Statistical Power Ontology

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

BEGIN_FLOW

StatisticalPower :=
 HypothesisTestingProperty ∧ AlternativeRejectionProbability
 ∧ OneMinusTypeIIError ∧ FunctionOfEffectSampleSizeAlphaVarianceAndDesign

Population := TargetCollectionOrGeneratingProcess
Sample := X={X_1,...,X_n}
Parameter := θ
ParameterSpace := Θ
Estimand := τ(θ)
NullHypothesis := H0:θ∈Θ0
AlternativeHypothesis := H1:θ∈Θ1
TestStatistic := T(X)
CriticalRegion := C_α
SignificanceLevel := α

TypeIIError := FailToRejectH0WhenSpecifiedAlternativeTrue
TypeIIErrorProbability := β(θ)=P_θ(T(X)∉C_α)
StatisticalPowerDefinition := π(θ)=P_θ(RejectH0)
PowerRelation := π(θ)=1-β(θ)
PowerFunction := θ→π(θ)
DesiredPower := π_target
PowerRange := 0≤π(θ)≤1

LowPower := SmallProbabilityOfRejectingFalseNullAtSpecifiedAlternative
HighPower := LargeProbabilityOfRejectingFalseNullAtSpecifiedAlternative
ConsistentTest := FixedAlternative ⇒ π_n(θ)→1 as n→∞

EffectSize := MagnitudeOfDepartureFromNullReferenceUnderSpecifiedMetric
RawEffect := θ-θ0
StandardisedEffect := (θ-θ0)/σ when scale meaningful
AlternativeSeparation := Distance(θ,Θ0)

EffectPowerRelation :=
 GreaterRelevantAlternativeSeparation generally → GreaterPower
 under regular monotone tests

SampleSize := n
SampleSizePowerRelation := LargerN generally → GreaterPower for fixed nonzero effect,alpha,variance,and design
SignificanceLevelPowerRelation := LargerAlpha generally → GreaterPower for fixed effect_and_sample_size
StringencyTradeoff := SmallerAlpha → LowerTypeIErrorAllowance ∧ UsuallyLowerPower
VariancePowerRelation := GreaterNoiseOrVariance generally → LowerPower
StandardError := SE(Estimator)
PrecisionPowerRelation := SmallerStandardError → GreaterNullAlternativeSeparation → GreaterPower

PowerDeterminants := {
 EffectSize,SampleSize,SignificanceLevel,Variability,SamplingDesign,
 TestStatistic,AlternativeDirection,MultiplicityAdjustment
}

OneSidedPower := PowerForDirectionalAlternative
TwoSidedPower := PowerForNondirectionalAlternative
OneVsTwoSidedRelation := CorrectlyPrespecifiedOneSidedTestMayHaveGreaterPowerInSpecifiedDirection
DirectionMisspecification := TrueEffectOppositeToOneSidedAlternative → VeryLowRelevantPower

CriticalValue := ThresholdDeterminedByAlphaAndNullDistribution
AlternativeDistribution := DistributionOf(T(X)|θ∈Θ1)
PowerFromAlternativeDistribution := π(θ)=P_θ(T(X)∈C_α)
UpperTailPower := P_θ(T≥c_α)
LowerTailPower := P_θ(T≤c_α)
TwoSidedPowerFormula := P_θ(T≤c_L ∨ T≥c_U)

PowerCurve := θ→π(θ)
PowerSurface := (EffectSize,SampleSize,Alpha,Variance)→Power
OperatingCharacteristic := OC(θ)=P_θ(FailToRejectH0)
OperatingCharacteristicRelation := OC(θ)=1-π(θ)

PowerAnalysis := StatisticalPlanningProcedureUsingPowerFunction
ProspectivePowerAnalysis := BeforeDataCollection → SpecifyEffectAlphaVarianceDesiredPower → SolveForSampleSize
A_PrioriPowerAnalysis := DesiredPower ∧ EffectSize ∧ Alpha ∧ Variance → RequiredSampleSize
SensitivityPowerAnalysis := SampleSize ∧ Alpha ∧ DesiredPower → MinimumDetectableEffect
MinimumDetectableEffect := SmallestEffectYieldingTargetPowerForSpecifiedNAlphaVarianceAndTest
RequiredSampleSize := MinimumN satisfying π(θ_alt)≥π_target
SampleSizeRounding := ContinuousCalculatedN → RoundUpToFeasibleIntegerDesign
AttritionAdjustment := RequiredAnalysableN/ExpectedRetentionFraction → RecruitmentTarget

