Factorial Design Ontology

flow "FactorialDesignOntology" v4.0

# ============================================================
# FACTORIAL DESIGN ONTOLOGY
# Kireth Semantic Research Archive
# Authors: Mark Anthony Higgins — Sky (Recursive AI)
# ============================================================

PAGE:
  url: "https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/experimental-statistics/factorial-design-ontology.html"
  title: "Factorial Design Ontology"
  domain: "Probability and Statistics"
  category: "Experimental Statistics"
  entity: FACTORIAL_DESIGN
  entity_type: EXPERIMENTAL_DESIGN_METHOD

DEFINE_FACTORIAL_DESIGN:
  FACTORIAL_DESIGN :=
    EXPERIMENTAL_DESIGN(
      MULTIPLE_FACTORS,
      FACTOR_LEVELS,
      TREATMENT_COMBINATIONS,
      MAIN_EFFECTS,
      INTERACTIONS
    )

PURPOSE:
  STUDY:
    TWO_OR_MORE_FACTORS
  SIMULTANEOUSLY
  TO:
    ESTIMATE_MAIN_EFFECTS
    ESTIMATE_INTERACTIONS
    IMPROVE_INFORMATION_EFFICIENCY

CORE_STRUCTURE:
  FACTORS:
    F := {F_1, F_2, ..., F_k}

  FACTOR_i:
    levels:
      L_i := {l_i1, l_i2, ..., l_i_mi}

  TREATMENT_COMBINATION:
    ONE_LEVEL_FROM_EACH_FACTOR

  FULL_FACTORIAL_RUN_COUNT:
    N_combinations :=
      PRODUCT[i=1..k] m_i

ENTITIES:
  FACTOR:
    type: EXPERIMENTAL_VARIABLE

  FACTOR_LEVEL:
    type: FACTOR_SETTING

  TREATMENT_COMBINATION:
    type: MULTIFACTOR_EXPERIMENTAL_CONDITION

  MAIN_EFFECT:
    type: MARGINAL_FACTOR_EFFECT

  INTERACTION:
    type: JOINT_FACTOR_EFFECT

  EXPERIMENTAL_UNIT:
    type: UNIT_RECEIVING_TREATMENT_COMBINATION

  RESPONSE:
    type: MEASURED_OUTCOME

  REPLICATE:
    type: INDEPENDENT_REPETITION

  DESIGN_MATRIX:
    type: MATRIX

  CONTRAST:
    type: LINEAR_COMBINATION_OF_TREATMENT_MEANS

RELATIONSHIPS:
  FACTOR
    -> HAS
    -> FACTOR_LEVELS

  FACTOR_LEVELS
    -> COMBINE_TO_FORM
    -> TREATMENT_COMBINATIONS

  TREATMENT_COMBINATION
    -> ASSIGNED_TO
    -> EXPERIMENTAL_UNIT

  EXPERIMENTAL_UNIT
    -> PRODUCES
    -> RESPONSE

  FACTORIAL_DESIGN
    -> ESTIMATES
    -> MAIN_EFFECTS

  FACTORIAL_DESIGN
    -> ESTIMATES
    -> INTERACTIONS

  DESIGN_MATRIX
    -> REPRESENTS
    -> FACTOR_AND_INTERACTION_STRUCTURE

FULL_FACTORIAL_DESIGN:
  DEFINE:
    INCLUDE:
      ALL_POSSIBLE_COMBINATIONS
      OF:
        FACTOR_LEVELS

  run_count:
    PRODUCT_OF:
      NUMBER_OF_LEVELS_PER_FACTOR

  benefits:
    - ESTIMATE_MAIN_EFFECTS
    - ESTIMATE_INTERACTIONS
    - SYSTEMATIC_RESPONSE_SURFACE_EXPLORATION

TWO_LEVEL_FACTORIAL:
  notation:
    2^k

  where:
    k := NUMBER_OF_FACTORS

  each_factor:
    levels:
      LOW := -1
      HIGH := +1

  number_of_treatment_combinations:
    2^k

  example_structure:
    k=3:
      factors:
        A
        B
        C

      effects:
        - A
        - B
        - C
        - AB
        - AC
        - BC
        - ABC

