What this calculator does
Count regression is one half of Module 14 of Analysis Studio, Count, Ordinal, and Multinomial Regression, and it sits in the subscription set. It fits:
- Poisson regression for counts, with an exposure offset when cases are observed for different lengths of time or over different population sizes
- Negative binomial regression, the standard answer when the counts vary more than the Poisson model allows
Results come back as incidence rate ratios with intervals, so a coefficient reads as a multiplier on the expected count rather than as a log-scale number. The results screen reports the overdispersion check beside the fit, because the choice between the two models rests on it.
Promotion, stated with its date: through March 31, 2027, this module runs free on screen, like every module. Downloads and Copy Results stay with the subscription throughout. After the promotion, the module returns to the subscription set at $299 per year.
The method, in plain terms
A count outcome cannot go below zero and usually piles up at small values, so a straight line through the counts fits the wrong shape and can predict negative counts. The Poisson model instead works on the log of the expected count, which keeps every prediction positive, and it carries one strict condition: the variance of the counts equals their mean. Real counts often vary more than that, which is overdispersion, and the negative binomial model relaxes the condition by adding a dispersion parameter. An offset handles exposure: when one case is observed for a year and another for a month, the model compares rates rather than raw counts.
Worked example
Eight cases under a new program and eight under the standard one, counting events over the same observation window. The program counts are 4, 6, 5, 7, 3, 5, 6, 4, a mean of 5.0 events. The standard counts are 2, 3, 1, 4, 2, 3, 2, 1, a mean of 2.25 events.
- Incidence rate ratio: 2.22
- 95 percent interval: 1.27 to 3.88
- z: 2.81
- p: 0.0049
With a single grouping predictor, the fitted ratio is exactly the ratio of the two means, 5.0 over 2.25, and the regression adds what the raw ratio cannot state on its own: the interval, which says the data are consistent with anything from a modest 1.27-fold rate to nearly a fourfold one, and inconsistent with no difference at all.
When the calculator fits, and when it does not
It fits when:
- The outcome is a count of events: visits, defects, arrests, admissions, purchases.
- Cases differ in exposure, and the offset can carry observation time or population size.
- The counts vary more than the Poisson condition allows, in which case the negative binomial form takes over on the same screen.
It does not fit when:
- The outcome is ordered categories rather than a count. The ordinal models on the companion page for Module 14 answer that question.
- The outcome is a proportion of a known total. That is a binomial problem, and the logistic module carries it.
- Most cases are zero because two different processes produce the zeros and the counts. That structure needs a two-part model, and this module says so rather than fitting through it.
Run it
Module 14: Count, Ordinal, and Multinomial RegressionOpen in Analysis Studio →Load your data, pick the count outcome and its predictors, set the exposure variable if the cases were observed unevenly, and read the overdispersion check before the rate ratios.
Questions
Poisson or negative binomial: which should I fit? Fit the Poisson first and read the overdispersion check. If the counts vary about as much as their mean, the Poisson model stands. If they vary well beyond it, the negative binomial model is the honest fit, and its intervals will be the wider ones the data actually support.
What is an incidence rate ratio? The multiplier on the expected count, or on the rate when an offset is set, for a one-unit change in the predictor. A ratio of 2.22 says the expected count is a little more than twice as high; a ratio of 1 says no difference; a ratio below 1 says a lower rate.
Why does the offset matter? Without it, a case observed for a year and a case observed for a month are treated as if they had the same opportunity to produce events. The offset puts the observation window inside the model, so the comparison is between rates, which is the comparison intended.
Can the model handle a count with many zeros? Zeros are ordinary data for a count model, up to a point. When the zeros come from a separate process, a case that could never produce the event mixed in with cases that could, the single-equation fit misleads, and the results screen flags the excess rather than absorbing it silently.
Why not just run a linear regression on the counts? Because counts are bounded at zero and their variance grows with their level, the two facts the linear model assumes away. On counts with a reasonable mean the answers can look close; at small counts the linear fit predicts impossible values and understates its own uncertainty.