Null Hypothesis Ontology

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

BEGIN_FLOW

NullHypothesis :=
 StatisticalHypothesis ∧ ReferenceClaim ∧ NullParameterSpace
 ∧ BasisForTestStatisticDistribution ∧ BasisForTypeIErrorControl

Population := TargetCollectionOrGeneratingProcess
Sample := {X_1,...,X_n}
Parameter := θ
ParameterSpace := Θ
Estimand := τ(θ)
NullParameterSpace := Θ0⊆Θ
AlternativeParameterSpace := Θ1=Θ\Θ0

NullHypothesisDefinition := H0:θ∈Θ0
AlternativeHypothesisDefinition := H1:θ∈Θ1
HypothesisDisjointness := Θ0∩Θ1=∅
HypothesisExhaustiveness := Θ0∪Θ1=Θ when exhaustive

SimpleNullHypothesis := Θ0={θ0}
CompositeNullHypothesis := MultipleParameterValuesInΘ0
PointNullHypothesis := H0:θ=θ0
IntervalNullHypothesis := H0:θ∈[a,b]
OneSidedNullUpper := H0:θ≤θ0
OneSidedNullLower := H0:θ≥θ0
EqualityNull := H0:θ=θ0
NoDifferenceNull := H0:θ1-θ2=0
NoAssociationNull := H0:AssociationParameter=0
IndependenceNull := H0:VariablesAreIndependent
GoodnessOfFitNull := H0:DistributionBelongsToSpecifiedReferenceFamily
ModelRestrictionNull := H0:r(θ)=0
NestedModelNull := RestrictedModel⊂UnrestrictedModel

NullModel := StatisticalModel constrained_by H0
TestStatistic := T(X)
ObservedTestStatistic := t_obs=T(x)
NullDistribution := DistributionOf(T(X)|H0)

NullReferenceDistribution := {
 ExactNullDistribution,
 AsymptoticNullDistribution,
 ConditionalNullDistribution,
 RandomisationNullDistribution,
 PermutationNullDistribution,
 SimulatedNullDistribution
}

NullSimulation :=
 GenerateDataUnderH0 → ComputeTestStatisticReplicates → ApproximateNullDistribution

NullExpectation := E_H0[T]
NullVariance := Var_H0(T)
StandardisedNullStatistic := (T-E_H0[T])/sqrt(Var_H0(T))

SignificanceLevel := α
TypeIError := RejectH0WhenH0True
TypeIErrorProbability := P_θ(RejectH0) for θ∈Θ0
TestSize := sup_{θ∈Θ0}P_θ(RejectH0)
LevelAlphaConstraint := TestSize≤α

CriticalRegion := C={t:RejectH0}
CriticalRegionConstraint := sup_{θ∈Θ0}P_θ(T∈C)≤α
CriticalValue := BoundaryOfCriticalRegionDeterminedFromNullDistribution

PValue := NullBasedExtremenessProbabilityOfObservedStatistic
SimpleNullPValue := P_θ0(StatisticAtLeastAsExtremeAsObserved)

CompositeNullPValue :=
 RequiresNullWideCalibrationOrValidNuisanceHandling

PValueSemanticConstraint := PValue≠ProbabilityNullHypothesisTrue
RejectNull := ObservedStatisticInCriticalRegion ∨ PValue≤Alpha
FailToRejectNull := EvidenceInsufficientForRejectionAtSpecifiedAlpha
FailToRejectSemanticConstraint := FailToRejectH0 ↛ AcceptH0AsProvenTrue

AlternativeRelation := NullHypothesis contrasted_with AlternativeHypothesis
DirectionalAlternativeRelation := DirectionOfH1 determines tail_or_extremeness ordering
TwoSidedNullTest := EqualityNull versus TwoSidedAlternative
UpperTailedTest := H0:θ≤θ0 versus H1:θ>θ0
LowerTailedTest := H0:θ≥θ0 versus H1:θ<θ0

BoundaryNull := NullParameterValue lies_on Boundary(ParameterSpace)
BoundaryNullConstraint := StandardAsymptoticReference may fail at boundary

NuisanceParameter := ParameterAffectingNullDistributionButNotPrimaryTarget
NuisanceHandling := {Conditioning,Profiling,Maximisation,Integration,PlugInApproximation,Simulation}
CompositeNullCalibration := ControlTypeIErrorAcrossAllθInΘ0

