Calculator
Example Data Table
| Observation | Count |
|---|---|
| 1 | 2 |
| 2 | 0 |
| 3 | 3 |
| 4 | 1 |
| 5 | 4 |
| 6 | 2 |
| 7 | 1 |
| 8 | 3 |
| 9 | 2 |
| 10 | 2 |
Try the sample above with λ₀ = 2. This helps you test whether the observed event rate matches the claimed Poisson mean.
Formula Used
For a Poisson sample, the likelihood under a mean rate λ is based on the probability of the observed counts.
L(λ) = ∏ [e-λ λxᵢ / xᵢ!]
The log-likelihood is easier to compute.
log L(λ) = -nλ + (Σxᵢ)log(λ) - Σlog(xᵢ!)
The unrestricted estimate is the sample mean.
λ̂ = (Σxᵢ) / n
The likelihood ratio test statistic is:
LR = 2 × [log L(λ̂) - log L(λ₀)]
This statistic is compared with a chi-square distribution with 1 degree of freedom. A small p-value suggests the null mean does not fit the data well.
How to Use This Calculator
- Enter the observed counts in the count box.
- Type the null Poisson mean you want to test.
- Set the significance level, such as 0.05.
- Choose the decimal precision for output values.
- Press the calculate button.
- Review the likelihood values, test statistic, p-value, and decision.
- Download the result as CSV or PDF if needed.
Likelihood Ratio Test for Poisson Data
What this calculator does
This calculator tests whether a set of count observations follows a Poisson model with a specific mean. It uses a likelihood ratio test. That method compares two models. The first model fixes the mean at the null value. The second model lets the data choose the best mean. The difference between those two fits becomes the test statistic.
Why this method matters
Count data appears in many math problems. You may track arrivals, defects, clicks, calls, claims, or goals. A Poisson model is often the starting point. It assumes independent events and a stable average rate. The likelihood ratio test is useful because it measures how strongly the data prefers the fitted mean over the claimed mean. That makes it a good choice for formal rate checking.
How to read the output
The sample mean is the unrestricted estimate of the Poisson rate. The null mean is the value you want to test. The calculator also shows the log-likelihood under both models. A larger gap creates a larger likelihood ratio statistic. That usually leads to a smaller p-value. When the p-value falls below alpha, you reject the null hypothesis. When it stays above alpha, you fail to reject it.
Extra values included
The output also reports the sample variance and a dispersion ratio. A Poisson process often has variance close to the mean. When the dispersion ratio is much larger than one, overdispersion may be present. When it is much smaller than one, underdispersion may be present. These values do not replace a full model check, but they help you spot possible fit issues quickly.
Best practice
Use clean count data only. Negative values and decimal values do not belong in a Poisson sample. Keep the study design consistent. Make sure the observations reflect similar exposure or time windows. If exposures differ a lot, a rate model with offsets may be better. For basic count testing, this calculator gives a quick and practical way to evaluate evidence and export a neat summary for reporting.
FAQs
1. What does this test check?
It checks whether your observed count data is consistent with a Poisson distribution having the null mean you entered. It compares the fixed null model with the best-fitting mean from the sample.
2. What kind of data should I enter?
Enter non-negative whole numbers only. These should be observed event counts, such as calls per hour, defects per item, or arrivals per interval.
3. Why is the sample mean important?
For Poisson data, the sample mean is the maximum likelihood estimate of the rate. The test uses it as the unrestricted benchmark against the null mean.
4. What does a small p-value mean?
A small p-value means the null mean does not explain the observed data well. It suggests evidence against the claimed Poisson rate at the chosen significance level.
5. Why does the calculator show variance and dispersion?
They help you compare the spread of the data with the mean. Large differences may suggest overdispersion or underdispersion, which can affect Poisson model suitability.
6. Can I use decimals in the observed counts?
No. Poisson observations are counts, so they must be whole numbers. Decimals and negative values are invalid for this type of test.
7. What is the chi-square value used for?
The likelihood ratio statistic is compared with a chi-square distribution with one degree of freedom. This gives the p-value and supports the final hypothesis decision.
8. When should I avoid this calculator?
Avoid it when observations are not independent, when exposures differ greatly, or when the data shows strong dispersion issues that need a more advanced count model.