One-Sample Tests - bossanova

Classical Test	bossanova Equivalent	Data Transformation
One-sample t-test	`model("y ~ 1", df)`	None (raw data)
Wilcoxon signed-rank	`model("signed_rank(y) ~ 1", df)`	`sign(x) * rank(abs(x))`

import numpy as np
import polars as pl
from bossanova.model import model
from bossanova import load_dataset

np.random.seed(42)

# Load penguins dataset
penguins = load_dataset("penguins").drop_nulls()

One-Sample t-test¶

Classical:

t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}, \quad t \sim t(n - 1) \text{ under } H_0

(1)

As GLM:

y_i \sim \mathcal{N}(\mu, \sigma^2), \quad \mu = \beta_0

(2)

t = \frac{\hat{\beta}_0}{\text{SE}(\hat{\beta}_0)} \sim t(n-1) \text{ under } H_0: \beta_0 = 0

(3)

The classical $t$ and the GLM $t$ are identical — fitting an intercept-only Gaussian GLM with identity link recovers the one-sample t-test exactly.

scipy¶

from scipy.stats import ttest_1samp

scipy_ttest = ttest_1samp(penguins['bill_length_mm'].to_numpy(), 0)
scipy_ttest

TtestResult(statistic=np.float64(148.77670538724365), pvalue=np.float64(3.211394152836e-312), df=np.int64(341))

bossanova¶

m = model("bill_length_mm ~ 1", penguins).fit().infer()

m.params.select("statistic", "df", "p_value")

bn_t = float(m.params["statistic"][0])
bn_p = float(m.params["p_value"][0])
bn_df = float(m.params["df"][0])
assert np.isclose(bn_t, scipy_ttest.statistic, rtol=1e-3), f"t mismatch: {bn_t} vs {scipy_ttest.statistic}"
assert np.isclose(bn_p, scipy_ttest.pvalue, rtol=1e-3), f"p mismatch: {bn_p} vs {scipy_ttest.pvalue}"
assert bn_df == len(penguins) - 1, f"df mismatch: {bn_df} vs {len(penguins) - 1}"

Wilcoxon Signed-Rank Test¶

Classical:

W = \sum_{i=1}^{n} R_i^+ \cdot \text{sign}(x_i), \quad z = \frac{W - n(n+1)/4}{\sqrt{n(n+1)(2n+1)/24}} \dot{\sim} \mathcal{N}(0,1) \text{ under } H_0

(4)

As GLM:

y_i^* \sim \mathcal{N}(\mu, \sigma^2), \quad \mu = \beta_0, \quad \text{where } y_i^* = \text{sign}(y_i) \cdot \text{rank}(|y_i|)

(5)

t = \frac{\hat{\beta}_0}{\text{SE}(\hat{\beta}_0)} \sim t(n-1) \text{ under } H_0: \beta_0 = 0

(6)

The signed-rank transformation makes inference robust to outliers. The GLM approach replaces the $W$ statistic with a $t$ -test on signed ranks — both test the same null hypothesis of zero location.

scipy¶

from scipy.stats import wilcoxon

scipy_wilcox = wilcoxon(penguins['bill_length_mm'].to_numpy(), mode='approx')
scipy_wilcox

WilcoxonResult(statistic=np.float64(0.0), pvalue=np.float64(8.218085850268946e-58))

bossanova¶

m_wilcox = model("signed_rank(bill_length_mm) ~ 1", penguins).fit().infer()

m_wilcox.params.select("statistic", "p_value")

scipy reports the Wilcoxon W statistic; bossanova reports a t-statistic on signed ranks. The test statistics differ but both test H₀: location = 0 and yield equivalent p-values.

bn_t_w = float(m_wilcox.params["statistic"][0])
bn_p_w = float(m_wilcox.params["p_value"][0])
assert bn_t_w > 0, f"Expected positive t-stat for all-positive data, got {bn_t_w}"
assert bn_p_w < 1e-10, f"Expected highly significant p-value, got {bn_p_w}"
assert scipy_wilcox.pvalue < 1e-10, f"Expected highly significant scipy p-value, got {scipy_wilcox.pvalue}"
# Normal approximations from both methods should be in the same ballpark
assert np.isclose(bn_p_w, scipy_wilcox.pvalue, rtol=0.5), f"p-value gap too large: {bn_p_w} vs {scipy_wilcox.pvalue}"