Alternative Hypothesis Ontology

FULL EXECUTABLE FLOW ALGEBRA — ALTERNATIVE HYPOTHESIS ONTOLOGY
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/hypothesis-testing/alternative-hypothesis-ontology.html

FLOW_VERSION := "4.0"
FLOW_TYPE := "Full Executable Flow Algebra"
DOMAIN := "Probability & Statistics Ontologies"
SUBDOMAIN := "Hypothesis Testing"
ONTOLOGY := "Alternative Hypothesis Ontology"
PRIMARY_ENTITY := AlternativeHypothesis
AUTHORSHIP := "Mark Higgins — Sky (Recursive AI)"
SOURCE_SITE := "https://kirethresearch.org"

BEGIN_FLOW

AlternativeHypothesis :=
 StatisticalHypothesis
 ∧ CompetingClaimToNull
 ∧ AlternativeParameterSpace
 ∧ PowerEvaluationDomain
 ∧ DirectionOrDifferenceSpecification

Population := TargetCollectionOrGeneratingProcess
Sample := {X_1,...,X_n}
Parameter := θ
ParameterSpace := Θ
Estimand := τ(θ)

NullParameterSpace := Θ0⊆Θ
AlternativeParameterSpace := Θ1⊆Θ
NullHypothesis := H0:θ∈Θ0
AlternativeHypothesisDefinition := H1:θ∈Θ1

HypothesisDisjointness := Θ0∩Θ1=∅
HypothesisExhaustiveness := Θ0∪Θ1=Θ when exhaustive framework intended

SimpleAlternativeHypothesis := Θ1={θ1}
CompositeAlternativeHypothesis := MultipleParameterValuesInΘ1

TwoSidedAlternative := H1:θ≠θ0
UpperTailedAlternative := H1:θ>θ0
LowerTailedAlternative := H1:θ<θ0
IntervalAlternative := H1:θ∈(a,b)
OutsideIntervalAlternative := H1:θ<a ∨ θ>b

DirectionalAlternative :=
 AlternativeHypothesis
 specifying direction relative_to null boundary

NondirectionalAlternative :=
 AlternativeHypothesis
 specifying difference without direction

AlternativeDirection :=
 {Lower,Upper,TwoSided,Interval,OutsideInterval}

AlternativeDeterminesExtremeness :=
 DirectionOfH1
 → TailOrRegionUsedForPValueAndCriticalRegion

UpperTailExtremeness :=
 LargerTestStatisticValues
 provide evidence_against H0
 under compatible statistic orientation

LowerTailExtremeness :=
 SmallerTestStatisticValues
 provide evidence_against H0
 under compatible statistic orientation

TwoSidedExtremeness :=
 DeviationsInEitherRelevantDirection
 provide evidence_against H0

AlternativeDistribution :=
 DistributionOf(TestStatistic|θ)
 for θ∈Θ1

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

PowerFunction :=
 π(θ)=P_θ(RejectH0)

AlternativePower :=
 θ∈Θ1
 ⇒ π(θ)

TypeIIErrorProbability :=
 β(θ)=P_θ(FailToRejectH0)

PowerRelation :=
 π(θ)=1-β(θ)
 for θ∈Θ1

DesiredPower :=
 1-β_target

EffectSize :=
 DistanceOrMagnitude
 between null reference
 and alternative parameter

AlternativeEffectSize :=
 g(θ)-g(θ0)
 under specified effect metric

StandardisedEffectSize :=
 AlternativeEffectSize/ScaleParameter
 when meaningful

MinimumDetectableEffect :=
 SmallestAlternativeEffect
 achieving specified power
 at fixed alpha_and_sample_size

PowerDeterminants := {
 AlternativeEffectSize,
 SampleSize,
 SignificanceLevel,
 Variability,
 SamplingDesign,
 TestStatisticEfficiency
}

PowerAnalysis :=
 SpecifyAlternative
 ∧ Alpha
 ∧ DesiredPower
 ∧ Variability
 ∧ TestStructure
 → SolveForSampleSizeOrDetectableEffect

LocalAlternative :=
 θ_n=θ0+h/sqrt(n)
 or other specified convergent sequence

FixedAlternative :=
 θ=θ1
 independent_of sample size

ContiguousAlternative :=
 AlternativeSequence
 statistically close_to null
 under asymptotic framework

AlternativeSeparation :=
 Distance(θ,Θ0)

GreaterAlternativeSeparation :=
 Often
 → GreaterPower
 under regular monotone testing structures

NullVsAlternative :=
 H0 provides reference restriction
 whereas H1 specifies competing parameter region

