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Conditioning can remove a nuisance parameter and produce a clean exact reference distribution, but the conditioned margin may also carry information that the test deliberately sets aside.
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Chapter contents
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 conditioning choice
Route 2 · Work the table ledger
Route 3 · Transfer the comparison
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE CONDITIONING CHOICE
1. What conditioning changes
A statistical test needs a reference distribution under the null hypothesis. In a two-by-two table, the null of equal success probabilities still contains a common but unknown success probability. That common rate is a nuisance parameter: necessary for the sampling model, but not the contrast being tested.
Fisher’s exact test conditions on both margins. Once the row totals and the total number of successes are fixed, the cell count has a hypergeometric distribution that no longer contains the common success probability. The nuisance parameter disappears from the conditional calculation.
This is mathematically clean, but it changes the experiment being compared. The conditioned total is treated as fixed even if it was random in the original design. The central question is therefore not whether conditioning is “exact,” but whether the conditioned information is appropriate to discard for the scientific design.
CHAPTER 2 OF 12 · DEFINE THE CONDITIONING CHOICE
2. Ancillary, sufficient and design information
Conditioning is compelling when the conditioned statistic is fixed by design or carries no information about the parameter of interest. In matched or stratified studies, conditioning can honour the sampling mechanism. In a trial with fixed group sizes but a random total number of responses, the total may contain information about the nuisance rate and about how the alternative behaves.
A sufficient statistic for a nuisance parameter can remove it from the conditional distribution. That does not prove the conditional test is optimal for every decision. Sufficiency explains what is retained about the nuisance parameter; power depends on which sample points the rejection region can use.
Write down what was fixed before data collection. A margin fixed by design is different from a margin observed after the fact. The same table can therefore support different reference sets under different experiments.
CHAPTER 3 OF 12 · WORK THE TABLE LEDGER
3. Worked example: one observed table
Suppose 10 people receive treatment A and 10 receive treatment B. A has 8 successes and B has 3. The total number of successes is 11. Conditional on that total and both group sizes, the possible A-success counts follow a hypergeometric distribution.
| Quantity | Treatment A | Treatment B | Total |
|---|---|---|---|
| Success | 8 | 3 | 11 |
| Failure | 2 | 7 | 9 |
| Group total | 10 | 10 | 20 |
The conditional tail sums probabilities of tables with the same margins that are at least as extreme under a declared ordering. An unconditional test instead evaluates binomial sampling across possible success totals and controls the nuisance parameter by maximising or otherwise handling the null probability over its range.
CHAPTER 4 OF 12 · WORK THE TABLE LEDGER
4. Nuisance-parameter control without conditioning
An exact unconditional p-value can take the supremum of the tail probability over every allowed nuisance value. This protects Type I error without fixing the random success total. Barnard’s test is a classical example; Boschloo’s method uses Fisher’s p-value as a statistic and raises the conditional level while maintaining unconditional size.
The supremum can be computationally demanding because the tail set and nuisance maximum must be evaluated across a discrete sample space. Modern computation makes many two-by-two cases practical, but the algorithm, grid or optimisation tolerance must be reported.
“Unconditional” does not mean approximate or uncontrolled. It describes the reference experiment. Exactness still requires a finite-sample size calculation over the nuisance parameter.
CHAPTER 5 OF 12 · WORK THE TABLE LEDGER
5. Where power can be gained or lost
Conditioning narrows the reference set. That can remove nuisance dependence, yet it can also create a coarse p-value lattice and leave unused probability mass below the nominal level. An unconditional procedure may spend more of the available Type I error and therefore reject more alternatives.
The advantage is not universal. Mehta and Hilton showed that conclusions drawn from a single two-by-two comparison can be confounded by discreteness; in larger tables, the alleged power advantage may shrink. Power must be calculated for the actual design, statistic and alternative.
Compare tests at their actual sizes, not merely at a shared nominal label. A procedure using 3.1% actual size should not be condemned for lower power against one using 4.9% without acknowledging the different error budgets.
