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Parametric Tests

Parametric statistical tests that assume specific distributional properties of the data. These tests are generally more powerful than non-parametric alternatives when their assumptions are met.

Validation: All tests are validated against R implementations in the anofox-statistics crate.

ttest_ind

Independent samples t-test for comparing means of two groups.

Welch's t-test (default, equal_var=False) does not assume equal variances and is generally recommended. Student's t-test (equal_var=True) assumes equal variances and has slightly more power when the assumption holds.

ps.ttest_ind(
    x: Union[pl.Expr, str],
    y: Union[pl.Expr, str],
    alternative: str = "two-sided",  # "two-sided", "less", "greater"
    equal_var: bool = False,         # False = Welch's, True = Student's
) -> pl.Expr

Returns: Struct{statistic: Float64, p_value: Float64}

Assumptions: - Independent observations within and between groups - Data approximately normally distributed (robust to violations with n > 30) - For Student's t: equal variances (homoscedasticity)

Interpretation: - p_value < alpha: Reject null hypothesis of equal means - alternative="less": Test if mean(x) < mean(y) - alternative="greater": Test if mean(x) > mean(y)

Example: 

# Compare treatment vs control group means
df.select(ps.ttest_ind("treatment", "control", alternative="two-sided"))

# Per-group comparison
df.group_by("experiment").agg(
    ps.ttest_ind("treatment", "control").alias("test")
)

ttest_paired

Paired samples t-test for comparing means of two related measurements.

Tests whether the mean difference between paired observations differs from zero. More powerful than independent t-test when observations are naturally paired (e.g., before/after measurements on the same subjects).

ps.ttest_paired(
    x: Union[pl.Expr, str],
    y: Union[pl.Expr, str],
    alternative: str = "two-sided",  # "two-sided", "less", "greater"
) -> pl.Expr

Returns: Struct{statistic: Float64, p_value: Float64}

Assumptions: - Paired observations (same subjects measured twice) - Differences approximately normally distributed - Independence between pairs

When to use: - Before/after measurements on the same subjects - Matched pairs design - Repeated measures with two time points

Example: 

# Test if treatment changed scores
df.select(ps.ttest_paired("before", "after"))

# Directional test: did scores increase?
df.select(ps.ttest_paired("before", "after", alternative="less"))

brown_forsythe

Brown-Forsythe test for equality of variances (homoscedasticity).

A robust alternative to Levene's test that uses deviations from group medians instead of means. Recommended for checking variance homogeneity before using tests that assume equal variances.

ps.brown_forsythe(
    x: Union[pl.Expr, str],
    y: Union[pl.Expr, str],
) -> pl.Expr

Returns: Struct{statistic: Float64, p_value: Float64}

Interpretation: - p_value > 0.05: Variances can be considered equal - p_value < 0.05: Evidence of unequal variances (heteroscedasticity)

When to use: - Before ANOVA or Student's t-test to verify equal variance assumption - When comparing spread of two distributions - More robust than Levene's test for non-normal data

Example: 

# Check if variances are equal before using equal_var=True
variance_test = df.select(ps.brown_forsythe("group1", "group2"))

yuen_test

Yuen's test for trimmed means—a robust alternative to the independent t-test.

Compares trimmed means of two groups, reducing the influence of outliers. The test trims a proportion of observations from each tail before computing means and standard errors.

ps.yuen_test(
    x: Union[pl.Expr, str],
    y: Union[pl.Expr, str],
    trim: float = 0.2,  # Proportion to trim from each tail
) -> pl.Expr

Returns: Struct{statistic: Float64, p_value: Float64}

Parameters: - trim=0.2 (default): Removes 20% from each tail, using the middle 60% of data - trim=0.1: Removes 10% from each tail, using the middle 80% - trim=0.0: Equivalent to Student's t-test

When to use: - Data contains outliers or heavy tails - Distribution is skewed - You want robust inference without removing outliers manually

Note: The trimmed mean with 20% trimming is approximately 95% as efficient as the mean for normal data, but much more robust to contamination.

Example: 

# Robust comparison with default 20% trimming
df.select(ps.yuen_test("treatment", "control"))

# Less trimming for cleaner data
df.select(ps.yuen_test("treatment", "control", trim=0.1))

Choosing a Test

Situation Recommended Test
Two independent groups, normal data ttest_ind (Welch's)
Two independent groups, outliers present yuen_test
Paired/repeated measurements ttest_paired
Check variance equality brown_forsythe
Non-normal data Consider Non-Parametric Tests

See Also