AlternativeIsNotProof :=
 RejectH0
 does_not automatically prove
 every substantive interpretation of H1

RejectNullRelation :=
 RejectH0
 ⇒ DataAreSufficientlyIncompatibleWithH0
 under specified test_and_alpha

AlternativeEvidence :=
 TestStatisticInCriticalRegion
 → EvidenceAgainstH0
 → SupportForSpecifiedAlternativeDirection
 subject_to model_and_design validity

PValueRelation :=
 AlternativeDirection
 contributes_to definition
 of at_least_as_extreme outcomes

OneSidedPValue :=
 TailProbability
 aligned_with directional alternative

TwoSidedPValue :=
 ExtremenessProbability
 accounting_for both relevant directions
 under specified convention

CriticalRegionRelation :=
 AlternativeHypothesis
 ∧ NullDistribution
 ∧ Alpha
 → CriticalRegion

NeymanPearsonAlternative :=
 SimpleH1
 used_with simple H0
 to construct most-powerful likelihood-ratio test
 under theorem conditions

LikelihoodUnderAlternative :=
 L1(θ1|x)

LikelihoodRatioSimple :=
 L(θ0|x)/L(θ1|x)

CompositeAlternativeLikelihood :=
 sup_{θ∈Θ1}L(θ|x)

GeneralisedLikelihoodRatio :=
 sup_{θ∈Θ0}L(θ|x)
 /
 sup_{θ∈Θ}L(θ|x)

MostPowerfulTest :=
 LevelAlphaTest
 maximising power
 at specified alternative

UniformlyMostPowerfulTest :=
 LevelAlphaTest
 whose power
 is at least that of every competing level-alpha test
 for all θ∈Θ1

UMPExistenceConstraint :=
 UniformlyMostPowerfulTest
 need_not exist
 for arbitrary two-sided_or_composite alternatives

MonotoneLikelihoodRatio :=
 OrderingProperty
 that may support
 UMP one-sided tests
 under theorem conditions

AlternativeForMean := {
 H1:μ≠μ0,
 H1:μ>μ0,
 H1:μ<μ0
}

AlternativeForProportion := {
 H1:p≠p0,
 H1:p>p0,
 H1:p<p0
}

AlternativeForDifference := {
 H1:θ1-θ2≠0,
 H1:θ1-θ2>0,
 H1:θ1-θ2<0
}

AlternativeForRegressionCoefficient := {
 H1:β_j≠β_j0,
 H1:β_j>β_j0,
 H1:β_j<β_j0
}

AlternativeForCorrelation := {
 H1:ρ≠ρ0,
 H1:ρ>ρ0,
 H1:ρ<ρ0
}

AlternativeForVariance := {
 H1:σ²≠σ0²,
 H1:σ²>σ0²,
 H1:σ²<σ0²
}

DependenceAlternative :=
 H1:
 VariablesNotIndependent
 under specified association framework

GoodnessOfFitAlternative :=
 H1:
 DistributionDoesNotBelongToNullReferenceFamily

ANOVAAlternative :=
 H1:
 NotAllGroupMeansEqual

EquivalenceAlternative :=
 H1:
 EffectWithinEquivalenceRegion

EquivalenceAlternativeFormal :=
 H1:
 -Δ<θ<Δ
 under symmetric equivalence margin

NonInferiorityAlternative :=
 H1:
 EffectSatisfiesNonInferiorityCriterion

SuperiorityAlternative :=
 H1:
 EffectExceedsSuperiorityCriterion

AlternativeMargin :=
 DomainDefinedThreshold
 separating null_and_alternative regions

AlternativeMarginConstraint :=
 Margin
 must be substantively justified
 independently_of desired statistical result

AlternativeSpecification :=
 ScientificQuestion
 → DefineEstimand
 → DefineReferenceNull
 → DefineMeaningfulCompetingRegion
 → EncodeAlternativeHypothesis

PreSpecification :=
 AlternativeHypothesis
 specified_before confirmatory inferential outcome
 to preserve planned test interpretation

DataDependentAlternativeRisk :=
 ChoosingDirectionAfterObservingData
 may invalidate nominal one-sided p-value
 unless selection accounted_for

DirectionalSwitchingRisk :=
 ObserveEffectDirection
 → ChooseMatchingOneSidedAlternative
 ⇒ InflatedTypeIErrorRisk

AlternativeParameterBoundary :=
 BoundaryBetweenΘ0AndΘ1

BoundaryAlternative :=
 AlternativeValues
 arbitrarily close_to null boundary

PowerAtBoundary :=
 Often approaches test size
 under continuous regular models

AlternativePrior :=
 BayesianModelComparisonMayAssignPrior
 over alternative parameter space