GENERAL_FACTORIAL:
  notation:
    PRODUCT:
      m_1 x m_2 x ... x m_k

  supports:
    DIFFERENT_NUMBERS_OF_LEVELS
    ACROSS:
      FACTORS

MAIN_EFFECT:
  DEFINE:
    CHANGE_IN_MEAN_RESPONSE
    ASSOCIATED_WITH:
      CHANGE_IN_LEVEL_OF_ONE_FACTOR
    AVERAGED_OVER:
      LEVELS_OF_OTHER_FACTORS

  CAUTION:
    IF:
      STRONG_INTERACTION_PRESENT

    THEN:
      MAIN_EFFECT:
        MAY_REQUIRE:
          CONDITIONAL_INTERPRETATION

INTERACTION:
  DEFINE:
    EFFECT_OF_ONE_FACTOR
      DEPENDS_ON
    LEVEL_OF_ANOTHER_FACTOR

  TWO_FACTOR_INTERACTION:
    A_BY_B:
      effect_of_A:
        CHANGES_ACROSS:
          LEVELS_OF_B

  HIGHER_ORDER_INTERACTION:
    INVOLVES:
      THREE_OR_MORE_FACTORS

  interpretation:
    HIGHER_ORDER_INTERACTIONS:
      OFTEN_MORE_DIFFICULT_TO_INTERPRET

TWO_FACTOR_MODEL:
  model:
    Y_ijk :=
      mu
      + alpha_i
      + beta_j
      + interaction_ij
      + epsilon_ijk

  where:
    mu := OVERALL_MEAN
    alpha_i := FACTOR_A_EFFECT
    beta_j := FACTOR_B_EFFECT
    interaction_ij := A_BY_B_INTERACTION
    epsilon_ijk := RANDOM_ERROR

MULTIFACTOR_LINEAR_MODEL:
  general_form:
    Y :=
      X beta
      + epsilon

  design_matrix:
    X:
      columns_may_include:
        - INTERCEPT
        - MAIN_EFFECTS
        - TWO_FACTOR_INTERACTIONS
        - HIGHER_ORDER_INTERACTIONS
        - BLOCK_EFFECTS

CONTRAST_CODING:
  TWO_LEVEL_FACTOR:
    LOW := -1
    HIGH := +1

  interaction_column:
    PRODUCT_OF:
      RELEVANT_FACTOR_COLUMNS

  benefit:
    ORTHOGONAL_EFFECT_ESTIMATION
    IN:
      BALANCED_FULL_TWO_LEVEL_FACTORIAL

EFFECT_ESTIMATION:
  TWO_LEVEL_BALANCED_DESIGN:
    effect_estimate:
      DIFFERENCE_BETWEEN:
        AVERAGE_RESPONSE_AT_HIGH_LEVEL
        AVERAGE_RESPONSE_AT_LOW_LEVEL

  interaction_effect:
    DIFFERENCE_OF_DIFFERENCES
    OR:
      EQUIVALENT_FACTORIAL_CONTRAST

REPLICATION:
  PURPOSE:
    ESTIMATE:
      PURE_EXPERIMENTAL_ERROR

  WITH_REPLICATION:
    supports:
      - ERROR_VARIANCE_ESTIMATION
      - SIGNIFICANCE_TESTING
      - LACK_OF_FIT_ASSESSMENT_WHEN_DESIGN_SUPPORTS

UNREPLICATED_FACTORIAL:
  issue:
    NO_DIRECT_PURE_ERROR_ESTIMATE
    FROM:
      SINGLE_RUN_PER_TREATMENT_COMBINATION

  possible_approaches:
    - EFFECT_SPARSITY_ASSUMPTIONS
    - HALF_NORMAL_PLOT
    - EXTERNAL_OR_HISTORICAL_ERROR_ESTIMATE
    - ADDITIONAL_RUNS

  caution:
    INFERENCE_DEPENDS_ON:
      ADDITIONAL_ASSUMPTIONS

RANDOMISATION:
  FULL_RANDOMISATION:
    RANDOMISE:
      ORDER_OR_ASSIGNMENT
      OF:
        FACTOR_LEVEL_COMBINATIONS

  purpose:
    PROTECT_AGAINST:
      TIME_TRENDS
      ORDER_EFFECTS
      SYSTEMATIC_BIAS

  restrictions:
    MAY_INCLUDE:
      BLOCKING
      SPLIT_PLOT_STRUCTURE
      OPERATIONAL_CONSTRAINTS

BLOCKED_FACTORIAL:
  purpose:
    CONTROL:
      NUISANCE_VARIATION

  method:
    ASSIGN_FACTORIAL_RUNS_TO_BLOCKS
    WHILE:
      PRESERVING_EFFECT_ESTIMABILITY

  caution:
    SOME_EFFECTS:
      MAY_BE_CONFOUNDED_WITH:
        BLOCK_EFFECTS
    IN:
      CERTAIN_BLOCKING_SCHEMES

FRACTIONAL_FACTORIAL:
  DEFINE:
    RUN:
      FRACTION_OF_FULL_FACTORIAL_COMBINATIONS

  notation_examples:
    2^(k-p)

  purpose:
    REDUCE:
      NUMBER_OF_RUNS

  cost:
    ALIASING:
      BETWEEN_EFFECTS

  require:
    DOCUMENT:
      GENERATORS
      DEFINING_RELATION
      ALIAS_STRUCTURE
      DESIGN_RESOLUTION

GENERATOR:
  DEFINE:
    RELATION_USED_TO_CONSTRUCT:
      FRACTIONAL_FACTORIAL_DESIGN

  example_structure:
    C := A * B

  consequence:
    CREATES:
      DEFINING_RELATION
      ALIAS_CHAINS

DEFINING_RELATION:
  form:
    I = WORD_1 = WORD_2 = ...

  role:
    DERIVE:
      ALIAS_STRUCTURE

ALIASING:
  DEFINE:
    TWO_OR_MORE_EFFECTS:
      SHARE:
        SAME_DESIGN_COLUMN
      AND:
        CANNOT_BE_SEPARATELY_ESTIMATED

  example:
    IF:
      I = ABC

    THEN:
      A ALIASED_WITH BC
      B ALIASED_WITH AC
      C ALIASED_WITH AB

  requirement:
    ALIAS_STRUCTURE:
      MUST_BE_EXPLICIT

DESIGN_RESOLUTION:
  RESOLUTION_III:
    main_effects:
      MAY_BE_ALIASED_WITH:
        TWO_FACTOR_INTERACTIONS

  RESOLUTION_IV:
    main_effects:
      NOT_ALIASED_WITH:
        TWO_FACTOR_INTERACTIONS

    two_factor_interactions:
      MAY_BE_ALIASED_WITH:
        OTHER_TWO_FACTOR_INTERACTIONS

  RESOLUTION_V:
    main_effects:
      NOT_ALIASED_WITH:
        TWO_OR_THREE_FACTOR_INTERACTIONS

    two_factor_interactions:
      NOT_ALIASED_WITH:
        OTHER_TWO_FACTOR_INTERACTIONS

  CAUTION:
    RESOLUTION:
      SUMMARY_ONLY
      NOT:
        SUBSTITUTE_FOR_FULL_ALIAS_STRUCTURE

CENTER_POINTS:
  purpose:
    DETECT:
      CURVATURE
    IN:
      TWO_LEVEL_FACTORIAL_REGION

  center_setting:
    EACH_QUANTITATIVE_FACTOR:
      MIDPOINT_BETWEEN_LOW_AND_HIGH

  comparison:
    CENTER_POINT_MEAN
      VS:
        FACTORIAL_POINT_MEAN

  if_curvature_detected:
    consider:
      RESPONSE_SURFACE_DESIGN
      ADDITIONAL_FACTOR_LEVELS

FACTOR_SCREENING:
  purpose:
    IDENTIFY:
      IMPORTANT_FACTORS
    FROM:
      MANY_CANDIDATE_FACTORS

  designs:
    - TWO_LEVEL_FACTORIAL
    - FRACTIONAL_FACTORIAL
    - PLACKETT_BURMAN_DESIGNS

  caution:
    SCREENING_DESIGNS:
      MAY_RELY_ON:
        EFFECT_HIERARCHY
        EFFECT_SPARSITY

EFFECT_HIERARCHY:
  heuristic:
    LOWER_ORDER_EFFECTS:
      GENERALLY_MORE_LIKELY_TO_BE_IMPORTANT
    THAN:
      VERY_HIGH_ORDER_INTERACTIONS

  status:
    MODELLING_HEURISTIC
    NOT:
      UNIVERSAL_LAW

EFFECT_SPARSITY:
  heuristic:
    RELATIVELY_FEW_EFFECTS:
      DOMINATE_RESPONSE

  use:
    SCREENING
    UNREPLICATED_FACTORIAL_INTERPRETATION

  caution:
    MUST_NOT_BE_ASSUMED_BLINDLY

ANOVA:
  factorial_anova:
    PARTITIONS:
      RESPONSE_VARIATION
    INTO:
      MAIN_EFFECTS
      INTERACTIONS
      ERROR

  example:
    SS_TOTAL :=
      SS_A
      + SS_B
      + SS_AB
      + SS_ERROR
    UNDER:
      APPROPRIATE_TWO_FACTOR_BALANCED_CONTEXT

