Test Statistic Ontology

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

BEGIN_FLOW

TestStatistic :=
 StatisticalFunction ∧ FunctionOfSampleData ∧ EvidenceCompression
 ∧ NullReferenceComparableQuantity ∧ HypothesisTestingComponent

Population := TargetCollectionOrGeneratingProcess
Sample := X={X_1,...,X_n}
ObservedSample := x={x_1,...,x_n}
Parameter := θ
ParameterSpace := Θ
Estimand := τ(θ)
Statistic := T(X)
TestStatisticDefinition := T:SampleSpace→StatisticSpace
ObservedTestStatistic := t_obs=T(x)

NullHypothesis := H0:θ∈Θ0
AlternativeHypothesis := H1:θ∈Θ1
NullDistribution := DistributionOf(T(X)|H0)
AlternativeDistribution := DistributionOf(T(X)|H1)

TestStatisticPurpose :=
 MapObservedData → EvidenceSummary → CompareWithNullReference

EvidenceDirection :=
 OrderingOrRegion defining greater incompatibility with H0

UpperTailStatistic := LargerValuesIndicateGreaterEvidenceAgainstH0
LowerTailStatistic := SmallerValuesIndicateGreaterEvidenceAgainstH0
TwoSidedStatistic := DepartureInEitherRelevantDirectionIndicatesEvidenceAgainstH0
SignedStatistic := StatisticRetainsDirectionOfDeparture

StandardisedStatistic :=
 (Estimate-NullValue)/StandardError

StudentisedStatistic :=
 (Estimate-NullValue)/EstimatedStandardError

CriticalRegion := C⊆StatisticSpace
DecisionRule := T(x)∈C ⇒ RejectH0
CriticalValue := BoundaryValueOfCriticalRegion

PValue :=
 NullProbabilityOfStatisticValuesAtLeastAsExtremeAsObserved

PValueFromStatistic :=
 ObservedTestStatistic ∧ NullDistribution ∧ AlternativeDirection → PValue

SignificanceLevel := α
CriticalValueFromNull := NullDistribution ∧ Alpha → CriticalValue

StatisticMonotonicEquivalence :=
 StrictlyMonotonicTransformation
 ∧ ConsistentlyTransformedThresholds
 ⇒ EquivalentRejectionRegions

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

MinusTwoLogLikelihoodRatio := D=-2logΛ
LikelihoodRatioEvidence := SmallerΛ ↔ LargerD → GreaterEvidenceAgainstH0

WaldStatistic :=
 (θ_hat-θ0)/SE_hat(θ_hat)

SquaredWaldStatistic :=
 (θ_hat-θ0)²/Var_hat(θ_hat)

ScoreFunction := U(θ)=∂ℓ(θ)/∂θ
ScoreStatistic := U(θ0)^TI(θ0)^(-1)U(θ0)

ZStatistic :=
 (Estimate-NullValue)/KnownOrAsymptoticSE

TStatistic :=
 (Estimate-NullValue)/EstimatedSE

OneSampleMeanZStatistic :=
 (X_bar-μ0)/(σ/sqrt(n))

OneSampleMeanTStatistic :=
 (X_bar-μ0)/(S/sqrt(n))

TwoSampleWelchTStatistic :=
 (X_bar1-X_bar2-Δ0)/sqrt(S1²/n1+S2²/n2)

PairedTStatistic :=
 (D_bar-Δ0)/(S_D/sqrt(n))

ProportionZStatistic :=
 (p_hat-p0)/sqrt(p0(1-p0)/n)

ChiSquaredStatistic :=
 Σ_j(O_j-E_j)²/E_j

LikelihoodRatioChiSquared :=
 G²=2Σ_jO_jlog(O_j/E_j)

FStatistic :=
 VarianceLikeNumerator/VarianceLikeDenominator

ANOVAFStatistic :=
 MeanSquareBetween/MeanSquareWithin

RegressionCoefficientTStatistic :=
 (β_hat_j-β_j0)/SE(β_hat_j)

DevianceStatistic :=
 2[LogLikelihoodSaturated-LogLikelihoodFitted]

GoodnessOfFitStatistic :=
 Discrepancy(ObservedData,NullModelExpectation)

KolmogorovSmirnovStatistic :=
 sup_x|F_n(x)-F_0(x)|

TwoSampleKSStatistic :=
 sup_x|F_n(x)-G_m(x)|

SignStatistic := NumberOfPositiveDifferences
WilcoxonSignedRankStatistic := SumOfSignedOrPositiveRanks
MannWhitneyUStatistic := PairwiseOrderingOrRankSumStatistic
RankSumStatistic := SumOfRanksAssignedToOneSample