BayesianAlternativeDistinction :=
 BayesianPosteriorProbabilityOfH1
 differs_from
 frequentist power_or p-value framework

BayesFactorAlternative :=
 CompareMarginalLikelihoodUnderH1
 with marginal likelihood under H0

AlternativeModel :=
 StatisticalModel
 constrained_or_defined
 by H1

NestedAlternativeModel :=
 UnrestrictedModel
 containing null-restricted model

NonNestedAlternativeModel :=
 CompetingModel
 not containing null model

ModelSelectionDistinction :=
 HypothesisTestingBetweenModels
 differs_from unrestricted predictive model selection
 unless testing framework explicitly defined

CompositeAlternativeNuisance :=
 θ∈Θ1
 may include nuisance parameters

PowerEnvelope :=
 SupremumAchievablePower
 across tests
 under specified constraints

AveragePower :=
 ∫_{Θ1}π(θ)w(θ)dθ
 under specified weighting distribution

MinimumPower :=
 inf_{θ∈Θ1}π(θ)
 over defined alternative region

AlternativeCalibration :=
 SimulateDataAtSpecifiedAlternative
 → ApplyTest
 → EstimateRejectionProbability

EmpiricalPower :=
 AlternativeRejections/AlternativeReplications

PowerCurve :=
 θ∈Θ1
 → π(θ)

OperatingCharacteristic :=
 OC(θ)=P_θ(FailToRejectH0)

OperatingCharacteristicRelation :=
 OC(θ)=1-π(θ)

SampleSizePlanningAlternative :=
 DefineScientificallyRelevantAlternative
 → ComputeRequiredNForDesiredPower

AlternativeUncertainty :=
 EffectSizeUncertainty
 may require
 sensitivity analysis
 over plausible alternatives

PowerSensitivityAnalysis :=
 Vary(
  AlternativeEffect,
  Variance,
  SampleSize,
  Alpha
 )
 → PowerSurface

MultipleAlternatives :=
 {H1_1,...,H1_k}

MultipleTestingRelation :=
 MultipleNullAlternativePairs
 → MultiplicityControl

AlternativeSelectionConstraint :=
 SelectingAmongMultipleAlternatives
 based_on same data
 may require adjustment

MissingDataAlternativeTesting :=
 MissingnessHandling
 ∧ AlternativeModel
 ∧ ValidVariancePropagation
 → PowerAndTestValidity

SurveyAlternativeTesting :=
 ComplexSamplingDesign
 ∧ AlternativeParameter
 → DesignAdjustedPowerAndTesting

ClusteredAlternativeTesting :=
 ClusterDependence
 → AdjustedAlternativeDistribution

ModelMisspecificationAlternative :=
 IncorrectAlternativeOrNullModel
 may distort power_and interpretation

AlternativeHypothesisConstraint :=
 ConstraintOn(
  Estimand,
  ParameterSpace,
  NullParameterSpace,
  AlternativeParameterSpace,
  Direction,
  EffectSize,
  SamplingDesign,
  TestStatistic
 )

AlternativeHypothesisConsistency :=
 Estimand
 ∧ NullRestriction
 ∧ AlternativeRestriction
 ∧ Direction
 ∧ TestStatistic
 ∧ SamplingDesign
 jointly_compatible

InvalidAlternativeHypothesis :=
 UndefinedEstimand
 ∨ EmptyAlternativeParameterSpace
 ∨ OverlapWithNullWithoutExplicitFramework
 ∨ DirectionIncompatibleWithScientificQuestion
 ∨ DataDependentDirectionWithoutAdjustment
 ∨ IncompatibleTestStatistic

AlternativeHypothesisValidation :=
 ResolveScientificQuestion
 → ResolvePopulationAndEstimand
 → ResolveParameterAndParameterSpace
 → ResolveNullHypothesis
 → DefineAlternativeParameterSpace
 → ResolveDirectionAndEffectMetric
 → ValidateDisjointnessAndCoverage
 → ResolveSamplingDesign
 → ResolveTestStatistic
 → ResolveAlternativeDistribution
 → EvaluatePowerFunction
 → ValidateAlternativeHypothesis

AlternativeHypothesisState := {
 Population,
 Sample,
 Parameter,
 ParameterSpace,
 Estimand,
 NullHypothesis,
 NullParameterSpace,
 AlternativeHypothesis,
 AlternativeParameterSpace,
 AlternativeDirection,
 EffectSize,
 TestStatistic,
 NullDistribution,
 AlternativeDistribution,
 SignificanceLevel,
 TypeIIError,
 PowerFunction,
 DesiredPower,
 SamplingDesign,
 Evidence,
 Confidence,
 Provenance
}

