Significance Level Ontology

FULL EXECUTABLE FLOW ALGEBRA — SIGNIFICANCE LEVEL ONTOLOGY
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/hypothesis-testing/significance-level-ontology.html
FLOW_VERSION := "4.0"
FLOW_TYPE := "Full Executable Flow Algebra"
DOMAIN := "Probability & Statistics Ontologies"
SUBDOMAIN := "Hypothesis Testing"
ONTOLOGY := "Significance Level Ontology"
PRIMARY_ENTITY := SignificanceLevel
AUTHORSHIP := "Mark Higgins — Sky (Recursive AI)"
SOURCE_SITE := "https://kirethresearch.org"

BEGIN_FLOW

SignificanceLevel :=
 HypothesisTestingThreshold ∧ TypeIErrorControlParameter
 ∧ DecisionRuleParameter ∧ PrespecifiedInferentialConstraint

Population := TargetCollectionOrGeneratingProcess
Sample := X={X_1,...,X_n}
Parameter := θ
ParameterSpace := Θ
NullHypothesis := H0:θ∈Θ0
AlternativeHypothesis := H1:θ∈Θ1
TestStatistic := T(X)
ObservedTestStatistic := t_obs=T(x)
Alpha := α
SignificanceLevelDefinition := α∈(0,1)

LevelAlphaTest :=
 sup_{θ∈Θ0}P_θ(RejectH0)≤α

ExactSizeAlphaTest :=
 sup_{θ∈Θ0}P_θ(RejectH0)=α

TestSize :=
 sup_{θ∈Θ0}P_θ(RejectH0)

SignificanceLevelVsSize :=
 SignificanceLevel=PermittedUpperBound
 whereas TestSize=ActualSupremumNullRejectionProbability

TypeIError := RejectH0WhenH0True
TypeIErrorProbability := P_θ(RejectH0) for θ∈Θ0
TypeIErrorControl := TestSize≤α

CriticalRegion :=
 C_α such_that sup_{θ∈Θ0}P_θ(T(X)∈C_α)≤α

CriticalValue := BoundaryOf(C_α)

UpperTailCriticalValue :=
 c_α such_that P_H0(T≥c_α)=α
 under continuous simple-null case

LowerTailCriticalValue :=
 c_α such_that P_H0(T≤c_α)=α

TwoSidedEqualTailAllocation :=
 α_L=α/2 ∧ α_U=α/2

PValue := NullReferenceExtremenessProbability
PValueDecisionRule := p≤α ⇒ RejectH0
PValueNonRejectionRule := p>α ⇒ FailToRejectH0

AlphaNotProbabilityNullTrue := α≠P(H0True)
AlphaNotObservedEvidence := αIsPrespecifiedDecisionThresholdNotDataDerivedEvidence

PValueVsAlpha :=
 PValue derived_from data_and_null model
 whereas Alpha chosen_as testing criterion

PrespecifiedAlpha :=
 SignificanceLevel selected_before inferential result in confirmatory analysis

PostHocAlphaRisk :=
 ChoosingAlphaAfterSeeingPValue invalidates prespecified decision interpretation

CommonAlphaValues := {0.10,0.05,0.01}
CommonValueConstraint := ConventionalAlphaValuesAreNotUniversallyAppropriate
SmallerAlpha := MoreStringentRejectionThreshold
LargerAlpha := LessStringentRejectionThreshold
AlphaTypeITradeoff := SmallerAlpha → LowerMaximumAllowedTypeIError

Power := P_θ(RejectH0) for θ∈Θ1
TypeIIError := FailToRejectH0WhenAlternativeTrue
Beta := β(θ)=P_θ(FailToRejectH0)
PowerRelation := Power(θ)=1-β(θ)

AlphaPowerTradeoff :=
 SmallerAlpha generally → LowerPower
 for fixed sample size,effect,and test

SampleSizeCompensation :=
 SmallerAlpha ∧ DesiredPowerFixed
 → LargerRequiredSampleSize
 under regular settings