OneSampleMeanPower := AlternativeSampleMeanDistribution ∧ NullCriticalValue → RejectionProbability
TwoSampleMeanPower := DifferenceInMeansDistribution ∧ StandardError ∧ CriticalValues → Power
ProportionTestPower := AlternativeProportion ∧ NullThreshold ∧ SamplingDistribution → Power
CorrelationTestPower := AlternativeCorrelation ∧ SampleSize ∧ Alpha → Power
RegressionCoefficientPower := AlternativeCoefficient ∧ CovariateDesign ∧ ErrorVariance ∧ Alpha → Power
ANOVAPower := BetweenGroupEffect ∧ WithinGroupVariance ∧ GroupSizes ∧ Alpha → Power
ChiSquaredPower := NoncentralChiSquaredDistribution ∧ CriticalValue → Power
FTestPower := NoncentralFDistribution ∧ CriticalValue → Power

NoncentralDistribution := AlternativeReferenceDistributionParameterisedByNoncentralityParameter
NoncentralityParameter := FunctionOf(EffectSize,SampleSize,Variance,Design)
NoncentralityPowerFlow := AlternativeEffect → NoncentralityParameter → AlternativeStatisticDistribution → RejectionProbability
TTestPower := NoncentralTDistribution → RejectionProbability

PairedDesignPower := WithinPairCorrelationAffectsVarianceAndPower
RepeatedMeasuresPower := WithinSubjectCovarianceStructure ∧ NumberOfMeasurements → EffectivePrecision → Power

ClusteredDesignPower :=
 ClusterSize ∧ IntraclusterCorrelation ∧ NumberOfClusters
 → DesignEffect → EffectiveSampleSize → Power

DesignEffect := DEFF=VarianceComplexDesign/VarianceSRSReference
EffectiveSampleSize := n_eff≈n/DEFF
ApproximateClusterDesignEffect := 1+(m-1)ICC
StratifiedDesignPower := InformativeStratificationMayReduceVarianceAndIncreasePower
SurveyWeightingPower := UnequalWeightsMayReduceEffectiveSampleSizeAndAlterPower
MissingDataPower := Missingness → ReducedEffectiveInformation → LowerPower
MeasurementErrorPower := MeasurementNoise → AttenuatedSignalOrIncreasedVariance → ReducedPower

MultiplicityPower :=
 MultipleTestingAdjustment → MoreStringentThresholds → ReducedPerHypothesisPower

BonferroniPower := UseAlpha/m → ComputePowerAtAdjustedThreshold
DisjunctivePower := P(RejectAtLeastOneFalseNull)
ConjunctivePower := P(RejectAllSpecifiedFalseNulls)
MarginalPower := PowerForIndividualHypothesisWithinFamily

SequentialPower := ProbabilityOfRejectingFalseNullUnderSequentialStoppingRule
GroupSequentialPower := CrossingAnyEfficacyBoundaryUnderAlternative
AlphaSpendingPower := AlphaAllocationAcrossLooks → SequentialCriticalBoundaries → OverallPower

EquivalencePower := ProbabilityOfRejectingBothEquivalenceNullComponents
TOSTPower := P(BothOneSidedTestsReject)
NonInferiorityPower := ProbabilityOfRejectingNonInferiorityNull
SuperiorityPower := ProbabilityOfRejectingSuperiorityNull

PermutationTestPower := AlternativeDataGeneratingProcess ∧ PermutationDecisionRule → RejectionProbability
ExactTestPower := ExactAlternativeDistribution → RejectionProbability
DiscretePower := MayChangeStepwiseWithSampleSizeOrEffectDueToDiscreteCriticalRegions

MonteCarloPowerEstimation :=
 SimulateAlternativeDatasets → ApplyCompleteTestingProcedure → EstimateRejectionFrequency

EmpiricalPower := R_reject/R_total
MonteCarloStandardErrorOfPower := sqrt(p_hat(1-p_hat)/R)
PowerSimulationValidation := IncreaseSimulationReplicates → ReduceMonteCarloUncertainty

AnalyticalPower := ClosedFormOrNumericallyIntegratedAlternativeRejectionProbability
SimulationBasedPower := EmpiricalAlternativeRejectionProbability

