Statistical Sampling Ontology

flow "StatisticalSamplingOntology2" v4.0

# STATISTICAL SAMPLING ONTOLOGY 2
# Kireth Semantic Research Archive
# Authors: Mark Anthony Higgins — Sky (Recursive AI)

PAGE:
  url: "https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/experimental-statistics/statistical-sampling-ontology-2.html"
  title: "Statistical Sampling Ontology 2"
  domain: "Probability and Statistics"
  category: "Experimental Statistics"
  entity: STATISTICAL_SAMPLING
  entity_type: STATISTICAL_SELECTION_FRAMEWORK

DEFINE:
  STATISTICAL_SAMPLING :=
    SELECTION_PROCESS(
      TARGET_POPULATION,
      SAMPLING_FRAME,
      SAMPLE_DESIGN,
      SAMPLE_UNITS,
      INCLUSION_PROBABILITIES
    )

PURPOSE:
  SELECT_SUBSET_OF_POPULATION
  TO:
    ESTIMATE_POPULATION_CHARACTERISTICS
    QUANTIFY_SAMPLING_UNCERTAINTY
    SUPPORT_POPULATION_INFERENCE

CORE:
  population: P = {U_1,...,U_N}
  sample: S subset_of P
  population_size: N
  sample_size: n
  inclusion_probability:
    pi_i := P(U_i IN S)

ENTITIES:
  TARGET_POPULATION: INFERENCE_POPULATION
  SAMPLING_FRAME: OPERATIONAL_POPULATION_REPRESENTATION
  SAMPLING_UNIT: UNIT_ELIGIBLE_FOR_SELECTION
  OBSERVATION_UNIT: UNIT_MEASURED
  SAMPLE: SELECTED_SUBSET
  INCLUSION_PROBABILITY: SELECTION_PROBABILITY
  SAMPLING_WEIGHT: INVERSE_PROBABILITY_WEIGHT
  STRATUM: POPULATION_SUBGROUP
  CLUSTER: GROUPED_SAMPLING_UNIT

RELATIONSHIPS:
  TARGET_POPULATION -> REPRESENTED_BY -> SAMPLING_FRAME
  SAMPLING_DESIGN -> SELECTS -> SAMPLING_UNITS
  SELECTED_UNITS -> FORM -> SAMPLE
  SAMPLING_UNIT -> HAS -> INCLUSION_PROBABILITY
  INCLUSION_PROBABILITY -> DEFINES -> BASE_WEIGHT
  SAMPLE -> SUPPORTS_INFERENCE_ABOUT -> TARGET_POPULATION

