Hypothesis Testing

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

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

BEGIN_FLOW

HypothesisTesting :=
 StatisticalInference ∧ NullHypothesis ∧ AlternativeHypothesis ∧ TestStatistic
 ∧ SamplingDistribution ∧ DecisionRule ∧ ErrorControl ∧ EvidenceAssessment

Population := TargetCollectionOrGeneratingProcess
Sample := {X_1,...,X_n}
Parameter := θ
ParameterSpace := Θ
NullHypothesis := H0:θ∈Θ0
AlternativeHypothesis := H1:θ∈Θ1
HypothesisPartition := Θ0∩Θ1=∅

SimpleHypothesis := SingleParameterValueOrDistribution
CompositeHypothesis := MultipleParameterValuesOrDistributions
TwoSidedAlternative := H1:θ≠θ0
UpperTailedAlternative := H1:θ>θ0
LowerTailedAlternative := H1:θ<θ0

TestStatistic := T(X)
ObservedTestStatistic := t_obs=T(x)
NullDistribution := DistributionOf(T(X)|H0)
AlternativeDistribution := DistributionOf(T(X)|H1)
CriticalRegion := StatisticValuesLeadingToRejection
NonRejectionRegion := ComplementOfCriticalRegion

SignificanceLevel :=
 α=P(RejectH0|H0True)
 under simple null
 or supremum control under composite null

TypeIError := RejectH0WhenH0True
TypeIErrorProbability := α
TypeIIError := FailToRejectH0WhenSpecifiedAlternativeTrue
TypeIIErrorProbability := β(θ)=P_θ(FailToRejectH0)
PowerFunction := π(θ)=P_θ(RejectH0)
PowerUnderAlternative := 1-β(θ)
SizeOfTest := sup_{θ∈Θ0}P_θ(RejectH0)
LevelAlphaTest := SizeOfTest≤α

PValue :=
 ProbabilityUnderH0
 of statistic at_least_as_extreme_as observed
 according_to specified test and alternative

PValueDecisionRule := p≤α ⇒ RejectH0
PValueNonRejection := p>α ⇒ FailToRejectH0
PValueInterpretationConstraint := PValue ≠ ProbabilityH0IsTrue
FailToRejectConstraint := FailToRejectH0 ≠ ProveH0True

OneSidedPValue := TailProbabilityInAlternativeDirection
TwoSidedPValue := ExtremenessProbabilityInBothRelevantDirections
CriticalValue := QuantileOfNullDistributionDefiningRejectionThreshold

TestFunction := φ(X)∈[0,1]
NonRandomisedTest := φ(X)∈{0,1}
PowerFunctionFormal := π(θ)=E_θ[φ(X)]

NeymanPearsonFramework :=
 NullAndAlternative ∧ TypeIErrorConstraint → MaximisePower

NeymanPearsonLemma :=
 SimpleH0 versus SimpleH1
 ⇒ LikelihoodRatioTestMostPowerfulAtSpecifiedLevel
 under theorem conditions

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

LikelihoodRatioStatistic := -2logΛ
LikelihoodRatioTest := RejectH0ForSufficientlyLargeMinus2LogLambda

WilksTheorem :=
 UnderRegularityConditions
 ⇒ -2logΛ→χ²_df

WaldStatistic := (θ_hat-θ0)/SE(θ_hat)
WaldTest := StandardisedEstimate → CompareWithReferenceDistribution
ScoreFunction := U(θ)=∂ℓ(θ)/∂θ
ScoreTest := EvaluateScoreAtNull ∧ ScaleByInformation → TestStatistic

LikelihoodInferenceTriad := {LikelihoodRatioTest,WaldTest,ScoreTest}

ZTest := StandardisedStatisticComparedWithStandardNormal
OneSampleMeanZStatistic := Z=(X_bar-μ0)/(σ/sqrt(n))
OneSampleTTest := T=(X_bar-μ0)/(S/sqrt(n))
OneSampleTReference := NormalIIDSampleUnderH0 ⇒ T~t_(n-1)
TwoSampleTTest := DifferenceInMeans/EstimatedStandardError
WelchTStatistic := (X_bar1-X_bar2-Δ0)/sqrt(S1²/n1+S2²/n2)
PairedTTest := ApplyOneSampleTTestToWithinPairDifferences

