ANOVA Ontology

flow "ANOVAOntology" v4.0

# ANOVA 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/anova-ontology.html"
  title: "ANOVA Ontology"
  domain: "Probability and Statistics"
  category: "Experimental Statistics"
  entity: ANALYSIS_OF_VARIANCE
  aliases: [ANOVA]
  entity_type: STATISTICAL_INFERENCE_METHOD

DEFINE:
  ANALYSIS_OF_VARIANCE :=
    STATISTICAL_FRAMEWORK(
      VARIANCE_PARTITIONING,
      GROUP_MEAN_COMPARISON,
      FACTOR_EFFECTS,
      RESIDUAL_VARIATION,
      F_TESTING
    )

PURPOSE:
  DECOMPOSE OBSERVED_VARIABILITY
  INTO:
    EXPLAINED_VARIATION
    RESIDUAL_VARIATION
  TO:
    TEST_FACTOR_EFFECTS
    ESTIMATE_EFFECTS
    QUANTIFY_UNCERTAINTY

ENTITIES:
  RESPONSE_VARIABLE: NUMERIC_OUTCOME
  FACTOR: CATEGORICAL_EXPLANATORY_VARIABLE
  FACTOR_LEVEL: GROUP_OR_CONDITION
  GROUP_MEAN: LOCATION_STATISTIC
  GRAND_MEAN: OVERALL_LOCATION_STATISTIC
  SUM_OF_SQUARES: VARIATION_MEASURE
  DEGREES_OF_FREEDOM: DIMENSIONAL_QUANTITY
  MEAN_SQUARE: VARIANCE_ESTIMATE
  F_STATISTIC: TEST_STATISTIC
  RESIDUAL: UNEXPLAINED_DEVIATION

RELATIONSHIPS:
  FACTOR -> HAS -> FACTOR_LEVELS
  FACTOR_LEVEL -> CONTAINS -> OBSERVATIONS
  OBSERVATIONS -> DEFINE -> GROUP_MEANS
  ALL_OBSERVATIONS -> DEFINE -> GRAND_MEAN
  TOTAL_VARIATION -> PARTITIONS_INTO -> MODEL_VARIATION
  TOTAL_VARIATION -> PARTITIONS_INTO -> ERROR_VARIATION
  SUM_OF_SQUARES -> DIVIDED_BY -> DEGREES_OF_FREEDOM
  MEAN_SQUARE_EFFECT -> COMPARED_WITH -> MEAN_SQUARE_ERROR
  COMPARISON -> PRODUCES -> F_STATISTIC

ONE_WAY_ANOVA:
  model:
    Y_ij := mu + tau_i + epsilon_ij

  null_hypothesis:
    H0: mu_1 = mu_2 = ... = mu_k

  alternative:
    AT_LEAST_ONE_GROUP_MEAN_DIFFERS

VARIANCE_DECOMPOSITION:
  SS_TOTAL :=
    SUM_i SUM_j (Y_ij - Y_grand)^2

  SS_BETWEEN :=
    SUM_i n_i (Y_i_bar - Y_grand)^2

  SS_WITHIN :=
    SUM_i SUM_j (Y_ij - Y_i_bar)^2

  identity:
    SS_TOTAL = SS_BETWEEN + SS_WITHIN

DEGREES_OF_FREEDOM:
  df_total := N - 1
  df_between := k - 1
  df_within := N - k

  identity:
    df_total = df_between + df_within

MEAN_SQUARES:
  MS_BETWEEN := SS_BETWEEN / df_between
  MS_WITHIN := SS_WITHIN / df_within

F_TEST:
  F := MS_BETWEEN / MS_WITHIN

  under_H0:
    F -> F_DISTRIBUTION
    UNDER:
      CLASSICAL_MODEL_ASSUMPTIONS

P_VALUE:
  definition:
    P(F_REFERENCE >= F_OBSERVED | H0)

