Core equations
LR = L(H1 | x) / L(H0 | x)
LLR = ln L(H1 | x) − ln L(H0 | x)Posterior odds = Prior odds × LR
Posterior probability = odds / (1 + odds)G² = 2 × (ℓ1 − ℓ0)
p = P(χ²df ≥ G²)LLRₙ = Σᵢ wᵢ[ln f1(xᵢ) − ln f0(xᵢ)]The application performs primary calculations in natural-log space. This approach reduces underflow when datasets contain extremely small probabilities.
Recommended workflow
1. Select a mode
Use distribution mode for raw observations. Use nested mode for fitted model log likelihoods.
2. Define both hypotheses
Choose distributions and parameters, or enter direct likelihood information.
3. Review evidence
Inspect the LLR, posterior probability, test statistic, charts, and observation table.
Interpretation notes
A positive LLR favors H1. A negative LLR favors H0. Zero indicates equal likelihood under both hypotheses.
Likelihood evidence and a frequentist p-value answer different questions. The displayed evidence labels are practical guides, not universal scientific standards.
For nested-model testing, regular chi-square calibration can fail near parameter boundaries. Mixture distributions or simulation may then be more appropriate.
Custom formulas should represent nonnegative mass or density values. Users remain responsible for normalization and model assumptions.
Common questions
Why calculate in the log domain?
Products of many small probabilities can underflow. Log likelihoods turn products into stable sums.
What does a positive LLR mean?
It means the entered evidence is more likely under H1 than H0.
Can densities exceed one?
Yes. Continuous densities may exceed one, while integrated probability remains bounded by one.
Is the LLR the same as a Bayes factor?
For simple fully specified hypotheses, the likelihood ratio matches the Bayes factor. Composite models require integrated likelihoods.
When is the chi-square p-value appropriate?
It is commonly used for regular nested models under standard large-sample assumptions.
What do observation weights do?
They multiply each observation’s log contribution before the total is formed.
How are sequential boundaries calculated?
When left blank, Wald-style boundaries are derived from Type I and Type II targets.
Can I use different distributions under H0 and H1?
Yes. Each hypothesis has an independent distribution selector and parameter set.
Does the custom parser execute PHP?
No. It uses a restricted mathematical parser with an explicit function allowlist.