LikelihoodUnderNull := L0=sup_{θ∈Θ0}L(θ|x)
UnrestrictedLikelihood := L1=sup_{θ∈Θ}L(θ|x)
LikelihoodRatio := Λ=L0/L1
LikelihoodRatioNullTest := SmallΛ ∨ LargeMinus2LogΛ → EvidenceAgainstH0
WaldNullTest := Distance(Estimate,NullRestriction)/EstimatorUncertainty
ScoreNullTest := EvaluateLikelihoodDerivativeAtNull

NullInConfidenceSetRelation :=
 CompatibleLevelAlphaTest
 ↔ CompatibleOneMinusAlphaConfidenceSet
 under test-inversion duality

TestInversion := ConfidenceSet={θ0:H0(θ0)NotRejected}

NullEffectValue := ParameterValueRepresentingReferenceEffect
NullEffectConstraint := ZeroIsNotUniversallyNullValue
RatioNullValue := RatioEffectMeasureOftenUsesReference1
DifferenceNullValue := DifferenceEffectMeasureOftenUsesReference0
CorrelationNullValue := CorrelationParameterMayUseReference0
RegressionCoefficientNull := H0:β_j=β_j0
MeanNull := H0:μ=μ0
ProportionNull := H0:p=p0
VarianceNull := H0:σ²=σ0²
DistributionNull := H0:F=F0 or F∈SpecifiedFamily
ANOVANull := H0:μ1=μ2=...=μk

EquivalenceNull := H0:EffectOutsideEquivalenceRegion
NonInferiorityNull := H0:EffectFailsNonInferiorityCriterion
SuperiorityNull := H0:EffectDoesNotExceedSuperiorityCriterion

NullSpecification :=
 ScientificQuestion → DefineEstimand → DefineReferenceClaim → TranslateToParameterConstraint

DataDependentNullRisk :=
 SelectingNullAfterObservingData
 may invalidate nominal inference unless selection accounted_for

MultipleNullHypotheses := {H0_1,...,H0_m}
MultiplicityRelation := MultipleNullHypotheses → DefineFamily → ControlChosenErrorRate
FamilyWiseError := ProbabilityOfRejectingAtLeastOneTrueNull
TrueNull := TrueParameterContainedInNullRestriction
FalseNull := TrueParameterExcludedFromNullRestriction
PowerAgainstFalseNull := P_θ(RejectH0) for θ∈Θ1

NullCalibration :=
 UnderH0 → SimulateOrDeriveRejectionRate → CompareWithAlpha

EmpiricalNullRejectionRate := NumberOfNullRejections/NumberOfNullReplications

ModelMisspecificationUnderNull :=
 IncorrectNullReferenceModel → PValueAndTypeIErrorDistortionRisk

SamplingDesignConstraint :=
 NullDistribution must reflect actual randomisation_or sampling design

DependenceConstraint :=
 DependenceStructure must be represented in null variance_or reference distribution

NullHypothesisConstraint :=
 ConstraintOn(
  Estimand,ParameterSpace,NullParameterSpace,AlternativeParameterSpace,
  StatisticalModel,SamplingDesign,TestStatistic,SignificanceLevel
 )

NullHypothesisConsistency :=
 Estimand ∧ NullRestriction ∧ AlternativeRestriction ∧ TestStatistic
 ∧ NullDistribution ∧ SamplingDesign
 jointly_compatible

InvalidNullHypothesis :=
 UndefinedEstimand
 ∨ EmptyNullParameterSpace
 ∨ OverlappingNullAndAlternativeWithoutExplicitFramework
 ∨ IncompatibleTestStatistic
 ∨ InvalidNullDistribution
 ∨ UnresolvedNuisanceParameter
 ∨ DataDependentSpecificationWithoutAdjustment

NullHypothesisValidation :=
 ResolveScientificQuestion
 → ResolvePopulationAndEstimand
 → ResolveParameterAndParameterSpace
 → DefineNullRestriction
 → DefineAlternativeRestriction
 → ValidateDisjointnessAndCoverage
 → ResolveStatisticalModel
 → ResolveSamplingDesign
 → ResolveTestStatistic
 → DeriveNullDistribution
 → ResolveNuisanceParameters
 → ValidateTypeIErrorControl
 → ValidateNullHypothesis