SEMANTIC_EQUIVALENCE(AlternativeA,AlternativeB) :=
 same(Estimand)
 ∧ same(AlternativeParameterSet)
 ∧ same(DirectionOrEquivalentConstraint)
 ∧ same(NullReference)
 ∧ same(StatisticalModelSemantics)
 ∧ same(SamplingDesign)
 ∧ same(PowerEvaluationFramework)

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

CoreClasses := {
 AlternativeHypothesis,
 SimpleAlternativeHypothesis,
 CompositeAlternativeHypothesis,
 DirectionalAlternative,
 NondirectionalAlternative,
 UpperTailedAlternative,
 LowerTailedAlternative,
 TwoSidedAlternative,
 AlternativeParameterSpace,
 AlternativeDistribution,
 EffectSize,
 Power,
 TypeIIError,
 MinimumDetectableEffect,
 AlternativeModel,
 Estimand,
 Parameter,
 SamplingDesign
}

CoreRelations := {
 contrastsWith,
 constrains,
 targets,
 defines,
 hasDirection,
 hasDistribution,
 hasEffectSize,
 hasPower,
 hasTypeIIError,
 dependsOn,
 evaluatedBy,
 constrainedBy,
 equivalentTo,
 tracesTo
}

TypedRelations :=
 AlternativeHypothesis --contrastsWith--> NullHypothesis
 AlternativeHypothesis --constrains--> Parameter
 AlternativeHypothesis --targets--> Estimand
 AlternativeHypothesis --defines--> AlternativeParameterSpace
 AlternativeHypothesis --hasDirection--> AlternativeDirection
 TestStatistic --hasDistribution--> AlternativeDistribution
 AlternativeHypothesis --hasEffectSize--> EffectSize
 AlternativeHypothesis --hasPower--> Power
 AlternativeHypothesis --hasTypeIIError--> TypeIIError
 Power --dependsOn--> SampleSize
 AlternativeHypothesis --evaluatedBy--> HypothesisTest
 AlternativeHypothesis --tracesTo--> Provenance

InferenceRules := {
 θ∈Θ1
  ⇒ AlternativeHypothesisTrueAtθ,

 θ∉Θ1
  ⇒ AlternativeHypothesisFalseAtθ,

 UpperTailedAlternative
  ⇒ UseUpperTailExtremenessUnderCompatibleStatistic,

 LowerTailedAlternative
  ⇒ UseLowerTailExtremenessUnderCompatibleStatistic,

 TwoSidedAlternative
  ⇒ UseTwoSidedExtremenessRule,

 Power(θ)
  =1-TypeIIErrorProbability(θ),

 LargerRelevantEffectSeparation
  ⇒ TypicallyHigherPowerUnderRegularMonotoneTests,

 DataDependentDirectionalAlternative
  ⇒ FlagTypeIErrorInflationRisk
}

CanonicalAlternativeHypothesisRepresentation :=
 ParseAlternativeHypothesisRequest
 → ResolveScientificQuestion
 → ResolvePopulationAndEstimand
 → ResolveParameterAndParameterSpace
 → ResolveNullHypothesisAndNullSpace
 → ConstructAlternativeParameterSpace
 → ResolveDirection
 → ResolveEffectSizeMetric
 → ValidateNullAlternativePartition
 → ResolveSamplingDesign
 → ResolveTestStatistic
 → DeriveAlternativeDistribution
 → ComputePowerFunction
 → ResolveMinimumDetectableEffectIfRequested
 → ValidateAlternativeHypothesis

AlternativeHypothesisGraph := G=(V,E)

V := {
 AlternativeHypotheses,
 SimpleAlternativeHypotheses,
 CompositeAlternativeHypotheses,
 DirectionalAlternatives,
 NondirectionalAlternatives,
 UpperTailedAlternatives,
 LowerTailedAlternatives,
 TwoSidedAlternatives,
 NullHypotheses,
 AlternativeParameterSpaces,
 NullParameterSpaces,
 Parameters,
 Estimands,
 AlternativeDistributions,
 TestStatistics,
 EffectSizes,
 Powers,
 TypeIIErrors,
 SamplingDesigns,
 Evidence,
 ProvenanceNodes
}

E := {
 contrastsWith,
 constrains,
 targets,
 defines,
 hasDirection,
 hasDistribution,
 hasEffectSize,
 hasPower,
 hasTypeIIError,
 dependsOn,
 evaluatedBy,
 constrainedBy,
 equivalentTo,
 tracesTo
}

