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Two small Poisson counts can be compared exactly by conditioning on their total, but exposure, two-sided ordering and the choice between conditional and unconditional coverage all change the answer.
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Choose the route closest to the question, or read from Chapter 1 for the complete argument. The teaching chapters remain open; these grouped routes form the index.
Route 1 · Define the rate-ratio model
Route 2 · Work the exact ledger
Route 3 · Transfer the coverage choice
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE RATE-RATIO MODEL
1. Two Poisson counts hide one rate-ratio question
Suppose event counts Y1 and Y0 arise from independent Poisson processes with rates lambda1 and lambda0 observed for exposures t1 and t0. The parameter of interest is the rate ratio rho = lambda1/lambda0. A ratio above one means the first process has the higher event rate after exposure is accounted for.
Small counts make normal approximations fragile. Exact methods work with the count distribution itself, but there is more than one exact route because lambda0 remains a nuisance parameter unless it is removed or maximised over.
CHAPTER 2 OF 12 · DEFINE THE RATE-RATIO MODEL
2. Conditioning turns the problem into a binomial model
Condition on the total N = Y1 + Y0. Given N, Y1 is binomial with success probability q = t1 rho/(t0 + t1 rho). The common intensity disappears. A confidence interval for q can therefore be transformed back through rho = t0 q/[t1(1 − q)].
This is an exact conditional construction. Its validity is conditional on the observed total and the independent Poisson model. It should not be confused with an unconditional method that evaluates the joint Poisson sample space.
CHAPTER 3 OF 12 · WORK THE EXACT LEDGER
3. Worked example: seven events versus three
Take y1 = 7 over 120 person-years and y0 = 3 over 100 person-years. The observed rate ratio is (7/120)/(3/100) = 1.944. Under rho = 1, q0 = 120/(120 + 100) = 0.5455.
An exact two-sided binomial test for observing 7 of the 10 total events in group 1 gives p approximately 0.3639 under this null. The equal-tail 95% binomial interval for q is approximately [0.3475, 0.9333], which transforms to a rate-ratio interval of approximately [0.444, 11.653].
| Ledger item | Calculation | Result |
|---|---|---|
| Group 1 rate | 7 / 120 | 0.0583 |
| Group 0 rate | 3 / 100 | 0.0300 |
| Observed rate ratio | 0.0583 / 0.0300 | 1.944 |
| Conditional exact p-value at rho = 1 | Binomial(10, 0.5455) | 0.3639 |
| Transformed 95% interval | q limits mapped to rho | [0.444, 11.653] |
CHAPTER 4 OF 12 · WORK THE EXACT LEDGER
4. Inversion must use the same two-sided rule
To invert the conditional test, propose a value of rho, convert it to q, compute the chosen exact p-value and retain rho when that p-value exceeds alpha. Repeat across the positive parameter space and solve the transition points.
Probability-ordered, central and Blaker-style two-sided rules can retain different values. The exact confidence-interval inversion guide explains why an interval borrowed from another rule can disagree with the test.
CHAPTER 5 OF 12 · WORK THE EXACT LEDGER
5. Conditional exactness can be conservative
Once N is fixed, the binomial sample space is discrete. A non-randomised rejection region cannot usually spend exactly five per cent of probability for every q. Coverage therefore often exceeds the nominal level, and intervals can be wide when N is small.
Conditioning also discards information in the random total about the nuisance rate. That removal gives a clean calculation, but it may cost power compared with a well-designed unconditional method.
CHAPTER 6 OF 12 · TRANSFER THE COVERAGE CHOICE
6. Unconditional exact methods keep the full sample space
An unconditional test enumerates pairs (Y1, Y0), orders them by a statistic and controls rejection probability over the nuisance parameter. Efficient-score, likelihood-ratio or conditional-p-value orderings can improve power, but each requires a nuisance maximisation or another valid device.
| Method | Sample space used | Main audit burden |
|---|---|---|
| Conditional exact | Y1 given N | Tail rule and transformation |
| Unconditional exact | Joint pairs (Y1, Y0) | Nuisance maximisation and truncation |
| Plug-in approximation | Joint model at estimated nuisance | Coverage error at small counts |
The Berger–Boos guide shows one way to restrict nuisance maximisation while retaining validity.
CHAPTER 7 OF 12 · TRANSFER THE COVERAGE CHOICE
7. Changed context: unequal exposure is part of the null
If the first group has twice the exposure, equal rates do not imply q = 0.5. They imply q = t1/(t0 + t1). Ignoring exposure converts a rate comparison into a count comparison and can reverse the conclusion.
Transfer the worked example to t1 = 200 and t0 = 100 while keeping counts 7 and 3. The observed rate ratio becomes 1.167, and the conditional null probability becomes two thirds. The data did not change, but the scientific comparison did.
CHAPTER 8 OF 12 · TRANSFER THE COVERAGE CHOICE
8. Changed context: zero counts need careful limits
If one group records zero events, the maximum-likelihood rate ratio may be zero or infinite. An exact interval can still be finite on one side and unbounded on the other. Adding 0.5 to every count produces a convenient estimate, not an exact guarantee.
Report the exposure, zero-event handling and whether the interval is one-sided or two-sided. A zero count is evidence of low incidence over the observed exposure, not proof of a zero underlying rate.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and misconceptions
Exact methods do not repair overdispersion, event dependence, changing risk over time or exposure measurement error. A Poisson model assumes an intensity structure appropriate to the design.
Conditional and unconditional exact procedures can both be valid while returning different widths and power. Choosing the smaller observed p-value after seeing the data is method shopping. Specify the ordering and interval pairing in advance.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Evidence and verification route
Han compares conditional and unconditional exact tests for independent incidence rates in Comparing Two Independent Incidence Rates. Price and Bonett examine exact coverage for rate-ratio intervals in Estimating the Ratio of Two Poisson Rates00100-0).
Fay’s Two-Sided Exact Tests and Matching Confidence Intervals for Discrete Data gives the conditional binomial transformation and matching-test logic. Verification should enumerate the conditional sample space, check the rho-to-q transformation, audit tail ordering and, for unconditional work, document nuisance bounds and truncation error.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Why is the interval so wide with a ratio near two?
Only ten total events were observed. The point estimate is unstable and compatible with a broad range of true ratios.
Does conditioning assume equal exposure?
No. Exposure enters q and the transformation back to rho.
Is an unconditional test automatically better?
No. It may gain power, but its statistic, nuisance control and computation still need validation.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Final transfer check and connected reading
Recalculate the observed ratio when t1 changes from 120 to 200 person-years while counts remain 7 and 3. Explain why the ratio falls from 1.944 to 1.167 and why equal rates now imply q = 2/3.
Then state what must be audited before calling an unconditional interval exact. Continue with the Probability root, conditional tests, exact interval inversion, the Gold Standard of Education and How X Works.
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