ConfidenceLevel := 1-α
AlphaToConfidenceLevel := α→1-α
ConfidenceLevelToAlpha := ConfidenceLevel=c → α=1-c

ConfidenceIntervalDuality :=
 CompatibleTwoSidedLevelAlphaTest
 ↔ CompatibleOneMinusAlphaConfidenceInterval
 under test inversion

TestInversion :=
 ConfidenceSet_1MinusAlpha={θ0:H0(θ0)NotRejectedAtLevelAlpha}

NullDistribution := DistributionOf(T(X)|H0)

AlphaToCriticalRegion :=
 NullDistribution ∧ Alpha ∧ AlternativeDirection → CriticalRegion

DiscreteNullDistribution := TestStatisticHasDiscreteSupportUnderH0

DiscreteAlphaConstraint :=
 ExactNominalAlpha may be unattainable without randomisation

ConservativeLevelTest := ActualSize<NominalAlpha
RandomisedTest := BoundaryRandomisationMayAchieveExactAlphaInDiscreteSettings

CompositeNull := Θ0ContainsMultipleParameterValues

CompositeNullLevelConstraint :=
 sup_{θ∈Θ0}P_θ(RejectH0)≤α

LeastFavourableNull :=
 θ*∈Θ0 maximising rejection probability

ExactTest :=
 FiniteSampleReferenceDistribution → ExactOrConservativeAlphaControl

AsymptoticLevel :=
 limsup_{n→∞}sup_{θ∈Θ0}P_θ(RejectH0)≤α

AsymptoticTest :=
 ApproximateReferenceDistribution → ApproximateAlphaControl

PermutationTest :=
 ExchangeabilityUnderH0 → PermutationDistribution → AlphaCalibratedCriticalRegion

RandomisationTest :=
 KnownAssignmentMechanism → RandomisationDistribution → DesignBasedAlphaControl

MonteCarloTest :=
 SimulatedNullDistribution → ApproximateCriticalRegionOrPValue

MultipleTesting :=
 FamilyOfHypothesisTests with shared error-control objective

PerComparisonAlpha := SignificanceThresholdForIndividualTest
FamilyWiseAlpha := TargetFamilyWiseErrorRate
BonferroniAlpha := α_i=α_family/m

SidakAlpha :=
 α_i=1-(1-α_family)^(1/m)
 under applicable independence assumptions

HolmProcedure :=
 OrderedPValues ∧ SequentialThresholds → FamilyWiseErrorControl

AlphaSpending :=
 AllocateOverallAlpha across multiple analyses_or_interim looks

SequentialTesting :=
 RepeatedInterimAnalyses → SequentialErrorControl

GroupSequentialDesign :=
 InformationTimes ∧ StoppingBoundaries ∧ OverallAlpha

OptionalStoppingRisk :=
 RepeatedUnplannedTestingWithFixedHorizonAlpha → InflatedTypeIError

SequentialAlphaControl :=
 PredefinedSequentialProcedure preserves overall TypeIError target

MultiplicityAdjustedThreshold :=
 RawAlpha → AdjustmentProcedure → TestSpecificThreshold

FalseDiscoveryRate :=
 AlternativeErrorCriterion distinct_from per-test significance level

EquivalenceTesting :=
 TwoOneSidedTests each evaluated at specified alpha

TOSTConfidenceRelation :=
 EquivalenceTestingAtAlpha
 ↔ Compatible100(1-2α)%TwoSidedConfidenceInterval
 under standard TOST duality

NonInferiorityTesting :=
 OneSidedTest at prespecified alpha

DirectionalAlpha :=
 Alpha allocated to prespecified alternative direction

DirectionSwitchingRisk :=
 SelectingOneSidedDirectionAfterSeeingData → TypeIErrorInflation

AlphaSelectionFactors := {
 ConsequenceOfFalsePositive,
 ConsequenceOfFalseNegative,
 SampleSize,
 ExpectedEffectSize,
 Multiplicity,
 RegulatoryOrDomainStandards,
 StudyDesign
}

