Constraints on the expected counts

The goodness-of-fit test is easiest when the null hypothesis specifies all Poisson distribution means, \(\{E_i\}\). In practice however, the null hypothesis values of the \(\{E_i\}\) involve unknown parameters that must be estimated from the data. The simplest example is

Estimating the unknown parameters makes the chi-squared test statistic smaller. If \(c\) parameters are estimated from the data and used to get values for the \(\{E_i\}\), we say that there are \(c\) constraints on the \(\{E_i\}\) and

\[ X^2 \;\;=\;\; \sum_{i=1}^k {\frac{\left(O_i - E_i\right)^2}{E_i}} \;\; \underset{\text{approx}}{\sim} \;\; \ChiSqrDistn(k - c \text{ df}) \]

Example

This table shows the number of heart attacks in a city in each of ten weeks.

Week 1 2 3 4 5 6 7 8 9 10
Count   6 11 13 10 21 8 16 6 9 19

Test whether the heart attacks have occurred at random with a constant rate over this period.

(Solved in full version)