ChiSquaredGoodnessOfFit := Σ_j(O_j-E_j)²/E_j
ChiSquaredIndependenceTest := ContingencyTable → ExpectedCountsUnderIndependence → ChiSquaredStatistic
FTest := RatioOrQuadraticFormStatisticComparedWithFReference
ANOVATest := BetweenGroupVariation/WithinGroupVariation → FStatistic

ExactTest := NullDistributionComputedExactlyUnderFiniteSampleModel
ApproximateTest := ReferenceDistributionFromAsymptoticOrNumericalApproximation
PermutationTest := ExchangeabilityUnderH0 → Permute → RecomputeStatistic → EmpiricalNullDistribution
RandomisationTest := KnownAssignmentMechanism → ReassignAccordingToDesign → NullDistribution
BootstrapTest := ResamplingUnderNull → ApproximateNullDistribution → PValueOrCriticalRegion

SignTest := SignsOfDifferences → BinomialReference
WilcoxonSignedRankTest := SignedRanks → RankBasedPairedLocationTest
MannWhitneyTest := RankBasedTwoSampleComparison
KolmogorovSmirnovTest := SupremumDifferenceBetweenCDFs
ShapiroWilkTest := OrderStatisticBasedNormalityTest
JarqueBeraTest := SampleSkewness ∧ SampleKurtosis → AsymptoticChiSquaredStatistic

TestAssumptions := {
 SamplingDesign,Independence,DistributionalAssumptions,VarianceStructure,
 MeasurementScale,SampleSizeRequirements
}

TestSelection :=
 ResolveEstimand → ResolveDataStructure → ResolveHypotheses
 → ResolveSamplingDesign → ResolveAssumptions → SelectValidTest

EffectSize := QuantitativeMagnitudeOfRelevantDifferenceOrAssociation
HypothesisTestVsEffectSize := TestAssessesEvidenceAgainstNull whereas EffectSizeMeasuresMagnitude

ConfidenceIntervalRelation :=
 TwoSidedLevelAlphaTest ∧ CompatibleOneMinusAlphaCI
 → TestInversionDuality

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

StatisticalSignificance := TestDecisionMeetingSpecifiedAlphaThreshold
PracticalSignificance := EffectMagnitudeEvaluatedAgainstDomainImportance
StatisticalVsPracticalSignificance := StatisticalSignificance ↛ PracticalImportance

PowerDeterminants := {
 EffectSize,SampleSize,SignificanceLevel,Variability,SamplingDesign,TestEfficiency
}

PowerAnalysis :=
 Specify(EffectSize,Alpha,DesiredPower,Variability,Test)
 → SolveForSampleSizeOrDetectableEffect

AlphaPowerTradeoff :=
 LowerAlpha generally reduces power for fixed sample size_and_effect

MultipleTesting :=
 PerformMultipleHypothesisTests → IncreasedChanceOfFalseRejection

FamilyWiseErrorRate := P(AtLeastOneTypeIErrorAmongFamily)
BonferroniCorrection := PerTestAlpha=α_family/m
HolmProcedure := SequentiallyRejectiveBonferroniMethod
FalseDiscoveryRate := ExpectedFalseDiscoveryProportionAmongRejections
BenjaminiHochbergProcedure := OrderedPValues → FDRControllingThresholds

OptionalStoppingRisk :=
 RepeatedUnplannedTesting can inflate TypeIError under fixed-horizon methods

SequentialTesting :=
 DataArriveOverTime → PredefinedSequentialDecisionRule → ControlledErrorProperties

SequentialProbabilityRatioTest :=
 SequentialLikelihoodRatio → ContinueAcceptOrRejectDecision

EquivalenceTesting :=
 H0:EffectOutsideEquivalenceRegion
 versus H1:EffectWithinEquivalenceRegion