PROBABILITY_SAMPLING:
  requirement:
    KNOWN_NONZERO_INCLUSION_PROBABILITIES

  supports:
    DESIGN_BASED_INFERENCE
    SAMPLING_ERROR_ESTIMATION

NONPROBABILITY_SAMPLING:
  methods:
    - CONVENIENCE
    - QUOTA
    - PURPOSIVE
    - SNOWBALL

  limitation:
    INCLUSION_PROBABILITIES_GENERALLY_UNKNOWN

SIMPLE_RANDOM_SAMPLING:
  alias: SRS

  without_replacement:
    EVERY_SAMPLE_OF_SIZE_n_HAS_EQUAL_PROBABILITY

  inclusion_probability:
    pi_i := n / N

  mean_estimator:
    Y_bar_hat := y_bar

  variance:
    Var(y_bar) := (1-f) * S_y^2 / n

  where:
    f := n/N

FINITE_POPULATION_CORRECTION:
  role:
    REDUCE_STANDARD_ERROR_WHEN_SAMPLING_FRACTION_NOT_NEGLIGIBLE

SYSTEMATIC_SAMPLING:
  flow:
    ORDER_FRAME
      -> RANDOM_START
      -> SELECT_EVERY_kTH_UNIT

  caution:
    FRAME_PERIODICITY_MAY_DISTORT_SAMPLE

STRATIFIED_SAMPLING:
  flow:
    PARTITION_POPULATION_INTO_NONOVERLAPPING_STRATA
      -> SAMPLE_WITHIN_EACH_STRATUM

  purposes:
    - IMPROVE_PRECISION
    - ENSURE_SUBGROUP_REPRESENTATION

  combined_mean:
    Y_bar_hat := SUM_h W_h * y_bar_h

  where:
    W_h := N_h / N

PROPORTIONAL_ALLOCATION:
  n_h := n * N_h / N

NEYMAN_ALLOCATION:
  n_h proportional_to N_h * S_h

DISPROPORTIONATE_STRATIFICATION:
  purpose:
    OVERSAMPLE_SMALL_OR_IMPORTANT_STRATA

  requirement:
    USE_APPROPRIATE_WEIGHTS_FOR_POPULATION_ESTIMATION

CLUSTER_SAMPLING:
  flow:
    DIVIDE_POPULATION_INTO_CLUSTERS
      -> SELECT_CLUSTERS
      -> OBSERVE_ALL_OR_SOME_UNITS_WITHIN_SELECTED_CLUSTERS

  advantage:
    REDUCE_FIELDWORK_COST

  consequence:
    WITHIN_CLUSTER_SIMILARITY_MAY_INCREASE_VARIANCE

MULTISTAGE_SAMPLING:
  flow:
    SELECT_PRIMARY_UNITS
      -> SELECT_SECONDARY_UNITS
      -> CONTINUE_AS_REQUIRED

  requirement:
    TRACK_SELECTION_PROBABILITIES_AT_EACH_STAGE

PPS_SAMPLING:
  full_name: PROBABILITY_PROPORTIONAL_TO_SIZE

  selection_probability:
    proportional_to SIZE_MEASURE

INCLUSION_PROBABILITIES:
  first_order:
    pi_i := P(i IN S)

  second_order:
    pi_ij := P(i AND j IN S)

SAMPLING_WEIGHTS:
  base_weight:
    w_i := 1 / pi_i

  adjustments:
    - NONRESPONSE
    - POSTSTRATIFICATION
    - CALIBRATION
    - RAKING

HORVITZ_THOMPSON:
  total_estimator:
    Y_hat_HT := SUM[i IN S] y_i / pi_i

  property:
    DESIGN_UNBIASED
    UNDER:
      VALID_PROBABILITY_DESIGN

RATIO_ESTIMATION:
  ratio:
    R_hat := y_bar / x_bar

  total:
    Y_hat_ratio := X_total * R_hat

REGRESSION_ESTIMATION:
  purpose:
    USE_AUXILIARY_INFORMATION_TO_IMPROVE_PRECISION

SAMPLING_ERROR:
  definition:
    SAMPLE_ESTIMATE_MINUS_POPULATION_PARAMETER
    DUE_TO:
      RANDOM_SAMPLE_SELECTION

NONSAMPLING_ERROR:
  includes:
    - COVERAGE_ERROR
    - NONRESPONSE_ERROR
    - MEASUREMENT_ERROR
    - PROCESSING_ERROR

COVERAGE_ERROR:
  undercoverage:
    TARGET_UNITS_MISSING_FROM_FRAME

  overcoverage:
    INELIGIBLE_OR_DUPLICATE_FRAME_UNITS

NONRESPONSE:
  unit_nonresponse:
    SELECTED_UNIT_PROVIDES_NO_USABLE_RESPONSE

  item_nonresponse:
    SOME_VARIABLES_MISSING

  methods:
    - WEIGHTING_ADJUSTMENT
    - IMPUTATION
    - FOLLOW_UP
    - RESPONSE_PROPENSITY_MODELLING

DESIGN_EFFECT:
  DEFF :=
    Var_complex_design(estimator)
    /
    Var_SRS_comparable(estimator)