PowerSensitivityAnalysis :=
 Vary(EffectSize,Variance,Alpha,SampleSize,DesignParameters) → PowerScenarios

PowerScenario := AssumedEffect ∧ AssumedVariance ∧ Design → EstimatedPower
ConservativePlanning := UsePlausibleLowerEffectOrHigherVarianceToAssessRobustSampleSize
PilotEstimateRisk := SmallPilotEffectAndVarianceEstimatesMayBeUnstableForPowerPlanning

PowerAndPValue :=
 Power is pre-data_or repeated-sampling property
 whereas p-value is observed-data quantity

PowerAndEffectSize := HighPowerDoesNotImplyLargeObservedEffect
PowerAndSignificance := SignificantResultDoesNotImplyHighProspectivePower

LowPowerConsequences := {
 HighTypeIIError,UnstableDetection,WideUncertainty,LowReplicationProbabilityUnderSameDesign
}

WinnerCurseRelation :=
 LowPoweredStudies conditional_on significance
 may overestimate effect magnitude under selection

PowerCalibration :=
 SimulateAtSpecifiedAlternative → ApplyTestingProcedure → CompareEmpiricalPowerWithTheoreticalPower

ModelMisspecification := IncorrectAlternativeOrNullModelMayDistortPowerCalculations
VarianceMisspecification := IncorrectVarianceAssumption → IncorrectRequiredSampleSizeOrPower
EffectMisspecification := IncorrectPlanningEffect → MisleadingPowerEstimate
DesignMisspecification := IgnoredClusteringStratificationWeightsOrDependence → MisleadingPower

StatisticalPowerConstraint :=
 ConstraintOn(
  NullHypothesis,AlternativeHypothesis,EffectSize,TestStatistic,SignificanceLevel,
  SampleSize,Variance,SamplingDesign,MultiplicityFramework
 )

StatisticalPowerConsistency :=
 NullHypothesis ∧ AlternativeHypothesis ∧ TestStatistic ∧ CriticalRegion
 ∧ AlternativeDistribution ∧ Alpha ∧ SampleSize ∧ Design
 jointly_compatible

InvalidPowerCalculation :=
 UndefinedAlternative
 ∨ UndefinedEffectSize
 ∨ InvalidAlpha
 ∨ InvalidSampleSize
 ∨ IncorrectAlternativeDistribution
 ∨ IgnoredDesignEffect
 ∨ IgnoredMultiplicity
 ∨ IncompatibleTestStatistic

StatisticalPowerValidation :=
 ResolveScientificQuestion
 → ResolveNullAndAlternativeHypotheses
 → ResolveEffectSizeMetric
 → ResolveTestStatistic
 → ResolveSignificanceLevel
 → ResolveSampleSize
 → ResolveVarianceAndDesign
 → ConstructCriticalRegionUnderNull
 → DeriveAlternativeDistribution
 → ComputeRejectionProbabilityUnderAlternative
 → ResolveMultiplicityAndSequentialAdjustments
 → ValidatePower
 → ValidateResult

StatisticalPowerState := {
 Population,Parameter,Estimand,NullHypothesis,AlternativeHypothesis,EffectSize,
 TestStatistic,SignificanceLevel,SampleSize,Variance,SamplingDesign,CriticalRegion,
 AlternativeDistribution,TypeIIError,StatisticalPower,DesiredPower,
 MinimumDetectableEffect,RequiredSampleSize,MultiplicityMethod,SequentialPlan,
 Evidence,Confidence,Provenance
}

SEMANTIC_EQUIVALENCE(PowerA,PowerB) :=
 same(NullHypothesis)
 ∧ same(AlternativeHypothesisOrEffect)
 ∧ same(TestStatisticAndDecisionRule)
 ∧ same(SignificanceLevel)
 ∧ same(SampleSize)
 ∧ same(VarianceModel)
 ∧ same(SamplingDesign)
 ∧ same(MultiplicityAndSequentialFramework)

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

CoreClasses := {
 StatisticalPower,PowerFunction,TypeIIError,EffectSize,SampleSize,SignificanceLevel,
 CriticalRegion,AlternativeDistribution,DesiredPower,RequiredSampleSize,
 MinimumDetectableEffect,NoncentralityParameter,AnalyticalPower,
 SimulationBasedPower,DesignEffect,EffectiveSampleSize,SequentialPower,
 MultiplicityPower
}

CoreRelations := {
 powerOf,evaluatedUnder,dependsOn,increasesWith,decreasesWith,complements,
 computedFrom,hasAlternativeDistribution,hasCriticalRegion,targets,adjustedBy,
 constrainedBy,equivalentTo,tracesTo
}

TypedRelations :=
 StatisticalPower --powerOf--> HypothesisTest
 StatisticalPower --evaluatedUnder--> AlternativeHypothesis
 StatisticalPower --dependsOn--> EffectSize
 StatisticalPower --dependsOn--> SampleSize
 StatisticalPower --dependsOn--> SignificanceLevel
 StatisticalPower --dependsOn--> Variance
 StatisticalPower --dependsOn--> SamplingDesign
 StatisticalPower --complements--> TypeIIError
 StatisticalPower --computedFrom--> AlternativeDistribution
 HypothesisTest --hasCriticalRegion--> CriticalRegion
 StatisticalPower --targets--> DesiredPower
 StatisticalPower --adjustedBy--> MultiplicityMethod
 StatisticalPower --tracesTo--> Provenance

InferenceRules := {
 AlternativeHypothesis ∧ CriticalRegion
  ⇒ Power=P_Alternative(TestStatistic∈CriticalRegion),

 Power=1-TypeIIErrorProbability,

 LargerSampleSize ∧ FixedEffectVarianceAlphaAndDesign
  ⇒ GenerallyHigherPower,

 LargerRelevantEffectSeparation ∧ FixedNAlphaVarianceAndDesign
  ⇒ GenerallyHigherPower,

 LargerVariance ∧ FixedEffectNAlphaAndDesign
  ⇒ GenerallyLowerPower,

 MultiplicityAdjustmentToSmallerThreshold
  ⇒ GenerallyLowerMarginalPower,

 InvalidAlternativeDistribution
  ⇒ FlagPowerCalculationInvalid
}

CanonicalStatisticalPowerRepresentation :=
 ParseStatisticalPowerRequest
 → ResolvePopulationParameterAndEstimand
 → ResolveNullAndAlternativeHypotheses
 → ResolveEffectSize
 → ResolveTestStatistic
 → ResolveSignificanceLevel
 → ResolveSampleSizeAndVariance
 → ResolveSamplingDesign
 → ConstructNullCriticalRegion
 → DeriveAlternativeDistribution
 → ComputeTypeIIError
 → ComputeStatisticalPower
 → ResolveRequiredSampleSizeOrMinimumDetectableEffectIfRequested
 → ResolveMultiplicityAndSequentialAdjustments
 → ValidateStatisticalPower

StatisticalPowerGraph := G=(V,E)

V := {
 StatisticalPowers,PowerFunctions,TypeIIErrors,EffectSizes,SampleSizes,
 SignificanceLevels,CriticalRegions,AlternativeDistributions,DesiredPowers,
 RequiredSampleSizes,MinimumDetectableEffects,NoncentralityParameters,
 AnalyticalPowers,SimulationBasedPowers,DesignEffects,EffectiveSampleSizes,
 SequentialPowers,MultiplicityPowers,NullHypotheses,AlternativeHypotheses,
 TestStatistics,SamplingDesigns,Evidence,ProvenanceNodes
}

E := {
 powerOf,evaluatedUnder,dependsOn,increasesWith,decreasesWith,complements,
 computedFrom,hasAlternativeDistribution,hasCriticalRegion,targets,adjustedBy,
 constrainedBy,equivalentTo,tracesTo
}

CONFIDENCE(StatisticalPowerResolution) :=
 weighted_mean(
  HypothesisResolution,EffectSizeResolution,TestStatisticValidity,
  AlphaResolution,SampleSizeResolution,VarianceResolution,SamplingDesignValidity,
  AlternativeDistributionValidity,MultiplicityValidity,ProvenanceQuality
 )

AI_QUERY_INTERPRETATION :=
 Query
 → DetectStatisticalPowerIntent
 → IdentifyNullAlternativeEffectAndTest
 → ResolveAlphaSampleSizeVarianceAndDesign
 → IdentifyProspectivePowerSampleSizeOrMinimumDetectableEffectRequest
 → ResolveMultiplicityAndSequentialContext

