Why the Boundary Can Determine the Error Rate
A one-sided trial decision often declares benefit when the data look sufficiently favorable. If the true treatment effect moves toward benefit, favorable data may become more likely. Within the null region, that trend means false-positive probability rises toward the null value nearest the alternative. When the trend is guaranteed by the model and decision rule, the boundary gives the largest Type I error. The word when matters: a boundary check is justified by monotonicity, not merely by convention.
This tutorial studies a scalar parameter \(\theta\), a one-sided null \(H_0:\theta\leq\theta_0\), and an alternative \(H_a:\theta>\theta_0\). A Bayesian procedure supplies a decision rule; Type I error is then evaluated by repeated sampling under parameter values in the null region. This operating-characteristic probability is distinct from the posterior probability used to make the decision.
The key condition is monotonicity of the repeated-sampling rejection probability. A useful sufficient condition is that there is a statistic \(T\) such that the rule declares benefit for larger values of \(T\), and the sampling distribution of \(T\) shifts upward as \(\theta\) increases. Formally, if \(P_\theta(T\geq c)\) is nondecreasing in \(\theta\) for every relevant cutoff \(c\), then any fixed upper-tail rule \(D=\{T\geq c\}\) has nondecreasing rejection probability. For a discrete statistic, “larger” includes the rule’s actual integer cutoff; no continuity approximation is required.
For example, with independent binary outcomes having common response probability \(p\), the response count is stochastically increasing in \(p\). If benefit is declared for a count at least \(c\), then \(P_p(X\geq c)\) increases with \(p\). For null values this is the false-positive probability; for values above the null boundary it is the probability of declaring benefit under the alternative, commonly called power. The same increasing probability has different interpretations on the two sides of the boundary.
A Short Monotonicity Justification
For a binomial response count, monotonicity can be shown by constructing outcomes at two parameter values on the same probability space. This argument also makes clear why the decision rule must be an upper-tail rule.
Take independent \(U_1,\ldots,U_n\sim\mathrm{Uniform}(0,1)\). For each response probability \(p\), define \(Y_i(p)=1\) if \(U_i\leq p\), and \(Y_i(p)=0\) otherwise. Then \(X(p)=\sum_{i=1}^nY_i(p)\sim\mathrm{Binomial}(n,p)\).
If \(p_1\leq p_2\), then \(Y_i(p_1)\leq Y_i(p_2)\) for every \(i\), so \(X(p_1)\leq X(p_2)\) for every realization of the shared uniforms.
For a rule \(D=\{X\geq c\}\), \(X(p_1)\geq c\) implies \(X(p_2)\geq c\). Thus \(P_{p_1}(D)\leq P_{p_2}(D)\), and the largest rate over \(p\leq p_0\) occurs at \(p_0\).
The conclusion follows because, under this coupling, rejection at the smaller parameter value is contained in rejection at the larger one. Therefore its probability cannot be larger. The proof does not say that every Bayesian rule is monotone; it establishes the claim for a rule whose decision event has this upper-tail form under the stated sampling model.
Worked Binomial Example: From a Posterior Rule to a Count Cutoff
Consider a hypothetical single-arm trial with \(n=20\) patients and binary response. Assume patient responses are independent Bernoulli outcomes with common true response probability \(p\), and let \(X\) be the number of responses. The trial declares benefit if the posterior probability that \(p\) exceeds \(p_0=0.30\) is at least \(0.95\). Use a uniform prior \(p\sim\mathrm{Beta}(1,1)\). These assumptions specify both the posterior calculation and the repeated-sampling model used to assess Type I error.
Worked Example: Finding the Decision Threshold
After observing \(X=x\), conjugacy gives \(p\mid X=x\sim\mathrm{Beta}(x+1,21-x)\). For this uniform prior, the beta-binomial identity gives \(P(p>0.30\mid X=x)=P(Y\leq x)\), where \(Y\sim\mathrm{Binomial}(21,0.30)\). The rule therefore declares benefit when this cumulative probability is at least \(0.95\).
At \(x=9\), \(P(Y\leq9)\approx0.93243\), which is below \(0.95\). At \(x=10\), \(P(Y\leq10)\approx0.97361\), which is at least \(0.95\). Since the binomial cumulative probability \(P(Y\leq x)\) cannot decrease as \(x\) increases, the smallest qualifying count is 10. Thus the complete decision rule is \(D=\{X\geq10\}\).
The rule is an upper-tail count rule, and \(X\sim\mathrm{Binomial}(20,p)\) under the assumed sampling model. The coupling argument therefore applies: \(P_p(D)\) is nondecreasing in \(p\). For the composite null \(p\leq0.30\), its maximum is at \(p=0.30\).
Worked Example: Exact Type I Error at the Boundary
At the boundary \(p=0.30\), the count has distribution \(X\sim\mathrm{Binomial}(20,0.30)\). The boundary false-positive probability is
Thus the rule’s Type I error rate over \(p\leq0.30\) is approximately \(0.0480\), or \(4.80\%\). This interpretation concerns repeated trials whose true response probability is in the null region, with the largest rate attained at \(p=0.30\). It is not the posterior probability that the null hypothesis is true.
For comparison, at \(p=0.25\), the probability of reaching at least 10 responses is smaller. That probability is still a false-positive rate because \(0.25\) is in the null region. At \(p=0.35\), the same event probability is instead the power at that alternative value.