TOST := TwoOneSidedTests → EquivalenceDecision
NonInferiorityTesting := CompareEffectAgainstPrespecifiedNonInferiorityMargin
SuperiorityTesting := TestEffectAgainstPrespecifiedSuperiorityCriterion

CompositeNull := Θ0ContainsMultipleParameterValues
NuisanceParameter := ParameterAffectingNullDistributionButNotPrimaryTarget
ConditionalTest := ConditionOnStatisticToEliminateNuisanceParameter
ExactConditionalTest := ConditionalNullDistribution → ExactPValue

MonteCarloPValue :=
 SimulateNullReplicates → CountAtLeastAsExtreme → EstimatedPValue

MonteCarloPValueCorrection := (B_extreme+1)/(B+1)
DiscreteTest := DiscreteNullDistribution → DiscreteAttainableSignificanceLevels

MissingDataHypothesisTesting :=
 MissingnessHandling ∧ TestStatistic ∧ VariancePropagation → ValidOrApproximateTest

SurveyHypothesisTesting :=
 ComplexSamplingDesign ∧ DesignBasedVariance → DesignAdjustedTest

ClusterRobustTesting :=
 ClusteredData ∧ ClusterRobustCovariance → RobustTestStatistic

HeteroscedasticityRobustTesting :=
 HeteroscedasticData ∧ RobustVariance → AdjustedInference

TestCalibration :=
 UnderNullSimulation → EstimateRejectionFrequency → CompareWithNominalAlpha

EmpiricalTypeIError := NullRejections/NullReplications
EmpiricalPower := AlternativeRejections/AlternativeReplications

HypothesisTestingConstraint :=
 ConstraintOn(
  NullHypothesis,AlternativeHypothesis,TestStatistic,SamplingDistribution,
  SamplingDesign,SignificanceLevel,Assumptions,MultiplicityPlan
 )

HypothesisTestingConsistency :=
 Hypotheses ∧ TestStatistic ∧ NullDistribution ∧ DecisionRule ∧ Alpha ∧ SamplingDesign
 jointly_compatible

InvalidHypothesisTest :=
 UndefinedNull
 ∨ UndefinedAlternative
 ∨ InvalidTestStatistic
 ∨ IncorrectNullDistribution
 ∨ ViolatedCriticalAssumption
 ∨ IgnoredDependence
 ∨ IgnoredSamplingDesign
 ∨ UncontrolledMultiplicityWhenRequired

HypothesisTestValidation :=
 ResolveScientificQuestion
 → ResolvePopulationSampleAndEstimand
 → SpecifyNullHypothesis
 → SpecifyAlternativeHypothesis
 → ResolveSamplingDesign
 → SelectTestStatistic
 → DeriveOrResolveNullDistribution
 → ResolveSignificanceLevel
 → ComputeObservedStatistic
 → ComputePValueOrCriticalRegion
 → ApplyDecisionRule
 → QuantifyEffectAndUncertainty
 → ValidateAssumptionsAndMultiplicity
 → ValidateResult

HypothesisTestingState := {
 Population,Sample,Estimand,Parameter,NullHypothesis,AlternativeHypothesis,
 TestStatistic,ObservedTestStatistic,NullDistribution,AlternativeDistribution,
 SignificanceLevel,PValue,CriticalRegion,Decision,TypeIErrorRate,TypeIIErrorRate,
 Power,EffectSize,SamplingDesign,MultiplicityMethod,Assumptions,Evidence,
 Confidence,Provenance
}

SEMANTIC_EQUIVALENCE(TestA,TestB) :=
 same(NullHypothesis)
 ∧ same(AlternativeHypothesis)
 ∧ same(TestStatisticOrMonotonicEquivalent)
 ∧ same(NullReferenceDistribution)
 ∧ same(SignificanceLevel)
 ∧ same(SamplingDesign)
 ∧ same(MultiplicityFramework)