JarqueBeraStatistic :=
 n[S²/6+(K-3)²/24]
 under conventional skewness_and_kurtosis definitions

PermutationStatistic :=
 PrespecifiedStatistic recomputed_under permutations

RandomisationStatistic :=
 PrespecifiedOutcomeContrast recomputed_under assignment permutations

BootstrapTestStatistic :=
 Statistic recomputed_under null-compatible bootstrap samples

ExactStatistic :=
 Statistic whose finite-sample null distribution is derived exactly

AsymptoticStatistic :=
 Statistic whose null distribution is approximated by limiting theory

RankStatistic := FunctionOfSampleRanks
QuadraticFormStatistic := Z^TAZ

TestStatisticExpectation := E_H0[T]
TestStatisticVariance := Var_H0(T)
StandardisedNullStatistic := (T-E_H0[T])/sqrt(Var_H0(T))
NullCentredStatistic := T-E_H0[T]
StatisticSupport := SetOfPossibleValues(T)
DiscreteTestStatistic := StatisticSupportCountableOrFinite
ContinuousTestStatistic := StatisticSupportContinuous
VectorTestStatistic := T(X)∈R^k
ScalarTestStatistic := T(X)∈R

StatisticSelection :=
 NullHypothesis ∧ AlternativeHypothesis ∧ SamplingDesign ∧ Model
 → ChooseStatisticWithValidReferenceDistribution

StatisticSensitivity :=
 ChangeInStatistic induced_by perturbation_of observations

RobustTestStatistic :=
 Statistic designed_to limit sensitivity to contamination_or assumption deviations

InvariantTestStatistic :=
 Statistic unchanged_or equivariant under specified transformation group

AncillaryStatistic :=
 Statistic whose distribution does_not depend_on parameter under model

ConditioningStatistic :=
 Statistic conditioned_on to remove nuisance parameters
 or obtain exact null distribution

NuisanceParameter :=
 Parameter affecting statistic distribution but not primary target

Studentisation :=
 DivideByEstimatedScale → ReduceDependenceOnUnknownScale

Pivotalisation :=
 TransformStatistic → ReferenceDistributionLessDependentOnUnknownParameters

NullCalibration :=
 DeriveOrSimulateDistributionUnderH0 → DetermineCriticalRegionAndPValue

EmpiricalNullDistribution :=
 RepeatedNullStatistics → EmpiricalCDF

PermutationNullDistribution :=
 PermuteUnderExchangeability → StatisticReplicates

MonteCarloPValue :=
 (1+NumberOfSimulatedStatisticsAtLeastAsExtreme)/(B+1)

Ties :=
 EqualObservedValues may alter rank-statistic distribution

TieCorrection :=
 AdjustVarianceOrReferenceDistributionForTies

ContinuityCorrection :=
 DiscreteStatisticApproximatedByContinuousDistribution
 → OptionalBoundaryAdjustment

MissingDataStatistic :=
 MissingnessHandling → EffectiveSample → TestStatistic

WeightedTestStatistic :=
 Observations ∧ Weights → WeightedStatistic

SurveyTestStatistic :=
 ComplexSamplingDesign ∧ DesignBasedVariance → AdjustedStatistic

ClusterRobustStatistic :=
 EstimateContrast/ClusterRobustSE

HeteroscedasticityRobustStatistic :=
 EstimateContrast/RobustSE

SequentialTestStatistic :=
 StatisticUpdatedAsDataAccumulate

StoppingBoundary :=
 SequentialStatistic ∧ InformationTime → ContinueOrStopDecision

MultipleTestingStatistic :=
 CollectionOfTestStatistics associated_with hypothesis family

GlobalTestStatistic :=
 AggregateEvidence across multiple parameters_or_hypotheses

TestStatisticConstraint :=
 ConstraintOn(
  Data,Estimand,NullHypothesis,AlternativeHypothesis,StatisticalModel,
  SamplingDesign,StatisticDefinition,ReferenceDistribution
 )

TestStatisticConsistency :=
 Data ∧ Hypotheses ∧ StatisticDefinition ∧ NullDistribution
 ∧ AlternativeDirection ∧ SamplingDesign
 jointly_compatible

InvalidTestStatistic :=
 UndefinedDataMapping
 ∨ IncompatibleHypothesis
 ∨ InvalidReferenceDistribution
 ∨ UnresolvedNuisanceDependence
 ∨ IgnoredSamplingDesign
 ∨ DataDependentStatisticSelectionWithoutAdjustment
 ∨ ComputationalFailure

