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<article id="content">
<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:
    &#34;&#34;&#34;
    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.
    &#34;&#34;&#34;

    def __init__(self, control_experiment_data: Dict[str, list], 
                 treatments_experiment_data: Dict[str, Dict[str, list]],
                 results_key:str = None):
        &#34;&#34;&#34;
        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.,{&#34;control_exp&#34;: {&#34;metric1&#34;: [0.1, 0.2], &#34;metric2&#34;: [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., {&#34;exp1&#34;: {&#34;metric1&#34;: [0.1, 0.2], &#34;metric2&#34;: [0.3, 0.4]}, 
                       &#34;exp2&#34;: {&#34;metric1&#34;: [0.5, 0.6], &#34;metric2&#34;: [0.7, 0.8]}, ...}
        &#34;&#34;&#34;
        
        # if results_key is provided, use it to extract the relevant data from the control and treatment data
        # e.g., {&#34;exp1&#34;: {&#34;results&#34;: {&#34;metric1&#34;: [0.1, 0.2], &#34;metric2&#34;: [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):
        &#34;&#34;&#34;Validate the input data formats and structure.&#34;&#34;&#34;
        # Check that control and treatments are dictionaries
        if not isinstance(self.control_experiment_data, dict):
            raise TypeError(&#34;Control experiment data must be a dictionary&#34;)
        if not isinstance(self.treatments_experiment_data, dict):
            raise TypeError(&#34;Treatments experiment data must be a dictionary&#34;)
        
        # Check that control has at least one experiment
        if not self.control_experiment_data:
            raise ValueError(&#34;Control experiment data cannot be empty&#34;)

        # Check only one control
        if len(self.control_experiment_data) &gt; 1:
            raise ValueError(&#34;Only one control experiment is allowed&#34;)
            
        # Validate control experiment structure
        for control_id, control_metrics in self.control_experiment_data.items():
            if not isinstance(control_metrics, dict):
                raise TypeError(f&#34;Metrics for control experiment &#39;{control_id}&#39; must be a dictionary&#34;)
            
            # Check that the metrics dictionary is not empty
            if not control_metrics:
                raise ValueError(f&#34;Control experiment &#39;{control_id}&#39; has no metrics&#34;)
                
            # Validate that metric values are lists
            for metric, values in control_metrics.items():
                if not isinstance(values, list):
                    raise TypeError(f&#34;Values for metric &#39;{metric}&#39; in control experiment &#39;{control_id}&#39; must be a list&#34;)
        
        # Check treatments have at least one experiment
        if not self.treatments_experiment_data:
            raise ValueError(&#34;Treatments experiment data cannot be empty&#34;)
        
        # Validate treatment experiment structure
        for treatment_id, treatment_data in self.treatments_experiment_data.items():
            if not isinstance(treatment_data, dict):
                raise TypeError(f&#34;Data for treatment &#39;{treatment_id}&#39; must be a dictionary&#34;)
                
            # Check that the metrics dictionary is not empty
            if not treatment_data:
                raise ValueError(f&#34;Treatment &#39;{treatment_id}&#39; has no metrics&#34;)
            
            # 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&#39;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&#34;Treatment &#39;{treatment_id}&#39; has no metrics in common with any control experiment&#34;)
            
            # Check that treatment metrics are lists
            for metric, values in treatment_data.items():
                if not isinstance(values, list):
                    raise TypeError(f&#34;Values for metric &#39;{metric}&#39; in treatment &#39;{treatment_id}&#39; must be a list&#34;)
    
    def run_test(self, 
                test_type: str=&#34;welch_t_test&#34;, 
                alpha: float = 0.05, 
                **kwargs) -&gt; Dict[str, Dict[str, Any]]:
        &#34;&#34;&#34;
        Run the specified statistical test on the control and treatments data.