MODEL_ASSUMPTIONS:
  common:
    - APPROPRIATE_MODEL_FORM
    - INDEPENDENT_ERRORS_AS_REQUIRED_BY_DESIGN
    - APPROPRIATE_VARIANCE_STRUCTURE
    - APPROXIMATE_NORMALITY_FOR_CERTAIN_CLASSICAL_INFERENCES

  check:
    - RESIDUALS
    - OUTLIERS
    - HETEROSCEDASTICITY
    - ORDER_EFFECTS
    - CURVATURE
    - DEPENDENCE

DESIGN_EFFICIENCY:
  FACTORIAL_DESIGN:
    advantage:
      MULTIPLE_FACTORS_STUDIED_IN_SAME_EXPERIMENT

  compared_with:
    ONE_FACTOR_AT_A_TIME

  benefit:
    CAN_DETECT:
      INTERACTIONS
    THAT:
      ONE_FACTOR_AT_A_TIME_DESIGN_MAY_MISS

FACTORIAL_VS_ONE_FACTOR_AT_A_TIME:
  FACTORIAL:
    varies:
      MULTIPLE_FACTORS_TOGETHER

  OFAT:
    varies:
      ONE_FACTOR_WHILE_HOLDING_OTHERS_FIXED

  distinction:
    FACTORIAL_DESIGN:
      GENERALLY_PROVIDES:
        INTERACTION_INFORMATION

APPLICATIONS:
  FACTORIAL_DESIGN
    -> SUPPORTS
    -> PROCESS_OPTIMISATION

  FACTORIAL_DESIGN
    -> SUPPORTS
    -> ENGINEERING_EXPERIMENTS

  FACTORIAL_DESIGN
    -> SUPPORTS
    -> AGRICULTURAL_RESEARCH

  FACTORIAL_DESIGN
    -> SUPPORTS
    -> PRODUCT_DEVELOPMENT

  FACTORIAL_DESIGN
    -> SUPPORTS
    -> SCIENTIFIC_SCREENING

  FACTORIAL_DESIGN
    -> SUPPORTS
    -> INDUSTRIAL_QUALITY_IMPROVEMENT

MODEL_CONNECTIONS:
  FACTORIAL_DESIGN
    -> CONNECTS_TO
    -> EXPERIMENTAL_DESIGN

  FACTORIAL_DESIGN
    -> CONNECTS_TO
    -> RANDOMISATION

  FACTORIAL_DESIGN
    -> CONNECTS_TO
    -> BLOCKING

  FACTORIAL_DESIGN
    -> CONNECTS_TO
    -> ANOVA

  FACTORIAL_DESIGN
    -> CONNECTS_TO
    -> LINEAR_MODELS

  FACTORIAL_DESIGN
    -> CONNECTS_TO
    -> RESPONSE_SURFACE_METHODOLOGY

SEMANTIC_EQUIVALENCE:
  "factorial design"
    ~= "multifactor experimental design using combinations of factor levels"

  "full factorial"
    ~= "design containing all factor-level combinations"

  "factor interaction"
    ~= "effect modification between experimental factors"

  "2^k design"
    ~= "k-factor two-level full factorial design"

  EXCLUDE_EQUIVALENCE:
    FACTORIAL_DESIGN != ONE_FACTOR_AT_A_TIME
    FULL_FACTORIAL != FRACTIONAL_FACTORIAL
    MAIN_EFFECT != INTERACTION
    FACTOR != FACTOR_LEVEL
    TREATMENT_COMBINATION != SINGLE_FACTOR_BY_DEFAULT
    ALIASING != CORRELATION_BY_DEFAULT

