
	critical_value = stats.t.ppf(1 - alpha/2, df)
	margin_error = critical_value * se
	ci_lower = mean_diff - margin_error
	ci_upper = mean_diff + margin_error

	control_std = np.std(control, ddof=1)
	treatment_std = np.std(treatment, ddof=1)

	# Determine if the result is significant
	significant = p_value < alpha

	return {
	'test_type': 'Welch t-test (unequal variance)',
	'control_mean': control_mean,
	'treatment_mean': treatment_mean,
	'mean_difference': mean_diff,
	'percent_change': (mean_diff / control_mean * 100) if control_mean != 0 else float('inf'),
	't_statistic': t_stat,
	'p_value': p_value,
	'confidence_interval': (ci_lower, ci_upper),
	'confidence_level': 1 - alpha,
	'significant': significant,
	'degrees_of_freedom': df,
	'control_sample_size': len(control),
	'treatment_sample_size': len(treatment),
	'control_std': control_std,
	'treatment_std': treatment_std,
	'effect_size': cohen_d(control, treatment)
	}

	def _run_mann_whitney(self, control_values: list, treatment_values: list, alpha: float, **kwargs) -> Dict[str, Any]:
	"""Run Mann-Whitney U test (non-parametric test)."""
	# Convert to numpy arrays
	control = np.array(control_values, dtype=float)
	treatment = np.array(treatment_values, dtype=float)

	# Calculate basic statistics
	control_median = np.median(control)
	treatment_median = np.median(treatment)
	median_diff = treatment_median - control_median

	# Run the Mann-Whitney U test
	u_stat, p_value = stats.mannwhitneyu(control, treatment, alternative='two-sided')

	# Calculate common language effect size
	# (probability that a randomly selected value from treatment is greater than control)
	count = 0
	for tc in treatment:
	for cc in control:
	if tc > cc:
	count += 1
	cles = count / (len(treatment) * len(control))

	# Calculate approximate confidence interval using bootstrap
	try:
	from scipy.stats import bootstrap

	def median_diff_func(x, y):
	return np.median(x) - np.median(y)

	res = bootstrap((control, treatment), median_diff_func,
	confidence_level=1-alpha,
	n_resamples=1000,
	random_state=42)
	ci_lower, ci_upper = res.confidence_interval
	except ImportError:
	# If bootstrap is not available, return None for confidence interval
	ci_lower, ci_upper = None, None
	logger.warning("SciPy bootstrap not available, skipping confidence interval calculation")

	# Determine if the result is significant
	significant = p_value < alpha

	return {
	'test_type': 'Mann-Whitney U test',
	'control_median': control_median,
	'treatment_median': treatment_median,
	'median_difference': median_diff,
	'percent_change': (median_diff / control_median * 100) if control_median != 0 else float('inf'),
	'u_statistic': u_stat,
	'p_value': p_value,
	'confidence_interval': (ci_lower, ci_upper) if ci_lower is not None else None,
	'confidence_level': 1 - alpha,
	'significant': significant,
	'control_sample_size': len(control),
	'treatment_sample_size': len(treatment),
	'effect_size': cles
	}

	def _run_anova(self, control_values: list, treatment_values: list, alpha: float, **kwargs) -> Dict[str, Any]:
	"""Run one-way ANOVA test."""
	# For ANOVA, we typically need multiple groups, but we can still run it with just two
	# Convert to numpy arrays
	control = np.array(control_values, dtype=float)
	treatment = np.array(treatment_values, dtype=float)