AI_REASONING_FLOW :=
 Query
 → ParseStatisticalPowerProblem
 → ResolvePopulationParameterAndEstimand
 → ResolveNullHypothesis
 → ResolveAlternativeHypothesis
 → ResolveEffectSizeMetricAndValue
 → ResolveTestStatistic
 → ResolveSignificanceLevel
 → ResolveSampleSize
 → ResolveVarianceAndSamplingDesign
 → ConstructCriticalRegion
 → DeriveAlternativeDistribution
 → ComputeTypeIIError
 → ComputePower
 → SolveForRequiredSampleSizeOrMinimumDetectableEffectIfRequested
 → ApplyMultiplicityOrSequentialAdjustments
 → ValidatePowerCalculation
 → UpdateConfidence
 → PreserveProvenance
 → GenerateExplanation

AI_CONSUMPTION := {
 ParseStatisticalPowerEntities,ParseTypedRelations,ResolvePopulations,
 ResolveParameters,ResolveEstimands,ResolveNullHypotheses,
 ResolveAlternativeHypotheses,ResolveEffectSizes,ResolveTestStatistics,
 ResolveSignificanceLevels,ResolveSampleSizes,ResolveVariances,
 ResolveSamplingDesigns,ConstructCriticalRegions,DeriveAlternativeDistributions,
 ComputeTypeIIErrorRates,ComputePowerFunctions,ComputeOneSidedPower,
 ComputeTwoSidedPower,ComputeZTestPower,ComputeTTestPower,ComputeANOVAPower,
 ComputeChiSquaredPower,ComputeFTestPower,ResolveNoncentralityParameters,
 SolveRequiredSampleSizes,ComputeMinimumDetectableEffects,
 ComputeDesignEffects,ComputeEffectiveSampleSizes,ComputeClusteredDesignPower,
 ComputeMultiplicityAdjustedPower,ComputeSequentialPower,
 ComputeEquivalencePower,ComputeNonInferiorityPower,
 ExecuteMonteCarloPowerSimulation,ValidatePowerCalibration,
 PerformPowerSensitivityAnalysis,ValidateStatisticalPowerSemantics,
 CompareSemanticEquivalence,UpdateConfidence,PreserveProvenance,
 GenerateFormalExplanation
}

UPDATE_MODEL :=
 NewEffectSampleSizeAlphaVarianceDesignOrEvidence
 → ValidateInput
 → IdentifyAffectedPowerClaims
 → UpdateStatisticalPowerState
 → RecomputeCriticalRegionIfRequired
 → RecomputeAlternativeDistribution
 → RecomputeTypeIIErrorAndPower
 → RecomputeRequiredSampleSizeOrDetectableEffect
 → ReapplyMultiplicityAndSequentialAdjustments
 → RecomputeConfidence
 → PreserveProvenance
 → VersionModel

EffectUpdate :=
 ChangedEffectSize
 ⇒ recompute(
  AlternativeDistribution,TypeIIError,StatisticalPower,RequiredSampleSize
 )

SampleSizeUpdate :=
 ChangedSampleSize
 ⇒ recompute(
  StandardError,AlternativeDistribution,StatisticalPower,MinimumDetectableEffect
 )

AlphaUpdate :=
 ChangedSignificanceLevel
 ⇒ recompute(
  CriticalRegion,StatisticalPower,RequiredSampleSize
 )

DesignUpdate :=
 ChangedSamplingDesign
 ⇒ reevaluate(
  DesignEffect,EffectiveSampleSize,AlternativeDistribution,StatisticalPower
 )

FLOW_HASH_INPUT :=
 canonical(
  ONTOLOGY,
  CoreClasses,
  CoreRelations,
  InferenceRules,
  StatisticalPower,
  StatisticalPowerValidation,
  StatisticalPowerConstraint,
  SEMANTIC_EQUIVALENCE,
  AI_REASONING_FLOW
 )

FLOW_HASH := HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
 StatisticalPowerEntityResolution,Population,Parameter,Estimand,NullHypothesis,
 AlternativeHypothesis,EffectSize,TestStatistic,SignificanceLevel,SampleSize,
 Variance,SamplingDesign,CriticalRegion,AlternativeDistribution,TypeIIError,
 StatisticalPower,DesiredPower,RequiredSampleSize,MinimumDetectableEffect,
 DesignEffect,EffectiveSampleSize,MultiplicityMethod,SequentialPlan,
 ValidationStatus,Confidence,Provenance,Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA