Bernoulli Response Simulator
Generates binary outcomes Y_i ∈ {0, 1} with success probabilities governed by the sigmoid function p_i = 1 / (1 + e^-(β_0 + β_1 x_i)).
Statistics & Probability / Logistic Regression
FREE TO EXPLORELABORATORY 36 · STATISTICS & PROBABILITY
Logistic Regression, Logit Link Function, and Maximum Likelihood Odds Ratios
Model binary categorical outcomes, formulate the sigmoidal logistic link function, linearize log-odds, estimate parameters via Iteratively Reweighted Least Squares (IRLS), and verify the logistic affine log-odds response. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
LOG-ODDS LOGIT LINK RESPONSE LN(P / (1 - P))—
SAMPLING TELEMETRY
Ready to sample with Predictor x = 4.0.
Adjust Predictor x, then execute statistical trial.
Probabilistic samples are evaluated rigorously using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations. Continuous data structures support high-fidelity empirical probability functions.
Statistical assumptions and distribution limits
Logistic regression assumes that the log-odds of the outcome is a linear function of the continuous predictors. The empirical regression of log-odds logits against predictor coordinates confirms the logistic model with slope m = 0.8000, intercept b = -2.0000, and R² = 1.000000.
Statistical tools & sampling modules.
Sigmoid Probability Mapper
Transforms real linear predictors η = β_0 + β_1 x into bounded probabilities p ∈ (0, 1).
Logit Odds Link Inverter
Linearizes success probabilities into log-odds logit(p) = ln(p / (1 - p)) = β_0 + β_1 x.
IRLS Maximum Likelihood Optimizer
Solves logistic score equations via Newton-Raphson Iteratively Reweighted Least Squares: β_{t+1} = (X^T W X)^(-1) X^T W z.
Odds Ratio Exponentiator
Calculates adjusted odds ratios OR = exp(β_1) and evaluates profile-likelihood confidence intervals.
Log-Odds Linear Regression Engine
Performs linear regression of empirical log-odds versus x to extract slope β_1 = 0.800 and intercept β_0 = -2.000.
FOLLOW THE METHOD
A reason behind every step.
Model binary categorical outcomes, formulate the sigmoidal logistic link function, linearize log-odds, estimate parameters via Iteratively Reweighted Least Squares (IRLS), and verify the logistic affine log-odds response. Standardized parameter settings: 4.0, 6.0, 8.0, 10.0, 12.0, 14.0 units.
01
Set Model Parameters and Predictor Domain
Configure true logistic parameters β_0 = -2.000 and β_1 = 0.800 on the binary classification dashboard.
02
Synthesize Large Binary Dataset
Draw 50,000 observations with predictor x uniformly distributed on [2, 16]. Evaluate true probabilities p = 1 / (1 + e^(-(0.8x - 2.0))).
03
Generate Bernoulli Categorical Outcomes
Assign Y_i = 1 if U_i ≤ p_i and Y_i = 0 otherwise. Confirm the characteristic S-shaped sigmoidal response curve.
04
Execute IRLS Maximum Likelihood Estimation
Run the Iteratively Reweighted Least Squares algorithm until parameter convergence (||β_{t+1} - β_t|| < 1e-6). Verify estimated coefficients.
05
Step Predictor Across Standardized Values x = 4 to 14
Bin data into predictor slices centered at x ∈ {4.0, 6.0, 8.0, 10.0, 12.0, 14.0}, computing empirical log-odds y = ln(p̂ / (1 - p̂)).
06
Perform Linear Regression on Log-Odds Response
Fit a linear regression line y = mx + b to coordinate pairs (x, y). Confirm that empirical slope m = 0.8000 and intercept b = -2.0000 validate the logit link.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Logistic Regression mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
ln(Odds) = 0.8000 · x - 2.0000
| # | Continuous Predictor Coordinate x (units) | Log-Odds Logit Link Response ln(p / (1 - p)) (Log-Odds) | Continuous Predictor x | Log-Odds Logit ln(Odds) | Remove |
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