	# Run one-way ANOVA
	f_stat, p_value = stats.f_oneway(control, treatment)

	# Calculate effect size (eta-squared)
	total_values = np.concatenate([control, treatment])
	grand_mean = np.mean(total_values)

	ss_total = np.sum((total_values - grand_mean) ** 2)
	ss_between = (len(control) * (np.mean(control) - grand_mean) ** 2 +
	len(treatment) * (np.mean(treatment) - grand_mean) ** 2)

	eta_squared = ss_between / ss_total if ss_total > 0 else 0

	# Determine if the result is significant
	significant = p_value < alpha

	return {
	'test_type': 'One-way ANOVA',
	'f_statistic': f_stat,
	'p_value': p_value,
	'significant': significant,
	'control_sample_size': len(control),
	'treatment_sample_size': len(treatment),
	'effect_size': eta_squared,
	'effect_size_type': 'eta_squared'
	}

	def _run_chi_square(self, control_values: list, treatment_values: list, alpha: float, **kwargs) -> Dict[str, Any]:
	"""Run Chi-square test for categorical data."""
	# For chi-square, we assume the values represent counts in different categories
	# Convert to numpy arrays
	control = np.array(control_values, dtype=float)
	treatment = np.array(treatment_values, dtype=float)

	# Check if the arrays are the same length (same number of categories)
	if len(control) != len(treatment):
	raise ValueError("Control and treatment must have the same number of categories for chi-square test")

	# Run chi-square test
	contingency_table = np.vstack([control, treatment])
	chi2_stat, p_value, dof, expected = stats.chi2_contingency(contingency_table)

	# Calculate Cramer's V as effect size
	n = np.sum(contingency_table)
	min_dim = min(contingency_table.shape) - 1
	cramers_v = np.sqrt(chi2_stat / (n * min_dim)) if n * min_dim > 0 else 0

	# Determine if the result is significant
	significant = p_value < alpha

	return {
	'test_type': 'Chi-square test',
	'chi2_statistic': chi2_stat,
	'p_value': p_value,
	'degrees_of_freedom': dof,
	'significant': significant,
	'effect_size': cramers_v,
	'effect_size_type': 'cramers_v'
	}

	def check_assumptions(self, metric: str) -> Dict[str, Dict[str, Any]]:
	"""
	Check statistical assumptions for the given metric across all treatments.

	Args:
	metric (str): The metric to check assumptions for.

	Returns:
	dict: Dictionary with results of assumption checks for each treatment.
	"""
	if metric not in self.control_experiment_data:
	raise ValueError(f"Metric '{metric}' not found in control data")

	results = {}
	control_values = np.array(self.control_experiment_data[metric], dtype=float)

	# Check normality of control
	control_shapiro = stats.shapiro(control_values)
	control_normality = {
	'test': 'Shapiro-Wilk',
	'statistic': control_shapiro[0],
	'p_value': control_shapiro[1],
	'normal': control_shapiro[1] >= 0.05
	}

	for treatment_id, treatment_data in self.treatments_experiment_data.items():
	if metric not in treatment_data:
	logger.warning(f"Metric '{metric}' not found in treatment '{treatment_id}'")
	continue

	treatment_values = np.array(treatment_data[metric], dtype=float)

	# Check normality of treatment
	treatment_shapiro = stats.shapiro(treatment_values)
	treatment_normality = {
	'test': 'Shapiro-Wilk',
	'statistic': treatment_shapiro[0],
	'p_value': treatment_shapiro[1],
	'normal': treatment_shapiro[1] >= 0.05
	}

	# Check homogeneity of variance
	levene_test = stats.levene(control_values, treatment_values)
	variance_homogeneity = {
	'test': 'Levene',
	'statistic': levene_test[0],
	'p_value': levene_test[1],
	'equal_variance': levene_test[1] >= 0.05
	}

	# Store results and convert to JSON serializable types
	results[treatment_id] = convert_to_serializable({
	'control_normality': control_normality,
	'treatment_normality': treatment_normality,
	'variance_homogeneity': variance_homogeneity,
	'recommended_test': self._recommend_test(control_normality['normal'],
	treatment_normality['normal'],
	variance_homogeneity['equal_variance'])
	})

