DataFrame-level Stats Examples

Basic examples of statistical test methods.

Used Libraries

The following methods selectively use these libraries:

• pandas

• numpy

• plotly

• scipy

• statsmodels

• pingouin

• scikit-learn

import plotly.io as pio
pio.renderers.default = "notebook"

Normality Test

normality() - Tests for normal distribution

from frameon import load_dataset, FrameOn as fo

titanic = fo(load_dataset('titanic'))
titanic.stats.normality(
    dv='age',
    between='class',
    method='jarque_bera',
    show_qqplot=True
)

⚠️ Found 177 (19.87%) missing values in dependent variable 'age'
⚠️ Group 'First' in factor 'class' has 30 (13.89%) missing values in 'age'
⚠️ Group 'Second' in factor 'class' has 11 (5.98%) missing values in 'age'
⚠️ Group 'Third' in factor 'class' has 136 (27.70%) missing values in 'age'
Removed 177 rows (19.87% of data) with missing values in: age, class

Normality Test Report

- Dependent Variable: age
- Grouping Variable: class
- Alpha Level: 0.05
- Test Method: jarque_bera

🔹 Hypothesis Formulation

- H0: The data follows a normal distribution
- H1: The data does not follow a normal distribution

	W	pval	normal	Reject H0
group
Third	27.48	0.00	False	True
First	1.57	0.46	True	False
Second	0.80	0.67	True	False

Warning: Non-normal distributions detected in groups: Third

Levene’s Test

levene() - Tests equality of variances across groups.

from frameon import load_dataset, FrameOn as fo

titanic = fo(load_dataset('titanic'))
titanic.stats.levene(
    dv='age',
    between='class'
)

⚠️ Found 177 (19.87%) missing values in dependent variable 'age'
⚠️ Group 'First' in factor 'class' has 30 (13.89%) missing values in 'age'
⚠️ Group 'Second' in factor 'class' has 11 (5.98%) missing values in 'age'
⚠️ Group 'Third' in factor 'class' has 136 (27.70%) missing values in 'age'
Removed 177 rows (19.87% of data) with missing values in: age, class

Levene's Test Report

- Dependent Variable: age
- Groups: Third vs First
- Alpha Level: 0.05

	W	pval	equal_var
levene	5.620164	0.003787	False

Conclusion: At alpha = 0.05, we reject the null hypothesis that the variances are equal (p=0.0038).

T-Test

ttest() - Compares means between two groups using independent samples t-test.

from frameon import load_dataset, FrameOn as fo

titanic = fo(load_dataset('titanic'))
titanic.stats.ttest(
    dv='age',
    between='alive',
)

⚠️ Found 177 (19.87%) missing values in dependent variable 'age'
⚠️ Group 'no' in factor 'alive' has 125 (22.77%) missing values in 'age'
⚠️ Group 'yes' in factor 'alive' has 52 (15.20%) missing values in 'age'
Removed 177 rows (19.87% of data) with missing values in: age, alive

T-Test Report

- Dependent Variable: age
- Between Factor: alive (no vs yes)
- Alpha Level: 0.05
- Alternative: two-sided

🔹 Hypothesis Formulation

- H0: The mean of no equals the mean of yes
- H1: The mean of no differs from the mean of yes

🔹 Descriptive Statistics

Group	count	mean	std	min	max
no	424	30.63	14.17	1.0	74.0
yes	290	28.34	14.95	0.42	80.0

🔹 Homogeneity of Variance (Levene's Test)

	W	pval	equal_var
levene	1.195	0.275	True

- Using Student's t-test - equal variances assumed (pval = 0.275 ≥ 0.05)

🔹 T-Test Results

	T	dof	alternative	p-val	CI95%	cohen-d	BF10	power
T-test	2.067	712	two-sided	0.039	[0.11 4.45]	0.157	0.685	0.541

🔹 Conclusion: At alpha = 0.05, we reject the null hypothesis (p-val = 0.0391).