TestStatisticValidation :=
 ResolvePopulationSampleAndEstimand
 → ResolveNullAndAlternativeHypotheses
 → ResolveStatisticalModel
 → ResolveSamplingDesign
 → DefineStatisticFunction
 → ValidateStatisticDomain
 → ComputeObservedStatistic
 → DeriveOrResolveNullDistribution
 → ResolveAlternativeDirection
 → ResolveCriticalRegionAndPValue
 → ValidateAssumptions
 → ValidateResult

TestStatisticState := {
 Population,Sample,Parameter,Estimand,NullHypothesis,AlternativeHypothesis,
 StatisticalModel,SamplingDesign,StatisticDefinition,ObservedTestStatistic,
 StatisticSupport,NullDistribution,AlternativeDistribution,NullExpectation,
 NullVariance,AlternativeDirection,CriticalRegion,PValue,Assumptions,
 Evidence,Confidence,Provenance
}

SEMANTIC_EQUIVALENCE(StatisticA,StatisticB) :=
 same(HypothesisTestingTarget)
 ∧ same(DataSemantics)
 ∧ monotonic_or_decision_equivalent(StatisticMapping)
 ∧ same(NullReferenceOrdering)
 ∧ same(SamplingDesign)
 ∧ same(DecisionRegions)

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

CoreClasses := {
 TestStatistic,ObservedTestStatistic,StandardisedStatistic,StudentisedStatistic,
 LikelihoodRatioStatistic,WaldStatistic,ScoreStatistic,ZStatistic,TStatistic,
 ChiSquaredStatistic,FStatistic,RankStatistic,PermutationStatistic,ExactStatistic,
 AsymptoticStatistic,NullDistribution,AlternativeDistribution,CriticalRegion,
 PValue,SamplingDesign
}

CoreRelations := {
 computedFrom,tests,targets,hasNullDistribution,hasAlternativeDistribution,
 hasSupport,hasExpectation,hasVariance,standardisedBy,comparedWith,determines,
 calibratedBy,constrainedBy,equivalentTo,tracesTo
}

TypedRelations :=
 TestStatistic --computedFrom--> Sample
 TestStatistic --tests--> NullHypothesis
 TestStatistic --targets--> Estimand
 TestStatistic --hasNullDistribution--> NullDistribution
 TestStatistic --hasAlternativeDistribution--> AlternativeDistribution
 TestStatistic --hasSupport--> StatisticSupport
 TestStatistic --hasExpectation--> NullExpectation
 TestStatistic --hasVariance--> NullVariance
 StudentisedStatistic --standardisedBy--> EstimatedStandardError
 ObservedTestStatistic --comparedWith--> NullDistribution
 TestStatistic --determines--> CriticalRegion
 TestStatistic --calibratedBy--> SamplingDesign
 TestStatistic --tracesTo--> Provenance

InferenceRules := {
 Sample ∧ StatisticFunction
  ⇒ ObservedTestStatistic=T(Sample),

 ObservedTestStatistic ∧ NullDistribution ∧ ExtremenessRule
  ⇒ PValue,

 NullDistribution ∧ Alpha
  ⇒ CriticalRegion,

 ObservedTestStatistic∈CriticalRegion
  ⇒ RejectNullHypothesis,

 StrictlyMonotonicStatisticTransformation ∧ TransformedCriticalRegion
  ⇒ DecisionEquivalentTest,

 InvalidNullDistribution
  ⇒ FlagTestStatisticInvalid,

 DataDependentStatisticSelectionWithoutAdjustment
  ⇒ FlagSelectionBiasRisk
}

CanonicalTestStatisticRepresentation :=
 ParseTestStatisticRequest
 → ResolvePopulationSampleAndEstimand
 → ResolveNullAndAlternativeHypotheses
 → ResolveStatisticalModelAndSamplingDesign
 → DefineStatisticFunction
 → ResolveScalarVectorDiscreteContinuousOrRankStructure
 → ComputeObservedStatistic
 → ResolveNullDistribution
 → ResolveAlternativeDistributionIfRequired
 → ResolveExtremenessOrdering
 → ComputeCriticalRegionAndPValue
 → ValidateTestStatistic

TestStatisticGraph := G=(V,E)

V := {
 TestStatistics,ObservedTestStatistics,StandardisedStatistics,StudentisedStatistics,
 LikelihoodRatioStatistics,WaldStatistics,ScoreStatistics,ZStatistics,TStatistics,
 ChiSquaredStatistics,FStatistics,RankStatistics,PermutationStatistics,
 ExactStatistics,AsymptoticStatistics,NullDistributions,AlternativeDistributions,
 CriticalRegions,PValues,Samples,Estimands,SamplingDesigns,Evidence,ProvenanceNodes
}

