Parametric Tests

Tests that assume normal distributions and work with means.

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Normality Tests

Test whether data follows a normal distribution.

Shapiro-Wilk Test

Most powerful test for small to medium samples (n < 5000).

Parameters

Parameter	Type	Required	Default	Description
value	DOUBLE	Yes	-	Values to test

Output

Field	Type	Description
statistic	DOUBLE	W statistic (closer to 1 = more normal)
p_value	DOUBLE	p-value
is_normal	BOOLEAN	p > 0.05

Example

SELECT anofox_stats_shapiro_wilk_agg(value) as result
FROM data;

Jarque-Bera Test

Based on skewness and kurtosis. Good for large samples.

Parameters

Parameter	Type	Required	Default	Description
value	DOUBLE	Yes	-	Values to test

Output

Field	Type	Description
statistic	DOUBLE	JB statistic
p_value	DOUBLE	p-value
skewness	DOUBLE	Sample skewness
kurtosis	DOUBLE	Sample excess kurtosis

Example

SELECT anofox_stats_jarque_bera_agg(value) as result
FROM data;

D'Agostino K² Test

Omnibus test combining skewness and kurtosis tests.

Parameters

Parameter	Type	Required	Default	Description
value	DOUBLE	Yes	-	Values to test

Output

Field	Type	Description
statistic	DOUBLE	K² statistic
p_value	DOUBLE	p-value
z_skewness	DOUBLE	z-score for skewness
z_kurtosis	DOUBLE	z-score for kurtosis

Example

SELECT anofox_stats_dagostino_k2_agg(value) as result
FROM data;

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Group Comparison Tests

Student's t-Test

Compare means of 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
kind	VARCHAR	welch	welch or student (equal variances)
confidence_level	DOUBLE	0.95	Confidence level for CI
mu	DOUBLE	0.0	Hypothesized difference

Output

Field	Type	Description
statistic	DOUBLE	t-statistic
p_value	DOUBLE	p-value
df	DOUBLE	Degrees of freedom
mean_diff	DOUBLE	Difference in means
ci_lower	DOUBLE	Confidence interval lower
ci_upper	DOUBLE	Confidence interval upper
cohens_d	DOUBLE	Effect size

Example

SELECT
    (result).p_value,
    (result).mean_diff,
    (result).cohens_d as effect_size
FROM (
    SELECT anofox_stats_t_test_agg(
        conversion_rate,
        treatment_group
    ) as result
    FROM ab_test_data
);

One-Way ANOVA

Compare means across 3+ groups.

Parameters

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

Output

Field	Type	Description
f_statistic	DOUBLE	F-statistic
p_value	DOUBLE	p-value
df_between	BIGINT	Between-groups df
df_within	BIGINT	Within-groups df
ss_between	DOUBLE	Sum of squares between
ss_within	DOUBLE	Sum of squares within
eta_squared	DOUBLE	Effect size

Example

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

Yuen's Trimmed Mean Test

Robust alternative to t-test with trimmed means.

Parameters

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

Options MAP:

Option	Type	Default	Description
trim	DOUBLE	0.2	Proportion to trim from each tail
alternative	VARCHAR	two_sided	two_sided, less, greater

Example

SELECT anofox_stats_yuen_agg(
    value,
    group_id,
    MAP {'trim': '0.2'}
) as result
FROM data;

Brown-Forsythe Test

Test equality of variances (more robust than Levene's).

Parameters

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

Example

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

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Proportion Tests

Tests for proportions and binomial data.

One-Sample Proportion Test

Parameters

Parameter	Type	Required	Default	Description
successes	INTEGER	Yes	-	Number of successes
trials	INTEGER	Yes	-	Number of trials
p0	DOUBLE	Yes	-	Null hypothesis proportion
options	MAP	No	-	Configuration options

Example

SELECT anofox_stats_prop_test_one_agg(
    successes,
    trials,
    0.5
) as result
FROM data;

Two-Sample Proportion Test

Parameters

Parameter	Type	Required	Default	Description
successes	INTEGER	Yes	-	Number of successes
trials	INTEGER	Yes	-	Number of trials
group_id	INTEGER	Yes	-	Group identifier (0 or 1)
options	MAP	No	-	Configuration options

Example

SELECT anofox_stats_prop_test_two_agg(
    successes,
    trials,
    group_id
) as result
FROM data;

Binomial Test

Exact test for proportions.

Parameters

Parameter	Type	Required	Default	Description
successes	INTEGER	Yes	-	Number of successes
trials	INTEGER	Yes	-	Number of trials
p0	DOUBLE	Yes	-	Null hypothesis proportion
options	MAP	No	-	Configuration options

Example

SELECT anofox_stats_binom_test_agg(
    successes,
    trials,
    0.5
) as result
FROM data;

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Equivalence Tests (TOST)

Two One-Sided Tests for equivalence.

TOST t-Test

Parameters

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

Example

SELECT anofox_stats_tost_t_test_agg(
    value,
    group_id,
    0.5
) as result
FROM data;

TOST Paired

Parameters

Parameter	Type	Required	Default	Description
value1	DOUBLE	Yes	-	First measurement
value2	DOUBLE	Yes	-	Second measurement
delta	DOUBLE	Yes	-	Equivalence margin
options	MAP	No	-	Configuration options

Example

SELECT anofox_stats_tost_paired_agg(
    value1,
    value2,
    0.5
) as result
FROM paired_data;

TOST Correlation

Parameters

Parameter	Type	Required	Default	Description
x	DOUBLE	Yes	-	First variable
y	DOUBLE	Yes	-	Second variable
rho0	DOUBLE	Yes	-	Null correlation
delta	DOUBLE	Yes	-	Equivalence margin

Example

SELECT anofox_stats_tost_correlation_agg(
    x,
    y,
    0.0,
    0.1
) as result
FROM data;

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