Advanced statistics extends beyond introductory methods to handle complex real-world scenarios. When assumptions are violated, samples are small, data is missing, or you need causal inference, advanced techniques become essential. This guide covers the most practical advanced methods.

This comprehensive guide covers advanced statistical techniques with practical applications.

Section 1: Non-Parametric Methods

Non-parametric tests don’t assume normal distribution. Use when assumptions violated or data ordinal.

Advantages of Non-Parametric Tests

  • ✅ No normality assumption - Works with skewed data
  • ✅ Robust to outliers - Based on ranks, not values
  • ✅ Works for ordinal data - Rankings, Likert scales
  • ✅ Small samples - Often acceptable with n < 30

Disadvantages

  • ❌ Less powerful - Harder to detect true effects (with normal data)
  • ❌ Less informative - Tests location, not mean specifically
  • ❌ Complex CI calculation - No simple formulas

Common Non-Parametric Tests

Mann-Whitney U Test (vs Independent t-test)

  • Compares two independent groups using ranks
  • Alternative when normality violated
  • Tests if distributions differ

Wilcoxon Signed-Rank Test (vs Paired t-test)

  • Compares two paired/repeated measurements
  • Uses ranks of differences
  • Better for non-normal data

Kruskal-Wallis Test (vs One-way ANOVA)

  • Compares 3+ independent groups
  • Non-parametric alternative to ANOVA
  • Based on ranks

Friedman Test (vs Repeated measures ANOVA)

  • Compares 3+ repeated measurements
  • For blocked designs
  • Based on ranks within blocks

Spearman Rank Correlation (vs Pearson Correlation)

  • Measures association using ranks
  • Robust to outliers and non-linearity
  • No normality assumption

When to Use Non-Parametric

Definite use:

  • Ordinal data (ratings, rankings)
  • Clearly non-normal distribution
  • Extreme outliers present
  • Very small sample (n < 10)

Consider:

  • Sample size 10-30 with moderate non-normality
  • “Prefer robustness over power” philosophy

Unnecessary:

  • Large samples (n > 30) with any distribution
  • Normal data
  • Extreme outliers are genuine (investigate first)

Interactive Calculators: Non-Parametric Tests

[Integrated calculator links from your existing tools]

Section 2: Bootstrap and Resampling Methods

Bootstrap is a computer-intensive method that resamples from data to estimate sampling distributions.

How Bootstrap Works

Process:

  1. Start with original sample of n observations
  2. Randomly sample n observations with replacement from original sample
  3. Calculate statistic (mean, SD, correlation, etc.)
  4. Repeat steps 2-3 many times (1000-10000 iterations)
  5. Use distribution of bootstrap statistics to estimate uncertainty

Key insight: Sampling distribution estimated from data, not formulas

Bootstrap Confidence Intervals

Percentile Method:

  • 95% CI = [2.5th percentile, 97.5th percentile] of bootstrap statistics

Advantages:

  • Works for any statistic (mean, median, trimmed mean, etc.)
  • No normality assumption needed
  • Very flexible

Example:

  • Original sample median: 50
  • 1000 bootstrap resamples
  • Bootstrap medians range from 45 to 55
  • 95% CI ≈ [45, 55]

Permutation Tests

Alternative to parametric tests when assumptions violated

Process:

  1. Combine all data from both groups
  2. Randomly permute (shuffle) into two groups same size as original
  3. Calculate test statistic (difference in means, etc.)
  4. Repeat 1000+ times
  5. Calculate proportion of permutations with statistic ≥ observed

This proportion is the p-value.

Advantages:

  • Exact p-values (not approximations)
  • No distributional assumptions
  • Valid for any test statistic
  • Works with small samples

Monte Carlo Simulation

Computational method that simulates random data to study behavior

Example: Checking power of t-test

  1. Assume population with mean 100, SD 15
  2. Repeatedly simulate samples of size 50
  3. For each sample, perform t-test against μ = 95
  4. Calculate proportion rejecting H₀
  5. That proportion = power

Uses:

  • Evaluate test properties
  • Check robustness to assumption violations
  • Plan studies with complex designs
  • Estimate sampling distributions

Section 3: Bayesian Statistics

Bayesian approach incorporates prior beliefs about parameters, updates with data.

