Correlation Tests

AnoFox provides 5 correlation methods spanning linear (Pearson), rank-based (Spearman, Kendall), nonlinear (distance correlation), and reliability (Intraclass Correlation Coefficient with 3 model types). Distance correlation is uniquely powerful: it equals zero if and only if the two variables are statistically independent -- unlike Pearson, which only detects linear relationships. ICC supports one-way, two-way random, and two-way mixed models for inter-rater reliability assessment. All correlation functions return p-values, confidence intervals, and sample sizes alongside the correlation coefficient.

Correlation is a statistical measure that quantifies the strength and direction of the relationship between two variables. A correlation of +1 indicates a perfect positive relationship, -1 a perfect negative relationship, and 0 no relationship. Different correlation methods capture different types of relationships: linear (Pearson), monotonic (Spearman, Kendall), or general dependence (distance correlation).

---

Linear Correlation

Pearson Correlation

Linear correlation with significance test.

Parameters

Parameter	Type	Required	Default	Description
x	DOUBLE	Yes	-	First variable
y	DOUBLE	Yes	-	Second variable
options	MAP	No	-	Configuration options

Output

Field	Type	Description
r	DOUBLE	Pearson correlation (-1 to 1)
p_value	DOUBLE	p-value
t_statistic	DOUBLE	t-statistic
ci_lower	DOUBLE	CI lower bound
ci_upper	DOUBLE	CI upper bound
n	BIGINT	Sample size

Example

SELECT anofox_stats_pearson_agg(x, y) as result
FROM data;

Interpretation:

r value	Strength
0.0 - 0.3	Weak
0.3 - 0.7	Moderate
0.7 - 1.0	Strong

---

Rank Correlations

Spearman Rank Correlation

Monotonic relationship (robust to outliers).

Parameters

Parameter	Type	Required	Default	Description
x	DOUBLE	Yes	-	First variable
y	DOUBLE	Yes	-	Second variable
options	MAP	No	-	Configuration options

Output

Field	Type	Description
rho	DOUBLE	Spearman's rho
p_value	DOUBLE	p-value
n	BIGINT	Sample size

Example

SELECT anofox_stats_spearman_agg(x, y) as result
FROM data;

Kendall's Tau

Rank correlation (handles ties well).

Parameters

Parameter	Type	Required	Default	Description
x	DOUBLE	Yes	-	First variable
y	DOUBLE	Yes	-	Second variable
options	MAP	No	-	Configuration options

Output

Field	Type	Description
tau	DOUBLE	Kendall's tau
p_value	DOUBLE	p-value
n	BIGINT	Sample size

Example

SELECT anofox_stats_kendall_agg(x, y) as result
FROM data;

---

Nonlinear Correlation

Distance Correlation

Detects nonlinear relationships using the distance correlation measure.

Parameters

Parameter	Type	Required	Default	Description
x	DOUBLE	Yes	-	First variable
y	DOUBLE	Yes	-	Second variable

Output

Field	Type	Description
dcor	DOUBLE	Distance correlation (0 to 1)
dcov	DOUBLE	Distance covariance
n	BIGINT	Sample size

Example

SELECT anofox_stats_distance_cor_agg(x, y) as result
FROM data;

Key property: Distance correlation = 0 if and only if X and Y are independent (unlike Pearson which only detects linear relationships).

---

Reliability Measures

Intraclass Correlation (ICC)

Reliability/agreement between raters.

Parameters

Parameter	Type	Required	Default	Description
value	DOUBLE	Yes	-	Rating value
rater_id	INTEGER	Yes	-	Rater identifier
subject_id	INTEGER	Yes	-	Subject identifier
options	MAP	No	-	Model configuration

Options MAP:

Option	Type	Default	Description
model	VARCHAR	two_way_random	one_way, two_way_random, two_way_mixed

Example

SELECT anofox_stats_icc_agg(
    value,
    rater_id,
    subject_id,
    MAP {'model': 'two_way_random'}
) as result
FROM rating_data;

Interpretation:

ICC value	Reliability
< 0.5	Poor
0.5 - 0.75	Moderate
0.75 - 0.9	Good
> 0.9	Excellent

---

Choosing a Correlation Method

Scenario	Recommended
Linear relationship, normal data	Pearson
Outliers present	Spearman
Ordinal data	Spearman or Kendall
Many ties	Kendall
Nonlinear relationship	Distance Correlation
Rater agreement	ICC

---