Mann-Whitney U Test

mwu() - Non-parametric test for comparing two independent groups.

from frameon import load_dataset, FrameOn as fo

titanic = fo(load_dataset('titanic'))
titanic.stats.mwu(
    dv='age',
    between='alive'
)

⚠️ Found 177 (19.87%) missing values in dependent variable 'age'
⚠️ Group 'no' in factor 'alive' has 125 (22.77%) missing values in 'age'
⚠️ Group 'yes' in factor 'alive' has 52 (15.20%) missing values in 'age'
Removed 177 rows (19.87% of data) with missing values in: age, alive

Mann-Whitney U Test Report

- Dependent Variable: age
- Between Factor: alive (no vs yes)
- Alpha Level: 0.05
- Alternative: two-sided

🔹 Hypothesis Formulation

- H0: The distribution of no equals the distribution of yes
- H1: The distribution of no differs from the distribution of yes

🔹 Descriptive Statistics

Group	count	median	mean	std	min	max
no	424	28.0	30.63	14.17	1.0	74.0
yes	290	28.0	28.34	14.95	0.42	80.0

🔹 Mann-Whitney U Test Results

	U-val	alternative	p-val	RBC	CLES
MWU	65278.0	two-sided	0.16	0.062	0.531

Effect Sizes:
- Rank-Biserial Correlation (RBC) = 0.06 (small effect)
- Common Language Effect Size (CLES) = 0.53 (small effect)

🔹 Conclusion: At alpha = 0.05, we fail to reject the null hypothesis (p-val = 0.1605).

ANOVA

anova() - One-way analysis of variance for comparing means across multiple groups.

from frameon import load_dataset, FrameOn as fo

titanic = fo(load_dataset('titanic'))
titanic.stats.anova(
    dv='age',
    between='class'
)

⚠️ Found 177 (19.87%) missing values in dependent variable 'age'
⚠️ Group 'First' in factor 'class' has 30 (13.89%) missing values in 'age'
⚠️ Group 'Second' in factor 'class' has 11 (5.98%) missing values in 'age'
⚠️ Group 'Third' in factor 'class' has 136 (27.70%) missing values in 'age'
Removed 177 rows (19.87% of data) with missing values in: age, class

One-way ANOVA Report

- Dependent Variable: age
- Between Factors: class
- Alpha Level: 0.05

🔹 Hypothesis Formulation

- H0: All group means are equal (Third, First, Second)
- H1: At least one group mean differs among Third, First, Second

🔹 Descriptive Statistics

Group	count	mean	std	min	max	median
First	186	38.23	14.8	0.92	80.0	37.0
Second	173	29.88	14.0	0.67	70.0	29.0
Third	355	25.14	12.5	0.42	74.0	24.0

🔹 Homogeneity of Variance Check

	W	pval	equal_var
levene	5.62	0.004	False

- Using Welch's ANOVA - Levene's test showed unequal variances (pval = 0.004 < 0.05)

🔹 Welch ANOVA Results

Source	ddof1	ddof2	F	p-unc	np2
class	2	359.444	53.355	0.0	0.139

🔹 Post-hoc Analysis

Games-Howell for unequal variances

A	B	mean(A)	mean(B)	diff	se	T	df	pval	hedges
First	Second	38.233	29.878	8.356	1.52	5.496	356.897	0.0	0.578
First	Third	38.233	25.141	13.093	1.272	10.293	325.228	0.0	0.981
Second	Third	29.878	25.141	4.737	1.254	3.777	308.829	0.001	0.364

🔹 Conclusion: Significant effects found (p-unc = {aov['p-unc'].iloc[0]:.3f})

Significant pairwise comparisons:
- First vs Second: pval = 0.000
- First vs Third: pval = 0.000
- Second vs Third: pval = 0.001

Kruskal-Wallis Test

kruskal() - Non-parametric alternative to one-way ANOVA.

from frameon import load_dataset, FrameOn as fo

titanic = fo(load_dataset('titanic'))
titanic.stats.kruskal(
    dv='age',
    between='class'
)

⚠️ Found 177 (19.87%) missing values in dependent variable 'age'
⚠️ Group 'First' in factor 'class' has 30 (13.89%) missing values in 'age'
⚠️ Group 'Second' in factor 'class' has 11 (5.98%) missing values in 'age'
⚠️ Group 'Third' in factor 'class' has 136 (27.70%) missing values in 'age'
Removed 177 rows (19.87% of data) with missing values in: age, class

Kruskal-Wallis Test Report

- Dependent Variable: age
- Grouping Variable: class
- Groups: Third, First, Second
- Alpha Level: 0.05

