Bernoullli Documentation

Supported Input File Format:

Comma-Separated Values (.csv). Ensure UTF-8 character encoding to preserve scientific symbols and special characters.

Dataset Hygiene Checklist:

• Tidy Structure: Each column represents a single variable; each row represents a single subject or observation.
• Single Header Row: Column names must appear strictly on Row 1. Do not use merged header rows or subtitle banners.
• Clear Column Names: Use alphanumeric snake_case (e.g. plant_height_cm, water_treatment). Avoid punctuation (# % $ / ), spaces, or leading numbers.
• Missing Values: Represent missing values consistently with empty cells or NA. Avoid mixing placeholder strings (such as "missing", "N/A", "999", or "-").

1. Distributional Normality

Evaluated using the Shapiro-Wilk test alongside empirical Quantile-Quantile (Q-Q) plots. Note that a non-significant result ($p > 0.05$) indicates insufficient evidence to reject normality, rather than mathematical proof of a normal distribution. For small samples exhibiting marked skewness, non-parametric alternatives or power transformations are suggested.

2. Homogeneity of Variance (Homoscedasticity)

Tested across groups using Levene's test (median-centered for robustness against non-normality). If variances differ significantly across groups, Welch's correction is applied to protect against inflated Type I error rates.

3. Multicollinearity in Regression

For multi-predictor linear models, Bernoullli computes the Variance Inflation Factor (VIF). Predictors with $ ext{VIF} > 5.0$ are flagged for potential collinear redundancy, which can artificially inflate coefficient standard errors.

4. Influential Outliers & Leverage

Diagnostic residuals and Cook's Distance are plotted to help researchers identify potential data entry errors or highly influential points that unduly pull regression slopes.

The Okabe–Ito Categorical Palette:

Color-vision deficiencies affect ~8% of men and 0.5% of women. Bernoullli employs the scientific Okabe–Ito palette as its default for discrete comparative groups:

No Red–Green Treatment Pairings: Red and green collapse into indistinguishable tones under deuteranopia and protanopia. Bernoullli assigns distinct, distinguishable colors consistently across all figures in a session.

Canvas & Formatting Integrity:

• Solid White Canvas: Plots are always rendered with opaque white backgrounds (bg = 'white') to prevent transparent image inversion when viewed in Microsoft Word Dark Mode.
• Solid Baseline Axes: Coordinate boundaries feature clean, dark baseline axes (#1e293b) to ground the visual data.
• Clear Error Bar Specifications: Charts explicitly declare whether error bars represent Standard Error ($Mean pm 1 SE$), Standard Deviation ($Mean pm 1 SD$), or 95% Confidence Intervals.

User Responsibility & Academic Judgment:

Bernoullli assists in computing statistical models and evaluating mathematical assumptions, but it does not replace scientific expertise or domain judgment:

• Authorship: You are the author of your research. You must verify that the chosen statistical methodology appropriately addresses your research question.
• Data Accuracy: Bernoullli computes results based on the data you supply. Erroneous data inputs or improperly measured variables will yield mathematically valid but substantively meaningless conclusions.
• Institutional Compliance: Ensure your use of statistical software tools complies with your department's thesis guidelines and journal disclosure requirements.