	return results

	def _recommend_test(self, control_normal: bool, treatment_normal: bool, equal_variance: bool) -> str:
	"""Recommend a statistical test based on assumption checks."""
	if control_normal and treatment_normal:
	if equal_variance:
	return 't_test'
	else:
	return 'welch_t_test'
	else:
	return 'mann_whitney'</code></pre>
	</details>
	<h3>Methods</h3>
	<dl>
	<dt id="tinytroupe.experimentation.statistical_tests.StatisticalTester.check_assumptions"><code class="name flex">
	<span>def <span class="ident">check_assumptions</span></span>(<span>self, metric: str) ‑> Dict[str, Dict[str, Any]]</span>
	</code></dt>
	<dd>
	<div class="desc"><p>Check statistical assumptions for the given metric across all treatments.</p>
	<h2 id="args">Args</h2>
	<dl>
	<dt><strong><code>metric</code></strong> :&ensp;<code>str</code></dt>
	<dd>The metric to check assumptions for.</dd>
	</dl>
	<h2 id="returns">Returns</h2>
	<dl>
	<dt><code>dict</code></dt>
	<dd>Dictionary with results of assumption checks for each treatment.</dd>
	</dl></div>
	<details class="source">
	<summary>
	<span>Expand source code</span>
	</summary>
	<pre><code class="python">def check_assumptions(self, metric: str) -> Dict[str, Dict[str, Any]]:
	"""
	Check statistical assumptions for the given metric across all treatments.

	Args:
	metric (str): The metric to check assumptions for.

	Returns:
	dict: Dictionary with results of assumption checks for each treatment.
	"""
	if metric not in self.control_experiment_data:
	raise ValueError(f"Metric '{metric}' not found in control data")

	results = {}
	control_values = np.array(self.control_experiment_data[metric], dtype=float)

	# Check normality of control
	control_shapiro = stats.shapiro(control_values)
	control_normality = {
	'test': 'Shapiro-Wilk',
	'statistic': control_shapiro[0],
	'p_value': control_shapiro[1],
	'normal': control_shapiro[1] >= 0.05
	}

	for treatment_id, treatment_data in self.treatments_experiment_data.items():
	if metric not in treatment_data:
	logger.warning(f"Metric '{metric}' not found in treatment '{treatment_id}'")
	continue

	treatment_values = np.array(treatment_data[metric], dtype=float)

	# Check normality of treatment
	treatment_shapiro = stats.shapiro(treatment_values)
	treatment_normality = {
	'test': 'Shapiro-Wilk',
	'statistic': treatment_shapiro[0],
	'p_value': treatment_shapiro[1],
	'normal': treatment_shapiro[1] >= 0.05
	}

	# Check homogeneity of variance
	levene_test = stats.levene(control_values, treatment_values)
	variance_homogeneity = {
	'test': 'Levene',
	'statistic': levene_test[0],
	'p_value': levene_test[1],
	'equal_variance': levene_test[1] >= 0.05
	}

	# Store results and convert to JSON serializable types
	results[treatment_id] = convert_to_serializable({
	'control_normality': control_normality,
	'treatment_normality': treatment_normality,
	'variance_homogeneity': variance_homogeneity,
	'recommended_test': self._recommend_test(control_normality['normal'],
	treatment_normality['normal'],
	variance_homogeneity['equal_variance'])
	})

	return results</code></pre>
	</details>
	</dd>
	<dt id="tinytroupe.experimentation.statistical_tests.StatisticalTester.run_test"><code class="name flex">
	<span>def <span class="ident">run_test</span></span>(<span>self, test_type: str = 'welch_t_test', alpha: float = 0.05, **kwargs) ‑> Dict[str, Dict[str, Any]]</span>
	</code></dt>
	<dd>
	<div class="desc"><p>Run the specified statistical test on the control and treatments data.</p>
	<h2 id="args">Args</h2>
	<dl>
	<dt><strong><code>test_type</code></strong> :&ensp;<code>str</code></dt>
	<dd>Type of statistical test to run.
	Options: 't_test', 'welch_t_test', 'mann_whitney', 'anova', 'chi_square'</dd>
	<dt><strong><code>alpha</code></strong> :&ensp;<code>float</code></dt>
	<dd>Significance level, defaults to 0.05</dd>
	<dt><strong><code>**kwargs</code></strong></dt>
	<dd>Additional arguments for specific test types.</dd>
	</dl>
	<h2 id="returns">Returns</h2>
	<dl>
	<dt><code>dict</code></dt>
	<dd>Dictionary containing the results of the statistical tests for each treatment (vs the one control).
	Each key is the treatment ID and each value is a dictionary with test results.</dd>
	</dl></div>
	<details class="source">
	<summary>
	<span>Expand source code</span>
	</summary>
	<pre><code class="python">def run_test(self,
	test_type: str="welch_t_test",
	alpha: float = 0.05,
	**kwargs) -> Dict[str, Dict[str, Any]]:
	"""
	Run the specified statistical test on the control and treatments data.

	Args:
	test_type (str): Type of statistical test to run.
	Options: 't_test', 'welch_t_test', 'mann_whitney', 'anova', 'chi_square'
	alpha (float): Significance level, defaults to 0.05
	**kwargs: Additional arguments for specific test types.

	Returns:
	dict: Dictionary containing the results of the statistical tests for each treatment (vs the one control).
	Each key is the treatment ID and each value is a dictionary with test results.
	"""
	supported_tests = {
	't_test': self._run_t_test,
	'welch_t_test': self._run_welch_t_test,
	'mann_whitney': self._run_mann_whitney,
	'anova': self._run_anova,
	'chi_square': self._run_chi_square
	}

	if test_type not in supported_tests:
	raise ValueError(f"Unsupported test type: {test_type}. Supported types: {list(supported_tests.keys())}")

	results = {}
	for control_id, control_data in self.control_experiment_data.items():
	# get all metrics from control data
	metrics = set()
	metrics.update(control_data.keys())
	for treatment_id, treatment_data in self.treatments_experiment_data.items():
	results[treatment_id] = {}

	for metric in metrics:
	# Skip metrics not in treatment data
	if metric not in treatment_data:
	logger.warning(f"Metric '{metric}' not found in treatment '{treatment_id}'")
	continue

	control_values = control_data[metric]
	treatment_values = treatment_data[metric]

	# Skip if either control or treatment has no values
	if len(control_values) == 0 or len(treatment_values) == 0:
	logger.warning(f"Skipping metric '{metric}' for treatment '{treatment_id}' due to empty values")
	continue

	# Run the selected test and convert to JSON serializable types
	test_result = supported_tests[test_type](control_values, treatment_values, alpha, **kwargs)
	results[treatment_id][metric] = convert_to_serializable(test_result)

	return results</code></pre>
	</details>
	</dd>
	</dl>
	</dd>
	</dl>
	</section>
	</article>
	<nav id="sidebar">
	<h1>Index</h1>
	<div class="toc">
	<ul></ul>
	</div>
	<ul id="index">
	<li><h3>Super-module</h3>
	<ul>
	<li><code><a title="tinytroupe.experimentation" href="index.html">tinytroupe.experimentation</a></code></li>
	</ul>
	</li>
	<li><h3><a href="#header-functions">Functions</a></h3>
	<ul class="">
	<li><code><a title="tinytroupe.experimentation.statistical_tests.cohen_d" href="#tinytroupe.experimentation.statistical_tests.cohen_d">cohen_d</a></code></li>
	<li><code><a title="tinytroupe.experimentation.statistical_tests.convert_to_serializable" href="#tinytroupe.experimentation.statistical_tests.convert_to_serializable">convert_to_serializable</a></code></li>
	</ul>
	</li>
	<li><h3><a href="#header-classes">Classes</a></h3>
	<ul>
	<li>
	<h4><code><a title="tinytroupe.experimentation.statistical_tests.StatisticalTester" href="#tinytroupe.experimentation.statistical_tests.StatisticalTester">StatisticalTester</a></code></h4>
	<ul class="">
	<li><code><a title="tinytroupe.experimentation.statistical_tests.StatisticalTester.check_assumptions" href="#tinytroupe.experimentation.statistical_tests.StatisticalTester.check_assumptions">check_assumptions</a></code></li>
	<li><code><a title="tinytroupe.experimentation.statistical_tests.StatisticalTester.run_test" href="#tinytroupe.experimentation.statistical_tests.StatisticalTester.run_test">run_test</a></code></li>
	</ul>
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