Categorical Tests

AnoFox provides 9 functions for categorical data analysis: 5 independence and goodness-of-fit tests (chi-square independence, chi-square GOF, G-test, Fisher's exact for 2x2 tables, McNemar's for paired before/after data) and 4 effect size measures (Cramer's V, Phi coefficient, contingency coefficient, Cohen's Kappa for inter-rater agreement). Fisher's exact test is the correct choice for small sample 2x2 tables where chi-square approximations break down. Cohen's Kappa returns values on a -1 to 1 scale where 0.8+ indicates almost perfect agreement between raters.

Categorical tests are statistical methods for analyzing data where variables take on discrete categories rather than continuous values. These tests determine whether observed frequencies differ significantly from expected frequencies (goodness-of-fit), whether two categorical variables are related (independence), or whether raters agree on their classifications (agreement).

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Independence Tests

Chi-Square Test of Independence

Tests whether two categorical variables are associated.

Parameters

Parameter	Type	Required	Default	Description
row_var	INTEGER	Yes	-	Row category
col_var	INTEGER	Yes	-	Column category
options	MAP	No	-	yates_correction (default: false)

Output

Field	Type	Description
chi2	DOUBLE	Chi-square statistic
p_value	DOUBLE	p-value
df	BIGINT	Degrees of freedom
cramers_v	DOUBLE	Effect size

Example

SELECT anofox_stats_chisq_test_agg(
    row_var,
    col_var
) as result
FROM contingency_data;

Chi-Square Goodness of Fit

Parameters

Parameter	Type	Required	Default	Description
observed	INTEGER	Yes	-	Observed counts
expected	DOUBLE	Yes	-	Expected counts

Example

SELECT anofox_stats_chisq_gof_agg(
    observed,
    expected
) as result
FROM frequency_data;

G-Test (Log-Likelihood Ratio)

Alternative to chi-square, better for small samples.

Parameters

Parameter	Type	Required	Default	Description
row_var	INTEGER	Yes	-	Row category
col_var	INTEGER	Yes	-	Column category

Example

SELECT anofox_stats_g_test_agg(
    row_var,
    col_var
) as result
FROM data;

Fisher's Exact Test

Exact test for 2x2 tables.

Parameters

Parameter	Type	Required	Default	Description
row_var	INTEGER	Yes	-	Row category
col_var	INTEGER	Yes	-	Column category
options	MAP	No	-	alternative setting

Example

SELECT anofox_stats_fisher_exact_agg(
    row_var,
    col_var
) as result
FROM data;

McNemar's Test

Paired categorical data (before/after).

Parameters

Parameter	Type	Required	Default	Description
var1	INTEGER	Yes	-	Before measurement
var2	INTEGER	Yes	-	After measurement
options	MAP	No	-	Configuration options

Example

SELECT anofox_stats_mcnemar_agg(
    var1,
    var2
) as result
FROM paired_categorical;

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Effect Size Measures

Quantify the magnitude of associations.

Cramér's V

Effect size for chi-square test.

SELECT anofox_stats_cramers_v_agg(row_var, col_var) as v
FROM data;

Interpretation:

V value	Effect Size
0.0 - 0.1	Negligible
0.1 - 0.3	Small
0.3 - 0.5	Medium
> 0.5	Large

Phi Coefficient

Effect size for 2x2 tables.

SELECT anofox_stats_phi_coefficient_agg(row_var, col_var) as phi
FROM data;

Contingency Coefficient

Normalized association measure.

SELECT anofox_stats_contingency_coef_agg(row_var, col_var) as c
FROM data;

Cohen's Kappa

Inter-rater agreement measure.

Parameters

Parameter	Type	Required	Default	Description
rater1	INTEGER	Yes	-	First rater's classification
rater2	INTEGER	Yes	-	Second rater's classification

Output

Field	Type	Description
kappa	DOUBLE	Cohen's kappa (-1 to 1)
p_value	DOUBLE	p-value
se	DOUBLE	Standard error
ci_lower	DOUBLE	CI lower
ci_upper	DOUBLE	CI upper

Example

SELECT anofox_stats_cohen_kappa_agg(
    rater1,
    rater2
) as result
FROM rating_data;

Interpretation:

Kappa	Agreement
< 0	Less than chance
0.0 - 0.2	Slight
0.2 - 0.4	Fair
0.4 - 0.6	Moderate
0.6 - 0.8	Substantial
0.8 - 1.0	Almost perfect

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Choosing a Test

Scenario	Recommended
2+ categories, large samples	Chi-Square
2x2 table, small samples	Fisher's Exact
Before/after paired data	McNemar's
Prefer likelihood-based	G-Test
Quantify association	Cramér's V
Rater agreement	Cohen's Kappa

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