Logistic Regression Calculator

Binary Logistic Regression

Binary Dependent Variable (Y ∈ {0, 1}) Enter binary 0 or 1 observations separated by commas or spaces

Please enter valid binary 0 or 1 values for Y.

Predictor X₁

Significance Level (α) Default threshold is 0.05 (5%)

Significance level α must be strictly between 0 and 1 (e.g. 0.05).

Binary Probabilistic Modeling

What Is Logistic Regression?

Binary Logistic Regression is a generalized linear model (GLM) used to predict the probability of a binary outcome ($Y \in \{0, 1\}$) given one or more continuous or categorical predictor variables ($X_1, X_2, \dots$).

Instead of modeling the raw response directly, logistic regression models the log-odds (logit) of the event using the sigmoid function, ensuring predicted probabilities are strictly bounded between $0.0$ and $1.0$.

When to Use It

Binary Classification: Predicting success/failure, pass/fail, disease presence/absence, or churn.
Odds Ratio Estimation: Exponentiating coefficients ($\exp(\beta_j)$) to measure multiplicative changes in event odds.
Medical Risk Scoring: Estimating patient survival or condition likelihood based on biomarker levels.
Hypothesis Testing: Evaluating whether specific predictors exert significant partial effects on binary likelihood.

The Maximum Likelihood Formulas

$$\text{Logit Link: } \ln\left(\frac{p}{1 - p}\right) = \beta_0 + \beta_1 X_1 + \dots + \beta_k X_k$$ $$\text{Sigmoid Probability: } p_i = \frac{1}{1 + \exp(-(\beta_0 + \beta_1 X_{1i} + \dots))}$$ $$\text{Likelihood Ratio Test: } G = -2(LL_0 - LL) \sim \chi^2(k), \quad \text{Odds Ratio} = e^{\beta_j}$$

Where $LL$ is the maximized log-likelihood of the fitted model, $LL_0$ is the null log-likelihood with intercept only, and $G$ tests overall model significance.

Worked Example

Scenario: Regressing binary outcome $Y = [0, 0, 0, 0, 1, 0, 1, 1, 1, 1]$ on $X = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]$ ($n=10, k=1$).

Step 1: MLE Convergence Estimates ($\boldsymbol{\beta}$):
$\hat{\beta}_0 = -3.2201, \quad \hat{\beta}_1 = 0.7618, \quad SE(\hat{\beta}_1) = 0.3810$.
Model: $\ln\left(\frac{p}{1-p}\right) = -3.2201 + 0.7618 X$.

Step 2: Odds Ratio ($e^{\beta}$):
$OR = e^{0.7618} = 2.1422 \quad (95\% \text{ CI: } [1.015, 4.520])$.

Step 3: Likelihood Ratio Omnibus Test ($G$):
$G = -2(-6.9185 - (-3.5120)) = 6.8124 \implies p = 0.0090$.

Conclusion: The logistic model is statistically significant overall ($p = 0.0090$), indicating $X$ reliably predicts outcome likelihood.

MLE Fit Complete

Logistic Regression Results

Estimated Logit Equation

ln(p / (1 - p)) = -3.2201 + 0.7618 X₁

Overall Model Is Statistically Significant (G = 6.81, p = 0.0090)

McFadden R² 0.4925

Log-Likelihood (LL) -3.5120

Likelihood Ratio (G) 6.8124

Model p-value 0.0090

Akaike AIC 11.0240

Sample Size (n) 10

Logistic Coefficients & Odds Ratios

Predictor	Coefficient (β)	Std Error (SE)	Wald z	p-value	Odds Ratio (e^β)	95% CI for OR	Sig?

Logistic Regression Calculator

Binary Logistic Regression

Logistic Regression Results

Logistic Coefficients & Odds Ratios

Step-by-Step Calculation

Interpretation

Fitted Logistic Sigmoid Probability Curve

Perfect for students, researchers, and professionals looking to build real statistical skills.