Checking the Boundary With Simulation
A simulation can check whether estimated rejection rates across null values behave as the monotonicity argument predicts. For each fixed \(p\), generate many independent trial datasets from the specified binomial model, apply the same decision rule, and record whether each replicate declares benefit. If \(M\) is the number of replicates and \(R\) is the number of declarations, the estimated rate is \(\hat{q}=R/M\), with Monte Carlo standard error \(SE=\sqrt{\hat{q}(1-\hat{q})/M}\). This standard error describes simulation noise, not uncertainty about the trial’s true response probability.
Worked Example: Comparing Simulated Null Values
For illustration, suppose a hypothetical simulation uses \(M=200{,}000\) independent replicates at each value of \(p\), generates \(X\sim\mathrm{Binomial}(20,p)\), and applies the rule \(X\geq10\). The null values below are all at or below \(0.30\); the replicate counts and estimates are illustrative simulated output.
| True \(p\) | Declarations \(R\) | Estimated rate \(R/M\) | Approximate Monte Carlo \(SE\) |
|---|---|---|---|
| 0.20 | 520 | 0.00260 | 0.000114 |
| 0.25 | 2,770 | 0.01385 | 0.000261 |
| 0.28 | 6,110 | 0.03055 | 0.000385 |
| 0.30 | 9,610 | 0.04805 | 0.000478 |
For example, at \(p=0.28\), \(\hat{q}=6110/200000=0.03055\), and \(SE=\sqrt{0.03055(1-0.03055)/200000}\approx0.000385\). The displayed estimates increase across the null values and the boundary estimate is close to the exact value \(0.0479608\). Small differences from exact probabilities are expected from Monte Carlo variation. This grid supports the boundary conclusion but, by itself, cannot prove monotonicity at every possible null value; the coupling argument provides that justification for this model and rule.
A practical check should also verify that the simulation implements the intended data-generating mechanism and decision rule. Here each replicate must contain 20 independent Bernoulli responses with the same \(p\), and benefit must be declared exactly when \(X\geq10\). A wrong cutoff, a changed sample size, or dependence between outcomes would invalidate the comparison.
When Boundary Evaluation Is and Is Not Enough
The boundary result is valuable because it can replace an exhaustive search over a continuum of null values with one evaluation, provided the assumptions are defensible. It also gives a useful simulation diagnostic: a pronounced decrease in estimated rejection rates as a one-sided null parameter approaches its boundary may signal Monte Carlo noise, a coding error, or a failure of the assumed monotonicity.
There are two main ways to assess the error rate. Exact calculation evaluates the sampling probability directly when the model and decision event allow it, as in the binomial tail above. Simulation estimates the same probability by repeated draws and remains usable when exact enumeration is impractical. Exact calculation avoids Monte Carlo error but depends on a tractable model; simulation is flexible but has finite-replicate uncertainty. When a reliable exact benchmark exists, comparing simulation with it is a useful implementation check.
Boundary-only evaluation is not justified merely because the null is written as an inequality. It can fail when the rejection region is not an upper-tail set in a stochastically increasing statistic, when the probability model does not shift monotonically with the parameter, or when additional nuisance parameters affect rejection probability. A two-sided rule may also have its largest null rejection probability away from the boundary. In such cases, evaluating only the nearest null value could understate the maximum Type I error.
- Check the direction. Confirm that larger values of the relevant statistic favor the alternative and that the rule’s rejection event is ordered accordingly.
- Check the model. Confirm that the sampling distribution is stochastically increasing in the parameter over the full null region.
- Separate error from power. A declaration probability is Type I error only at null parameter values; above the boundary it is power.
- Use simulation as a check, not a proof. A finite grid can miss a local maximum between grid points. A mathematical monotonicity argument or sufficiently thorough assessment of the full null region is needed to justify the supremum.
A common mistake is to reason, “the boundary is the most favorable null value, so it must have the largest false-positive probability.” Favorability alone is not a proof. The correction is to identify a monotone statistic and decision event, establish stochastic ordering under the model, and then conclude that \(P_\theta(D)\leq P_{\theta_0}(D)\) for every \(\theta\leq\theta_0\). If any part of that chain fails, the boundary may not govern.
Check Your Understanding
Answer each question, then compare with the solutions.
- A rule declares benefit when \(X\geq8\), where \(X\sim\mathrm{Binomial}(12,p)\), and the null is \(p\leq0.40\). Under the independent common-probability binomial model, where is the largest Type I error rate attained?
- For the rule in question 1, if the true value is \(p=0.45\), is \(P_{0.45}(X\geq8)\) a Type I error rate or power?
- Suppose a simulation at a null value has \(M=50{,}000\) replicates and \(R=1{,}500\) declarations. Calculate the estimated rate and its Monte Carlo standard error.
- Give one reason a boundary-only evaluation might fail for a one-sided null.
Solutions. (1) The event is an upper-tail count event and the binomial count is stochastically increasing in \(p\). Therefore the rejection probability is nondecreasing in \(p\), so the largest rate over \(p\leq0.40\) is attained at \(p=0.40\). (2) Since \(0.45>0.40\), the probability of declaring benefit is power at \(p=0.45\), not Type I error. (3) The estimate is \(\hat{q}=1500/50000=0.0300\). Its Monte Carlo standard error is \(\sqrt{0.0300(1-0.0300)/50000}\approx0.000763\). (4) For example, the rule may not be an upper-tail event in a stochastically increasing statistic, or nuisance parameters may cause the rejection probability to peak away from the boundary. In either case, monotonicity over the null region has not been established.