  constraint:
    P_VALUE != PROBABILITY_H0_IS_TRUE

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

  components:
    - FACTOR_A_MAIN_EFFECT
    - FACTOR_B_MAIN_EFFECT
    - A_BY_B_INTERACTION
    - RESIDUAL_ERROR

INTERACTION:
  definition:
    EFFECT_OF_FACTOR_A
      DEPENDS_ON
    LEVEL_OF_FACTOR_B

  rule:
    IF_INTERACTION_SUBSTANTIAL:
      INTERPRET_MAIN_EFFECTS_CONDITIONALLY

FACTORIAL_ANOVA:
  supports:
    - MULTIPLE_FACTORS
    - MAIN_EFFECTS
    - INTERACTIONS

RANDOMISED_BLOCK_ANOVA:
  model:
    Y_ij := mu + tau_i + beta_j + epsilon_ij

  requirement:
    PRESERVE_BLOCK_STRUCTURE_IN_ANALYSIS

REPEATED_MEASURES_ANOVA:
  structure:
    SAME_EXPERIMENTAL_UNIT_MEASURED_REPEATEDLY

  implication:
    WITHIN_UNIT_RESPONSES_CORRELATED

  requirement:
    MODEL_WITHIN_UNIT_DEPENDENCE

  classical_issue:
    SPHERICITY

  corrections_or_alternatives:
    - GREENHOUSE_GEISSER
    - HUYNH_FELDT
    - MULTIVARIATE_APPROACH
    - MIXED_EFFECTS_MODEL

FIXED_RANDOM_MIXED_EFFECTS:
  FIXED_EFFECTS:
    inference_about:
      SPECIFIC_INCLUDED_LEVELS

  RANDOM_EFFECTS:
    levels_viewed_as:
      SAMPLE_FROM_LARGER_POPULATION

    purpose:
      ESTIMATE_VARIANCE_COMPONENTS

  MIXED_EFFECTS:
    combines:
      FIXED_EFFECTS
      RANDOM_EFFECTS

ASSUMPTIONS:
  CLASSICAL_ANOVA:
    - APPROPRIATE_MODEL_FORM
    - INDEPENDENCE_AS_REQUIRED_BY_DESIGN
    - APPROPRIATE_RESIDUAL_VARIANCE_STRUCTURE
    - APPROXIMATE_RESIDUAL_NORMALITY_FOR_CLASSICAL_SMALL_SAMPLE_INFERENCE

  note:
    NORMALITY_CONCERNS:
      ERROR_OR_RESIDUAL_STRUCTURE

DIAGNOSTICS:
  RESIDUAL_VS_FITTED:
    checks:
      - HETEROSCEDASTICITY
      - MODEL_FORM

  NORMAL_QQ:
    checks:
      RESIDUAL_NORMALITY

  RESIDUAL_ORDER:
    checks:
      ORDER_OR_TIME_DEPENDENCE

  OUTLIER_DIAGNOSTICS:
    checks:
      EXTREME_OR_INFLUENTIAL_OBSERVATIONS

HOMOGENEITY_OF_VARIANCE:
  standard_model:
    Var(epsilon_ij) := sigma^2

  checks:
    - RESIDUAL_PLOTS
    - LEVENE_TEST
    - BROWN_FORSYTHE_TEST

  if_violated:
    consider:
      - WELCH_ANOVA
      - TRANSFORMATION
      - HETEROSCEDASTIC_MODEL
      - ROBUST_METHOD

WELCH_ANOVA:
  purpose:
    COMPARE_GROUP_MEANS
    WITHOUT_EQUAL_VARIANCE_ASSUMPTION

POST_HOC_COMPARISONS:
  methods:
    - TUKEY_HSD
    - BONFERRONI
    - HOLM
    - DUNNETT

  requirement:
    CONTROL_MULTIPLE_COMPARISON_ERROR
    ACCORDING_TO:
      INFERENTIAL_GOAL

PLANNED_CONTRASTS:
  form:
    L := SUM_i c_i mu_i

  common_constraint:
    SUM_i c_i = 0

  purpose:
    TEST_PRE_SPECIFIED_SCIENTIFIC_COMPARISONS

EFFECT_SIZE:
  ETA_SQUARED:
    eta_squared := SS_EFFECT / SS_TOTAL

  PARTIAL_ETA_SQUARED:
    partial_eta_squared :=
      SS_EFFECT / (SS_EFFECT + SS_ERROR_RELEVANT)

  OMEGA_SQUARED:
    role:
      POPULATION_EFFECT_SIZE_ESTIMATION

CONFIDENCE_INTERVALS:
  report_for:
    - GROUP_MEANS
    - MEAN_DIFFERENCES
    - CONTRASTS
    - EFFECT_SIZES_WHEN_APPROPRIATE

