Non-parametric Methods

Non-parametric methods are a class of statistical procedures that do not rely on assumptions about the probability distributions of the populations from which the data are drawn. Often referred to as distribution-free methods, they stand in contrast to parametric approaches which assume data follows a specific distribution (e.g., the normal distribution). Instead of operating on parameters like the mean and standard deviation, non-parametric techniques typically use ranks or medians, making them particularly robust and suitable for analyzing ordinal data, skewed data, or data with outliers where the assumptions for parametric tests are not met.

1. Introduction to Non-parametric Statistics
   1. Defining Non-parametric Methods
      1. Distribution-free Properties
      2. Robustness to Outliers
      3. Role of Ranks and Order Statistics
      4. Applicability to Various Data Types
   2. Parametric vs. Non-parametric Approaches
      1. Core Assumptions of Parametric Tests
         1. Normality Assumption
         2. Homogeneity of Variance
         3. Independence of Observations
         4. Interval or Ratio Scale Requirements
      2. Assumptions of Non-parametric Tests
         1. Minimal Distributional Assumptions
         2. Ordinal Scale Requirements
         3. Independence Considerations
      3. Consequences of Assumption Violations
         1. Type I Error Rate Inflation
         2. Loss of Statistical Power
         3. Invalid Confidence Intervals
      4. Decision Criteria for Method Selection
         1. Sample Size Considerations
         2. Data Distribution Assessment
         3. Measurement Scale Evaluation
   3. Advantages and Disadvantages
      1. Advantages of Non-parametric Methods
         1. Wider Applicability
         2. Robustness to Outliers
         3. Effectiveness with Small Samples
         4. Flexibility with Skewed Data
      2. Disadvantages of Non-parametric Methods
         1. Lower Statistical Power
         2. Less Precise Estimates
         3. Limited Interpretability
         4. Computational Complexity
   4. Scales of Measurement
      1. Nominal Scale
         1. Definition and Properties
         2. Examples and Applications
         3. Appropriate Non-parametric Tests
      2. Ordinal Scale
         1. Definition and Properties
         2. Examples and Applications
         3. Appropriate Non-parametric Tests
      3. Interval Scale
         1. Definition and Properties
         2. Use in Non-parametric Context
      4. Ratio Scale
         1. Definition and Properties
         2. Use in Non-parametric Context