Bayesian vs Frequentist

Aspect Bayesian Frequentist
Probability Subjective (degree of belief) Long-run frequency
Parameters Random variables Fixed, unknown
Data Fixed (observed) Random (would repeat)
Inference Posterior distribution Confidence intervals, p-values

Bayes’ Theorem

Formula:

P(θ|Data) = P(Data|θ) × P(θ) / P(Data)

where:
P(θ|Data) = Posterior (updated belief)
P(Data|θ) = Likelihood (data given parameter)
P(θ) = Prior (before seeing data)
P(Data) = Evidence (normalizing constant)

Bayesian Interpretation

Posterior Distribution:

  • Represents all uncertainty about parameter
  • Can calculate probability parameter in range (unlike frequentist CI)

Credible Interval:

  • Bayesian equivalent to confidence interval
  • True probability parameter is in interval (unlike frequentist interpretation)

Advantages of Bayesian

  • ✅ Direct probability statements - “95% probability μ is between X and Y”
  • ✅ Incorporates prior knowledge - Uses previous research
  • ✅ Sequential analysis - Naturally adapts as data collected
  • ✅ Small samples - Prior can stabilize estimates
  • ✅ Complex models - Handles hierarchical structures

Disadvantages

  • ❌ Prior selection - Subjective, affects results
  • ❌ Computational complexity - Often requires MCMC sampling
  • ❌ Requires software - Not as simple as frequentist tests
  • ❌ Posterior depends on prior - Sensitivity analysis needed

When to Use Bayesian

Ideal for:

  • Incorporating expert opinion or prior research
  • Sequential decision-making
  • Complex hierarchical models
  • Small sample sizes

Less ideal for:

  • Objective, uninformed analysis
  • Simple studies
  • When prior impossible to specify
  • Regulatory settings wanting objectivity

Section 4: Missing Data

Types of Missingness

Missing Completely at Random (MCAR):

  • Probability of missing doesn’t depend on any variables
  • Missing at random
  • Example: Lab error losing blood sample

Missing at Random (MAR):

  • Probability of missing depends on observed variables (but not unobserved)
  • Common in practice
  • Example: Higher income individuals less likely to report income

Missing Not at Random (MNAR):

  • Probability depends on unobserved variables themselves
  • Most problematic
  • Example: Sicker patients skip survey

Consequences of Missing Data

Complete Case Analysis (Listwise Deletion):

  • Only analyze observations with no missing values
  • Can introduce bias (if MAR or MNAR)
  • Reduces sample size and power
  • ❌ Often inappropriate

Imputation Methods:

  • Estimate missing values from observed data
  • Multiple imputation: Create multiple datasets with different plausible values
  • ✅ Better than deletion for MAR
  • Requires careful implementation

Imputation Approaches

  1. Mean Imputation: Replace with column mean (underestimates variance)
  2. Forward/Backward Fill: Use last/next value (for time series)
  3. Multiple Imputation: Create m datasets, analyze each, combine results
  4. Predictive Mean Matching: Use regression to predict missing values
  5. K-Nearest Neighbors: Use similar cases to impute values

Best Practice: Multiple imputation for MAR data

Section 5: Multivariate Analysis

Multivariate methods analyze multiple variables simultaneously.

Types of Multivariate Methods

Dimensionality Reduction:

  • Principal Component Analysis (PCA): Create uncorrelated components from correlated variables
  • Factor Analysis: Identify latent factors underlying variables

Classification:

  • Discriminant Analysis: Classify observations into groups
  • Logistic Regression: Predict binary outcome from predictors

Clustering:

  • K-Means: Group observations into clusters
  • Hierarchical Clustering: Dendrograms of similarity

Prediction:

  • Multiple Regression: Predict continuous outcome
  • Path Analysis: Model relationships between variables

When Multivariate Analysis Needed

  • Many variables simultaneously
  • Variables intercorrelated
  • Want to understand multivariate relationships
  • Need to reduce dimensionality

Section 6: Causal Inference

Challenge: Causation from Observational Data

Randomized Controlled Trial (Gold Standard):

  • Random assignment → Comparable groups
  • Can infer causation
  • Usually expensive and time-consuming

Observational Data:

  • No random assignment → Groups may differ
  • Many alternative explanations for associations
  • Causal inference more challenging but possible

Causal Diagrams (DAGs)

Directed Acyclic Graphs (DAGs) map causal relationships:

Treatment → Outcome
    ↑          ↓
 Confounder

Confounder: Variable affecting both treatment and outcome

Confounding Control Methods

1. Stratification:

  • Analyze treatment-outcome separately within confounder groups
  • Compare within-group effects

2. Regression Adjustment:

  • Include confounder as predictor
  • Estimates treatment effect “controlling for” confounder

3. Matching:

  • Match treated and control on confounder values
  • Compare matched pairs

4. Propensity Score Methods:

  • Model probability of treatment given covariates
  • Use to create balanced groups
  • Common in epidemiology

Causal Identification Assumptions

1. Unconfoundedness (Ignorability):

  • No unmeasured confounders
  • Strongest, hardest to verify
  • Assumption, not testable

2. Overlap:

  • Both treated and untreated across covariate ranges
  • Check with propensity score overlap

3. Consistency (SUTVA):

  • One version of treatment
  • No interference between units

Cannot infer causation without meeting assumptions

Section 7: Special Topics

Time Series Analysis

Analysis of data collected over time

Challenges:

  • Observations not independent (autocorrelation)
  • Trends, seasonal patterns
  • Future unpredictable

Methods:

  • ARIMA modeling
  • Exponential smoothing
  • Decomposition (trend, seasonal, residual)
  • Forecasting

Survival Analysis

Time to event data (failure time, survival time)

Challenges:

  • Censored data (event didn’t occur during study)
  • Non-normal distributions
  • Hazard rates vary over time

Methods:

  • Kaplan-Meier curves
  • Cox proportional hazards regression
  • Log-rank tests

Quality Control and Process Monitoring

Statistical process control for manufacturing

Methods:

  • Control charts (X-bar, R, p-charts)
  • Sequential sampling
  • Process capability analysis

Experimental Design Advanced Topics

Blocking, Factorials, Response Surface Methods:

  • Block designs for nuisance variables
  • Factorial experiments (multiple factors)
  • Response surface methods (optimization)

Section 8: Regularization and Model Selection

The Problem: Overfitting

As model complexity increases:

  • Training error decreases
  • Test error first decreases, then increases (overfitting)

Regularization Methods

Ridge Regression (L2 Penalty):

  • Shrinks coefficients toward zero
  • Reduces variance, adds slight bias
  • Good when many correlated predictors

Lasso Regression (L1 Penalty):

  • Shrinks some coefficients to exactly zero
  • Variable selection built-in
  • Good for feature selection

Elastic Net:

  • Combines Ridge and Lasso
  • Flexible penalty structure

Model Selection

Cross-Validation:

  • Divide data into training and test sets
  • Train model on training, evaluate on test
  • Repeat multiple times
  • Choose model with best test error

AIC/BIC:

  • Information criteria balancing fit and complexity
  • Lower values better
  • Useful for comparing non-nested models

Section 9: Reproducibility and Best Practices

Reproducible Research

Good practices:

  • ✅ Document all decisions and analyses
  • ✅ Pre-register studies before data collection
  • ✅ Share data and code
  • ✅ Report all analyses (not just significant ones)
  • ✅ Use version control (Git)
  • ✅ Write dynamic reports (R Markdown, Jupyter)

Reproducibility Crisis

Problem: Many results don’t replicate

Causes:

  • Publication bias (only significant results published)
  • P-hacking (multiple comparisons without adjustment)
  • Underpowered studies
  • Poor documentation

Solutions:

  • Pre-registration
  • Larger sample sizes
  • Open science practices
  • Replication studies

Section 10: Advanced Concepts Summary

Topic Use When Key Idea
Bootstrap No formula available Resample to estimate uncertainty
Bayesian Incorporating prior knowledge Update beliefs with data
Non-parametric Violate assumptions Rank-based alternatives
Multivariate Multiple related variables Analyze simultaneously
Causal Observational data only Control confounding carefully
Time Series Sequential time data Account for autocorrelation
Survival Time-to-event data Handle censoring
Regularization Many predictors Shrink coefficients

Best Practices for Advanced Analysis

  1. ✅ Understand assumptions - Know what could go wrong
  2. ✅ Visualize data first - See the patterns
  3. ✅ Start simple - Complex model only if needed
  4. ✅ Validate results - Cross-validation or hold-out test set
  5. ✅ Check sensitivity - Change assumptions slightly, do conclusions hold?
  6. ✅ Document thoroughly - Future you will need to understand
  7. ✅ Get expert help - Advanced methods need statistical expertise

Common Mistakes

  1. ❌ Using advanced method when simple method sufficient
  2. ❌ Ignoring assumptions for advanced methods
  3. ❌ Deleting missing data without investigation
  4. ❌ Claiming causation from observational data
  5. ❌ Not validating models on held-out data
  6. ❌ Selecting method after seeing data
  7. ❌ Insufficiently documenting analyses

Learning Path

From Intermediate to Advanced:

  1. Master fundamentals - Descriptive, inferential, correlation, regression
  2. Learn diagnostics - Check assumptions, identify violations
  3. Study alternatives - Non-parametric, robust methods
  4. Explore specialties - Bayesian, time series, causal, multivariate
  5. Practice reproducibility - Write better code and documentation
  6. Apply to real data - Messy data teaches most

Summary

Advanced statistics extends your toolkit for:

  • Complex data structures
  • Violated assumptions
  • Causal inference
  • Prediction and forecasting
  • Multivariate relationships

Choose methods based on data characteristics and research questions, not just statistical sophistication.