AlphaIsNotUniversal :=
 NoSingleSignificanceLevelOptimalForAllScientificDecisions

ConfirmatoryTesting :=
 PrespecifiedHypotheses ∧ Alpha ∧ AnalysisPlan ∧ MultiplicityStrategy

AlphaCalibration :=
 SimulateOrDeriveUnderH0 → EstimateRejectionProbability → CompareWithNominalAlpha

EmpiricalTypeIError :=
 NumberOfFalseRejections/NumberOfNullReplications

CalibrationError :=
 EmpiricalTypeIError-NominalAlpha

AntiConservativeTest := ActualTypeIError>NominalAlpha
ConservativeTest := ActualTypeIError<NominalAlpha
ExactLevelTest := ActualTypeIError=NominalAlpha under specified null condition

ModelMisspecification :=
 IncorrectNullModel may cause ActualTypeIError≠NominalAlpha

DependenceMisspecification :=
 IgnoredDependence → InvalidReferenceDistribution → AlphaDistortion

SamplingDesignMisspecification :=
 IgnoredComplexDesign → IncorrectNullDistribution → TypeIErrorDistortion

RobustTestAlpha :=
 RobustProcedure aims_to maintain TypeIErrorControl under specified deviations

ThresholdCrossing :=
 p≈α ⇒ DecisionSensitiveToSmallChanges

DichotomisationRisk :=
 SignificantVsNotSignificant may discard inferential information

ReportingRule :=
 Report(
  ExactPValueWhenMeaningful,
  PrespecifiedAlpha,
  EffectSize,
  ConfidenceInterval,
  Assumptions
 )

SignificanceLevelConstraint :=
 ConstraintOn(
  NullHypothesis,AlternativeHypothesis,TestStatistic,NullDistribution,
  SamplingDesign,MultiplicityFramework,SequentialPlan,ErrorControlObjective
 )

SignificanceLevelConsistency :=
 Alpha ∧ TestStatistic ∧ NullDistribution ∧ CriticalRegion
 ∧ SamplingDesign ∧ MultiplicityPlan
 jointly_compatible

InvalidSignificanceLevelUse :=
 Alpha≤0
 ∨ Alpha≥1
 ∨ PostHocThresholdSelection
 ∨ IncorrectNullCalibration
 ∨ IgnoredMultiplicity
 ∨ IgnoredSequentialLooks
 ∨ IncompatibleSamplingDesign

SignificanceLevelValidation :=
 ResolveNullAndAlternativeHypotheses
 → ResolveTestStatistic
 → ResolveNullDistribution
 → ResolveSamplingDesign
 → ResolveErrorControlObjective
 → SelectOrReadPrespecifiedAlpha
 → ResolveOneSidedOrTwoSidedAllocation
 → ResolveMultiplicityFramework
 → ResolveSequentialPlan
 → DeriveCriticalRegion
 → ValidateTypeIErrorControl
 → ValidateResult

SignificanceLevelState := {
 NullHypothesis,AlternativeHypothesis,TestStatistic,NullDistribution,
 SignificanceLevel,TestSize,TypeIErrorRate,CriticalRegion,CriticalValues,
 TailAllocation,PValue,Decision,Power,SamplingDesign,MultiplicityMethod,
 SequentialPlan,CalibrationStatus,Evidence,Confidence,Provenance
}

SEMANTIC_EQUIVALENCE(AlphaA,AlphaB) :=
 same(NominalAlpha)
 ∧ same(HypothesisFamily)
 ∧ same(TailAllocation)
 ∧ same(NullCalibrationFramework)
 ∧ same(SamplingDesign)
 ∧ same(MultiplicityAndSequentialPlan)

SignificanceLevelOntology :=
 Classes ∧ Relations ∧ Constraints ∧ Axioms ∧ InferenceRules