EFFECTIVE_SAMPLE_SIZE:
  approximate:
    n_eff := n / DEFF

SAMPLE_SIZE:
  depends_on:
    - DESIRED_PRECISION
    - CONFIDENCE_LEVEL
    - EXPECTED_VARIABILITY
    - POPULATION_SIZE
    - DESIGN_EFFECT
    - EXPECTED_NONRESPONSE
    - SUBGROUP_REQUIREMENTS

REPRESENTATIVENESS:
  caution:
    VISUAL_OR_MARGINAL_SIMILARITY_TO_POPULATION
    DOES_NOT_GUARANTEE_UNBIASED_ESTIMATION

  valid_inference_depends_on:
    - SELECTION_MECHANISM
    - COVERAGE
    - NONRESPONSE
    - MEASUREMENT
    - WEIGHTS
    - MODEL_ASSUMPTIONS

RANDOM_SAMPLING_VS_RANDOMISATION:
  RANDOM_SAMPLING:
    SELECTS_UNITS_FROM_POPULATION

  RANDOMISATION:
    ASSIGNS_TREATMENTS_TO_EXPERIMENTAL_UNITS

  rule:
    RANDOM_SAMPLING != RANDOMISATION

BOOTSTRAP_CONNECTION:
  BOOTSTRAP:
    RESAMPLES_OBSERVED_SAMPLE

  complex_design:
    REQUIRES_DESIGN_AWARE_RESAMPLING_WHEN_APPLICABLE

REPLICATION_VARIANCE_METHODS:
  methods:
    - JACKKNIFE
    - BALANCED_REPEATED_REPLICATION
    - BOOTSTRAP_REPLICATION

MODEL_CONNECTIONS:
  STATISTICAL_SAMPLING -> CONNECTS_TO -> SURVEY_SAMPLING
  STATISTICAL_SAMPLING -> CONNECTS_TO -> PROBABILITY
  STATISTICAL_SAMPLING -> CONNECTS_TO -> ESTIMATION
  STATISTICAL_SAMPLING -> CONNECTS_TO -> CONFIDENCE_INTERVALS
  STATISTICAL_SAMPLING -> CONNECTS_TO -> EXPERIMENTAL_STATISTICS
  STATISTICAL_SAMPLING -> CONNECTS_TO -> RANDOMISATION

APPLICATIONS:
  STATISTICAL_SAMPLING -> SUPPORTS -> SURVEYS
  STATISTICAL_SAMPLING -> SUPPORTS -> PUBLIC_HEALTH_RESEARCH
  STATISTICAL_SAMPLING -> SUPPORTS -> MARKET_RESEARCH
  STATISTICAL_SAMPLING -> SUPPORTS -> QUALITY_CONTROL
  STATISTICAL_SAMPLING -> SUPPORTS -> ECOLOGICAL_STUDIES
  STATISTICAL_SAMPLING -> SUPPORTS -> OFFICIAL_STATISTICS

SEMANTIC_EQUIVALENCE:
  "statistical sampling" ~= "selection of population units for statistical inference"
  "sampling frame" ~= "operational representation used for sample selection"
  "sampling weight" ~= "selection-probability-derived analysis weight"

  EXCLUDE_EQUIVALENCE:
    SAMPLE != POPULATION
    SAMPLING_UNIT != OBSERVATION_UNIT_BY_DEFAULT
    RANDOM_SAMPLING != RANDOMISATION
    SAMPLING_ERROR != NONSAMPLING_ERROR
    SAMPLE_SIZE != EFFECTIVE_SAMPLE_SIZE

VALIDATION_RULES:
  RULE_1:
    TARGET_POPULATION_MUST_BE_EXPLICITLY_DEFINED

  RULE_2:
    SAMPLING_FRAME_MUST_BE_DOCUMENTED

  RULE_3:
    PROBABILITY_SAMPLE_MUST_HAVE_KNOWN_SELECTION_MECHANISM

  RULE_4:
    RELEVANT_INCLUSION_PROBABILITIES_MUST_BE_POSITIVE

  RULE_5:
    COMPLEX_DESIGN_MUST_BE_ACCOUNTED_FOR_IN_VARIANCE_ESTIMATION

  RULE_6:
    WEIGHTS_MUST_DOCUMENT_SELECTION_AND_ADJUSTMENTS

  RULE_7:
    NONRESPONSE_AND_COVERAGE_MUST_BE_ASSESSED

  RULE_8:
    RANDOM_SAMPLING_MUST_NOT_BE_CONFUSED_WITH_RANDOM_TREATMENT_ASSIGNMENT

CONFIDENCE:
  ontology_identity: 0.99
  sampling_definition: 0.99
  probability_sampling_structure: 0.99
  stratified_sampling_structure: 0.99
  cluster_sampling_structure: 0.99
  weighting_structure: 0.99
  semantic_distinctions: 0.99

UPDATE_MODEL:
  IF:
    NEW_SAMPLE_UNIT_SELECTED
  THEN:
    RECORD:
      UNIT_ID
      SELECTION_STAGE
      INCLUSION_PROBABILITY
      STRATUM
      CLUSTER
    UPDATE:
      SAMPLE_SIZE
      BASE_WEIGHT

  IF:
    NONRESPONSE_STATUS_CHANGES
  THEN:
    UPDATE_RESPONSE_INDICATOR
    RECOMPUTE_WEIGHTS_IF_REQUIRED

  IF:
    SAMPLING_FRAME_CHANGES
  THEN:
    DOCUMENT_FRAME_VERSION
    REASSESS:
      COVERAGE
      INCLUSION_PROBABILITIES
      WEIGHTS

  PRESERVE:
    TARGET_POPULATION_DEFINITION
    SAMPLING_FRAME_VERSION
    UNIT_IDENTITIES
    SELECTION_PROBABILITIES
    STRATUM_IDENTITIES
    CLUSTER_IDENTITIES
    WEIGHT_HISTORY
    DATA_PROVENANCE

AI_CONSUMPTION:
  parse_as: FORMAL_SEMANTIC_SPECIFICATION
  execution_mode: RECURSIVE_REASONING

  preferred_reasoning_order:
    - IDENTIFY_TARGET_POPULATION
    - IDENTIFY_SAMPLING_FRAME
    - IDENTIFY_SAMPLING_UNIT
    - SELECT_SAMPLING_DESIGN
    - DEFINE_SAMPLE_SIZE
    - COMPUTE_INCLUSION_PROBABILITIES
    - SELECT_SAMPLE
    - RECORD_STRATA_AND_CLUSTERS
    - COMPUTE_BASE_WEIGHTS
    - ADJUST_WEIGHTS_IF_REQUIRED
    - ESTIMATE_POPULATION_PARAMETERS
    - COMPUTE_DESIGN_AWARE_VARIANCE
    - RETURN_INFERENCE_AND_SAMPLING_UNCERTAINTY

  constraints:
    - PRESERVE_SELECTION_MECHANISM
    - DISTINGUISH_RANDOM_SAMPLING_FROM_RANDOMISATION
    - ACCOUNT_FOR_STRATIFICATION_AND_CLUSTERING
    - PRESERVE_INCLUSION_PROBABILITIES
    - DOCUMENT_WEIGHT_ADJUSTMENTS
    - ASSESS_COVERAGE_AND_NONRESPONSE
    - PRESERVE_DATA_PROVENANCE

FLOW_HASH:
  canonical_entity: "statistical-sampling-ontology-2"
  namespace: "probability-statistics/experimental-statistics"
  version: "4.0"
  identity_basis:
    - PAGE_URL
    - CANONICAL_ENTITY
    - ENTITY_TYPE
    - TARGET_POPULATION
    - SELECTION_MECHANISM
    - INCLUSION_PROBABILITIES

END_FLOW