        Args:
            test_type (str): Type of statistical test to run. 
                Options: &#39;t_test&#39;, &#39;welch_t_test&#39;, &#39;mann_whitney&#39;, &#39;anova&#39;, &#39;chi_square&#39;
            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.
        &#34;&#34;&#34;
        supported_tests = {
            &#39;t_test&#39;: self._run_t_test,
            &#39;welch_t_test&#39;: self._run_welch_t_test,
            &#39;mann_whitney&#39;: self._run_mann_whitney,
            &#39;anova&#39;: self._run_anova,
            &#39;chi_square&#39;: self._run_chi_square
        }
        
        if test_type not in supported_tests:
            raise ValueError(f&#34;Unsupported test type: {test_type}. Supported types: {list(supported_tests.keys())}&#34;)
            
        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&#34;Metric &#39;{metric}&#39; not found in treatment &#39;{treatment_id}&#39;&#34;)
                        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&#34;Skipping metric &#39;{metric}&#39; for treatment &#39;{treatment_id}&#39; due to empty values&#34;)
                        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) -&gt; Dict[str, Any]:
        &#34;&#34;&#34;Run Student&#39;s t-test (equal variance assumed).&#34;&#34;&#34;
        # 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 &lt; alpha
        
        return {
            &#39;test_type&#39;: &#39;Student t-test (equal variance)&#39;,
            &#39;control_mean&#39;: control_mean,
            &#39;treatment_mean&#39;: treatment_mean,
            &#39;mean_difference&#39;: mean_diff,
            &#39;percent_change&#39;: (mean_diff / control_mean * 100) if control_mean != 0 else float(&#39;inf&#39;),
            &#39;t_statistic&#39;: t_stat,
            &#39;p_value&#39;: p_value,
            &#39;confidence_interval&#39;: (ci_lower, ci_upper),
            &#39;confidence_level&#39;: 1 - alpha,
            &#39;significant&#39;: significant,
            &#39;control_sample_size&#39;: len(control),
            &#39;treatment_sample_size&#39;: len(treatment),
            &#39;control_std&#39;: control_std,
            &#39;treatment_std&#39;: treatment_std,
            &#39;effect_size&#39;: cohen_d(control, treatment)
        }
    
    def _run_welch_t_test(self, control_values: list, treatment_values: list, alpha: float, **kwargs) -&gt; Dict[str, Any]:
        &#34;&#34;&#34;Run Welch&#39;s t-test (unequal variance).&#34;&#34;&#34;
        # 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&#39;s t-test
        t_stat, p_value = stats.ttest_ind(control, treatment, equal_var=False)
        
        # Calculate confidence interval (for Welch&#39;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 &gt; 0 else float(&#39;inf&#39;)
        
        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 &lt; alpha
        
        return {
            &#39;test_type&#39;: &#39;Welch t-test (unequal variance)&#39;,
            &#39;control_mean&#39;: control_mean,
            &#39;treatment_mean&#39;: treatment_mean,
            &#39;mean_difference&#39;: mean_diff,
            &#39;percent_change&#39;: (mean_diff / control_mean * 100) if control_mean != 0 else float(&#39;inf&#39;),
            &#39;t_statistic&#39;: t_stat,
            &#39;p_value&#39;: p_value,
            &#39;confidence_interval&#39;: (ci_lower, ci_upper),
            &#39;confidence_level&#39;: 1 - alpha,
            &#39;significant&#39;: significant,
            &#39;degrees_of_freedom&#39;: df,
            &#39;control_sample_size&#39;: len(control),
            &#39;treatment_sample_size&#39;: len(treatment),
            &#39;control_std&#39;: control_std,
            &#39;treatment_std&#39;: treatment_std,
            &#39;effect_size&#39;: cohen_d(control, treatment)
        }
    
    def _run_mann_whitney(self, control_values: list, treatment_values: list, alpha: float, **kwargs) -&gt; Dict[str, Any]:
        &#34;&#34;&#34;Run Mann-Whitney U test (non-parametric test).&#34;&#34;&#34;
        # 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=&#39;two-sided&#39;)
        
        # 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 &gt; 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(&#34;SciPy bootstrap not available, skipping confidence interval calculation&#34;)
        
