Advanced Statistical Inference and Hypothesis Testing

Parametric tests, such as t-tests and ANOVA, rely on assumptions about the data distribution (e.g., normality). When these assumptions are violated, non-parametric tests provide robust alternatives. These tests don't assume a specific distribution. Examples include the Mann-Whitney U test (for comparing two independent samples), the Wilcoxon signed-rank test (for comparing two related samples), and the Kruskal-Wallis test (for comparing multiple independent groups).

Example: Imagine analyzing customer satisfaction scores. If the data is not normally distributed (e.g., skewed), a Mann-Whitney U test would be more appropriate than a t-test to compare satisfaction scores between two different marketing campaigns. The output of these tests involves rank-based statistics and p-values to make inferences. Always interpret the p-value in the context of the null hypothesis and effect size.

Code Snippet (Python with SciPy):

from scipy.stats import mannwhitneyu

group1 = [10, 12, 14, 16, 18]
group2 = [5, 7, 9, 11, 13, 15]

# Perform Mann-Whitney U test
statistic, p_value = mannwhitneyu(group1, group2, alternative='two-sided')

print(f"Mann-Whitney U statistic: {statistic}")
print(f"P-value: {p_value}")

Bayesian statistics offers an alternative framework for inference, focusing on updating beliefs (prior) based on observed data (likelihood) to obtain a posterior distribution. This approach allows for incorporating prior knowledge and providing a more intuitive interpretation of results. Bayesian methods are particularly useful when dealing with complex models, limited data, or when incorporating expert knowledge is beneficial. Libraries like PyMC3 and Stan provide powerful tools for Bayesian inference.

Key Concepts:
* Prior: The initial belief about a parameter before observing data.
* Likelihood: The probability of observing the data given a specific value of the parameter.
* Posterior: The updated belief about the parameter after observing the data (Prior x Likelihood).

Example: Suppose we want to estimate the probability of a user clicking an ad. We might start with a prior belief based on historical click-through rates. After observing data from a new campaign (likelihood), we update our belief to obtain a posterior distribution, which reflects the combined information from our prior and the observed data.

Code Snippet (Python with PyMC3):

import pymc3 as pm
import numpy as np

# Simulate some data (Bernoulli trials)
observed_successes = 30
total_trials = 100

with pm.Model() as model:
    # Define prior (e.g., uniform)
    theta = pm.Uniform('theta', lower=0, upper=1)

    # Define likelihood (Bernoulli)
    y = pm.Binomial('y', n=total_trials, p=theta, observed=observed_successes)

    # Perform MCMC sampling
    trace = pm.sample(2000, tune=1000, random_seed=42)

pm.traceplot(trace)
pm.summary(trace)

Power analysis helps determine the sample size needed to detect a statistically significant effect with a certain level of confidence (typically 80% or 90%). It's crucial for experimental design to avoid underpowered studies, which may fail to detect real effects (Type II error).

Key Concepts:
* Power: The probability of correctly rejecting the null hypothesis when it is false (1 - β).
* Effect Size: The magnitude of the effect you want to detect (e.g., Cohen's d).
* Significance Level (α): The probability of making a Type I error (false positive). (Typically 0.05).
* Sample Size: The number of observations in your study.

Example: If you want to detect a small difference in the average performance of two training programs (small effect size), you'll need a larger sample size than if you anticipate a large effect. Tools like statsmodels in Python can help perform power analysis.

Code Snippet (Python with statsmodels):

import statsmodels.stats.power as smp

# Parameters
effect_size = 0.5  # Example: Cohen's d
alpha = 0.05
power = 0.8

# Calculate required sample size for a two-sample t-test
analysis = smp.TTestIndPower()
n_samples = analysis.solve_power(effect_size=effect_size, alpha=alpha, power=power, alternative='two-sided')

print(f"Required sample size per group: {n_samples:.0f}")

When performing multiple hypothesis tests, the probability of making a Type I error (false positive) increases. Multiple hypothesis correction methods address this issue by adjusting the significance level.

Methods:
* Bonferroni Correction: Multiplies each p-value by the number of tests.
* Benjamini-Hochberg (False Discovery Rate - FDR): Controls the expected proportion of false positives among rejected hypotheses. (More powerful than Bonferroni)

Example: If you perform 100 independent hypothesis tests, and you use an alpha of 0.05, you expect to see 5 false positives. Multiple hypothesis correction is vital in fields like genomics, where thousands of tests are performed simultaneously.

Code Snippet (Python with statsmodels):

import statsmodels.stats.multitest as smm
import numpy as np

# Example p-values (from multiple tests)
p_values = np.array([0.01, 0.03, 0.04, 0.005, 0.08])

# Apply Benjamini-Hochberg correction
reject, p_adjusted, _, _ = smm.multipletests(p_values, method='fdr_bh')

print("Original p-values:", p_values)
print("Adjusted p-values:", p_adjusted)
print("Rejected hypotheses:", reject)