BALANCED_DESIGN:
  advantages:
    - SIMPLER_VARIANCE_DECOMPOSITION
    - ORTHOGONAL_EFFECTS_IN_MANY_DESIGNS
    - EASIER_INTERPRETATION

UNBALANCED_DESIGN:
  issue:
    EFFECTS_MAY_NOT_BE_ORTHOGONAL

  sums_of_squares:
    - TYPE_I
    - TYPE_II
    - TYPE_III

  requirement:
    CHOICE_MUST_MATCH:
      MODEL
      HYPOTHESES
      CODING
      DESIGN

TYPE_I:
  alias: SEQUENTIAL_SUMS_OF_SQUARES
  role:
    EACH_TERM_TESTED_AFTER_PREVIOUS_TERMS

TYPE_II:
  role:
    COMMONLY_TEST_MAIN_EFFECT_AFTER_OTHER_MAIN_EFFECTS
    BUT_NOT_AFTER_INTERACTIONS_CONTAINING_IT

  caution:
    EXACT_DEFINITION_SOFTWARE_DEPENDENT

TYPE_III:
  role:
    TEST_EFFECT_CONDITIONAL_ON_OTHER_MODEL_TERMS

  caution:
    INTERPRETATION_DEPENDS_ON:
      CONTRAST_CODING
      PARAMETERISATION

ANOVA_AS_LINEAR_MODEL:
  model:
    Y := X beta + epsilon

  relation:
    ANOVA -> IS_SPECIAL_CASE_OF -> GENERAL_LINEAR_MODEL

ANCOVA_CONNECTION:
  combines:
    CATEGORICAL_FACTORS
    CONTINUOUS_COVARIATES

NONPARAMETRIC_ALTERNATIVES:
  one_way_independent:
    KRUSKAL_WALLIS

  repeated_or_blocked:
    FRIEDMAN_TEST

  caution:
    ALTERNATIVES_HAVE_DIFFERENT_NULL_HYPOTHESES_AND_INTERPRETATIONS

APPLICATIONS:
  ANOVA -> SUPPORTS -> EXPERIMENTAL_COMPARISON
  ANOVA -> SUPPORTS -> AGRICULTURAL_EXPERIMENTS
  ANOVA -> SUPPORTS -> CLINICAL_RESEARCH
  ANOVA -> SUPPORTS -> INDUSTRIAL_EXPERIMENTS
  ANOVA -> SUPPORTS -> PSYCHOLOGICAL_RESEARCH
  ANOVA -> SUPPORTS -> A_B_N_TESTING

MODEL_CONNECTIONS:
  ANOVA -> CONNECTS_TO -> EXPERIMENTAL_DESIGN
  ANOVA -> CONNECTS_TO -> FACTORIAL_DESIGN
  ANOVA -> CONNECTS_TO -> BLOCKING
  ANOVA -> CONNECTS_TO -> F_DISTRIBUTION
  ANOVA -> CONNECTS_TO -> LINEAR_MODELS
  ANOVA -> CONNECTS_TO -> HYPOTHESIS_TESTING
  ANOVA -> CONNECTS_TO -> MULTIPLE_COMPARISONS

SEMANTIC_EQUIVALENCE:
  "ANOVA" ~= "analysis of variance"
  "between-group variation" ~= "variation attributable to group mean differences"
  "within-group variation" ~= "residual variation within groups"

  EXCLUDE_EQUIVALENCE:
    ANOVA != TEST_OF_VARIANCES_ONLY
    P_VALUE != PROBABILITY_H0_IS_TRUE
    MAIN_EFFECT != INTERACTION
    ANOVA != POST_HOC_TEST
    STATISTICAL_SIGNIFICANCE != PRACTICAL_SIGNIFICANCE
    GROUP_DIFFERENCE != CAUSAL_EFFECT_BY_DEFAULT

VALIDATION_RULES:
  RULE_1:
    DESIGN_STRUCTURE:
      MUST_BE_IDENTIFIED

  RULE_2:
    INDEPENDENCE_STRUCTURE:
      MUST_MATCH_DESIGN

  RULE_3:
    BLOCKING_OR_REPEATED_MEASURES:
      MUST_BE_INCLUDED_WHEN_PRESENT

  RULE_4:
    INTERACTIONS:
      MUST_BE_ASSESSED_BEFORE_SIMPLE_MAIN_EFFECT_INTERPRETATION

  RULE_5:
    RESIDUAL_ASSUMPTIONS:
      MUST_BE_CHECKED

  RULE_6:
    MULTIPLE_COMPARISONS:
      MUST_BE_CONTROLLED_WHEN_RELEVANT

  RULE_7:
    UNBALANCED_DESIGNS:
      MUST_DOCUMENT_SUM_OF_SQUARES_METHOD_AND_CODING

  RULE_8:
    SIGNIFICANT_OMNIBUS_TEST:
      DOES_NOT_IDENTIFY_WHICH_GROUPS_DIFFER

CONFIDENCE:
  ontology_identity: 0.99
  anova_definition: 0.99
  variance_decomposition: 0.99
  f_test_structure: 0.99
  factorial_relationship: 0.99
  linear_model_relationship: 0.99
  assumption_structure: 0.98

UPDATE_MODEL:
  IF:
    NEW_OBSERVATIONS_ARRIVE

  THEN:
    UPDATE:
      GROUP_MEANS
      GRAND_MEAN
      SUMS_OF_SQUARES
      DEGREES_OF_FREEDOM
      MEAN_SQUARES
      F_STATISTICS

  IF:
    NEW_FACTOR_OR_LEVEL_ADDED

  THEN:
    EXPAND:
      DESIGN_MATRIX
      MODEL_TERMS

    REASSESS:
      INTERACTIONS
      DEGREES_OF_FREEDOM
      ESTIMABILITY

  IF:
    DESIGN_STRUCTURE_CHANGES

  THEN:
    UPDATE:
      ERROR_TERM
      BLOCK_STRUCTURE
      DEPENDENCE_STRUCTURE

    REVALIDATE:
      F_TESTS
      ASSUMPTIONS
      CONTRASTS

  PRESERVE:
    FACTOR_IDENTITIES
    LEVEL_IDENTITIES
    OBSERVATION_IDENTITIES
    DESIGN_STRUCTURE
    CONTRAST_DEFINITIONS
    DATA_PROVENANCE

AI_CONSUMPTION:
  parse_as: FORMAL_SEMANTIC_SPECIFICATION
  execution_mode: RECURSIVE_REASONING

  preferred_reasoning_order:
    - IDENTIFY_RESPONSE_VARIABLE
    - IDENTIFY_FACTORS_AND_LEVELS
    - IDENTIFY_DESIGN_STRUCTURE
    - IDENTIFY_BLOCKING_OR_REPEATED_MEASURES
    - CONSTRUCT_LINEAR_MODEL
    - PARTITION_VARIATION
    - COMPUTE_DEGREES_OF_FREEDOM
    - COMPUTE_MEAN_SQUARES
    - COMPUTE_F_STATISTICS
    - CHECK_RESIDUAL_ASSUMPTIONS
    - EVALUATE_INTERACTIONS
    - APPLY_CONTRASTS_OR_POST_HOC_TESTS_IF_REQUIRED
    - ESTIMATE_EFFECT_SIZES
    - RETURN_INFERENCE_AND_UNCERTAINTY

  constraints:
    - ALIGN_ERROR_TERM_WITH_DESIGN
    - PRESERVE_BLOCK_AND_DEPENDENCE_STRUCTURE
    - DO_NOT_INTERPRET_P_VALUE_AS_PROBABILITY_H0_IS_TRUE
    - DO_NOT_EQUATE_STATISTICAL_SIGNIFICANCE_WITH_PRACTICAL_IMPORTANCE
    - CONSIDER_INTERACTIONS_BEFORE_MAIN_EFFECT_INTERPRETATION
    - REPORT_SUM_OF_SQUARES_METHOD_FOR_UNBALANCED_DESIGNS
    - PRESERVE_DATA_PROVENANCE

FLOW_HASH:
  canonical_entity: "anova"
  namespace: "probability-statistics/experimental-statistics"
  version: "4.0"
  identity_basis:
    - PAGE_URL
    - CANONICAL_ENTITY
    - ENTITY_TYPE
    - VARIANCE_DECOMPOSITION
    - F_STATISTIC
    - FACTOR_EFFECT_STRUCTURE

END_FLOW