        # Determine if the result is significant
        significant = p_value &lt; alpha
        
        return {
            &#39;test_type&#39;: &#39;Mann-Whitney U test&#39;,
            &#39;control_median&#39;: control_median,
            &#39;treatment_median&#39;: treatment_median,
            &#39;median_difference&#39;: median_diff,
            &#39;percent_change&#39;: (median_diff / control_median * 100) if control_median != 0 else float(&#39;inf&#39;),
            &#39;u_statistic&#39;: u_stat,
            &#39;p_value&#39;: p_value,
            &#39;confidence_interval&#39;: (ci_lower, ci_upper) if ci_lower is not None else None,
            &#39;confidence_level&#39;: 1 - alpha,
            &#39;significant&#39;: significant,
            &#39;control_sample_size&#39;: len(control),
            &#39;treatment_sample_size&#39;: len(treatment),
            &#39;effect_size&#39;: cles
        }
    
    def _run_anova(self, control_values: list, treatment_values: list, alpha: float, **kwargs) -&gt; Dict[str, Any]:
        &#34;&#34;&#34;Run one-way ANOVA test.&#34;&#34;&#34;
        # 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 &gt; 0 else 0
        
        # Determine if the result is significant
        significant = p_value &lt; alpha
        
        return {
            &#39;test_type&#39;: &#39;One-way ANOVA&#39;,
            &#39;f_statistic&#39;: f_stat,
            &#39;p_value&#39;: p_value,
            &#39;significant&#39;: significant,
            &#39;control_sample_size&#39;: len(control),
            &#39;treatment_sample_size&#39;: len(treatment),
            &#39;effect_size&#39;: eta_squared,
            &#39;effect_size_type&#39;: &#39;eta_squared&#39;
        }
    
    def _run_chi_square(self, control_values: list, treatment_values: list, alpha: float, **kwargs) -&gt; Dict[str, Any]:
        &#34;&#34;&#34;Run Chi-square test for categorical data.&#34;&#34;&#34;
        # 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(&#34;Control and treatment must have the same number of categories for chi-square test&#34;)
        
        # Run chi-square test
        contingency_table = np.vstack([control, treatment])
        chi2_stat, p_value, dof, expected = stats.chi2_contingency(contingency_table)
        
        # Calculate Cramer&#39;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 &gt; 0 else 0
        
        # Determine if the result is significant
        significant = p_value &lt; alpha
        
        return {
            &#39;test_type&#39;: &#39;Chi-square test&#39;,
            &#39;chi2_statistic&#39;: chi2_stat,
            &#39;p_value&#39;: p_value,
            &#39;degrees_of_freedom&#39;: dof,
            &#39;significant&#39;: significant,
            &#39;effect_size&#39;: cramers_v,
            &#39;effect_size_type&#39;: &#39;cramers_v&#39;
        }
    
    def check_assumptions(self, metric: str) -&gt; Dict[str, Dict[str, Any]]:
        &#34;&#34;&#34;
        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.
        &#34;&#34;&#34;
        if metric not in self.control_experiment_data:
            raise ValueError(f&#34;Metric &#39;{metric}&#39; not found in control data&#34;)
            
        results = {}
        control_values = np.array(self.control_experiment_data[metric], dtype=float)
        
        # Check normality of control
        control_shapiro = stats.shapiro(control_values)
        control_normality = {
            &#39;test&#39;: &#39;Shapiro-Wilk&#39;,
            &#39;statistic&#39;: control_shapiro[0],
            &#39;p_value&#39;: control_shapiro[1],
            &#39;normal&#39;: control_shapiro[1] &gt;= 0.05
        }
        
        for treatment_id, treatment_data in self.treatments_experiment_data.items():
            if metric not in treatment_data:
                logger.warning(f&#34;Metric &#39;{metric}&#39; not found in treatment &#39;{treatment_id}&#39;&#34;)
                continue
                
            treatment_values = np.array(treatment_data[metric], dtype=float)
            
            # Check normality of treatment
            treatment_shapiro = stats.shapiro(treatment_values)
            treatment_normality = {
                &#39;test&#39;: &#39;Shapiro-Wilk&#39;,
                &#39;statistic&#39;: treatment_shapiro[0],
                &#39;p_value&#39;: treatment_shapiro[1],
                &#39;normal&#39;: treatment_shapiro[1] &gt;= 0.05
            }
            
            # Check homogeneity of variance
            levene_test = stats.levene(control_values, treatment_values)
            variance_homogeneity = {
                &#39;test&#39;: &#39;Levene&#39;,
                &#39;statistic&#39;: levene_test[0],
                &#39;p_value&#39;: levene_test[1],
                &#39;equal_variance&#39;: levene_test[1] &gt;= 0.05
            }
            
            # Store results and convert to JSON serializable types
            results[treatment_id] = convert_to_serializable({
                &#39;control_normality&#39;: control_normality,
                &#39;treatment_normality&#39;: treatment_normality,
                &#39;variance_homogeneity&#39;: variance_homogeneity,
                &#39;recommended_test&#39;: self._recommend_test(control_normality[&#39;normal&#39;], 
                                                       treatment_normality[&#39;normal&#39;],
                                                       variance_homogeneity[&#39;equal_variance&#39;])
            })
            
        return results
    
    def _recommend_test(self, control_normal: bool, treatment_normal: bool, equal_variance: bool) -&gt; str:
        &#34;&#34;&#34;Recommend a statistical test based on assumption checks.&#34;&#34;&#34;
        if control_normal and treatment_normal:
            if equal_variance:
                return &#39;t_test&#39;
            else:
                return &#39;welch_t_test&#39;
        else:
            return &#39;mann_whitney&#39;


def cohen_d(x: Union[list, np.ndarray], y: Union[list, np.ndarray]) -&gt; float:
    &#34;&#34;&#34;
    Calculate Cohen&#39;s d effect size for two samples.
    
    Args:
        x: First sample
        y: Second sample
        
    Returns:
        float: Cohen&#39;s d effect size
    &#34;&#34;&#34;
    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&#39;s d
    return (my - mx) / pooled_sd if pooled_sd &gt; 0 else 0


def convert_to_serializable(obj):
    &#34;&#34;&#34;
    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
    &#34;&#34;&#34;
    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]) -&gt; float:
    &#34;&#34;&#34;
    Calculate Cohen&#39;s d effect size for two samples.
    
    Args:
        x: First sample
        y: Second sample
        
    Returns:
        float: Cohen&#39;s d effect size
    &#34;&#34;&#34;
    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&#39;s d
    return (my - mx) / pooled_sd if pooled_sd &gt; 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):
    &#34;&#34;&#34;
    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
    &#34;&#34;&#34;
    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], &hellip;}}</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]}, &hellip;}</dd>
</dl></div>
<details class="source">
<summary>
<span>Expand source code</span>
</summary>
<pre><code class="python">class StatisticalTester:
    &#34;&#34;&#34;
    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.
    &#34;&#34;&#34;

    def __init__(self, control_experiment_data: Dict[str, list], 
                 treatments_experiment_data: Dict[str, Dict[str, list]],
                 results_key:str = None):
        &#34;&#34;&#34;
        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.,{&#34;control_exp&#34;: {&#34;metric1&#34;: [0.1, 0.2], &#34;metric2&#34;: [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., {&#34;exp1&#34;: {&#34;metric1&#34;: [0.1, 0.2], &#34;metric2&#34;: [0.3, 0.4]}, 
                       &#34;exp2&#34;: {&#34;metric1&#34;: [0.5, 0.6], &#34;metric2&#34;: [0.7, 0.8]}, ...}
        &#34;&#34;&#34;
        
        # if results_key is provided, use it to extract the relevant data from the control and treatment data
        # e.g., {&#34;exp1&#34;: {&#34;results&#34;: {&#34;metric1&#34;: [0.1, 0.2], &#34;metric2&#34;: [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):
        &#34;&#34;&#34;Validate the input data formats and structure.&#34;&#34;&#34;
        # Check that control and treatments are dictionaries
        if not isinstance(self.control_experiment_data, dict):
            raise TypeError(&#34;Control experiment data must be a dictionary&#34;)
        if not isinstance(self.treatments_experiment_data, dict):
            raise TypeError(&#34;Treatments experiment data must be a dictionary&#34;)
        
        # Check that control has at least one experiment
        if not self.control_experiment_data:
            raise ValueError(&#34;Control experiment data cannot be empty&#34;)

        # Check only one control
        if len(self.control_experiment_data) &gt; 1:
            raise ValueError(&#34;Only one control experiment is allowed&#34;)
            
        # Validate control experiment structure
        for control_id, control_metrics in self.control_experiment_data.items():
            if not isinstance(control_metrics, dict):
                raise TypeError(f&#34;Metrics for control experiment &#39;{control_id}&#39; must be a dictionary&#34;)
            
            # Check that the metrics dictionary is not empty
            if not control_metrics:
                raise ValueError(f&#34;Control experiment &#39;{control_id}&#39; has no metrics&#34;)
                
            # Validate that metric values are lists
            for metric, values in control_metrics.items():
                if not isinstance(values, list):
                    raise TypeError(f&#34;Values for metric &#39;{metric}&#39; in control experiment &#39;{control_id}&#39; must be a list&#34;)
        
        # Check treatments have at least one experiment
        if not self.treatments_experiment_data:
            raise ValueError(&#34;Treatments experiment data cannot be empty&#34;)
        
        # Validate treatment experiment structure
        for treatment_id, treatment_data in self.treatments_experiment_data.items():
            if not isinstance(treatment_data, dict):
                raise TypeError(f&#34;Data for treatment &#39;{treatment_id}&#39; must be a dictionary&#34;)
                
            # Check that the metrics dictionary is not empty
            if not treatment_data:
                raise ValueError(f&#34;Treatment &#39;{treatment_id}&#39; has no metrics&#34;)
            
            # 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&#39;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&#34;Treatment &#39;{treatment_id}&#39; has no metrics in common with any control experiment&#34;)
            
            # Check that treatment metrics are lists
            for metric, values in treatment_data.items():
                if not isinstance(values, list):
                    raise TypeError(f&#34;Values for metric &#39;{metric}&#39; in treatment &#39;{treatment_id}&#39; must be a list&#34;)
    
    def run_test(self, 
                test_type: str=&#34;welch_t_test&#34;, 
                alpha: float = 0.05, 
                **kwargs) -&gt; Dict[str, Dict[str, Any]]:
        &#34;&#34;&#34;
        Run the specified statistical test on the control and treatments data.

        Args:
            test_type (str): Type of statistical test to run. 
                Options: &#39;t_test&#39;, &#39;welch_t_test&#39;, &#39;mann_whitney&#39;, &#39;anova&#39;, &#39;chi_square&#39;
            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.
        &#34;&#34;&#34;
        supported_tests = {
            &#39;t_test&#39;: self._run_t_test,
            &#39;welch_t_test&#39;: self._run_welch_t_test,
            &#39;mann_whitney&#39;: self._run_mann_whitney,
            &#39;anova&#39;: self._run_anova,
            &#39;chi_square&#39;: self._run_chi_square
        }
        
        if test_type not in supported_tests:
            raise ValueError(f&#34;Unsupported test type: {test_type}. Supported types: {list(supported_tests.keys())}&#34;)
            
        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&#34;Metric &#39;{metric}&#39; not found in treatment &#39;{treatment_id}&#39;&#34;)
                        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&#34;Skipping metric &#39;{metric}&#39; for treatment &#39;{treatment_id}&#39; due to empty values&#34;)
                        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) -&gt; Dict[str, Any]:
        &#34;&#34;&#34;Run Student&#39;s t-test (equal variance assumed).&#34;&#34;&#34;
        # 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 &lt; alpha
        