CHAPTER 6 OF 12 · TRANSFER THE COMPARISON
6. Changed context: a fixed-success design
Imagine a case-control design that deliberately selects exactly 11 successes and 9 failures, then records exposure groups. Here the success margin is fixed by sampling. Conditioning on it reflects the design rather than discarding a random outcome.
| Design | Group totals fixed? | Success total fixed? | Natural reference set |
|---|---|---|---|
| Randomised trial | Yes | No | Unconditional total may matter |
| Case-control sampling | Often no in the same sense | Yes by selection | Conditional margin is natural |
The transfer lesson is to reconstruct the data-generating process before selecting a test. A software menu cannot infer whether a margin was fixed scientifically or merely observed.
CHAPTER 7 OF 12 · TRANSFER THE COMPARISON
7. Two-sidedness remains a separate choice
Conditioning does not define “more extreme.” Two-sided Fisher calculations may order tables by hypergeometric probability, double a one-sided tail, or use a statistic such as the odds ratio. These choices can produce different p-values.
Unconditional tests also need a statistic and tail rule. The nuisance-parameter method controls size for the resulting rejection region; it does not make the extremeness definition self-evident.
Pre-specify the direction, statistic and tie treatment. If several versions are reported, name each rather than presenting one as the inevitable exact answer.
CHAPTER 8 OF 12 · TRANSFER THE COMPARISON
8. Mid-p and randomisation are additional layers
A conservative conditional test can be modified with a mid-p value or a randomised boundary. The mid-p guide explains the size–power compromise; the randomised exact-test guide explains exact boundary mixing.
These modifications are not the same as moving to an unconditional reference experiment. Conditioning chooses the sample space. Mid-p or randomisation chooses how to use probability mass at its boundary.
Keep the two decisions separate in the method record. Otherwise “less conservative exact test” can conceal several distinct procedures with different guarantees.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and misconceptions
Fisher’s exact test is not automatically wrong because it can be conservative. Its conditional logic may be exactly aligned with the design, and guaranteed size can be valuable. An unconditional test is not automatically superior because one power curve is higher.
Power calculations depend on the alternative, nuisance rate and statistic. Sparse data, imbalance and two-sided ordering can change the ranking. Post-hoc method shopping inflates the chance of a favourable conclusion.
No exact test repairs biased allocation, outcome misclassification or unplanned sequential looks. Exact probability calculations remain conditional on the model and data-collection process.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Evidence and verification route
Barnard’s foundational work is Significance Tests for 2×2 Tables. Boschloo’s raised conditional level is documented in Raised conditional level of significance for the 2×2-table.
Mehta and Hilton’s Exact Power of Conditional and Unconditional Tests shows why size and discreteness must be separated when comparing power. A modern design discussion is available in Conditional versus unconditional analyses of a two-by-two table.
Verification should enumerate the sample space, reproduce the conditional margins, maximise nuisance probabilities for the unconditional rule, and tabulate actual size and power across the planned parameter grid.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Does conditioning always lose information?
No. It may remove variation that is irrelevant or fixed by design. The cost depends on the sampling plan and inferential target.
Is Fisher’s test exact?
It is exact for its conditional reference distribution and controls conditional Type I error. That description does not settle whether the conditioning choice fits every design.
What is a nuisance parameter?
It is a parameter required by the model but not the primary quantity being tested, such as a common success probability under equality.
What should be reported?
Report the sampling design, conditioned statistics, test statistic, tail rule, nuisance treatment, nominal and actual size, software and optimisation settings.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Final transfer check and connected reading
Take the worked table and imagine that the total number of successes was fixed by recruitment. Explain why conditioning now mirrors the design. Then return to the randomised trial and explain what information becomes random again.
Compare the conditional and unconditional rules at equal actual Type I error across several common success rates. If the ranking changes, the learner has found why “more powerful” requires a domain rather than a slogan.
Continue with the Probability root, mid-p tests, randomised exact tests, the Gold Standard of Education and How X Works.
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