Non-Parametric Tests Overview

Non-Parametric Tests: Overview

• Parametric tests involve estimating population parameters, making assumptions about the shape of the data and assumptions about the scaling of the variables.
• Non-parametric tests do not make assumptions about the shape of the data or the scaling of the variables.

Assumptions about the Shape of the Data

• Parametric tests assume that the data is normally distributed.
• Non-parametric tests do not make a priori assumptions about the specific shape of the distribution.

Assumptions about the Scaling of the Variables

• Parametric tests assume that the outcome variable has been measured at interval or ratio level.
• Non-parametric tests do not make assumptions about the scaling of the variables.

Advantages of Non-Parametric Tests

• They do not require assumptions of normality and homogeneity of variances.
• Ideal for analysing data from small samples.
• Generally easier to calculate.
• Use of ranks reduces the effect of extreme outliers.

Ranking

• Ranking involves the ordering of a set of scores from the smallest to the largest.
• Provides a standard distribution of scores with standard characteristics.

Spearman's Rho

• Spearman's rho (rS) is calculated using Pearson's r formula, but with ranked data.
• Handy when:
  • The data naturally falls in ranks.
  • There are extreme scores in the sample.
  • There is a monotonic relationship between the variables.
• Can be used when you have two continuous variables, but one (or both) is badly skewed due to extreme scores.

Non-Parametric Correlation

• The goal of any non-parametric test is to establish overall differences between two (or possibly more) distributions, not to identify the differences between any particular parameters.
• The null hypothesis is more general, stating that the samples come from identical populations, not just populations with the same mean.

Summary

• Parametric tests involve estimating population parameters, making assumptions about the shape of the data and assumptions about the scaling of the variables.
• Non-parametric tests do not make these assumptions.
• Ranking is one approach used by non-parametric tests.
• Spearman's rho can be used to calculate correlations between variables with natural ranks, extreme scores, or monotonic relationships.
• Non-parametric tests are generally less powerful than parametric tests.

Parametric Tests

• Involve estimation of population parameters
• Make assumptions about the distribution of the population (e.g. normality)
• Make assumptions about the scaling of the variables (e.g. interval or ratio level)

Non-Parametric Tests

• Do not estimate population parameters
• Do not make assumptions about the distribution of the population (distribution-free)
• Do not make assumptions about the scaling of the variables
• Ideal for analyzing data from small samples or severely skewed data

Characteristics of Non-Parametric Tests

• Use ranks to reduce the effect of extreme values
• Generally easy to calculate and require less computation
• Can be used for categorical variables or data with natural ranks

Spearman's Rank Correlation

• Calculated using Pearson's correlation formula, but with ranked data
• Can be used when data has extreme scores or non-linear relationships
• Can be used for continuous variables with skewed data
• Has fewer assumptions than Pearson's correlation

Ranking

• Involves ordering a set of scores from smallest to largest
• Can be used to reduce the effect of extreme values
• Can create a linear relationship between variables with non-linear relationships

Key Differences between Parametric and Non-Parametric Tests

• Parametric tests are more powerful, but require more assumptions
• Non-Parametric tests are less powerful, but require fewer assumptions
• Parametric tests are typically preferred, but Non-Parametric tests are useful for certain types of data or situations

A Priori Assumptions

• A priori assumptions are statements presumed to be true without empirical evidence or further proof.
• These assumptions are based on theoretical deduction rather than observation of objective facts.
• They are made without assessing the facts, implying a lack of empirical support.
• A priori assumptions rely on theoretical reasoning rather than objective data.
• They are often characterized by a lack of detailed observation of objective elements.

Non-Parametric Test: Pearson's Rho

• Measures the correlation between two continuous variables
• Also known as Spearman's rank correlation coefficient (ρ)
• Does not require normality or equal intervals of the data

Calculation

• Rank data for each variable separately
• Calculate the difference between ranks (d) for each pair of observations
• Calculate the sum of the squared differences (Σd^2)
• Calculate the coefficient of determination (ρ) using the formula: ρ = 1 - (6 * Σd^2) / (n * (n^2 - 1))

Interpretation

• Values range from -1 (perfect negative correlation) to 1 (perfect positive correlation)
• A value of 0 indicates no correlation
• Coefficient is sensitive to outliers and non-linear relationships

Advantages

• Does not require normality or equal intervals of the data
• Can be used with ordinal data or continuous data with non-normal distributions
• Robust to outliers and non-linear relationships

Disadvantages

• Less powerful than parametric tests (e.g. Pearson's r) when data is normally distributed
• May not be suitable for small sample sizes

When to Use Pearson's Rho

• When data is ordinal or continuous with non-normal distributions
• When outliers are present in the data
• When the relationship between variables is non-linear