        return {
            &#39;test_type&#39;: &#39;Student t-test (equal variance)&#39;,
            &#39;control_mean&#39;: control_mean,
            &#39;treatment_mean&#39;: treatment_mean,
            &#39;mean_difference&#39;: mean_diff,
            &#39;percent_change&#39;: (mean_diff / control_mean * 100) if control_mean != 0 else float(&#39;inf&#39;),
            &#39;t_statistic&#39;: t_stat,
            &#39;p_value&#39;: p_value,
            &#39;confidence_interval&#39;: (ci_lower, ci_upper),
            &#39;confidence_level&#39;: 1 - alpha,
            &#39;significant&#39;: significant,
            &#39;control_sample_size&#39;: len(control),
            &#39;treatment_sample_size&#39;: len(treatment),
            &#39;control_std&#39;: control_std,
            &#39;treatment_std&#39;: treatment_std,
            &#39;effect_size&#39;: cohen_d(control, treatment)
        }
    
    def _run_welch_t_test(self, control_values: list, treatment_values: list, alpha: float, **kwargs) -&gt; Dict[str, Any]:
        &#34;&#34;&#34;Run Welch&#39;s t-test (unequal variance).&#34;&#34;&#34;
        # 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&#39;s t-test
        t_stat, p_value = stats.ttest_ind(control, treatment, equal_var=False)
        
        # Calculate confidence interval (for Welch&#39;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 &gt; 0 else float(&#39;inf&#39;)
        
        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 &lt; alpha
        
        return {
            &#39;test_type&#39;: &#39;Welch t-test (unequal variance)&#39;,
            &#39;control_mean&#39;: control_mean,
            &#39;treatment_mean&#39;: treatment_mean,
            &#39;mean_difference&#39;: mean_diff,
            &#39;percent_change&#39;: (mean_diff / control_mean * 100) if control_mean != 0 else float(&#39;inf&#39;),
            &#39;t_statistic&#39;: t_stat,
            &#39;p_value&#39;: p_value,
            &#39;confidence_interval&#39;: (ci_lower, ci_upper),
            &#39;confidence_level&#39;: 1 - alpha,
            &#39;significant&#39;: significant,
            &#39;degrees_of_freedom&#39;: df,
            &#39;control_sample_size&#39;: len(control),
            &#39;treatment_sample_size&#39;: len(treatment),
            &#39;control_std&#39;: control_std,
            &#39;treatment_std&#39;: treatment_std,
            &#39;effect_size&#39;: cohen_d(control, treatment)
        }
    
    def _run_mann_whitney(self, control_values: list, treatment_values: list, alpha: float, **kwargs) -&gt; Dict[str, Any]:
        &#34;&#34;&#34;Run Mann-Whitney U test (non-parametric test).&#34;&#34;&#34;
        # 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=&#39;two-sided&#39;)
        
        # 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 &gt; 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(&#34;SciPy bootstrap not available, skipping confidence interval calculation&#34;)
        
        # Determine if the result is significant
        significant = p_value &lt; alpha
        
        return {
            &#39;test_type&#39;: &#39;Mann-Whitney U test&#39;,
            &#39;control_median&#39;: control_median,
            &#39;treatment_median&#39;: treatment_median,
            &#39;median_difference&#39;: median_diff,
            &#39;percent_change&#39;: (median_diff / control_median * 100) if control_median != 0 else float(&#39;inf&#39;),
            &#39;u_statistic&#39;: u_stat,
            &#39;p_value&#39;: p_value,
            &#39;confidence_interval&#39;: (ci_lower, ci_upper) if ci_lower is not None else None,
            &#39;confidence_level&#39;: 1 - alpha,
            &#39;significant&#39;: significant,
            &#39;control_sample_size&#39;: len(control),
            &#39;treatment_sample_size&#39;: len(treatment),
            &#39;effect_size&#39;: cles
        }
    
    def _run_anova(self, control_values: list, treatment_values: list, alpha: float, **kwargs) -&gt; Dict[str, Any]:
        &#34;&#34;&#34;Run one-way ANOVA test.&#34;&#34;&#34;
        # 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 &gt; 0 else 0
        
        # Determine if the result is significant
        significant = p_value &lt; alpha
        
        return {
            &#39;test_type&#39;: &#39;One-way ANOVA&#39;,
            &#39;f_statistic&#39;: f_stat,
            &#39;p_value&#39;: p_value,
            &#39;significant&#39;: significant,
            &#39;control_sample_size&#39;: len(control),
            &#39;treatment_sample_size&#39;: len(treatment),
            &#39;effect_size&#39;: eta_squared,
            &#39;effect_size_type&#39;: &#39;eta_squared&#39;
        }
    
    def _run_chi_square(self, control_values: list, treatment_values: list, alpha: float, **kwargs) -&gt; Dict[str, Any]:
        &#34;&#34;&#34;Run Chi-square test for categorical data.&#34;&#34;&#34;
        # 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(&#34;Control and treatment must have the same number of categories for chi-square test&#34;)
        
        # Run chi-square test
        contingency_table = np.vstack([control, treatment])
        chi2_stat, p_value, dof, expected = stats.chi2_contingency(contingency_table)
        
        # Calculate Cramer&#39;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 &gt; 0 else 0
        
        # Determine if the result is significant
        significant = p_value &lt; alpha
        
        return {
            &#39;test_type&#39;: &#39;Chi-square test&#39;,
            &#39;chi2_statistic&#39;: chi2_stat,
            &#39;p_value&#39;: p_value,
            &#39;degrees_of_freedom&#39;: dof,
            &#39;significant&#39;: significant,
            &#39;effect_size&#39;: cramers_v,
            &#39;effect_size_type&#39;: &#39;cramers_v&#39;
        }
    
    def check_assumptions(self, metric: str) -&gt; Dict[str, Dict[str, Any]]:
        &#34;&#34;&#34;
        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.
        &#34;&#34;&#34;
        if metric not in self.control_experiment_data:
            raise ValueError(f&#34;Metric &#39;{metric}&#39; not found in control data&#34;)
            
        results = {}
        control_values = np.array(self.control_experiment_data[metric], dtype=float)
        
        # Check normality of control
        control_shapiro = stats.shapiro(control_values)
        control_normality = {
            &#39;test&#39;: &#39;Shapiro-Wilk&#39;,
            &#39;statistic&#39;: control_shapiro[0],
            &#39;p_value&#39;: control_shapiro[1],
            &#39;normal&#39;: control_shapiro[1] &gt;= 0.05
        }
        
        for treatment_id, treatment_data in self.treatments_experiment_data.items():
            if metric not in treatment_data:
                logger.warning(f&#34;Metric &#39;{metric}&#39; not found in treatment &#39;{treatment_id}&#39;&#34;)
                continue
                
            treatment_values = np.array(treatment_data[metric], dtype=float)
            
            # Check normality of treatment
            treatment_shapiro = stats.shapiro(treatment_values)
            treatment_normality = {
                &#39;test&#39;: &#39;Shapiro-Wilk&#39;,
                &#39;statistic&#39;: treatment_shapiro[0],
                &#39;p_value&#39;: treatment_shapiro[1],
                &#39;normal&#39;: treatment_shapiro[1] &gt;= 0.05
            }
            
            # Check homogeneity of variance
            levene_test = stats.levene(control_values, treatment_values)
            variance_homogeneity = {
                &#39;test&#39;: &#39;Levene&#39;,
                &#39;statistic&#39;: levene_test[0],
                &#39;p_value&#39;: levene_test[1],
                &#39;equal_variance&#39;: levene_test[1] &gt;= 0.05
            }
            
            # Store results and convert to JSON serializable types
            results[treatment_id] = convert_to_serializable({
                &#39;control_normality&#39;: control_normality,
                &#39;treatment_normality&#39;: treatment_normality,
                &#39;variance_homogeneity&#39;: variance_homogeneity,
                &#39;recommended_test&#39;: self._recommend_test(control_normality[&#39;normal&#39;], 
                                                       treatment_normality[&#39;normal&#39;],
                                                       variance_homogeneity[&#39;equal_variance&#39;])
            })
            
        return results
    
    def _recommend_test(self, control_normal: bool, treatment_normal: bool, equal_variance: bool) -&gt; str:
        &#34;&#34;&#34;Recommend a statistical test based on assumption checks.&#34;&#34;&#34;
        if control_normal and treatment_normal:
            if equal_variance:
                return &#39;t_test&#39;
            else:
                return &#39;welch_t_test&#39;
        else:
            return &#39;mann_whitney&#39;</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) -&gt; Dict[str, Dict[str, Any]]:
    &#34;&#34;&#34;
    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.
    &#34;&#34;&#34;
    if metric not in self.control_experiment_data:
        raise ValueError(f&#34;Metric &#39;{metric}&#39; not found in control data&#34;)
        
    results = {}
    control_values = np.array(self.control_experiment_data[metric], dtype=float)
    
    # Check normality of control
    control_shapiro = stats.shapiro(control_values)
    control_normality = {
        &#39;test&#39;: &#39;Shapiro-Wilk&#39;,
        &#39;statistic&#39;: control_shapiro[0],
        &#39;p_value&#39;: control_shapiro[1],
        &#39;normal&#39;: control_shapiro[1] &gt;= 0.05
    }
    
    for treatment_id, treatment_data in self.treatments_experiment_data.items():
        if metric not in treatment_data:
            logger.warning(f&#34;Metric &#39;{metric}&#39; not found in treatment &#39;{treatment_id}&#39;&#34;)
            continue
            
        treatment_values = np.array(treatment_data[metric], dtype=float)
        
        # Check normality of treatment
        treatment_shapiro = stats.shapiro(treatment_values)
        treatment_normality = {
            &#39;test&#39;: &#39;Shapiro-Wilk&#39;,
            &#39;statistic&#39;: treatment_shapiro[0],
            &#39;p_value&#39;: treatment_shapiro[1],
            &#39;normal&#39;: treatment_shapiro[1] &gt;= 0.05
        }
        
        # Check homogeneity of variance
        levene_test = stats.levene(control_values, treatment_values)
        variance_homogeneity = {
            &#39;test&#39;: &#39;Levene&#39;,
            &#39;statistic&#39;: levene_test[0],
            &#39;p_value&#39;: levene_test[1],
            &#39;equal_variance&#39;: levene_test[1] &gt;= 0.05
        }
        
        # Store results and convert to JSON serializable types
        results[treatment_id] = convert_to_serializable({
            &#39;control_normality&#39;: control_normality,
            &#39;treatment_normality&#39;: treatment_normality,
            &#39;variance_homogeneity&#39;: variance_homogeneity,
            &#39;recommended_test&#39;: self._recommend_test(control_normality[&#39;normal&#39;], 
                                                   treatment_normality[&#39;normal&#39;],
                                                   variance_homogeneity[&#39;equal_variance&#39;])
        })
        
    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=&#34;welch_t_test&#34;, 
            alpha: float = 0.05, 
            **kwargs) -&gt; Dict[str, Dict[str, Any]]:
    &#34;&#34;&#34;
    Run the specified statistical test on the control and treatments data.

    Args:
        test_type (str): Type of statistical test to run. 
            Options: &#39;t_test&#39;, &#39;welch_t_test&#39;, &#39;mann_whitney&#39;, &#39;anova&#39;, &#39;chi_square&#39;
        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.
    &#34;&#34;&#34;
    supported_tests = {
        &#39;t_test&#39;: self._run_t_test,
        &#39;welch_t_test&#39;: self._run_welch_t_test,
        &#39;mann_whitney&#39;: self._run_mann_whitney,
        &#39;anova&#39;: self._run_anova,
        &#39;chi_square&#39;: self._run_chi_square
    }
    
    if test_type not in supported_tests:
        raise ValueError(f&#34;Unsupported test type: {test_type}. Supported types: {list(supported_tests.keys())}&#34;)
        
    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&#34;Metric &#39;{metric}&#39; not found in treatment &#39;{treatment_id}&#39;&#34;)
                    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&#34;Skipping metric &#39;{metric}&#39; for treatment &#39;{treatment_id}&#39; due to empty values&#34;)
                    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>
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<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>
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<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>
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<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>
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