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

CoreClasses := {
 HypothesisTesting,NullHypothesis,AlternativeHypothesis,SimpleHypothesis,
 CompositeHypothesis,TestStatistic,NullDistribution,CriticalRegion,
 SignificanceLevel,PValue,TypeIError,TypeIIError,Power,LikelihoodRatioTest,
 WaldTest,ScoreTest,ZTest,TTest,ChiSquaredTest,FTest,PermutationTest,MultipleTesting
}

CoreRelations := {
 tests,contrastsWith,computedFrom,hasNullDistribution,hasAlternativeDistribution,
 hasPValue,hasSignificanceLevel,hasPower,hasTypeIError,hasTypeIIError,
 rejects,failsToReject,adjustedBy,constrainedBy,equivalentTo,tracesTo
}

TypedRelations :=
 HypothesisTest --tests--> NullHypothesis
 NullHypothesis --contrastsWith--> AlternativeHypothesis
 TestStatistic --computedFrom--> Sample
 TestStatistic --hasNullDistribution--> NullDistribution
 TestStatistic --hasAlternativeDistribution--> AlternativeDistribution
 HypothesisTest --hasPValue--> PValue
 HypothesisTest --hasSignificanceLevel--> SignificanceLevel
 HypothesisTest --hasPower--> Power
 HypothesisTest --hasTypeIError--> TypeIError
 HypothesisTest --hasTypeIIError--> TypeIIError
 MultipleTesting --adjustedBy--> MultiplicityMethod
 HypothesisTest --tracesTo--> Provenance

InferenceRules := {
 PValue≤Alpha ⇒ RejectH0,
 PValue>Alpha ⇒ FailToRejectH0,
 TypeIErrorProbability=Alpha under calibrated simple-null test,
 Power=1-TypeIIErrorProbability,
 FailToRejectH0 ↛ H0True,
 StatisticalSignificance ↛ PracticalSignificance,
 ValidTestInversion ⇒ ConfidenceSet={NonRejectedParameterValues},
 MultipleTestingWithoutAdjustment ⇒ FlagMultiplicityRisk
}

CanonicalHypothesisTestingRepresentation :=
 ParseHypothesisTestingRequest
 → ResolvePopulationSampleAndEstimand
 → ResolveNullAndAlternativeHypotheses
 → ResolveSamplingDesign
 → ResolveTestStatistic
 → ResolveNullAndAlternativeDistributions
 → ResolveSignificanceLevel
 → ComputeObservedTestStatistic
 → ComputePValueAndCriticalRegion
 → ApplyDecisionRule
 → ResolveTypeITypeIIAndPowerProperties
 → ResolveEffectSizeAndConfidenceIntervalIfRequested
 → ResolveMultiplicityAdjustment
 → ValidateHypothesisTest

HypothesisTestingGraph := G=(V,E)

V := {
 HypothesisTests,NullHypotheses,AlternativeHypotheses,TestStatistics,
 NullDistributions,AlternativeDistributions,CriticalRegions,SignificanceLevels,
 PValues,TypeIErrors,TypeIIErrors,Powers,EffectSizes,SamplingDesigns,
 LikelihoodRatioTests,WaldTests,ScoreTests,ZTests,TTests,ChiSquaredTests,
 FTests,PermutationTests,MultiplicityMethods,Evidence,ProvenanceNodes
}

E := {
 tests,contrastsWith,computedFrom,hasNullDistribution,hasAlternativeDistribution,
 hasPValue,hasSignificanceLevel,hasPower,hasTypeIError,hasTypeIIError,
 rejects,failsToReject,adjustedBy,constrainedBy,equivalentTo,tracesTo
}

CONFIDENCE(HypothesisTestingResolution) :=
 weighted_mean(
  HypothesisResolution,EstimandResolution,SamplingDesignResolution,
  TestStatisticValidity,NullDistributionValidity,SignificanceLevelResolution,
  AssumptionValidity,MultiplicityValidity,ProvenanceQuality
 )

AI_QUERY_INTERPRETATION :=
 Query
 → DetectHypothesisTestingIntent
 → IdentifyPopulationSampleEstimandAndHypotheses
 → ResolveOneSidedTwoSidedExactAsymptoticOrResamplingContext
 → ResolveTestStatisticNullDistributionAndAlpha
 → IdentifyRequestedPValueDecisionPowerEffectSizeOrMultiplicityAdjustment

AI_REASONING_FLOW :=
 Query
 → ParseHypothesisTestingProblem
 → ResolvePopulationSampleAndEstimand
 → SpecifyNullAndAlternativeHypotheses
 → ResolveSamplingDesign
 → SelectTestStatistic
 → ResolveNullDistribution
 → ResolveSignificanceLevel
 → ComputeObservedStatistic
 → ComputePValue
 → DetermineCriticalRegion
 → ApplyDecisionRule
 → ComputeOrResolvePowerAndErrorRatesIfRequested
 → ResolveEffectSizeAndConfidenceInterval
 → ApplyMultiplicityAdjustmentIfRequired
 → ValidateAssumptions
 → UpdateConfidence
 → PreserveProvenance
 → GenerateExplanation

AI_CONSUMPTION := {
 ParseHypothesisTestingEntities,ParseTypedRelations,ResolvePopulations,ResolveSamples,
 ResolveEstimands,ResolveParameters,ResolveNullHypotheses,ResolveAlternativeHypotheses,
 ResolveSamplingDesigns,ResolveTestStatistics,ResolveNullDistributions,
 ResolveAlternativeDistributions,ComputeObservedStatistics,ComputePValues,
 ResolveCriticalValues,ResolveCriticalRegions,ApplyDecisionRules,ComputeTypeIError,
 ComputeTypeIIError,ComputePower,PerformPowerAnalysis,ComputeEffectSizes,
 ResolveConfidenceIntervalDuality,ExecuteLikelihoodRatioTests,ExecuteWaldTests,
 ExecuteScoreTests,ExecuteZTests,ExecuteTTests,ExecuteChiSquaredTests,ExecuteFTests,
 ExecutePermutationTests,ExecuteBootstrapTests,ResolveSequentialTests,
 ResolveEquivalenceAndNonInferiorityTests,ApplyMultiplicityCorrections,
 ValidateHypothesisTestingSemantics,CompareSemanticEquivalence,UpdateConfidence,
 PreserveProvenance,GenerateFormalExplanation
}

UPDATE_MODEL :=
 NewDataHypothesisTestDesignAlphaOrEvidence
 → ValidateInput
 → IdentifyAffectedHypothesisTestingClaims
 → UpdateHypothesisTestingState
 → RecomputeObservedStatistic
 → RecomputeNullReferenceAndPValue
 → RecomputeCriticalRegionAndDecision
 → RecomputePowerAndErrorPropertiesIfRequired
 → ReapplyMultiplicityAdjustment
 → RevalidateAssumptions
 → RecomputeConfidence
 → PreserveProvenance
 → VersionModel

DataUpdate :=
 ChangedObservedData
 ⇒ recompute(ObservedTestStatistic,PValue,Decision,EffectSize)

HypothesisUpdate :=
 ChangedNullOrAlternative
 ⇒ reevaluate(TestStatistic,NullDistribution,PValue,CriticalRegion,Power)

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

FLOW_HASH_INPUT :=
 canonical(
  ONTOLOGY,
  CoreClasses,
  CoreRelations,
  InferenceRules,
  HypothesisTesting,
  HypothesisTestValidation,
  HypothesisTestingConstraint,
  SEMANTIC_EQUIVALENCE,
  AI_REASONING_FLOW
 )

FLOW_HASH := HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
 HypothesisTestingEntityResolution,
 Population,Sample,Estimand,Parameter,NullHypothesis,AlternativeHypothesis,
 TestStatistic,ObservedTestStatistic,NullDistribution,AlternativeDistribution,
 SignificanceLevel,PValue,CriticalRegion,Decision,TypeIErrorRate,TypeIIErrorRate,
 Power,EffectSize,ConfidenceIntervalRelation,SamplingDesign,MultiplicityMethod,
 AssumptionStatus,ValidationStatus,Confidence,Provenance,Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA