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A mid-p value uses half of the observed probability mass, reducing the conservatism of an ordinary exact tail without gaining the exact-size guarantee of a randomised test.
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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 · See the discrete tail
Route 2 · Work the mid-p ledger
Route 3 · Transfer the decision
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · SEE THE DISCRETE TAIL
1. Why an exact tail can be conservative
An exact test calculates probabilities under a stated null distribution. In a discrete sample space, however, the tail probability moves in jumps. If an observed outcome has strict-tail probability 0.032 and its own probability mass is 0.041, the ordinary inclusive exact p-value is 0.073. A 5% rule cannot reject, even though the evidence lies just beyond the strict tail.
This conservatism is not an error in the calculation. It comes from treating the whole observed mass as at least as extreme. The same lattice appeared in the companion guide on randomised exact tests. A randomised test can split the boundary operationally and achieve exact size. A mid-p test instead splits that mass in the reported p-value.
The distinction matters. “Exact” may describe the null probabilities while the decision rule remains conservative. A learner should therefore ask which tail was used, how ties were ordered and whether the observed mass was included fully, partly or through explicit randomisation.
CHAPTER 2 OF 12 · SEE THE DISCRETE TAIL
2. The mid-p definition and its meaning
For a one-sided test statistic where larger values are more extreme, write p> for the null probability of outcomes strictly more extreme than the observation and p= for the probability of outcomes tied with it. The ordinary inclusive p-value is p> + p=. The mid-p value is p> + 0.5p=.
The factor one half is a convention for centring a discrete p-value. It is not a data-dependent tuning parameter. Lancaster proposed the mid-probability idea as an alternative to auxiliary randomisation in discrete tests. It often reduces the excessive conservatism of an inclusive tail, but it does not ensure that rejection probability is at most the nominal level for every null parameter value.
That limit should travel with every calculation: mid-p is calculated from exact discrete probabilities, yet it is not an exact-level test in the strict repeated-sampling sense. Its attraction is a practical size–power compromise, not a theorem that it dominates every alternative.
CHAPTER 3 OF 12 · WORK THE MID-P LEDGER
3. Worked example: split the observed mass
Suppose a one-sided null distribution gives strict-tail mass 0.028 and observed-outcome mass 0.036. The inclusive exact p-value is 0.064. The mid-p value is 0.028 + 0.5 × 0.036 = 0.046. At a nominal 5% threshold, the two procedures reach different decisions.
| Quantity | Probability | Inclusive exact treatment | Mid-p treatment |
|---|---|---|---|
| Strictly more extreme outcomes | 0.028 | 0.028 | 0.028 |
| Observed outcome or tied class | 0.036 | 0.036 | 0.018 |
| Reported p-value | — | 0.064 | 0.046 |
Nothing in the data changed. The ordering and null model are identical; only the treatment of the observed mass differs. Reporting “p = 0.046” without the word mid-p hides that choice. A defensible result names the statistic, conditioning rule, tail convention and mid-p definition.
CHAPTER 4 OF 12 · WORK THE MID-P LEDGER
4. Size and power are properties of rules
A p-value is observed evidence; size and power describe a decision rule over repeated samples. To study the 5% mid-p rule, list every possible outcome under each null parameter, calculate its mid-p value and sum the null probabilities of outcomes whose mid-p is at most 0.05. That sum is the actual Type I error probability at that parameter.
Because the sample space is discrete, actual size can move above and below 0.05 across null parameter values. The ordinary inclusive exact rule is usually conservative; the mid-p rule usually rejects more often and can therefore have greater power. The cost is that its pointwise Type I error may exceed the nominal level.
The useful question is not “Which p-value is correct?” in isolation. It is “Which operating property matters for this decision, and has the method’s error behaviour been evaluated for the design and parameter range at hand?”
CHAPTER 5 OF 12 · WORK THE MID-P LEDGER
5. Two-sided tails need an explicit rule
For a one-sided ordered statistic, “more extreme” may be clear. A two-sided discrete test introduces another choice. Possible rules include doubling the smaller one-sided tail, ordering outcomes by null probability, or using a statistic such as an odds-ratio distance. These rules can select different sets of tables.
A mid-p adjustment does not resolve this ambiguity. It halves the probability attached to the observed tie class after an ordering has been chosen. Two analysts can both say “two-sided mid-p” and still obtain different answers if their extremeness rules differ.
Write the algorithm, not merely the label. Software documentation, conditional margins and the treatment of equally likely tables belong in the method record. This makes the result reproducible and stops a favourable tail rule from being chosen after seeing the data.
