Effect Size Ontology

FULL EXECUTABLE FLOW ALGEBRA — EFFECT SIZE ONTOLOGY
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/hypothesis-testing/effect-size-ontology.html

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

BEGIN_FLOW

EffectSize :=
 QuantitativeMagnitude
 ∧ DifferenceOrAssociationMeasure
 ∧ PopulationOrSampleEffectRepresentation
 ∧ StatisticalInferenceComponent

Population := TargetCollectionOrGeneratingProcess
Sample := X={X_1,...,X_n}
Parameter := θ
Estimand := τ(θ)
Estimate := τ_hat
NullValue := τ0

EffectSizeDefinition :=
 MagnitudeOrDirectionOfDepartureFromReference

RawEffectSize :=
 τ-τ0

EstimatedRawEffectSize :=
 τ_hat-τ0

AbsoluteEffectSize :=
 |τ-τ0|

SignedEffectSize :=
 τ-τ0

StandardisedEffectSize :=
 RawEffectSize/ScaleParameter
 when denominator is meaningful

UnstandardisedEffectSize :=
 EffectExpressedInOriginalMeasurementUnits

EffectDirection :=
 sign(EffectSize)
 when signed metric used

ZeroEffect :=
 EffectSize=0
 for difference-scale null

NullEffectConstraint :=
 NullValue depends_on effect metric

DifferenceScaleNull :=
 NullValue=0

RatioScaleNull :=
 NullValue=1

MeanDifference :=
 μ1-μ2

EstimatedMeanDifference :=
 X_bar1-X_bar2

StandardisedMeanDifference :=
 (μ1-μ2)/σ_reference

CohenD :=
 (X_bar1-X_bar2)/S_pooled
 under specified sample convention

PopulationStandardisedMeanDifference :=
 δ=(μ1-μ2)/σ

PooledStandardDeviation :=
 S_pooled
 =
 sqrt(
  ((n1-1)S1²+(n2-1)S2²)
  /
  (n1+n2-2)
 )

HedgesG :=
 SmallSampleBiasCorrectedStandardisedMeanDifference

HedgesCorrection :=
 J(df)
 ×
 CohenD

GlassDelta :=
 (X_bar_treatment-X_bar_control)
 /
 S_control

PairedStandardisedMeanDifference :=
 MeanDifference/ChosenWithinPairScale

PairedEffectConstraint :=
 StandardisationConvention
 must be explicitly identified

ProportionDifference :=
 p1-p2

RiskDifference :=
 p1-p0

RiskRatio :=
 p1/p0

RelativeRisk :=
 RiskRatio

Odds :=
 p/(1-p)

OddsRatio :=
 [p1/(1-p1)]
 /
 [p0/(1-p0)]

LogOddsRatio :=
 log(OddsRatio)

NumberNeededToTreat :=
 1/AbsoluteRiskDifference
 under applicable causal and sign conventions

NNTConstraint :=
 Interpretation depends_on baseline risk,
 direction,
 follow-up period,
 and causal assumptions

CorrelationEffect :=
 ρ

SampleCorrelationEffect :=
 r

PearsonCorrelation :=
 LinearAssociationMeasure

SpearmanCorrelation :=
 RankAssociationMeasure

CoefficientOfDetermination :=
 R²

R2Effect :=
 ProportionOfVarianceExplained
 under specified model interpretation

CohenF2 :=
 R²/(1-R²)
 for selected regression contexts

CohenF :=
 sqrt(
  η²/(1-η²)
 )
 under selected ANOVA relation

EtaSquared :=
 η²
 =
 SSB/SST

PartialEtaSquared :=
 SS_effect
 /
 (SS_effect+SS_error)

OmegaSquared :=
 BiasAdjustedVarianceExplainedMeasure
 under ANOVA framework

CramersV :=
 sqrt(
  χ²
  /
  (n×min(r-1,c-1))
 )

PhiCoefficient :=
 sqrt(χ²/n)
 for compatible 2x2 table setting

StandardisedRegressionCoefficient :=
 β_j
 expressed_on standardised scales

UnstandardisedRegressionEffect :=
 ChangeInExpectedOutcomePerUnitPredictorChange
 under regression model

RateRatio :=
 λ1/λ0

HazardRatio :=
 h1(t)/h0(t)
 under proportional-hazards interpretation when applicable