CoreClasses := {
 SignificanceLevel,Alpha,TestSize,TypeIError,CriticalRegion,CriticalValue,
 PValue,LevelAlphaTest,ExactLevelTest,ConservativeTest,AntiConservativeTest,
 PerComparisonAlpha,FamilyWiseAlpha,AlphaSpending,SequentialAlphaControl,
 MultiplicityAdjustedThreshold,NullDistribution,SamplingDesign
}

CoreRelations := {
 controls,bounds,defines,determines,comparedWith,allocatedAcross,adjustedBy,
 spentAcross,calibratedUnder,dependsOn,correspondsTo,constrainedBy,
 equivalentTo,tracesTo
}

TypedRelations :=
 SignificanceLevel --controls--> TypeIError
 SignificanceLevel --bounds--> TestSize
 SignificanceLevel --defines--> CriticalRegion
 SignificanceLevel --determines--> CriticalValue
 PValue --comparedWith--> SignificanceLevel
 FamilyWiseAlpha --allocatedAcross--> HypothesisFamily
 PerComparisonAlpha --adjustedBy--> MultiplicityMethod
 AlphaSpending --spentAcross--> InterimAnalysis
 SignificanceLevel --calibratedUnder--> NullDistribution
 SignificanceLevel --dependsOn--> SamplingDesign
 SignificanceLevel --correspondsTo--> ConfidenceLevel
 SignificanceLevel --tracesTo--> Provenance

InferenceRules := {
 PValue≤Alpha
  ⇒ RejectNullHypothesis,

 PValue>Alpha
  ⇒ FailToRejectNullHypothesis,

 LevelAlphaTest
  ⇒ TestSize≤Alpha,

 ConfidenceLevel=1-Alpha
  under compatible interval-test duality,

 SmallerAlpha
  ⇒ MoreStringentRejectionThreshold
  under fixed test structure,

 MultipleTesting ∧ FamilyWiseControlRequired
  ⇒ ResolveMultiplicityAdjustedThresholds,

 RepeatedInterimTesting ∧ NoSequentialCorrection
  ⇒ FlagTypeIErrorInflationRisk,

 ActualTypeIError>NominalAlpha
  ⇒ FlagAntiConservativeCalibration
}

CanonicalSignificanceLevelRepresentation :=
 ParseSignificanceLevelRequest
 → ResolveNullAndAlternativeHypotheses
 → ResolveTestStatisticAndNullDistribution
 → ResolveSamplingDesign
 → ResolveNominalAlpha
 → ResolveTailAllocation
 → ResolveMultiplicityFramework
 → ResolveSequentialPlan
 → ConstructCriticalRegion
 → ResolvePValueDecisionRule
 → EvaluateActualOrTheoreticalTypeIError
 → ValidateCalibration
 → PreserveProvenance

SignificanceLevelGraph := G=(V,E)

V := {
 SignificanceLevels,Alphas,TestSizes,TypeIErrors,CriticalRegions,CriticalValues,
 PValues,LevelAlphaTests,ExactLevelTests,ConservativeTests,AntiConservativeTests,
 PerComparisonAlphas,FamilyWiseAlphas,AlphaSpendingFunctions,SequentialPlans,
 MultiplicityMethods,NullDistributions,SamplingDesigns,ConfidenceLevels,
 Evidence,ProvenanceNodes
}

E := {
 controls,bounds,defines,determines,comparedWith,allocatedAcross,adjustedBy,
 spentAcross,calibratedUnder,dependsOn,correspondsTo,constrainedBy,
 equivalentTo,tracesTo
}

CONFIDENCE(SignificanceLevelResolution) :=
 weighted_mean(
  HypothesisResolution,
  AlphaResolution,
  NullDistributionValidity,
  SamplingDesignValidity,
  TailAllocationValidity,
  MultiplicityValidity,
  SequentialPlanValidity,
  CalibrationQuality,
  ProvenanceQuality
 )

