Interpreting a count model and a logistic review model
An insurer fits a Poisson count model with a log link and log exposure as a specified unit-coefficient offset. At a defined reference feature pattern, the fitted annual claim rate is 0.12. A particular policy contributes half a year of exposure, and a binary feature has a log coefficient equal to the natural logarithm of 1.5 with no interaction in this exercise. A separate logistic model predicts an investigation outcome rather than claim count. Its reference probability is 0.20 and another binary feature doubles the odds. A manager describes both feature effects as percentage-point probability increases. Before accepting that interpretation, the analyst must preserve the distinction among annual rate, expected count, mean ratio, odds ratio, and event probability.
1. Calculate the count-model predictions before and after the feature change.
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Half-year exposure gives expected count 0.12 times 0.5, or 0.06 at the reference pattern. Multiplying the annual rate by 1.5 gives 0.18, and the half-year expected count becomes 0.09. These are counts and rates, not a logistic probability calculation.
2. Calculate the new logistic probability and explain the difference.
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Reference odds are 0.20 divided by 0.80, or 0.25. Doubling gives odds 0.50, which convert to probability 0.50 divided by 1.50, or one third. Doubling odds does not double probability and does not add a fixed number of probability points.