1. Advanced Probability Theory
Probability theory forms the foundation of statistical analysis. Beyond basic axioms, understanding advanced concepts helps in real-world applications:
- Independence vs. Dependence
- Two events are independent if P(A∩B) = P(A)P(B)
- Dependent events require conditional probability calculations
- Real examples include genetic inheritance and disease transmission
- Random Variables
- Discrete: countable outcomes (like coin flips)
- Continuous: infinite possible values (like height)
- Expected value and variance properties
2. Statistical Inference in Detail
Statistical inference allows us to draw conclusions about populations from sample data:
- Sampling Distribution Properties
- Shape depends on underlying population distribution
- Standard error decreases with larger sample sizes
- Used for constructing confidence intervals
- Confidence Intervals
- 95% CI means if we repeat sampling many times, 95% of intervals contain true parameter
- Wider intervals indicate more uncertainty
- Trade-off between confidence level and precision
3. Advanced Hypothesis Testing Concepts
Understanding the nuances of hypothesis testing is crucial for accurate statistical analysis:
- Effect Size Interpretation
- Small effect: difficult to detect, may require large samples
- Medium effect: visible to careful observer
- Large effect: obvious to casual observer
- Multiple Testing Problems
- Family-wise error rate increases with number of tests
- Bonferroni correction: conservative but simple
- False Discovery Rate: modern alternative for many tests
4. Regression Analysis Fundamentals
Regression analysis helps understand relationships between variables:
- Simple Linear Regression
- Assumes linear relationship between X and Y
- Residual analysis crucial for validation
- R² measures proportion of variance explained
- Multiple Regression Considerations
- Multicollinearity between predictors
- Interaction effects between variables
- Model selection and validation techniques
5. Common Statistical Pitfalls
Being aware of common mistakes helps ensure reliable analysis:
- Data Collection Issues
- Selection bias in sampling
- Non-response bias in surveys
- Measurement error impact
- Analysis Mistakes
- Overlooking assumptions of statistical tests
- Inappropriate use of parametric tests
- Over-interpretation of correlations
Remember to always consider the context of your data and the practical significance of your findings when conducting statistical analyses.