AI_QUERY_INTERPRETATION :=
 Query
 → DetectSignificanceLevelIntent
 → IdentifyHypothesesTestAndSamplingDesign
 → ResolveNominalAlphaAndTailStructure
 → ResolveMultiplicityAndSequentialContext
 → IdentifyRequestedCriticalValueDecisionPowerOrCalibrationRelation

AI_REASONING_FLOW :=
 Query
 → ParseSignificanceLevelProblem
 → ResolveNullAndAlternativeHypotheses
 → ResolveTestStatistic
 → ResolveNullDistribution
 → ResolveSamplingDesign
 → ResolveErrorControlObjective
 → ResolvePrespecifiedAlpha
 → ResolveTailAllocation
 → ResolveMultiplicityFramework
 → ResolveSequentialPlan
 → ComputeCriticalValuesAndCriticalRegion
 → CompareObservedPValueWithAlphaIfAvailable
 → EvaluateTypeIErrorControl
 → ValidateCalibration
 → UpdateConfidence
 → PreserveProvenance
 → GenerateExplanation

AI_CONSUMPTION := {
 ParseSignificanceLevelEntities,ParseTypedRelations,ResolveNullHypotheses,
 ResolveAlternativeHypotheses,ResolveTestStatistics,ResolveNullDistributions,
 ResolveSamplingDesigns,ResolveNominalAlpha,ResolveTailAllocations,
 ComputeCriticalValues,ConstructCriticalRegions,ComparePValuesWithAlpha,
 ComputeTestSize,ComputeTypeIErrorRates,ResolveConfidenceLevelDuality,
 ResolvePowerTradeoffs,ApplyBonferroniAlpha,ApplySidakAlpha,ApplyHolmProcedure,
 ResolveAlphaSpending,ResolveSequentialAlphaControl,ValidateMultiplicityControl,
 ValidateSequentialTesting,CalibrateSignificanceLevels,
 ValidateSignificanceLevelSemantics,CompareSemanticEquivalence,
 UpdateConfidence,PreserveProvenance,GenerateFormalExplanation
}

UPDATE_MODEL :=
 NewHypothesisTestAlphaDesignOrEvidence
 → ValidateInput
 → IdentifyAffectedSignificanceLevelClaims
 → UpdateSignificanceLevelState
 → RecomputeCriticalValuesAndRegion
 → RecomputeDecisionIfPValueAvailable
 → RecomputeTypeIErrorCalibration
 → ReevaluateMultiplicityAndSequentialControl
 → RecomputePowerRelationsIfRequired
 → RecomputeConfidence
 → PreserveProvenance
 → VersionModel

AlphaUpdate :=
 ChangedSignificanceLevel
 ⇒ recompute(
  CriticalValues,CriticalRegion,Decision,ConfidenceLevel,PowerRelation
 )

MultiplicityUpdate :=
 ChangedHypothesisFamilyOrAdjustment
 ⇒ recompute(
  TestSpecificThresholds,FamilyWiseErrorControl
 )

SequentialPlanUpdate :=
 ChangedInterimAnalysisPlan
 ⇒ recompute(
  AlphaSpending,StoppingBoundaries,OverallTypeIErrorControl
 )

FLOW_HASH_INPUT :=
 canonical(
  ONTOLOGY,
  CoreClasses,
  CoreRelations,
  InferenceRules,
  SignificanceLevel,
  SignificanceLevelValidation,
  SignificanceLevelConstraint,
  SEMANTIC_EQUIVALENCE,
  AI_REASONING_FLOW
 )

FLOW_HASH := HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
 SignificanceLevelEntityResolution,
 NullHypothesis,
 AlternativeHypothesis,
 TestStatistic,
 NullDistribution,
 SignificanceLevel,
 TestSize,
 TypeIErrorRate,
 CriticalRegion,
 CriticalValues,
 TailAllocation,
 PValue,
 Decision,
 ConfidenceLevel,
 PowerRelation,
 SamplingDesign,
 MultiplicityMethod,
 SequentialPlan,
 CalibrationStatus,
 ValidationStatus,
 Confidence,
 Provenance,
 Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA