5.4 Parametric and non-parametric tests

Statistical inference and hypothesis testing are crucial in data science. This section dives into parametric and non-parametric tests, two key approaches for analyzing data and drawing conclusions.

Parametric tests assume specific distributions, while non-parametric tests are more flexible. Understanding their differences, assumptions, and applications helps you choose the right test for your data and research questions.

Parametric vs Non-parametric Tests

Key Differences and Assumptions

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• Parametric tests assume data follows specific probability distribution (typically normal distribution) while non-parametric tests make no such assumptions
• Parametric tests require interval or ratio level data, non-parametric tests work with ordinal or nominal data
• Parametric test assumptions include normality, homogeneity of variance, and independence of observations
• Non-parametric tests have fewer assumptions and offer more flexibility
• Parametric tests generally have more statistical power when assumptions are met, increasing likelihood of detecting true effects
• Non-parametric tests often used for small sample sizes or when data violates parametric assumptions

Considerations for Test Selection

• Nature of data impacts test choice (distribution, measurement level, sample size)
• Research question guides selection between comparing groups, examining relationships, or predicting outcomes
• Parametric tests provide more power when assumptions are met
• Non-parametric alternatives offer robustness when assumptions are violated
• Evaluate tradeoffs between power and flexibility based on specific study characteristics

Applying Parametric Tests

T-tests and ANOVA

• T-tests compare means between two groups
  • Independent samples t-test for separate groups
  • Paired samples t-test for related measurements
• One-way ANOVA compares means among three or more independent groups
• Two-way ANOVA examines effects of two independent variables on dependent variable
• F-test in ANOVA assesses overall model significance by comparing explained to unexplained variance
• Post-hoc tests (Tukey's HSD, Bonferroni correction) perform pairwise comparisons while controlling for multiple comparisons

Regression Analysis

• Explores relationship between dependent variable and independent variable(s)
• Simple linear regression uses one predictor
• Multiple regression incorporates multiple predictors
• Assumptions include normality of residuals, homoscedasticity, independence of observations
• Effect sizes (Cohen's d for t-tests, R-squared for regression) indicate magnitude of observed effects

Non-parametric Tests: Principles & Applications

Rank-based Tests

• Wilcoxon rank-sum test (Mann-Whitney U test) compares two independent groups, alternative to independent t-test
• Wilcoxon signed-rank test compares two related samples, alternative to paired t-test
• Kruskal-Wallis test compares three or more independent groups, alternative to one-way ANOVA
• These tests analyze ranks rather than raw values, increasing robustness to outliers and non-normal distributions

Other Non-parametric Approaches

• Chi-square test of independence examines relationship between two categorical variables
• Spearman's rank correlation coefficient measures monotonicity between two variables when Pearson's assumptions not met
• Non-parametric tests generally have lower statistical power than parametric counterparts
• More appropriate for ordinal data or when parametric assumptions violated

Selecting the Right Statistical Test

Variable and Data Characteristics

• Identify variable types (categorical, ordinal, interval, ratio) and number of groups/variables compared
• Determine if data meets parametric assumptions using diagnostic tests and visualizations
• Consider sample size impact on assumption violations and test selection
• Assess whether data paired or unpaired to choose between independent and dependent samples tests

Research Question and Study Design

• Evaluate research goals (group comparison, relationship examination, outcome prediction) to guide test selection
• Consider test sensitivity to specific alternative hypotheses of interest
• Analyze power requirements and available sample size to inform parametric vs non-parametric choice
• Consult statistical literature or statisticians when uncertain about optimal test selection