E := {
 computedFrom,tests,targets,hasNullDistribution,hasAlternativeDistribution,
 hasSupport,hasExpectation,hasVariance,standardisedBy,comparedWith,determines,
 calibratedBy,constrainedBy,equivalentTo,tracesTo
}

CONFIDENCE(TestStatisticResolution) :=
 weighted_mean(
  DataResolution,EstimandResolution,HypothesisResolution,
  StatisticDefinitionValidity,NullDistributionValidity,
  AlternativeDirectionValidity,SamplingDesignValidity,
  CalibrationQuality,ProvenanceQuality
 )

AI_QUERY_INTERPRETATION :=
 Query
 → DetectTestStatisticIntent
 → IdentifyPopulationSampleEstimandAndHypotheses
 → IdentifyStatisticFamily
 → ResolveNullReferenceAndAlternativeDirection
 → IdentifyRequestedStatisticPValueCriticalRegionOrDistribution

AI_REASONING_FLOW :=
 Query
 → ParseTestStatisticProblem
 → ResolvePopulationAndSample
 → ResolveParameterAndEstimand
 → ResolveNullAndAlternativeHypotheses
 → ResolveStatisticalModel
 → ResolveSamplingDesign
 → DefineOrSelectTestStatistic
 → ValidateStatisticDomain
 → ComputeObservedTestStatistic
 → DeriveOrResolveNullDistribution
 → ResolveAlternativeDistributionIfNeeded
 → ResolveExtremenessRule
 → ComputePValueAndCriticalRegion
 → ValidateAssumptionsAndCalibration
 → UpdateConfidence
 → PreserveProvenance
 → GenerateExplanation

AI_CONSUMPTION := {
 ParseTestStatisticEntities,ParseTypedRelations,ResolvePopulations,ResolveSamples,
 ResolveParameters,ResolveEstimands,ResolveNullHypotheses,
 ResolveAlternativeHypotheses,ResolveStatisticalModels,ResolveSamplingDesigns,
 ResolveStatisticDefinitions,ComputeObservedTestStatistics,
 ResolveNullDistributions,ResolveAlternativeDistributions,
 ComputeLikelihoodRatioStatistics,ComputeWaldStatistics,ComputeScoreStatistics,
 ComputeZStatistics,ComputeTStatistics,ComputeChiSquaredStatistics,
 ComputeFStatistics,ComputeRankStatistics,ComputePermutationStatistics,
 ResolveCriticalValues,ResolveCriticalRegions,ComputePValues,
 ApplyTieCorrections,ApplyContinuityCorrections,ResolveRobustStatistics,
 ValidateTestStatisticSemantics,CompareSemanticEquivalence,
 UpdateConfidence,PreserveProvenance,GenerateFormalExplanation
}

UPDATE_MODEL :=
 NewDataHypothesisModelDesignStatisticOrEvidence
 → ValidateInput
 → IdentifyAffectedTestStatisticClaims
 → UpdateTestStatisticState
 → RecomputeObservedStatistic
 → RecomputeNullDistributionIfRequired
 → RecomputeAlternativeDistributionIfRequired
 → RecomputeCriticalRegionAndPValue
 → RevalidateCalibrationAndAssumptions
 → RecomputeConfidence
 → PreserveProvenance
 → VersionModel

DataUpdate :=
 ChangedObservedData
 ⇒ recompute(
  ObservedTestStatistic,PValue,CriticalRegion
 )

StatisticUpdate :=
 ChangedStatisticDefinition
 ⇒ reevaluate(
  StatisticSupport,NullDistribution,AlternativeDistribution,
  ExtremenessOrdering,PValue,SemanticEquivalence
 )

DesignUpdate :=
 ChangedSamplingDesign
 ⇒ recompute(
  NullDistribution,VarianceCalibration,CriticalRegion,PValue
 )

FLOW_HASH_INPUT :=
 canonical(
  ONTOLOGY,
  CoreClasses,
  CoreRelations,
  InferenceRules,
  TestStatistic,
  TestStatisticValidation,
  TestStatisticConstraint,
  SEMANTIC_EQUIVALENCE,
  AI_REASONING_FLOW
 )

FLOW_HASH := HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
 TestStatisticEntityResolution,
 Population,Sample,Parameter,Estimand,NullHypothesis,AlternativeHypothesis,
 StatisticalModel,SamplingDesign,StatisticDefinition,ObservedTestStatistic,
 StatisticSupport,NullDistribution,AlternativeDistribution,NullExpectation,
 NullVariance,AlternativeDirection,CriticalRegion,PValue,CalibrationStatus,
 ValidationStatus,Confidence,Provenance,Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA