ノンパラメトリック統計の概要

Parametric tests require normally distributed populations with specific parameters like mean and standard deviation, while nonparametric tests, also called distribution-free tests, do not depend on any parameters or specific distributions. Nonparametric tests can be applied to categorical data and populations with unknown distributions, making them more flexible for diverse data types.

Use nonparametric tests when dealing with small sample sizes, skewed data, ordinal data like rankings, or nominal data like gender and eye color. They are also preferable when population assumptions cannot be met, such as when data lacks normal distribution or homogeneity of variance. Nonparametric methods are more robust to outliers and extreme values.

Nonparametric tests reduce quantitative data to qualitative data, such as signs or ranks, thereby losing information about magnitude. For example, recording ocean level changes as positive or negative signs instead of millimeters reduces detail. This limitation requires larger sample sizes or greater differences between test statistics and critical values to achieve the same statistical power as parametric tests.

Nonparametric tests can analyze ordinal data such as rankings and ratings, as well as nominal data like categories. They rely on rank-ordering or contingency tables rather than estimating population parameters. This broader applicability makes nonparametric methods suitable for situations where parametric methods would be unsuitable due to data type constraints.

An efficiency rating of 0.63 indicates that a nonparametric test requires 100 observations to achieve the same results as 63 observations from the corresponding parametric test, when all other factors are equal and strict parametric conditions are met. This demonstrates the reduced statistical power of nonparametric methods compared to their parametric counterparts.

Common nonparametric tests include the Wilcoxon rank-sum test, Kruskal-Wallis test, and Chi-square test. These tests analyze data without requiring specific distributional assumptions and are often easier to interpret since they rely on rank-ordering or contingency tables. They are widely used across social sciences, biology, economics, and medicine.

Nonparametric tests are more robust to outliers because they rely on rank-ordering and signs rather than actual data values. By converting quantitative measurements into ranks or categorical indicators, extreme values have less influence on the test outcome. This reduces the impact of outliers that might otherwise skew results in parametric analysis.