CHAPTER 6 OF 12 · TRANSFER THE DECISION
6. Changed context: a different lattice
Now suppose the strict-tail mass is 0.011 and the observed mass is 0.052. The inclusive p-value is 0.063 and the mid-p value is 0.037. The numerical gap is larger because the boundary mass is larger. Mid-p decisions are therefore sensitive to how coarse the null distribution is around the observation.
| Context | Strict tail p> | Observed mass p= | Inclusive p | Mid-p |
|---|---|---|---|---|
| Original example | 0.028 | 0.036 | 0.064 | 0.046 |
| Coarser boundary | 0.011 | 0.052 | 0.063 | 0.037 |
The transfer lesson is to inspect both components. Quoting only the final p-value makes it impossible to see whether the result comes from a strong strict tail or a large halved boundary. In sparse tables, that distinction can be material.
CHAPTER 7 OF 12 · TRANSFER THE DECISION
7. Mid-p is not the same as randomisation
A randomised exact test uses an independent random variable to decide whether to reject on a boundary and chooses its mixing probability to achieve a target size. Mid-p gives every observed tie class a fixed half-weight in the p-value. These are related ways of handling discreteness, but they answer different operational questions.
Mid-p does not require a coin flip in the realised decision. Randomisation does. Conversely, mid-p does not generally deliver an exact 5% long-run rejection rate. Randomisation can, when its boundary probability is calculated correctly for the null model.
This is a common misconception worth correcting explicitly: a mid-p of 0.046 is not a hidden claim that the test randomly rejected this observation with probability one half. It is a deterministic summary built from half the observed null mass.
CHAPTER 8 OF 12 · TRANSFER THE DECISION
8. Confidence intervals and repeated use
Inverting one-sided mid-p tests can produce mid-p confidence limits. These intervals are often shorter than conservative exact intervals. Their coverage, like the size of the corresponding tests, can oscillate around the nominal level rather than remaining above it for every parameter value.
Repeated applications add another layer. A method with mild pointwise size excess can produce more false positives when many hypotheses are screened. Mid-p does not replace multiplicity control, model checking or pre-specification. The decision workflow must still reflect the number and dependence of tests.
Use simulation or exact enumeration when the design is consequential. Record the parameter grid, null model, ordering and rejection threshold. Average performance over a convenient grid should not be mistaken for a worst-case guarantee.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. What the method can and cannot claim
Mid-p can claim that it uses the exact discrete null masses and halves the observed mass by a declared convention. It can often reduce conservatism and improve power relative to an inclusive exact rule. It can be a sensible compromise when strict conservatism has a meaningful cost.
It cannot claim pointwise exact size without a separate proof. It cannot rescue a misspecified sampling model, post-hoc stopping rule or biased data collection process. Nor can it turn an ambiguous two-sided ordering into an objective fact.
A practical evidence check is to report the inclusive p, mid-p and, where relevant, the randomised boundary calculation together. When their decisions differ, the result is method-sensitive and deserves interpretation rather than a single unqualified threshold label.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Evidence and verification route
The primary historical source is H. O. Lancaster’s 1961 paper Significance Tests in Discrete Distributions, which describes the mid-probability alternative to auxiliary randomisation. For a focused review of interval inversion, see Mid-p Confidence Intervals: A Brief Review.
Comparative work such as Comparison of exact tests for association in unordered contingency tables evaluates standard, mid-p and randomised versions rather than assuming one universal ranking. These sources support the main lesson: the power gain and size behaviour must be assessed together.
For verification, reproduce the probability ledger by enumeration, check that all null masses sum to one, state the tail ordering and compare the observed decision with neighbouring outcomes. A software result without this audit trail is difficult to transfer safely.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Is a mid-p value an approximation?
Its component probabilities may be computed exactly, but the half-mass convention relaxes the conservative exact-level guarantee. “Exact probabilities” and “exact size” are different claims.
Why use one half?
It centres the observed jump and has a long statistical history. It is a convention with useful average properties, not a fitted fraction chosen from the data.
Should mid-p always replace Fisher’s exact p-value?
No. The choice depends on the cost of conservatism, required error guarantees, conditioning assumptions and how the method performs for the actual design.
What should be reported?
Name the statistic, null distribution, conditioning, tail ordering, mid-p formula, nominal threshold and any size or coverage study used to justify the rule.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Final transfer check and connected reading
For a new discrete test, make a four-line ledger: strict-tail mass, observed tie mass, inclusive p-value and mid-p value. Then enumerate the decision rule under the null. If the mid-p rule exceeds the nominal size at some parameter values, decide whether that trade-off is acceptable before looking at the substantive conclusion.
Change the observed mass while keeping the strict tail fixed. If the mid-p decision changes, explain why the coarseness of the lattice—not new evidence beyond the observation—drove the change. This transfer check reveals whether the learner understands the mechanism.
Continue with the Probability root, the specialist guide to randomised exact tests and boundary mixing, the Gold Standard of Education and the How X Works hub.
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