LogHazardRatio :=
 log(HazardRatio)

EffectMetric :=
 FunctionMappingPopulationParametersToMagnitudeMeasure

EffectMetricSelection :=
 ScientificQuestion
 → OutcomeScale
 → StudyDesign
 → ChooseInterpretableEffectMetric

EffectScale := {
 DifferenceScale,
 RatioScale,
 StandardisedScale,
 ProbabilityScale,
 CorrelationScale,
 LogScale
}

EffectSizeVsPValue :=
 EffectSize measures magnitude
 whereas PValue measures null-reference extremeness

EffectSizeVsStatisticalSignificance :=
 LargeEffect may be nonsignificant with low precision
 ∧ SmallEffect may be significant with large sample

PracticalSignificance :=
 EffectMagnitude
 interpreted_against domain-relevant threshold

ClinicalOrDomainImportance :=
 ContextSpecificMeaning
 of effect magnitude

MinimumImportantEffect :=
 SmallestEffectConsideredMeaningful
 under domain definition

MinimumDetectableEffect :=
 SmallestEffectDetectable
 at specified alpha,power,n,and design

ImportanceVsDetectability :=
 MinimumImportantEffect
 ≠
 MinimumDetectableEffect

ConfidenceIntervalForEffect :=
 EffectEstimate
 ∧ StandardErrorOrLikelihood
 → IntervalEstimate

EffectUncertainty :=
 SamplingDistribution
 ∨ PosteriorDistribution
 ∨ ConfidenceInterval

EffectPrecision :=
 NarrowerUncertaintyInterval
 generally → GreaterPrecision

SamplingError :=
 EstimatedEffect-TrueEffect

EffectEstimatorBias :=
 E[EstimatedEffect]-TrueEffect

EffectMSE :=
 Variance(EffectEstimator)+Bias²

PowerRelation :=
 EffectSize
 ∧ SampleSize
 ∧ Alpha
 ∧ Variance
 ∧ Design
 → StatisticalPower

LargerEffectPowerRelation :=
 GreaterRelevantEffectMagnitude
 generally → GreaterPower
 under fixed n,alpha,variance,and design

SampleSizeRelation :=
 LargerN
 → GreaterPrecision
 → GreaterPowerForFixedNonzeroEffect

StandardisationBenefit :=
 AllowsComparisonAcrossScales
 when denominator and construct are comparable

StandardisationRisk :=
 StandardisedEffect
 may obscure original-unit meaning

ScaleDependence :=
 UnstandardisedEffect
 depends_on measurement units

TransformationRelation :=
 EffectOnTransformedScale
 may require back-transformation
 for interpretation

LogRatioTransformation :=
 RatioEffect
 → log(Ratio)
 for symmetric modelling and inference properties

BackTransformation :=
 LogEffect
 → exp(LogEffect)
 → RatioEffect

NonlinearTransformationConstraint :=
 EffectMagnitudeAndSymmetry
 may change under transformation

DirectionReversal :=
 RecodingGroupsOrOutcome
 may reverse signed effect

ReferenceGroupConstraint :=
 RatioAndDifferenceEffects
 require explicit reference group

BaselineRiskConstraint :=
 RelativeEffects
 may correspond_to different absolute effects
 at different baseline risks

ConfoundingRelation :=
 ObservedAssociationEffect
 may differ_from causal effect
 when confounding present

CausalEffect :=
 EffectDefinedUnderInterventionOrPotentialOutcomeFramework

AssociationalEffect :=
 EffectDefinedFromObservedStatisticalAssociation

CausalInterpretationConstraint :=
 StatisticalEffectSize
 does_not imply causal effect
 without causal identification assumptions

AverageTreatmentEffect :=
 E[Y(1)-Y(0)]

AverageTreatmentEffectOnTreated :=
 E[Y(1)-Y(0)|Treated]

StandardisedEffectMetaAnalysis :=
 StudySpecificStandardisedEffects
 → WeightedSynthesis

MetaAnalyticEffect :=
 WeightedCombinationOfStudyEffects

FixedEffectMetaAnalysis :=
 CommonEffectAssumption
 → PrecisionWeightedEstimate

RandomEffectsMetaAnalysis :=
 StudySpecificTrueEffects
 ∧ BetweenStudyHeterogeneity
 → MeanEffectDistribution

Heterogeneity :=
 VariationInTrueEffectsAcrossStudies

TauSquared :=
 BetweenStudyVariance

I2 :=
 ProportionOfObservedVariationAttributedToHeterogeneity
 under meta-analytic convention

EffectSizeWeight :=
 OftenInverseVarianceWeight
 in meta-analysis

SmallStudyBiasRisk :=
 StudySizeRelatedSelectionOrReporting
 may distort pooled effect

WinnerCurseRelation :=
 SelectionOnStatisticalSignificance
 may inflate observed effect magnitude

RegressionToMean :=
 ExtremeObservedEffects
 may be less extreme on replication

ShrinkageEffectEstimate :=
 RawEstimate
 → PartialPoolingOrRegularisation
 → StabilisedEstimate

RobustEffectEstimate :=
 EffectEstimator
 designed_to reduce sensitivity
 to outliers_or_model deviations

BootstrapEffectInterval :=
 ResampleData
 → RecomputeEffect
 → BootstrapDistribution
 → Interval

PermutationEffectDistinction :=
 PermutationTest
 evaluates null-reference evidence
 whereas effect estimate quantifies magnitude

BayesianEffectSize :=
 PosteriorDistributionOfEffectMetric

PosteriorEffectProbability :=
 P(Effect>Threshold|Data)
 under Bayesian model

ROPE :=
 RegionOfPracticalEquivalence
 under Bayesian decision framework

EquivalenceMargin :=
 PrespecifiedEffectBounds
 defining practically negligible region

EquivalenceTestingRelation :=
 EffectEstimate
 ∧ ConfidenceInterval
 ∧ EquivalenceMargins
 → EquivalenceAssessment

NonInferiorityMargin :=
 PrespecifiedMaximumAcceptableLoss

SuperiorityMargin :=
 PrespecifiedMinimumRelevantGain

EffectSizeClassification :=
 SmallMediumLargeLabels
 may be domain-dependent
 and should_not be treated as universal truths

CohenBenchmarkConstraint :=
 ConventionalBenchmarks
 are heuristics
 not universal domain standards

EffectSizeReporting :=
 Report(
  EffectMetric,
  PointEstimate,
  ConfidenceInterval,
  UnitsOrScale,
  ReferenceGroup,
  Direction,
  Assumptions
 )

EffectSizeConstraint :=
 ConstraintOn(
  Estimand,
  EffectMetric,
  MeasurementScale,
  ReferenceValue,
  ReferenceGroup,
  StandardisationScale,
  StatisticalModel,
  SamplingDesign
 )

EffectSizeConsistency :=
 Estimand
 ∧ Metric
 ∧ Scale
 ∧ Reference
 ∧ Estimator
 ∧ SamplingDesign
 jointly_compatible

InvalidEffectSize :=
 UndefinedEstimand
 ∨ UndefinedMetric
 ∨ InvalidDenominator
 ∨ IncompatibleScale
 ∨ MissingReferenceGroup
 ∨ IncorrectStandardisation
 ∨ UnsupportedCausalInterpretation

EffectSizeValidation :=
 ResolveScientificQuestion
 → ResolvePopulationAndEstimand
 → ResolveOutcomeAndPredictorScales
 → ResolveReferenceValueOrGroup
 → SelectEffectMetric
 → ResolveEstimator
 → ComputeEffectEstimate
 → ResolveStandardisationIfRequired
 → QuantifyUncertainty
 → ValidateDirectionAndScale
 → ValidateInterpretation
 → ValidateResult

EffectSizeState := {
 Population,
 Sample,
 Parameter,
 Estimand,
 EffectMetric,
 EffectScale,
 ReferenceValue,
 ReferenceGroup,
 TrueEffectSize,
 EstimatedEffectSize,
 StandardisationScale,
 StandardError,
 ConfidenceInterval,
 Direction,
 PracticalImportanceThreshold,
 MinimumDetectableEffect,
 SamplingDesign,
 Assumptions,
 Evidence,
 Confidence,
 Provenance
}

SEMANTIC_EQUIVALENCE(EffectA,EffectB) :=
 same(Estimand)
 ∧ same(EffectMetric)
 ∧ same(ReferenceValueOrGroup)
 ∧ same(MeasurementScale)
 ∧ same(StandardisationConvention)
 ∧ same(DirectionConvention)
 ∧ same(PopulationAndDesignContext)

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

CoreClasses := {
 EffectSize,
 RawEffectSize,
 StandardisedEffectSize,
 MeanDifference,
 StandardisedMeanDifference,
 CohenD,
 HedgesG,
 RiskDifference,
 RiskRatio,
 OddsRatio,
 CorrelationEffect,
 EtaSquared,
 PartialEtaSquared,
 CramersV,
 RegressionEffect,
 RateRatio,
 HazardRatio,
 MinimumImportantEffect,
 MinimumDetectableEffect,
 EffectUncertainty
}

CoreRelations := {
 measures,
 targets,
 computedFrom,
 standardisedBy,
 references,
 hasDirection,
 hasScale,
 hasUncertainty,
 comparedWith,
 transformedBy,
 interpretedAgainst,
 dependsOn,
 constrainedBy,
 equivalentTo,
 tracesTo
}

TypedRelations :=
 EffectSize --measures--> EffectMagnitude
 EffectSize --targets--> Estimand
 EstimatedEffectSize --computedFrom--> Sample
 StandardisedEffectSize --standardisedBy--> ScaleParameter
 EffectSize --references--> ReferenceValue
 EffectSize --hasDirection--> EffectDirection
 EffectSize --hasScale--> EffectScale
 EffectSize --hasUncertainty--> ConfidenceInterval
 EffectSize --comparedWith--> MinimumImportantEffect
 EffectSize --transformedBy--> EffectTransformation
 EffectSize --interpretedAgainst--> DomainThreshold
 EffectSize --dependsOn--> SamplingDesign
 EffectSize --tracesTo--> Provenance

InferenceRules := {
 EffectEstimate ∧ NullValue
  ⇒ RawEffect=EffectEstimate-NullValue,

 MeanDifference ∧ PooledStandardDeviation
  ⇒ StandardisedMeanDifference=MeanDifference/PooledStandardDeviation,

 Risk1 ∧ Risk0
  ⇒ RiskDifference=Risk1-Risk0,

 Risk1 ∧ Risk0
  ⇒ RiskRatio=Risk1/Risk0,

 Odds1 ∧ Odds0
  ⇒ OddsRatio=Odds1/Odds0,

 RatioEffect
  ⇒ LogEffect=log(RatioEffect),

 LargerRelevantEffectMagnitude ∧ FixedNAlphaVarianceAndDesign
  ⇒ GenerallyHigherPower,

 UnsupportedCausalAssumptions
  ⇒ FlagCausalEffectInterpretationInvalid
}

CanonicalEffectSizeRepresentation :=
 ParseEffectSizeRequest
 → ResolvePopulationAndSample
 → ResolveParameterAndEstimand
 → ResolveMeasurementScale
 → ResolveReferenceValueOrGroup
 → SelectEffectMetric
 → ComputeEffectEstimate
 → ResolveStandardisation
 → ResolveDirection
 → QuantifyUncertainty
 → CompareWithDomainImportanceThresholdIfProvided
 → ResolvePowerRelationIfRequested
 → ValidateEffectSize

EffectSizeGraph := G=(V,E)

V := {
 EffectSizes,
 RawEffectSizes,
 StandardisedEffectSizes,
 MeanDifferences,
 CohenDs,
 HedgesGs,
 RiskDifferences,
 RiskRatios,
 OddsRatios,
 Correlations,
 EtaSquaredEffects,
 CramersVs,
 RegressionEffects,
 RateRatios,
 HazardRatios,
 MinimumImportantEffects,
 MinimumDetectableEffects,
 ConfidenceIntervals,
 Estimands,
 Samples,
 Populations,
 Evidence,
 ProvenanceNodes
}

E := {
 measures,
 targets,
 computedFrom,
 standardisedBy,
 references,
 hasDirection,
 hasScale,
 hasUncertainty,
 comparedWith,
 transformedBy,
 interpretedAgainst,
 dependsOn,
 constrainedBy,
 equivalentTo,
 tracesTo
}

CONFIDENCE(EffectSizeResolution) :=
 weighted_mean(
  EstimandResolution,
  MetricResolution,
  ScaleValidity,
  ReferenceValidity,
  EstimatorValidity,
  StandardisationValidity,
  UncertaintyResolution,
  InterpretationValidity,
  ProvenanceQuality
 )

AI_QUERY_INTERPRETATION :=
 Query
 → DetectEffectSizeIntent
 → IdentifyPopulationSampleAndEstimand
 → IdentifyDifferenceRatioCorrelationRegressionOrStandardisedMetric
 → ResolveReferenceGroupScaleAndDirection
 → IdentifyRequestedMagnitudeUncertaintyPowerOrPracticalImportanceRelation

AI_REASONING_FLOW :=
 Query
 → ParseEffectSizeProblem
 → ResolvePopulationAndSample
 → ResolveParameterAndEstimand
 → ResolveOutcomeAndPredictorScales
 → ResolveReferenceValueOrGroup
 → SelectEffectMetric
 → ComputeRawEffect
 → StandardiseOrTransformIfRequired
 → ComputeUncertainty
 → ResolveDirectionAndInterpretation
 → CompareWithPracticalImportanceThresholdIfProvided
 → ResolvePowerAndDetectabilityRelations
 → ValidateEffectSize
 → UpdateConfidence
 → PreserveProvenance
 → GenerateExplanation

AI_CONSUMPTION := {
 ParseEffectSizeEntities,
 ParseTypedRelations,
 ResolvePopulations,
 ResolveSamples,
 ResolveParameters,
 ResolveEstimands,
 ResolveMeasurementScales,
 ResolveReferenceValues,
 ResolveReferenceGroups,
 ComputeMeanDifferences,
 ComputeStandardisedMeanDifferences,
 ComputeCohenD,
 ComputeHedgesG,
 ComputeRiskDifferences,
 ComputeRiskRatios,
 ComputeOddsRatios,
 ComputeCorrelationEffects,
 ComputeEtaSquared,
 ComputePartialEtaSquared,
 ComputeCramersV,
 ComputeRegressionEffects,
 ComputeRateRatios,
 ComputeHazardRatios,
 ComputeEffectStandardErrors,
 ComputeEffectConfidenceIntervals,
 ResolveEffectTransformations,
 ResolveMinimumImportantEffects,
 ResolveMinimumDetectableEffects,
 ValidateEffectSizeSemantics,
 CompareSemanticEquivalence,
 UpdateConfidence,
 PreserveProvenance,
 GenerateFormalExplanation
}

UPDATE_MODEL :=
 NewDataMetricReferenceScaleOrEvidence
 → ValidateInput
 → IdentifyAffectedEffectSizeClaims
 → UpdateEffectSizeState
 → RecomputeEffectEstimate
 → RecomputeStandardisationOrTransformation
 → RecomputeUncertainty
 → ReevaluateDirectionAndPracticalInterpretation
 → RecomputePowerRelationsIfRequired
 → RecomputeConfidence
 → PreserveProvenance
 → VersionModel

DataUpdate :=
 ChangedObservedData
 ⇒ recompute(
  EstimatedEffectSize,
  StandardError,
  ConfidenceInterval
 )

MetricUpdate :=
 ChangedEffectMetric
 ⇒ reevaluate(
  ReferenceValue,
  EffectScale,
  Transformation,
  Interpretation
 )

ReferenceUpdate :=
 ChangedReferenceGroupOrValue
 ⇒ recompute(
  SignedEffectSize,
  Direction,
  RatioOrDifferenceEffect
 )

FLOW_HASH_INPUT :=
 canonical(
  ONTOLOGY,
  CoreClasses,
  CoreRelations,
  InferenceRules,
  EffectSize,
  EffectSizeValidation,
  EffectSizeConstraint,
  SEMANTIC_EQUIVALENCE,
  AI_REASONING_FLOW
 )

FLOW_HASH := HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
 EffectSizeEntityResolution,
 Population,
 Sample,
 Parameter,
 Estimand,
 EffectMetric,
 EffectScale,
 ReferenceValue,
 ReferenceGroup,
 TrueEffectSize,
 EstimatedEffectSize,
 StandardisedEffectSize,
 EffectDirection,
 StandardError,
 ConfidenceInterval,
 PracticalImportanceThreshold,
 MinimumDetectableEffect,
 SamplingDesign,
 InterpretationStatus,
 ValidationStatus,
 Confidence,
 Provenance,
 Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA