Lecture 11
Using the Poisson Distribution
to Test Randomness in Count Data
ABD 3e Chapter 8, Section 4
Learning Objectives
By the end of this lecture, you should be able to:
- Identify when count data are appropriately modeled by a Poisson process.
- State the assumptions of the Poisson model.
- Explain how expected frequencies are generated under a Poisson null model.
- Conduct and interpret a chi-square goodness-of-fit test for count data.
- Use the variance-to-mean ratio to assess clumping or dispersion.
The Poisson Distribution models random counts in time or space
- The Poisson distribution models the number of events that occur in fixed blocks of time or space
- events occur independently of one another
- events occur at a constant average rate
- events are equally likely at any instant in time or point in space
- If these assumptions hold, event counts follow a Poisson distribution.
Clumped and dispersed patterns violate Poisson assumptions
If events are not independent or the rate is not constant, counts will not follow a Poisson distribution.
Instead, events may appear:
- Clumped (distance between points within clumps is less than distance between points in different clumps)
- Dispersed (points are near each other less often than you would expect by random)
The Poisson distribution therefore represents a model of random spatial or temporal structure.
Random spatial counts produce a characteristic Poisson shape
We use the Poisson distribution to test for randomness
So far, we have described a model.
Now we use the Poisson distribution as a null hypothesis:
This is a Poisson Goodness-of-Fit Test
Workflow when testing whether counts follow a Poisson distribution:
- State hypotheses
- Estimate the rate parameter (μ) from the data
- Compute expected probabilities under the Poisson model
- Convert probabilities to expected frequencies
- Check chi-square assumptions
- Compute chi-square statistic
- Determine degrees of freedom
- Make a statistical decision
- Interpret biologically
Assumptions of the Poisson Model
For counts to follow a Poisson distribution:
- Events occur independently
- The average rate ( \(\mu\) ) is constant
- Events are rare relative to the interval size
- Two events cannot occur at exactly the same instant
If these assumptions fail, counts may be clumped or dispersed.
Case Study: Are Mass Extinctions Random Through Time?
Biological question:
Do extinctions occur randomly through geological time, or are there intervals with unusually high extinction rates?
If extinctions are random, counts per time interval should follow a Poisson distribution.
Raup and Sapkowski (1982)
- Raup and Sepkoski (1982) measured extinction rate of marine invertebrates in 76 blocks of time
Observed Extinction Counts Across 76 Time Intervals
These are the observed counts we will compare to Poisson expectations.
![]()
Raup DM, Sepkoski JT. 1982. Mass extinctions in the marine fossil record. Science 215: 1501-1503.
Estimate the mean rate ( \(\mu\) ) from the observed data
Step 1: Estimate the Poisson rate parameter ( \(\mu\) ) under \(H_0\).
We estimate \(\mu\) using the sample mean number of extinctions per interval.
| 0 |
0 |
| 1 |
13 |
| 2 |
15 |
| 3 |
16 |
| 4 |
7 |
| 5 |
10 |
| 6 |
4 |
| etc. |
… |
\[
\bar{X}=\frac{(0\times0)+(1\times13)+(2\times15)+\dots}{76}
\\
= 4.21
\]
We can use this expected mean \(X\) in place of \(\mu\) in the formula for the Poisson distribution to generate the expected frequencies
Use the Poisson Model to Generate Expected Frequencies
\[
\operatorname{Pr}[X \text{ successes}] = \frac{e^{-\mu}\mu^{X}}{X!}
\]
\[
\operatorname{Pr}[3 \text{ extinctions}] = \frac{e^{-4.21} \cdot 4.21^{3}}{3!}
\]
\[
\operatorname{Pr}[3 \text{ extinctions}] = 0.1846
\]
\[
\operatorname{E}[3 \text{ extinctions}] = 76 \times 0.1846 = 14.03
\]
- Step 2: Use the estimated μ to compute expected probabilities.
- Example: The probability that a block of time has exactly 3 extinctions is 0.1846
- Step 3: Convert probabilities into expected frequencies.
- Example: Therefore the expected number of blocks containing 3 extinctions is 14.03
- Now repeat these steps for each number of extinctions to obtain all expected frequencies
Comparing Observed and Expected Extinction Frequencies
Step 4: Compare observed counts to expected counts.
If differences are large relative to sampling variation, we reject \(H_0\).
![]()
The frequency distribution of the number of extinctions (histogram) compared with the frequencies expected from the Poisson distribution having the same mean (curve).
![]()
Chi-Square Assumptions Are Violated with Sparse Expected Counts
Before computing the chi-square statistic, we must check assumptions:
These data violate those assumptions.
Grouping Categories Resolves Chi-Square Assumption Violations
To satisfy chi-square assumptions, we combine adjacent categories.
This reduces small expected counts and allows valid inference.
Degrees of Freedom Determine the Chi-Square Decision
\[
df=(\operatorname{number of categories})-1-(\operatorname{number of parameters})
\\
df = 8 - 1 - 1 = 6
\]
Critical value for chi-square with 𝑑𝑓=6 at 𝛼=0.05 : 𝟏𝟐.𝟓𝟗
Our chi-square test statistic: 𝟐𝟑.𝟗𝟑
Do we reject \(H_0\) or not?
Rejecting \(H_0\) suggests extinction is not random
Statistical conclusion:
- Extinction counts do not fit a Poisson distribution.
- We do not know the exact alternative distribution, just that a Poisson model is inadequate
Biological interpretation:
Extinctions are not occurring at a constant random rate.
There are intervals with unusually high extinction rates.
This pattern is consistent with episodic mass extinction events.
Variance-to-Mean Ratio Reveals Clumping or Dispersion
An alternative diagnostic for randomness:
Compare the variance to the mean.
Variance ≈ Mean → Consistent with Poisson
Variance > Mean → Clumped pattern
Variance < Mean → Dispersed pattern
This provides a quick check before formal testing.