🔹 Hypothesis Formulation

- H0: The distributions of all groups are equal (Third, First, Second)
- H1: At least one group distribution differs among Third, First, Second

🔹 Descriptive Statistics

Group	count	median	mean	std	min	max
First	186	37.0	38.23	14.8	0.92	80.0
Second	173	29.0	29.88	14.0	0.67	70.0
Third	355	24.0	25.14	12.5	0.42	74.0

🔹 Kruskal-Wallis Test Results

Source	ddof1	H	p-unc
class	2	95.995	0.0

🔹 Post-hoc Analysis (Dunn's Test)

A	B	p-corr
First	Second	0.0
First	Third	0.0
Second	Third	0.0

🔹 Conclusion: Significant difference found (p-unc = 0.000)

Significant pairwise comparisons:
- First vs Second: pval = 0.000
- First vs Third: pval = 0.000
- Second vs Third: pval = 0.000

Chi-Square Test

chi2_independence() - Tests categorical variable independence

from frameon import load_dataset, FrameOn as fo

titanic = fo(load_dataset('titanic'))
titanic.stats.chi2_independence(
    x='class',
    y='alive',
    test='pearson'
)

Categorical Association Report

- class vs alive
- Total observations: 891


🔹 Hypothesis Formulation

- H0: class and alive are independent (Third, First, Second vs no, yes)
- H1: class and alive are associated (Third, First, Second vs no, yes)

🔹 Contingency Table

alive	no	yes
class
First	80	136
Second	97	87
Third	372	119

🔹 Statistical Tests

test	lambda	chi2	dof	pval	cramer	power	cramer strength
pearson	1.0	102.889	2.0	0.0	0.34	1.0	Moderate

🔹 Conclusion: At alpha = 0.05, we reject the null hypothesis of independence (pearson test, p = 0.000).
No significant association found between class and alive.

Bootstrap

bootstrap() - Computes bootstrap confidence intervals

from frameon import load_dataset, FrameOn as fo

titanic = fo(load_dataset('titanic'))
titanic.stats.bootstrap(
    dv='age',
    between='alive',
    reference_group='no',
    statistic='mean_diff',
    plot=True
)

⚠️ Found 177 (19.87%) missing values in dependent variable 'age'
⚠️ Group 'no' in factor 'alive' has 125 (22.77%) missing values in 'age'
⚠️ Group 'yes' in factor 'alive' has 52 (15.20%) missing values in 'age'
Removed 177 rows (19.87% of data) with missing values in: age, alive

method	statistic	n_resamples	standard_error	ci	alternative
BCa	2.28	9999	1.11	[0.16, 4.53]	two-sided

Linear Regression

ols() - Ordinary least squares regression

from frameon import load_dataset, FrameOn as fo

diamonds = fo(load_dataset('diamonds'))
diamonds.stats.ols(
    formula='price ~ carat + C(cut, Treatment("Ideal"))',
    cov_type='HC3',
    show_plots=True
)

Variance Inflation Factors

Variable	VIF
carat	2.08
C(cut, Treatment("Ideal"))[T.Premium]	1.491
C(cut, Treatment("Ideal"))[T.Very Good]	1.352
C(cut, Treatment("Ideal"))[T.Good]	1.158
C(cut, Treatment("Ideal"))[T.Fair]	1.079

OLS Regression Results

Dep. Variable:	price	R-squared:	0.856
Model:	OLS	Adj. R-squared:	0.856
Method:	Least Squares	F-statistic:	2.092e+04
Date:	Mon, 13 Oct 2025	Prob (F-statistic):	0.00
Time:	03:11:16	Log-Likelihood:	-4.7142e+05
No. Observations:	53940	AIC:	9.429e+05
Df Residuals:	53934	BIC:	9.429e+05
Df Model:	5
Covariance Type:	HC3

	coef	std err	t	P>|t|	[0.025	0.975]
Intercept	-2074.5457	16.092	-128.920	0.000	-2106.086	-2043.006
C(cut, Treatment("Ideal"))[T.Fair]	-1800.9240	51.723	-34.819	0.000	-1902.301	-1699.547
C(cut, Treatment("Ideal"))[T.Good]	-680.5921	22.873	-29.755	0.000	-725.423	-635.761
C(cut, Treatment("Ideal"))[T.Premium]	-361.8468	17.209	-21.026	0.000	-395.577	-328.116
C(cut, Treatment("Ideal"))[T.Very Good]	-290.7886	16.684	-17.430	0.000	-323.488	-258.089
carat	7871.0821	24.892	316.207	0.000	7822.293	7919.871