NullHypothesisState := {
 Population,Sample,Parameter,ParameterSpace,Estimand,NullHypothesis,
 NullParameterSpace,AlternativeHypothesis,AlternativeParameterSpace,
 StatisticalModel,SamplingDesign,TestStatistic,NullDistribution,
 SignificanceLevel,CriticalRegion,PValue,Decision,TypeIErrorControl,
 NuisanceParameters,Evidence,Confidence,Provenance
}

SEMANTIC_EQUIVALENCE(NullA,NullB) :=
 same(Estimand)
 ∧ same(NullParameterSet)
 ∧ same(ParameterisationOrEquivalentConstraint)
 ∧ same(StatisticalModelSemantics)
 ∧ same(SamplingDesign)
 ∧ same(TestCalibrationFramework)

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

CoreClasses := {
 NullHypothesis,SimpleNullHypothesis,CompositeNullHypothesis,PointNullHypothesis,
 IntervalNullHypothesis,NullParameterSpace,NullModel,NullDistribution,
 NullRestriction,CriticalRegion,SignificanceLevel,TypeIError,PValue,
 NuisanceParameter,BoundaryNull,Estimand,Parameter,StatisticalModel,SamplingDesign
}

CoreRelations := {
 constrains,targets,contrastsWith,defines,hasNullDistribution,testedBy,
 hasCriticalRegion,hasSignificanceLevel,hasTypeIError,hasPValue,contains,
 dependsOn,constrainedBy,equivalentTo,tracesTo
}

TypedRelations :=
 NullHypothesis --constrains--> Parameter
 NullHypothesis --targets--> Estimand
 NullHypothesis --contrastsWith--> AlternativeHypothesis
 NullHypothesis --defines--> NullParameterSpace
 TestStatistic --hasNullDistribution--> NullDistribution
 NullHypothesis --testedBy--> HypothesisTest
 HypothesisTest --hasCriticalRegion--> CriticalRegion
 HypothesisTest --hasSignificanceLevel--> SignificanceLevel
 HypothesisTest --hasTypeIError--> TypeIError
 HypothesisTest --hasPValue--> PValue
 NullParameterSpace --contains--> NullParameterValue
 NullDistribution --dependsOn--> StatisticalModel
 NullHypothesis --tracesTo--> Provenance

InferenceRules := {
 θ∈Θ0 ⇒ NullHypothesisTrueAtθ,
 θ∉Θ0 ⇒ NullHypothesisFalseAtθ,
 PValue≤Alpha ⇒ RejectNullHypothesis,
 PValue>Alpha ⇒ FailToRejectNullHypothesis,
 FailToRejectNullHypothesis ↛ NullHypothesisProven,
 CompositeNull ⇒ RequireNullWideTypeIErrorControl,
 BoundaryNull ⇒ CheckNonstandardAsymptotics,
 InvalidNullDistribution ⇒ FlagNullTestInvalid
}

CanonicalNullHypothesisRepresentation :=
 ParseNullHypothesisRequest
 → ResolveScientificQuestion
 → ResolvePopulationAndEstimand
 → ResolveParameterAndParameterSpace
 → ConstructNullParameterSpace
 → ConstructAlternativeParameterSpace
 → ValidateHypothesisPartition
 → ResolveStatisticalModelAndSamplingDesign
 → ResolveTestStatistic
 → DeriveNullDistribution
 → ResolveNuisanceParameters
 → ResolveAlphaAndCriticalRegion
 → ComputePValueIfDataProvided
 → ApplyDecisionRule
 → ValidateNullHypothesis

NullHypothesisGraph := G=(V,E)

V := {
 NullHypotheses,SimpleNullHypotheses,CompositeNullHypotheses,PointNullHypotheses,
 IntervalNullHypotheses,AlternativeHypotheses,NullParameterSpaces,
 AlternativeParameterSpaces,Parameters,Estimands,NullModels,NullDistributions,
 TestStatistics,CriticalRegions,SignificanceLevels,TypeIErrors,PValues,
 NuisanceParameters,SamplingDesigns,Evidence,ProvenanceNodes
}