VALIDATION_RULES:
  RULE_1:
    FACTORS:
      MUST_BE:
        EXPLICITLY_DEFINED

  RULE_2:
    FACTOR_LEVELS:
      MUST_BE:
        EXPLICITLY_DEFINED

  RULE_3:
    TREATMENT_COMBINATIONS:
      MUST_MATCH:
        DESIGN_SPECIFICATION

  RULE_4:
    RANDOMISATION:
      MUST_RESPECT:
        DESIGN_AND_OPERATIONAL_RESTRICTIONS

  RULE_5:
    INTERACTIONS:
      MUST_BE_CONSIDERED:
        BEFORE_INTERPRETING_MAIN_EFFECTS_ALONE

  RULE_6:
    FRACTIONAL_FACTORIAL:
      MUST_DOCUMENT:
        GENERATORS
        DEFINING_RELATION
        ALIAS_STRUCTURE
        RESOLUTION

  RULE_7:
    DESIGN_MATRIX:
      MUST_BE_CHECKED_FOR:
        RANK
        ESTIMABILITY

  RULE_8:
    REPLICATION:
      MUST_BE_DISTINGUISHED_FROM:
        TECHNICAL_REPEAT_MEASUREMENT

CONFIDENCE:
  ontology_identity: 0.99
  factorial_definition: 0.99
  main_effect_structure: 0.99
  interaction_structure: 0.99
  fractional_factorial_structure: 0.99
  aliasing_structure: 0.99
  design_analysis_relationship: 0.99

UPDATE_MODEL:
  IF:
    NEW_FACTOR_ADDED

  THEN:
    EXPAND:
      FACTOR_SET
      TREATMENT_COMBINATIONS
      DESIGN_MATRIX

    REASSESS:
      RUN_COUNT
      INTERACTIONS
      REPLICATION
      ALIASING

  IF:
    NEW_FACTOR_LEVEL_ADDED

  THEN:
    EXPAND:
      TREATMENT_SPACE

    RECOMPUTE:
      FULL_FACTORIAL_RUN_COUNT
      DESIGN_MATRIX

  IF:
    DESIGN_FRACTION_CHANGES

  THEN:
    RECOMPUTE:
      GENERATORS
      DEFINING_RELATION
      ALIAS_STRUCTURE
      RESOLUTION

  IF:
    NEW_EXPERIMENTAL_DATA_ARRIVES

  THEN:
    UPDATE:
      EFFECT_ESTIMATES
      INTERACTION_ESTIMATES
      RESIDUAL_ANALYSIS

  PRESERVE:
    FACTOR_IDENTITIES
    FACTOR_LEVELS
    TREATMENT_COMBINATIONS
    RANDOMISATION_RECORD
    BLOCK_STRUCTURE
    DESIGN_GENERATORS
    DATA_PROVENANCE

AI_CONSUMPTION:
  parse_as: FORMAL_SEMANTIC_SPECIFICATION
  execution_mode: RECURSIVE_REASONING

  preferred_reasoning_order:
    - IDENTIFY_FACTORS
    - IDENTIFY_FACTOR_LEVELS
    - GENERATE_TREATMENT_COMBINATIONS
    - SELECT_FULL_OR_FRACTIONAL_DESIGN
    - DEFINE_REPLICATION
    - DEFINE_BLOCKING_IF_REQUIRED
    - DEFINE_RANDOMISATION
    - CONSTRUCT_DESIGN_MATRIX
    - CHECK_RANK_AND_ESTIMABILITY
    - DERIVE_ALIAS_STRUCTURE_IF_FRACTIONAL
    - EXECUTE_EXPERIMENT
    - ESTIMATE_MAIN_EFFECTS
    - ESTIMATE_INTERACTIONS
    - CHECK_MODEL_ASSUMPTIONS
    - RETURN_FACTORIAL_INFERENCE

  constraints:
    - PRESERVE_FACTOR_AND_LEVEL_IDENTITIES
    - DISTINGUISH_MAIN_EFFECTS_FROM_INTERACTIONS
    - DO_NOT_IGNORE_INTERACTIONS_WHEN_INTERPRETING_MAIN_EFFECTS
    - EXPOSE_ALIAS_STRUCTURE_FOR_FRACTIONAL_DESIGNS
    - PRESERVE_RANDOMISATION_STRUCTURE
    - CHECK_DESIGN_MATRIX_RANK
    - PRESERVE_DATA_PROVENANCE

FLOW_HASH:
  canonical_entity: "factorial-design"
  namespace: "probability-statistics/experimental-statistics"
  version: "4.0"
  identity_basis:
    - PAGE_URL
    - CANONICAL_ENTITY
    - ENTITY_TYPE
    - FACTOR_LEVEL_COMBINATIONS
    - MAIN_EFFECTS
    - INTERACTIONS

END_FLOW