Goodness of fit for discrete distributions — 9231 FPS
Estimate the Poisson mean and account for the fitted parameter. Build expected frequencies, calculate the statistic and adjust degrees of freedom.
Notes and key methods
For a Poisson goodness-of-fit test, estimate λ from the sample mean when it is unknown. Calculate expected counts from Poisson interval probabilities, including a final class that covers the complete upper tail.
With k retained classes and one estimated parameter, the degrees of freedom are k−2. Combine classes before testing if any expected count is below 5.
Under a goodness-of-fit null model, expected frequencies are \(E_i=np_i\). The statistic is \(\chi^2=\sum(O_i-E_i)^2/E_i\).
After combining categories so every expected frequency is at least 5, use k−1−m degrees of freedom, where k is the number of retained categories and m is the number of parameters estimated from these same data. The p-value is an upper chi-squared tail.
Check the conditions
The fitted mean and grouped counts must describe the same sample.
Include all observations in the upper-tail class.
Categories cover the whole outcome space and do not overlap.
Combine sparse expected categories before calculating the final degrees of freedom.
Three worked examples
Example 1 · Fitting a Poisson model for a chi-squared test
Independent count observations give
Count class
0
1
2
3 or more
Observed
31
46
56
41
There are 174 observations with total count 322. Fit a Poisson distribution by estimating its mean; no classes need combining.
Use a 5% significance level for a written conclusion. A distribution calculator may be used for numerical p-values.
Find the fitted Poisson mean.
Find the expected frequency in the class 3 or more.
Find the chi-squared statistic.
Find the p-value with the fitted-parameter adjustment.
The p-value is \(0.129757468640534\). Since it exceeds 0.05, do not reject H₀: the sample gives insufficient evidence that the fitted Poisson model does not fit. This does not give the probability that H₀ is true.
Example 2 · Chi-squared goodness of fit
A null model assigns equal probabilities to four outcomes, with no fitted parameters. Observed counts are \(26,18,30,14\).
Use a 5% significance level for a written conclusion. A distribution calculator may be used for numerical p-values.
Find the expected frequency in each category.
Find the chi-squared statistic.
Find the p-value.
For a separate fit with k=7 retained classes and one estimated parameter, find the degrees of freedom.
The p-value is \(0.0636948051236637\). Since it exceeds 0.05, do not reject H₀: the sample gives insufficient evidence that the specified categorical distribution does not fit. This does not give the probability that H₀ is true.
Example 3 · Fitting a Poisson model for a chi-squared test
Independent count observations give
Count class
0
1
2
3 or more
Observed
33
48
58
43
There are 182 observations with total count 336. Fit a Poisson distribution by estimating its mean; no classes need combining.
Use a 5% significance level for a written conclusion. A distribution calculator may be used for numerical p-values.
Find the fitted Poisson mean.
Find the expected frequency in the class 3 or more.
Find the chi-squared statistic.
Find the p-value with the fitted-parameter adjustment.
The p-value is \(0.127336519233962\). Since it exceeds 0.05, do not reject H₀: the sample gives insufficient evidence that the fitted Poisson model does not fit. This does not give the probability that H₀ is true.
Common mistakes
The final class probability is a tail sum, not the probability of its first value only.
Estimating λ uses one additional degree of freedom.
Use expected, not observed, frequency in the denominator.
Subtract the number of fitted parameters when determining degrees of freedom.