Omnibus:	14616.138	Durbin-Watson:	1.027
Prob(Omnibus):	0.000	Jarque-Bera (JB):	150962.278
Skew:	1.007	Prob(JB):	0.00
Kurtosis:	10.944	Cond. No.	8.39

Notes:
[1] Standard Errors are heteroscedasticity robust (HC3)

Robust Linear Model

rlm() - Regression with robust standard errors

from frameon import load_dataset, FrameOn as fo

diamonds = fo(load_dataset('diamonds'))
diamonds.stats.rlm(
    formula='price ~ carat + C(cut, Treatment("Ideal"))',
)

Variance Inflation Factors

Variable	VIF
carat	2.08
C(cut, Treatment("Ideal"))[T.Premium]	1.491
C(cut, Treatment("Ideal"))[T.Very Good]	1.352
C(cut, Treatment("Ideal"))[T.Good]	1.158
C(cut, Treatment("Ideal"))[T.Fair]	1.079

Robust linear Model Regression Results

Dep. Variable:	price	No. Observations:	53940
Model:	RLM	Df Residuals:	53934
Method:	IRLS	Df Model:	5
Norm:	HuberT
Scale Est.:	mad
Cov Type:	H1
Date:	Mon, 13 Oct 2025
Time:	03:11:17
No. Iterations:	23

	coef	std err	z	P>|z|	[0.025	0.975]
Intercept	-1834.9416	8.689	-211.172	0.000	-1851.972	-1817.911
C(cut, Treatment("Ideal"))[T.Fair]	-1263.4005	24.022	-52.593	0.000	-1310.483	-1216.318
C(cut, Treatment("Ideal"))[T.Good]	-490.9289	14.651	-33.507	0.000	-519.645	-462.213
C(cut, Treatment("Ideal"))[T.Premium]	-231.6302	10.192	-22.727	0.000	-251.606	-211.654
C(cut, Treatment("Ideal"))[T.Very Good]	-220.2565	10.525	-20.926	0.000	-240.886	-199.627
carat	7219.3427	8.535	845.812	0.000	7202.614	7236.072

If the model instance has been used for another fit with different fit parameters, then the fit options might not be the correct ones anymore .

Generalized Linear Model

glm() - Generalized linear regression

from frameon import load_dataset, FrameOn as fo
import statsmodels.api as sm

titanic = fo(load_dataset('titanic'))
titanic.stats.glm(
    formula='survived ~ age + sex + pclass',
    family=sm.families.Binomial()
)

Variance Inflation Factors

Variable	VIF
pclass	3.359
age	2.924
sex[T.male]	2.859

Generalized Linear Model Regression Results

Dep. Variable:	survived	No. Observations:	714
Model:	GLM	Df Residuals:	710
Model Family:	Binomial	Df Model:	3
Link Function:	Logit	Scale:	1.0000
Method:	IRLS	Log-Likelihood:	-323.65
Date:	Mon, 13 Oct 2025	Deviance:	647.29
Time:	03:11:17	Pearson chi2:	768.
No. Iterations:	5	Pseudo R-squ. (CS):	0.3587
Covariance Type:	HC3

	coef	std err	t	P>|t|	[0.025	0.975]
Intercept	5.0560	0.526	9.607	0.000	4.023	6.089
sex[T.male]	-2.5221	0.203	-12.444	0.000	-2.920	-2.124
age	-0.0369	0.008	-4.610	0.000	-0.053	-0.021
pclass	-1.2885	0.148	-8.718	0.000	-1.579	-0.998

Quantile Regression

quantreg() - Quantile regression

from frameon import load_dataset, FrameOn as fo

diamonds = fo(load_dataset('diamonds'))
diamonds.stats.quantreg(
    formula='price ~ carat + C(cut, Treatment("Ideal"))',
    q=0.95
)

Variance Inflation Factors

Variable	VIF
carat	2.08
C(cut, Treatment("Ideal"))[T.Premium]	1.491
C(cut, Treatment("Ideal"))[T.Very Good]	1.352
C(cut, Treatment("Ideal"))[T.Good]	1.158
C(cut, Treatment("Ideal"))[T.Fair]	1.079

QuantReg Regression Results

Dep. Variable:	price	Pseudo R-squared:	0.7297
Model:	QuantReg	Bandwidth:	220.4
Method:	Least Squares	Sparsity:	6958.
Date:	Mon, 13 Oct 2025	No. Observations:	53940
Time:	03:11:18	Df Residuals:	53934
		Df Model:	5

	coef	std err	t	P>|t|	[0.025	0.975]
Intercept	-2220.0986	14.000	-158.574	0.000	-2247.540	-2192.658
C(cut, Treatment("Ideal"))[T.Fair]	-886.8169	40.760	-21.757	0.000	-966.707	-806.927
C(cut, Treatment("Ideal"))[T.Good]	-291.6197	24.700	-11.807	0.000	-340.031	-243.208
C(cut, Treatment("Ideal"))[T.Premium]	-102.2676	17.318	-5.905	0.000	-136.211	-68.325
C(cut, Treatment("Ideal"))[T.Very Good]	137.0000	17.739	7.723	0.000	102.231	171.769
carat	1.063e+04	16.741	635.114	0.000	1.06e+04	1.07e+04

Ordered Model

ordered_model() - Regression for ordered categorical dependent variables.

from frameon import load_dataset, FrameOn as fo
import pandas as pd

diamonds = fo(load_dataset('diamonds'))
cut_order = ['Fair', 'Good', 'Very Good', 'Premium', 'Ideal']
diamonds['cut'] = pd.Categorical(diamonds['cut'], categories=cut_order, ordered=True)
diamonds.stats.ordered_model(
    formula='cut ~ carat + price',
    distr='logit'
)

Optimization terminated successfully.
         Current function value: 1.349193
         Iterations: 28
         Function evaluations: 33
         Gradient evaluations: 33

OrderedModel Results

Dep. Variable:	cut	Log-Likelihood:	-72775.
Model:	OrderedModel	AIC:	1.456e+05
Method:	Maximum Likelihood	BIC:	1.456e+05
Date:	Mon, 13 Oct 2025
Time:	03:11:19
No. Observations:	53940
Df Residuals:	53934
Df Model:	2

	coef	std err	z	P>|z|	[0.025	0.975]
carat	-2.1468	0.045	-47.500	0.000	-2.235	-2.058
price	0.0002	5.37e-06	38.579	0.000	0.000	0.000
Fair/Good	-4.4558	0.033	-136.610	0.000	-4.520	-4.392
Good/Very Good	0.4227	0.015	28.551	0.000	0.394	0.452
Very Good/Premium	0.3132	0.009	36.365	0.000	0.296	0.330
Premium/Ideal	0.0793	0.008	10.283	0.000	0.064	0.094

Mixed Linear Model

mixedlm() - Linear regression with random effects.

from frameon import load_dataset, FrameOn as fo

penguins = fo(load_dataset('penguins'))
penguins.stats.mixedlm(
    "body_mass_g ~ bill_length_mm + C(sex)",
    groups="island"
)

Variance Inflation Factors

Variable	VIF
bill_length_mm	2.173
C(sex)[T.MALE]	2.173

Model:	MixedLM	Dependent Variable:	body_mass_g
No. Observations:	333	Method:	REML
No. Groups:	6	Scale:	0.0000
Min. group size:	1	Log-Likelihood:	2410.1918
Max. group size:	1	Converged:	Yes
Mean group size:	1.0

	Coef.	Std.Err.	z	P>|z|	[0.025	0.975]
Intercept	3876.707	330.617	11.726	0.000	3228.709	4524.705
C(sex)[T.MALE]	171.865	19.946	8.617	0.000	132.771	210.958
bill_length_mm	-8.892	8.505	-1.046	0.296	-25.562	7.777
Group Var	518.642	279357.703

Feature Importance Analysis

feature_importance_analysis() - Analyzes feature importance for data analysis.

from frameon import load_dataset, FrameOn as fo

iris = fo(load_dataset('iris'))
iris.stats.feature_importance_analysis(
    target_column='species',
    feature_columns=iris.columns.difference(['species', 'species_id']).tolist()
)