E := {
 constrains,targets,contrastsWith,defines,hasNullDistribution,testedBy,
 hasCriticalRegion,hasSignificanceLevel,hasTypeIError,hasPValue,contains,
 dependsOn,constrainedBy,equivalentTo,tracesTo
}

CONFIDENCE(NullHypothesisResolution) :=
 weighted_mean(
  EstimandResolution,ParameterResolution,NullRestrictionValidity,
  AlternativeRestrictionValidity,StatisticalModelResolution,SamplingDesignResolution,
  NullDistributionValidity,TypeIErrorCalibration,ProvenanceQuality
 )

AI_QUERY_INTERPRETATION :=
 Query
 → DetectNullHypothesisIntent
 → IdentifyPopulationParameterAndEstimand
 → IdentifyReferenceClaimAndAlternative
 → ResolveSimpleCompositePointDirectionalOrIntervalNull
 → ResolveTestStatisticNullDistributionAndAlpha

AI_REASONING_FLOW :=
 Query
 → ParseNullHypothesisProblem
 → ResolveScientificQuestion
 → ResolvePopulationAndSample
 → ResolveParameterAndEstimand
 → DefineNullParameterSpace
 → DefineAlternativeParameterSpace
 → ValidateHypothesisStructure
 → ResolveStatisticalModel
 → ResolveSamplingDesign
 → SelectTestStatistic
 → DeriveNullDistribution
 → ResolveNuisanceParameters
 → ResolveSignificanceLevel
 → ComputePValueAndCriticalRegionIfDataProvided
 → ApplyDecisionRule
 → ValidateTypeIErrorControl
 → UpdateConfidence
 → PreserveProvenance
 → GenerateExplanation

AI_CONSUMPTION := {
 ParseNullHypothesisEntities,ParseTypedRelations,ResolvePopulations,ResolveSamples,
 ResolveParameters,ResolveEstimands,ResolveParameterSpaces,
 ConstructNullParameterSpaces,ConstructAlternativeParameterSpaces,
 ValidateHypothesisPartitions,ResolveStatisticalModels,ResolveSamplingDesigns,
 ResolveTestStatistics,DeriveNullDistributions,ResolveNuisanceParameters,
 ResolveSignificanceLevels,ResolveCriticalRegions,ComputePValues,ApplyDecisionRules,
 ValidateTypeIErrorControl,ResolveLikelihoodRatioNullTests,ResolveWaldNullTests,
 ResolveScoreNullTests,ResolveConfidenceSetDuality,ResolveMultiplicityFrameworks,
 ValidateNullHypothesisSemantics,CompareSemanticEquivalence,UpdateConfidence,
 PreserveProvenance,GenerateFormalExplanation
}

UPDATE_MODEL :=
 NewHypothesisDataModelDesignOrEvidence
 → ValidateInput
 → IdentifyAffectedNullHypothesisClaims
 → UpdateNullHypothesisState
 → ReconstructNullAndAlternativeParameterSpaces
 → RecomputeNullDistribution
 → RecomputeCriticalRegionAndPValueIfRequired
 → RevalidateTypeIErrorControl
 → RecomputeConfidence
 → PreserveProvenance
 → VersionModel

NullRestrictionUpdate :=
 ChangedNullRestriction
 ⇒ recompute(
  NullParameterSpace,AlternativeParameterSpace,NullDistribution,
  CriticalRegion,PValue,Decision
 )

ModelUpdate :=
 ChangedStatisticalModel
 ⇒ reevaluate(
  NullDistribution,NuisanceParameters,Calibration,Validity
 )

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

FLOW_HASH_INPUT :=
 canonical(
  ONTOLOGY,
  CoreClasses,
  CoreRelations,
  InferenceRules,
  NullHypothesis,
  NullHypothesisValidation,
  NullHypothesisConstraint,
  SEMANTIC_EQUIVALENCE,
  AI_REASONING_FLOW
 )

FLOW_HASH := HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
 NullHypothesisEntityResolution,Population,Sample,Parameter,ParameterSpace,Estimand,
 NullHypothesis,NullParameterSpace,AlternativeHypothesis,AlternativeParameterSpace,
 StatisticalModel,SamplingDesign,TestStatistic,NullDistribution,SignificanceLevel,
 CriticalRegion,PValue,Decision,TypeIErrorControl,NuisanceParameterStatus,
 ValidationStatus,Confidence,Provenance,Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA