Nonparametric Tests

AnoFox provides 7 nonparametric test functions that make no assumptions about the underlying data distribution. The suite includes rank-based group comparisons (Mann-Whitney U for 2 groups, Kruskal-Wallis for 3+ groups), paired comparisons (Wilcoxon signed-rank), robust alternatives (Brunner-Munzel without equal variance assumption), exact testing (permutation t-test with configurable iterations up to 10,000), and full distribution comparison (energy distance, Maximum Mean Discrepancy). These tests are the correct choice when normality tests reject or when working with ordinal data, heavy-tailed distributions, or small sample sizes.

Nonparametric tests are distribution-free statistical methods that make no assumptions about the shape of the underlying data distribution. They work with ranks or permutations instead of raw values, making them robust to outliers, skewness, and non-normal data. Use these tests when normality tests reject or when working with ordinal data.

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Group Comparison Tests

Mann-Whitney U Test

Test for stochastic superiority between two groups.

Parameters

Parameter	Type	Required	Default	Description
value	DOUBLE	Yes	-	Measurement values
group_id	INTEGER	Yes	-	Group identifier (0 or 1)
options	MAP	No	-	Configuration options

Options MAP:

Option	Type	Default	Description
alternative	VARCHAR	two_sided	two_sided, less, greater
correction	BOOLEAN	true	Apply continuity correction
confidence_level	DOUBLE	0.95	Confidence level

Output

Field	Type	Description
u_statistic	DOUBLE	U statistic
p_value	DOUBLE	p-value
z_score	DOUBLE	z approximation
effect_size	DOUBLE	Rank-biserial correlation

Example

SELECT anofox_stats_mann_whitney_u_agg(
    value,
    group_id
) as result
FROM data;

Kruskal-Wallis Test

Test whether samples originate from the same distribution across 3+ groups.

Parameters

Parameter	Type	Required	Default	Description
value	DOUBLE	Yes	-	Measurement values
group_id	INTEGER	Yes	-	Group identifier

Output

Field	Type	Description
h_statistic	DOUBLE	H statistic
p_value	DOUBLE	p-value
df	BIGINT	Degrees of freedom
epsilon_squared	DOUBLE	Effect size

Example

SELECT anofox_stats_kruskal_wallis_agg(
    value,
    group_id
) as result
FROM data;

Wilcoxon Signed-Rank Test

Paired samples comparison.

Parameters

Parameter	Type	Required	Default	Description
value1	DOUBLE	Yes	-	First measurement
value2	DOUBLE	Yes	-	Paired measurement
options	MAP	No	-	Configuration options

Example

SELECT anofox_stats_wilcoxon_signed_rank_agg(
    value1,
    value2
) as result
FROM paired_data;

Brunner-Munzel Test

Robust comparison without equal variance assumption.

Parameters

Parameter	Type	Required	Default	Description
value	DOUBLE	Yes	-	Measurement values
group_id	INTEGER	Yes	-	Group identifier
options	MAP	No	-	Configuration options

Example

SELECT anofox_stats_brunner_munzel_agg(
    value,
    group_id
) as result
FROM data;

Permutation t-Test

Exact test using permutation distribution.

Parameters

Parameter	Type	Required	Default	Description
value	DOUBLE	Yes	-	Measurement values
group_id	INTEGER	Yes	-	Group identifier
options	MAP	No	-	n_permutations (default: 10000)

Example

SELECT anofox_stats_permutation_t_test_agg(
    value,
    group_id,
    MAP {'n_permutations': '10000'}
) as result
FROM data;

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Distribution Comparison

Compare entire distributions.

Energy Distance

Parameters

Parameter	Type	Required	Default	Description
value	DOUBLE	Yes	-	Values
group_id	INTEGER	Yes	-	Group identifier

Example

SELECT anofox_stats_energy_distance_agg(
    value,
    group_id
) as result
FROM data;

Maximum Mean Discrepancy (MMD)

Parameters

Parameter	Type	Required	Default	Description
value	DOUBLE	Yes	-	Values
group_id	INTEGER	Yes	-	Group identifier
options	MAP	No	-	Kernel settings

Example

SELECT anofox_stats_mmd_agg(
    value,
    group_id
) as result
FROM data;

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