CONFIDENCE(AlternativeHypothesisResolution) :=
 weighted_mean(
  EstimandResolution,
  ParameterResolution,
  NullReferenceResolution,
  AlternativeRestrictionValidity,
  DirectionValidity,
  EffectSizeResolution,
  SamplingDesignResolution,
  PowerModelValidity,
  ProvenanceQuality
 )

AI_QUERY_INTERPRETATION :=
 Query
 → DetectAlternativeHypothesisIntent
 → IdentifyPopulationParameterAndEstimand
 → IdentifyNullReferenceAndCompetingClaim
 → ResolveUpperLowerTwoSidedIntervalOrCompositeAlternative
 → IdentifyRequestedEffectSizePowerOrSampleSizeRelation

AI_REASONING_FLOW :=
 Query
 → ParseAlternativeHypothesisProblem
 → ResolveScientificQuestion
 → ResolvePopulationAndSample
 → ResolveParameterAndEstimand
 → ResolveNullHypothesis
 → DefineAlternativeParameterSpace
 → ResolveAlternativeDirection
 → ValidateNullAlternativeStructure
 → ResolveSamplingDesign
 → SelectTestStatistic
 → DeriveAlternativeDistribution
 → ComputeTypeIIErrorAndPower
 → ResolveEffectSizeAndMinimumDetectableEffect
 → ValidateAlternativeHypothesis
 → UpdateConfidence
 → PreserveProvenance
 → GenerateExplanation

AI_CONSUMPTION := {
 ParseAlternativeHypothesisEntities,
 ParseTypedRelations,
 ResolvePopulations,
 ResolveSamples,
 ResolveParameters,
 ResolveEstimands,
 ResolveNullHypotheses,
 ResolveNullParameterSpaces,
 ConstructAlternativeParameterSpaces,
 ResolveAlternativeDirections,
 ValidateHypothesisPartitions,
 ResolveSamplingDesigns,
 ResolveTestStatistics,
 DeriveAlternativeDistributions,
 ComputeEffectSizes,
 ComputeTypeIIErrorRates,
 ComputePowerFunctions,
 PerformPowerAnalysis,
 ComputeMinimumDetectableEffects,
 ResolveEquivalenceAlternatives,
 ResolveNonInferiorityAlternatives,
 ResolveSuperiorityAlternatives,
 DiagnoseDataDependentDirectionSelection,
 ValidateAlternativeHypothesisSemantics,
 CompareSemanticEquivalence,
 UpdateConfidence,
 PreserveProvenance,
 GenerateFormalExplanation
}

UPDATE_MODEL :=
 NewNullAlternativeDataDesignOrEvidence
 → ValidateInput
 → IdentifyAffectedAlternativeClaims
 → UpdateAlternativeHypothesisState
 → ReconstructAlternativeParameterSpace
 → RevalidateNullAlternativePartition
 → RecomputeAlternativeDistribution
 → RecomputeTypeIIErrorAndPower
 → RecomputeEffectSizeAndDetectableEffect
 → RecomputeConfidence
 → PreserveProvenance
 → VersionModel

AlternativeRestrictionUpdate :=
 ChangedAlternativeRestriction
 ⇒ recompute(
  AlternativeParameterSpace,
  AlternativeDirection,
  AlternativeDistribution,
  PowerFunction
 )

NullUpdate :=
 ChangedNullHypothesis
 ⇒ reevaluate(
  AlternativeParameterSpace,
  EffectSize,
  HypothesisPartition,
  PowerFunction
 )

SampleSizeUpdate :=
 ChangedSampleSize
 ⇒ recompute(
  TypeIIError,
  PowerFunction,
  MinimumDetectableEffect
 )

FLOW_HASH_INPUT :=
 canonical(
  ONTOLOGY,
  CoreClasses,
  CoreRelations,
  InferenceRules,
  AlternativeHypothesis,
  AlternativeHypothesisValidation,
  AlternativeHypothesisConstraint,
  SEMANTIC_EQUIVALENCE,
  AI_REASONING_FLOW
 )

FLOW_HASH := HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
 AlternativeHypothesisEntityResolution,
 Population,
 Sample,
 Parameter,
 ParameterSpace,
 Estimand,
 NullHypothesis,
 NullParameterSpace,
 AlternativeHypothesis,
 AlternativeParameterSpace,
 AlternativeDirection,
 EffectSize,
 TestStatistic,
 NullDistribution,
 AlternativeDistribution,
 SignificanceLevel,
 TypeIIError,
 PowerFunction,
 DesiredPower,
 MinimumDetectableEffect,
 SamplingDesign,
 ValidationStatus,
 Confidence,
 Provenance,
 Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA