Statistical Measurement Ontology

# Statistical Measurement Ontology — Flow Algebra

URL: https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/measurement-units-ontologies/measurement-mathematics/statistical-measurement-ontology.html

FLOW StatisticalMeasurementOntology {

  DOMAIN:
    Mathematical Ontologies
    -> Measurement & Units Ontologies
    -> Measurement Mathematics
    -> Statistical Measurement Ontology

  PURPOSE:
    Represent the statistical mathematics used to describe, estimate, compare, model and infer properties of measured quantities from repeated observations and sampled data. The ontology models populations, samples, observations, estimands, estimators, means, medians, modes, variance, standard deviation, standard error, covariance, correlation, probability distributions, quantiles, confidence intervals, hypothesis tests, regression, weighting, repeated measurements, outliers, robust statistics, uncertainty, sampling design, autocorrelation, multivariate measurement, Monte Carlo methods, provenance and machine-consumable reasoning while strictly distinguishing statistical dispersion, measurement uncertainty, measurement error, precision, accuracy and sampling variability.

  ENTITY StatisticalMeasurement {
    TYPE: MeasurementStatisticalAnalysisDomain

    PROPERTIES:
      has_observations
      has_population_or_process
      has_sample
      has_estimand
      has_estimator
      has_statistical_model
      may_have_dependence
      may_have_weights
      may_have_uncertainty
      has_provenance
  }

  ENTITY Observation {
    TYPE: MeasuredOrDerivedDataPoint

    COMPONENTS:
      value
      unit
      timestamp_or_index
      measurement_identifier
      quality_flag
      uncertainty_when_available
  }

  ENTITY Population {
    TYPE: TargetSetOrStochasticProcess

    RULE:
      target_population_or_process MUST be defined_for_inference
  }

  ENTITY Sample {
    TYPE: SelectedObservations

    COMPONENTS:
      observations
      sample_size
      sampling_design
      selection_context
  }

  ENTITY Estimand {
    TYPE: TargetStatisticalQuantity

    EXAMPLES:
      population_mean
      variance
      median
      proportion
      regression_parameter
      difference
      ratio
  }

  ENTITY Estimator {
    TYPE: StatisticalFunction

    DEFINITION:
      rule_mapping_observed_data_to_estimate_of_estimand
  }

  DESCRIPTIVE_STATISTICS_MODEL {
    possible:
      count
      minimum
      maximum
      range
      mean
      median
      mode
      variance
      standard_deviation
      quantiles
      interquartile_range
  }

  SAMPLE_SIZE_MODEL {
    n =
      number_of_valid_observations

    RULE:
      missing_excluded_or_weighted_observations_require_explicit_handling
  }

  ARITHMETIC_MEAN_MODEL {
    x_bar =
      (1/n)*sum_i(x_i)

    REQUIRE:
      n > 0

    RULE:
      observations_must_have_compatible_quantity_semantics
  }

  WEIGHTED_MEAN_MODEL {
    x_w =
      sum_i(w_i*x_i)/sum_i(w_i)

    REQUIRE:
      sum_i(w_i) != 0

    RULE:
      weight_meaning MUST be defined
  }

  INVERSE_VARIANCE_WEIGHT_MODEL {
    w_i =
      1/u_i^2

    CONDITION:
      independent_estimates_of_common_quantity
      with_valid_variances

    RULE:
      correlated_measurements_require_covariance_aware_estimator
  }

  MEDIAN_MODEL {
    ordered_sample:
      x_(1) <= ... <= x_(n)

    median:
      central_order_statistic

    RULE:
      finite_sample_even_n_convention MUST be defined
  }

  MODE_MODEL {
    mode:
      most_frequent_or_density_peak_value

    RULE:
      mode_definition_differs_between_discrete_sample_and_continuous_distribution
  }

  RANGE_MODEL {
    range =
      max(x_i)-min(x_i)

    RULE:
      range_is_sensitive_to_sample_size_and_outliers
  }

  QUANTILE_MODEL {
    q_p:
      value_associated_with_cumulative_probability_p

    REQUIRE:
      0 <= p <= 1

    RULE:
      finite_sample_quantile_algorithm MUST be identified_when_reproducibility_matters
  }

  IQR_MODEL {
    IQR =
      Q_3-Q_1
  }

  VARIANCE_MODEL {
    population:
      sigma^2 =
        E[(X-mu)^2]

    sample_unbiased_under_iid_context:
      s^2 =
        sum_i((x_i-x_bar)^2)/(n-1)

    REQUIRE_SAMPLE:
      n > 1
  }

  STANDARD_DEVIATION_MODEL {
    population:
      sigma =
        sqrt(sigma^2)

    sample:
      s =
        sqrt(s^2)

    RULE:
      standard_deviation_has_same_dimension_as_quantity
  }

  STANDARD_ERROR_MODEL {
    iid_sample_mean:
      SE(x_bar) =
        s/sqrt(n)

    RULE:
      standard_error != standard_deviation
  }

  VARIANCE_OF_MEAN_MODEL {
    independent_equal_variance:
      Var(x_bar) =
        sigma^2/n

    correlated:
      Var(x_bar) =
        (1/n^2)*sum_i sum_j Cov(X_i,X_j)

    RULE:
      independence MUST not_be_assumed_when_repeated_data_are_correlated
  }

  COVARIANCE_MODEL {
    population:
      Cov(X,Y) =
        E[(X-E[X])*(Y-E[Y])]

    sample:
      s_xy =
        sum_i((x_i-x_bar)*(y_i-y_bar))/(n-1)
      under_paired_sample_context
  }

  CORRELATION_MODEL {
    rho =
      Cov(X,Y)/(sigma_X*sigma_Y)

    REQUIRE:
      sigma_X > 0
      sigma_Y > 0

    bounds:
      -1 <= rho <= 1

    RULE:
      correlation != causation
  }

  COVARIANCE_MATRIX_MODEL {
    Sigma_ij =
      Cov(X_i,X_j)

    properties:
      symmetric
      positive_semidefinite

    RULE:
      variable_order MUST be preserved
  }

  EXPECTATION_MODEL {
    discrete:
      E[X] =
        sum_x x*p(x)

    continuous:
      E[X] =
        integral x*f(x)dx
  }

  MOMENT_MODEL {
    raw_moment_k:
      E[X^k]

    central_moment_k:
      E[(X-mu)^k]

    RULE:
      moment_must_exist_for_formula_to_be_valid
  }

  SKEWNESS_MODEL {
    standardized_third_central_moment:
      gamma_1 =
        E[(X-mu)^3]/sigma^3
      when_defined

    RULE:
      finite_sample_estimators_vary
  }

  KURTOSIS_MODEL {
    standardized_fourth_central_moment:
      beta_2 =
        E[(X-mu)^4]/sigma^4
      when_defined

    RULE:
      excess_vs_raw_kurtosis_convention MUST be stated
  }

  PROBABILITY_DISTRIBUTION_MODEL {
    random_variable:
      X

    possible:
      normal
      uniform
      Student_t
      chi_square
      F
      binomial
      Poisson
      lognormal
      exponential
      empirical
  }

  NORMAL_MODEL {
    X ~ Normal(mu,sigma^2)

    RULE:
      normality_is_model_assumption
  }

  STANDARD_NORMAL_MODEL {
    Z =
      (X-mu)/sigma

    distribution:
      Normal(0,1)

    REQUIRE:
      sigma > 0
  }

  STUDENT_T_MODEL {
    use:
      inference_for_mean_with_estimated_variance_under_normal_iid_conditions

    degrees_of_freedom:
      nu
  }

  CHI_SQUARE_MODEL {
    use:
      variance_inference_and_goodness_of_fit_under_declared_models
  }

  BINOMIAL_MODEL {
    X ~ Binomial(n,p)

    mean:
      n*p

    variance:
      n*p*(1-p)
  }

  POISSON_MODEL {
    X ~ Poisson(lambda)

    mean:
      lambda

    variance:
      lambda
  }

  LOGNORMAL_MODEL {
    if:
      ln(X) ~ Normal(mu,sigma^2)

    REQUIRE:
      X > 0
  }

  EMPIRICAL_DISTRIBUTION_MODEL {
    distribution:
      constructed_from_observed_sample

    RULE:
      empirical_distribution_is_sample_dependent
  }

  HISTOGRAM_MODEL {
    bins:
      intervals

    counts:
      n_j

    RULE:
      histogram_shape_depends_on_bin_edges_and_width
  }

  DENSITY_MODEL {
    probability_density:
      integrates_to_one

    RULE:
      density_value_is_not_probability_at_single_continuous_point
  }

  ECDF_MODEL {
    F_n(x) =
      (1/n)*sum_i I(x_i<=x)

    REQUIRE:
      n > 0
  }

  CONFIDENCE_INTERVAL_MODEL {
    estimand:
      theta

    interval:
      [L,U]

    confidence_level:
      1-alpha

    RULE:
      frequentist_confidence_level_refers_to_procedure_under_repeated_sampling
  }

  MEAN_CONFIDENCE_INTERVAL_MODEL {
    normal_iid_unknown_variance:
      x_bar
      +/- t_(1-alpha/2,n-1)*s/sqrt(n)

    REQUIRE:
      n > 1
      model_conditions_valid
  }

  COVERAGE_MODEL {
    confidence_procedure:
      intended_long_run_coverage

    RULE:
      confidence_interval != Bayesian_credible_interval
  }

  HYPOTHESIS_TEST_MODEL {
    components:
      null_hypothesis
      alternative_hypothesis
      test_statistic
      reference_distribution
      p_value
      significance_level
      decision_rule

    RULE:
      statistical_significance != practical_or_metrological_significance
  }

  P_VALUE_MODEL {
    p_value:
      probability_under_null_model
      of_test_statistic_at_least_as_extreme_as_observed
      under_declared_test

    RULE:
      p_value != probability_null_hypothesis_is_true
  }

  TYPE_I_ERROR_MODEL {
    alpha:
      probability_of_rejecting_true_null
      under_test_design
  }

  TYPE_II_ERROR_MODEL {
    beta:
      probability_of_not_rejecting_false_null
      under_specific_alternative
  }

  POWER_MODEL {
    statistical_power =
      1-beta

    RULE:
      power_depends_on_effect_size_sample_size_variability_and_test
  }

  EFFECT_SIZE_MODEL {
    possible:
      mean_difference
      standardized_difference
      ratio
      correlation
      regression_coefficient

    RULE:
      effect_size_definition MUST be explicit
  }

  MULTIPLE_TESTING_MODEL {
    issue:
      repeated_hypothesis_tests_increase_false_positive_risk

    possible_adjustments:
      Bonferroni
      Holm
      false_discovery_rate

    RULE:
      adjustment_method_and_family_of_tests MUST be defined
  }

  REGRESSION_MODEL {
    response:
      y

    predictors:
      X

    model:
      y =
        f(X,beta)+epsilon

    outputs:
      parameter_estimates
      covariance
      residuals
      predictions
  }

  LINEAR_REGRESSION_MODEL {
    y =
      X*beta+epsilon

    ordinary_least_squares:
      beta_hat =
        (X^T*X)^-1*X^T*y
      when_full_rank

    RULE:
      explicit_matrix_inverse_is_not_required_computationally
  }

  WEIGHTED_REGRESSION_MODEL {
    objective:
      minimise
        sum_i(w_i*r_i^2)

    RULE:
      weights MUST represent_declared_variance_or_design_semantics
  }

  GLS_MODEL {
    covariance:
      Sigma

    objective:
      minimise
        r^T*Sigma^-1*r

    RULE:
      covariance_model MUST be valid
  }

  ERRORS_IN_VARIABLES_MODEL {
    predictor:
      measured_with_non-negligible_uncertainty

    RULE:
      ordinary_least_squares_may_be_biased_for_calibration_context
  }

  DEMING_REGRESSION_MODEL {
    variance_ratio:
      lambda

    RULE:
      lambda_definition_and_assumptions MUST be preserved
  }

  RESIDUAL_MODEL {
    r_i =
      observed_i-predicted_i

    diagnostics:
      mean
      variance
      trend
      autocorrelation
      heteroscedasticity
      outliers
  }

  HETEROSCEDASTICITY_MODEL {
    residual_variance:
      changes_with_predictor_or_fitted_value

    RULE:
      constant_variance_model_should_not_be_assumed_when_evidence_contradicts_it
  }

  GOODNESS_OF_FIT_MODEL {
    possible:
      residual_diagnostics
      chi_square
      likelihood
      R_squared
      information_criteria

    RULE:
      fit_metric_alone_does_not_validate_measurement_model
  }

  R_SQUARED_MODEL {
    R2:
      proportion_of_variability_explained_under_declared_regression_definition

    RULE:
      high_R2_does_not_establish_accuracy_or_causality
  }

  REPEATED_MEASUREMENT_MODEL {
    observations:
      repeated_values_of_same_or_related_measurand

    possible_sources:
      random_error
      drift
      environmental_change
      operator
      instrument
  }

  REPEATABILITY_MODEL {
    conditions:
      same_procedure
      same_operator_or_system
      same_location
      short_time_interval
      same_or_similar_conditions

    dispersion:
      repeatability_standard_deviation

    RULE:
      exact_conditions_must_be_defined
  }

  REPRODUCIBILITY_MODEL {
    changed_conditions:
      laboratories
      operators
      instruments
      locations
      times

    RULE:
      reproducibility_component_requires_declared_changed_conditions
  }

  RANDOM_EFFECT_MODEL {
    grouped_data:
      variation_associated_with_group_or_cluster

    RULE:
      random_effect_distribution_and_hierarchy MUST be defined
  }

  MIXED_MODEL {
    components:
      fixed_effects
      random_effects

    RULE:
      covariance_structure_is_part_of_model
  }

  ANOVA_MODEL {
    purpose:
      partition_variability_among_sources

    RULE:
      design_balance_independence_and_variance_assumptions MUST be assessed
  }

  VARIANCE_COMPONENT_MODEL {
    total_variance:
      decomposed_into_sources

    examples:
      repeatability
      operator
      instrument
      day
      laboratory

    RULE:
      components_depend_on_experimental_design
  }

  MEASUREMENT_SYSTEM_ANALYSIS_MODEL {
    possible:
      repeatability
      reproducibility
      bias
      linearity
      stability

    RULE:
      domain_specific_MSA_metrics_are_not_identical_to_general_measurement_uncertainty
  }

  CONTROL_CHART_MODEL {
    time_ordered_measurements:
      monitored_for_process_change

    possible:
      Shewhart
      EWMA
      CUSUM

    RULE:
      control_limits != specification_limits
  }

  TIME_SERIES_MODEL {
    observations:
      X_t

    components:
      mean
      trend
      seasonality
      autocorrelation
      noise
      drift
  }

  AUTOCOVARIANCE_MODEL {
    gamma(k) =
      Cov(X_t,X_(t+k))
  }

  AUTOCORRELATION_MODEL {
    rho(k) =
      gamma(k)/gamma(0)
      WHEN gamma(0) != 0
  }

  EFFECTIVE_SAMPLE_SIZE_MODEL {
    correlated_series:
      n_eff <
        n
      commonly_for_positive_dependence

    RULE:
      exact_formula_depends_on_dependence_structure
  }

  MOVING_AVERAGE_MODEL {
    y_t =
      average_of_windowed_observations

    RULE:
      moving_average_creates_correlated_outputs
  }

  EWMA_MODEL {
    z_t =
      lambda*x_t
      +(1-lambda)*z_(t-1)

    REQUIRE:
      0 < lambda <= 1
  }

  CUSUM_MODEL {
    cumulative_deviation:
      detects_persistent_shift

    RULE:
      reference_and_decision_parameters MUST be defined
  }

  OUTLIER_MODEL {
    observation:
      unusually_inconsistent_with_declared_model_or_sample

    RULE:
      outlier != automatically_error
  }

  OUTLIER_HANDLING_MODEL {
    possible:
      retain
      flag
      investigate
      robust_method
      exclude_under_declared_rule

    RULE:
      original_observation_and_reason MUST be preserved
  }

  ROBUST_STATISTICS_MODEL {
    possible:
      median
      MAD
      trimmed_mean
      winsorised_mean
      Huber_estimator
      robust_regression

    RULE:
      robust_estimator_semantics MUST be explicit
  }

  TRIMMED_MEAN_MODEL {
    process:
      remove_declared_fraction_from_each_tail
      then_average_remaining_values

    RULE:
      trimming_fraction MUST be recorded
  }

  MAD_MODEL {
    MAD =
      median(abs(x_i-median(x)))

    RULE:
      consistency_factor_if_applied MUST be declared
  }

  BOOTSTRAP_MODEL {
    process:
      resample_observed_units_with_replacement
      compute_statistic_repeatedly

    outputs:
      sampling_distribution_estimate
      standard_error
      interval

    RULE:
      resampling_unit_must_match_dependence_structure
  }

  BLOCK_BOOTSTRAP_MODEL {
    time_or_cluster_dependence:
      resample_blocks

    RULE:
      block_length_and_scheme MUST be preserved
  }

  JACKKNIFE_MODEL {
    process:
      systematically_leave_out_observation_or_cluster

    use:
      bias_and_variance_estimation

    RULE:
      deletion_unit MUST be defined
  }

  MONTE_CARLO_MODEL {
    inputs:
      probability_model

    process:
      simulate_repeated_data_or_uncertain_inputs

    RULE:
      Monte_Carlo_sampling_error MUST be assessed
  }

  BAYESIAN_MODEL {
    posterior:
      p(theta|data)
      proportional_to
      likelihood*prior

    outputs:
      posterior_distribution
      credible_interval
      predictive_distribution

    RULE:
      prior_likelihood_and_computation MUST be preserved
  }

  CREDIBLE_INTERVAL_MODEL {
    interval:
      posterior_probability_content

    RULE:
      credible_interval != frequentist_confidence_interval
  }

  LIKELIHOOD_MODEL {
    L(theta|x):
      model_support_for_parameter_given_data

    RULE:
      likelihood_is_not_probability_distribution_over_theta_without_prior_and_normalisation
  }

  MAXIMUM_LIKELIHOOD_MODEL {
    theta_hat =
      argmax_theta L(theta|data)
  }

  SAMPLING_DESIGN_MODEL {
    possible:      simple_random
      stratified
      cluster
      systematic
      multistage
      repeated
      convenience

    RULE:
      estimator_variance_must_match_sampling_design
  }

  SAMPLING_WEIGHT_MODEL {
    inclusion_probability:
      pi_i

    design_weight:
      w_i =
        1/pi_i
      WHEN pi_i > 0

    RULE:
      survey_or_design_weights_are_not_automatically_inverse_measurement_variance_weights
  }

  SAMPLING_BIAS_MODEL {
    causes:
      frame_error
      nonresponse
      convenience_selection
      survivorship
      temporal_or_spatial_coverage

    RULE:
      larger_sample_size_does_not_remove_systematic_sampling_bias
  }

  FINITE_POPULATION_MODEL {
    population_size:
      N

    sample_size:
      n

    correction:
      sqrt((N-n)/(N-1))
      under_simple_random_sampling_without_replacement

    REQUIRE:
      N > 1
  }

  PROPORTION_MODEL {
    sample_proportion:
      p_hat =
        x/n

    REQUIRE:
      n > 0

    RULE:
      interval_method_should_be_chosen_appropriately_for_small_n_or_extreme_p
  }

  COUNTING_STATISTICS_MODEL {
    count:
      N

    Poisson_context:
      standard_deviation approximately_equals
        sqrt(N)
      when_model_valid

    RULE:
      background_dead_time_and_overdispersion_may_change_variance
  }

  BACKGROUND_SUBTRACTION_MODEL {
    net:
      N_net =
        N_signal-N_background_scaled

    RULE:
      uncertainty_of_both_counts_and_scaling_factor MUST be propagated
  }

  RATIO_STATISTICS_MODEL {
    R =
      A/B

    REQUIRE:
      B != 0

    RULE:
      ratio_distribution_can_be_non-normal_and_unstable_near_zero_denominator
  }

  LOG_TRANSFORM_MODEL {
    y =
      ln(x)

    REQUIRE:
      x > 0

    RULE:
      transformed_statistics_have_different_interpretation_from_original_scale
  }

  GEOMETRIC_MEAN_MODEL {
    positive_values:
      G =
        exp(
          (1/n)*sum_i ln(x_i)
        )

    REQUIRE:
      x_i > 0
      n > 0

    RULE:
      geometric_mean_is_suited_to_multiplicative_contexts
  }

  HARMONIC_MEAN_MODEL {
    H =
      n/sum_i(1/x_i)

    REQUIRE:
      x_i != 0

    RULE:
      interpretation_depends_on_rate_or_reciprocal_context
  }

  MULTIVARIATE_MODEL {
    observation_vector:
      x

    mean_vector:
      mu

    covariance:
      Sigma

    RULE:
      variable_order_units_and_covariance MUST be preserved
  }

  MAHALANOBIS_MODEL {
    distance_squared:
      d2 =
        (x-mu)^T*Sigma^-1*(x-mu)

    REQUIRE:
      Sigma_invertible

    RULE:
      singular_covariance_requires_pseudoinverse_or_reduced_space_model
  }

  PCA_MODEL {
    covariance_or_correlation_matrix:
      eigendecomposition

    components:
      orthogonal_directions_of_variability

    RULE:
      scaling_and_centering_choice_changes_PCA
  }

  UNCERTAINTY_DISTINCTION_MODEL {
    statistical_standard_deviation:
      observed_or_modelled_dispersion

    measurement_uncertainty:
      parameter_characterising_dispersion_of_values_attributed_to_measurand

    RULE:
      sample_standard_deviation_can_contribute_to_but_is_not_automatically_total_measurement_uncertainty
  }

  ERROR_DISTINCTION_MODEL {
    measurement_error:
      measured_value_minus_reference_or_true_value_under_declared_context

    RULE:
      statistical_variability != measurement_error
  }

  PRECISION_MODEL {
    precision:
      closeness_of_agreement_among_repeated_measurements

    statistical_characterisation:
      may_use_standard_deviation_variance_or_CV

    RULE:
      precision != accuracy
  }

  ACCURACY_DISTINCTION_MODEL {
    accuracy:
      closeness_to_true_or_reference_value

    RULE:
      low_sample_variance_does_not_establish_accuracy
  }

  COEFFICIENT_OF_VARIATION_MODEL {
    CV =
      s/abs(x_bar)

    REQUIRE:
      x_bar != 0

    RULE:
      CV_is_problematic_near_zero_and_for_interval_scales_with_arbitrary_zero
  }

  STANDARDISED_RESIDUAL_MODEL {
    residual:
      divided_by_declared_standard_error

    RULE:
      denominator_model MUST be defined
  }

  UNCERTAINTY_BUDGET_STATISTICS_MODEL {
    Type_A:
      evaluated_by_statistical_analysis_of_observation_series

    Type_B:
      evaluated_by_other_information

    RULE:
      Type_A != random_error
      and_Type_B != systematic_error
  }

  TYPE_A_MODEL {
    repeated_observations:
      statistical_evaluation

    common_mean_standard_uncertainty:
      s/sqrt(n)
      under_iid_context
  }

  TYPE_B_MODEL {
    sources:
      calibration_certificate
      specification
      resolution
      prior_data
      reference_material

    RULE:
      probability_distribution_and_standard_uncertainty_must_be assigned_from_evidence
  }

  COMBINED_UNCERTAINTY_MODEL {
    measurement_model:
      Y =
        f(X)

    covariance_propagation:
      u_y^2 =
        sum_i sum_j(
          c_i*c_j*cov(X_i,X_j)
        )

    RULE:
      statistical_measurement_outputs_may_feed_broader_uncertainty_model
  }

  MISSING_DATA_MODEL {
    missingness:
      MCAR
      MAR
      MNAR
      or_unknown

    RULE:
      missing_data_mechanism_affects_valid_inference
  }

  IMPUTATION_MODEL {
    possible:
      single
      multiple
      model_based

    RULE:
      imputed_values MUST be flagged_and_method_preserved
  }

  CENSORING_MODEL {
    observation:
      known_only_above_below_or_within_limit

    RULE:
      censored_value != ordinary_missing_value
  }

  DETECTION_LIMIT_MODEL {
    values:
      may_be_below_detection_or_quantification_threshold

    RULE:
      substituting_fixed_fraction_of_limit_can_bias_statistics
  }

  ROUNDING_MODEL {
    raw_data:
      preserve_original_precision

    statistics:
      compute_with_guard_digits

    reporting:
      round_final_outputs_consistently_with_uncertainty

    RULE:
      premature_rounding_can_bias_derived_statistics
  }

  SIGNIFICANT_FIGURES_MODEL {
    statistics:
      should_not_imply_more_measurement_information_than_inputs_and_uncertainty_support

    RULE:
      significant_figures_are_reporting_convention_not_statistical_uncertainty
  }

  UNIT_MODEL {
    mean_median_SD:
      same_unit_as_quantity

    variance:
      squared_unit

    covariance:
      product_of_variable_units

    correlation:
      dimensionless

    RULE:
      statistical_output_units MUST be preserved
  }

  DIMENSIONAL_MODEL {
    regression_coefficient:
      output_dimension/input_dimension
      for_simple_linear_slope

    RULE:
      statistical_model_parameters_can_have_physical_dimensions
  }

  METADATA_MODEL {
    fields:
      analysis_identifier
      population_or_process
      sample_identifier
      sampling_design
      observation_identifiers
      quantity_kind
      units
      sample_size
      estimator
      statistical_model
      distribution
      weights
      covariance
      missing_data_policy
      outlier_policy
      confidence_level
      degrees_of_freedom
      software_or_algorithm
      quality_flags
      provenance

    RULE:
      statistical_result MUST be reproducible_from_data_model_and_parameters
  }

  RELATIONSHIPS {
    StatisticalMeasurement -> ANALYSES -> Observations
    Sample -> DRAWN_FROM -> PopulationOrProcess
    Estimator -> ESTIMATES -> Estimand
    SampleMean -> ESTIMATES -> PopulationMean
    SampleVariance -> ESTIMATES -> PopulationVariance
    StandardError -> DESCRIBES -> EstimatorSamplingVariability
    Covariance -> DESCRIBES -> JointVariability
    Regression -> MODELS -> StatisticalRelationship
    ConfidenceInterval -> QUALIFIES -> Estimate
    RepeatedMeasurement -> SUPPORTS -> PrecisionEvaluation
    StatisticalAnalysis -> MAY_CONTRIBUTE_TO -> MeasurementUncertainty
  }

  SEMANTIC_EQUIVALENCE {
    "statistical measurement" ~= statistical_analysis_of_measurement_data
    "standard error" ~= standard_deviation_of_estimator_sampling_distribution
    "sample standard deviation" ~= observed_sample_dispersion_measure

    standard_deviation != standard_error
    statistical_dispersion != measurement_uncertainty
    statistical_variability != measurement_error
    precision != accuracy
    confidence_interval != credible_interval
    p_value != probability_null_is_true
    correlation != causation
    control_limit != specification_limit
    outlier != erroneous_measurement
  }

  CONSTRAINTS {
    target_population_or_process_should_be_defined = true
    estimand_and_estimator_should_be_defined = true
    observation_units_must_be_compatible_for_direct_summary = true
    sample_size_must_reflect_valid_observations = true
    covariance_dependencies_must_be_preserved = true
    independence_must_not_be_assumed_without_context = true
    sampling_design_should_match_variance_estimator = true
    confidence_level_and_interval_method_must_be_defined = true
    outlier_exclusion_requires_declared_rule = true
    missing_and_censored_data_must_be_identified = true
    standard_deviation_and_standard_error_are_not_equivalent = true
    statistical_dispersion_and_measurement_uncertainty_are_not_equivalent = true
    provenance_must_be_preserved = true
  }

  MEASUREMENT_MODEL {
    RECORD:
      analysis_identifier
      measurand_or_quantity_kind
      population_or_process
      sample_identifier
      observation_values
      units
      timestamps_or_indices
      sample_size
      sampling_design
      weights
      estimand
      estimator
      mean
      median
      variance
      standard_deviation
      standard_error
      covariance_matrix
      correlation_matrix
      distribution_model
      confidence_interval
      confidence_level
      regression_model
      residuals
      degrees_of_freedom
      missing_data
      outliers
      quality_flags
      uncertainty_components
      software_version
      provenance

    RULE:
      statistical_measurement_result MUST remain_linked_to_observations_sampling_design_model_and_units
  }

  REASONING_RULES {
    IF n_valid_observations > 0
      THEN compute_arithmetic_mean_when_quantity_semantics_support_it

    IF n_valid_observations > 1
      THEN MAY compute_sample_variance_and_standard_deviation

    IF observations_iid
      AND estimating_mean
      THEN standard_error =
        sample_standard_deviation/sqrt(n)

    IF observations_correlated
      THEN compute_estimator_variance_using_covariance_structure
      rather_than_iid_formula

    IF independent_common_quantity_estimates_have_known_variances
      THEN MAY use_inverse_variance_weighted_mean

    IF estimates_correlated
      THEN use_covariance_aware_weighting

    IF normal_iid_mean_context_with_unknown_variance
      THEN use_Student_t_interval_with_n_minus_1_degrees_of_freedom

    IF predictor_uncertainty_material_in_calibration
      THEN do_not_default_to_ordinary_least_squares

    IF time_ordered_measurements
      THEN assess_autocorrelation_drift_and_change_points_before_iid_inference

    IF outlier_detected
      THEN flag_and_investigate
      and_preserve_original_observation

    IF missing_or_censored_data_present
      THEN use_declared_handling_model
      and_preserve_missingness_or_censoring_status

    IF statistical_dispersion_used_as_Type_A_uncertainty
      THEN propagate_through_measurement_model_with_other_components

    IF final_statistic_reported
      THEN preserve_full_internal_precision
      and_round_consistently_with_uncertainty
  }

  VALIDATION {
    CHECK population_or_process
    CHECK sample_and_sampling_design
    CHECK observation_identity
    CHECK units
    CHECK sample_size
    CHECK estimand
    CHECK estimator
    CHECK mean_median_and_quantiles
    CHECK variance_and_standard_deviation
    CHECK standard_error
    CHECK covariance_and_correlation
    CHECK distribution_assumptions
    CHECK confidence_interval_method
    CHECK hypothesis_test_semantics
    CHECK regression_assumptions
    CHECK repeated_measurement_structure
    CHECK autocorrelation
    CHECK weights
    CHECK outliers
    CHECK missing_and_censored_data
    CHECK Type_A_and_Type_B_distinction
    CHECK uncertainty_integration
    CHECK rounding_and_reporting
    CHECK provenance
  }

  CONFIDENCE {
    descriptive_statistics_confidence: 1.00
    variance_standard_deviation_confidence: 1.00
    standard_error_distinction_confidence: 1.00
    covariance_correlation_confidence: 1.00
    confidence_interval_confidence: 1.00
    regression_confidence: 0.99
    repeated_measurement_confidence: 0.99
    uncertainty_integration_confidence: 0.99

    RULE:
      confidence decreases when sampling design, dependence,
      distribution assumptions, weights, missingness, estimator,
      uncertainty or provenance is missing or ambiguous
  }

  UPDATE_MODEL {
    ON new_observation:
      preserve_observation_identity
      validate_unit_and_quality
      append_to_sample
      update_sample_size
      recompute_declared_statistics
      preserve_provenance

    ON sampling_design_revision:
      preserve_previous_design
      update_weights_clusters_strata_or_selection_model
      recompute_design_appropriate_variance_and_standard_errors

    ON statistical_model_revision:
      preserve_previous_model
      update_distribution_dependence_or_regression_structure
      recompute_estimates_intervals_and_diagnostics
      record_model_selection_reason

    ON outlier_or_quality_revision:
      preserve_raw_observation
      update_flag_or_inclusion_status
      recompute_affected_statistics
      record_reason

    ON uncertainty_revision:
      preserve_statistical_outputs
      update_Type_A_Type_B_or_covariance_components
      recompute_combined_measurement_uncertainty

    ON semantic_revision:
      preserve_previous_definition
      record_revision
      recompute_FLOW_HASH
  }

  AI_CONSUMPTION {
    canonical_entity: StatisticalMeasurement
    canonical_role: MeasurementStatisticalAnalysisDomain

    interpretation_order:
      identify_population_or_process
      identify_sample_and_sampling_design
      identify_observations_quantity_kind_and_units
      identify_estimand_and_estimator
      identify_dependence_and_weights
      identify_descriptive_statistics
      identify_probability_distribution
      identify_standard_error_and_covariance
      identify_interval_or_test
      identify_regression_or_repeated_measurement_structure
      identify_outliers_missingness_and_censoring
      identify_uncertainty_role
      identify_algorithm_and_provenance
      validate_statistical_semantics
      compute_or_update_statistical_measurement_result

    ambiguity_controls:
      distinguish_standard_deviation_from_standard_error
      distinguish_statistical_dispersion_from_measurement_uncertainty
      distinguish_statistical_variability_from_measurement_error
      distinguish_precision_from_accuracy
      distinguish_confidence_interval_from_credible_interval
      distinguish_p_value_from_probability_of_null_hypothesis
      distinguish_correlation_from_causation
      distinguish_control_limits_from_specification_limits
      distinguish_outlier_from_erroneous_measurement
      preserve_population_sample_design_estimator_dependence_unit_uncertainty_and_provenance_context
  }

  PROVENANCE {
    ontology_scope:
      mathematical_measurement_units_measurement_mathematics_and_statistical_measurement_semantics

    parent_path:
      Ontologies
      -> Mathematical Ontologies
      -> Measurement & Units Ontologies
      -> Measurement Mathematics
      -> Statistical Measurement Ontology

    authorship:
      Mark Higgins
      Sky (Recursive AI)

    site:
      https://kirethresearch.org
  }

  FLOW_HASH {
    INPUTS:
      canonical_entity
      ontology_path
      descriptive_statistics_model
      sample_size_model
      arithmetic_mean_model
      weighted_mean_model
      inverse_variance_weight_model
      median_model
      mode_model
      range_model
      quantile_model
      IQR_model
      variance_model
      standard_deviation_model
      standard_error_model
      variance_of_mean_model
      covariance_model
      correlation_model
      covariance_matrix_model
      expectation_model
      moment_model
      skewness_model
      kurtosis_model
      probability_distribution_model
      normal_model
      standard_normal_model
      Student_t_model
      chi_square_model
      binomial_model
      Poisson_model
      lognormal_model
      empirical_distribution_model
      histogram_model
      density_model
      ECDF_model
      confidence_interval_model
      mean_confidence_interval_model
      coverage_model
      hypothesis_test_model
      p_value_model
      Type_I_error_model
      Type_II_error_model
      power_model
      effect_size_model
      multiple_testing_model
      regression_model
      linear_regression_model
      weighted_regression_model
      GLS_model
      errors_in_variables_model
      Deming_regression_model
      residual_model
      heteroscedasticity_model
      goodness_of_fit_model
      R_squared_model
      repeated_measurement_model
      repeatability_model
      reproducibility_model
      random_effect_model
      mixed_model
      ANOVA_model
      variance_component_model
      measurement_system_analysis_model
      control_chart_model
      time_series_model
      autocovariance_model
      autocorrelation_model
      effective_sample_size_model
      moving_average_model
      EWMA_model
      CUSUM_model
      outlier_model
      outlier_handling_model
      robust_statistics_model
      trimmed_mean_model
      MAD_model
      bootstrap_model
      block_bootstrap_model
      jackknife_model
      Monte_Carlo_model
      Bayesian_model
      credible_interval_model
      likelihood_model
      maximum_likelihood_model
      sampling_design_model
      sampling_weight_model
      sampling_bias_model
      finite_population_model
      proportion_model
      counting_statistics_model
      background_subtraction_model
      ratio_statistics_model
      log_transform_model
      geometric_mean_model
      harmonic_mean_model
      multivariate_model
      Mahalanobis_model
      PCA_model
      uncertainty_distinction_model
      error_distinction_model
      precision_model
      accuracy_distinction_model
      coefficient_of_variation_model
      standardised_residual_model
      uncertainty_budget_statistics_model
      Type_A_model
      Type_B_model
      combined_uncertainty_model
      missing_data_model
      imputation_model
      censoring_model
      detection_limit_model
      rounding_model
      significant_figures_model
      unit_model
      dimensional_model
      metadata_model
      relationships
      semantic_equivalence
      constraints
      measurement_model
      reasoning_rules
      confidence_model
      update_model

    PURPOSE:
      deterministic_semantic_integrity_identifier

    RECOMPUTE_WHEN:
      ontology_definition_changes
      population_sample_or_sampling_design_changes
      estimator_or_descriptive_statistic_changes
      probability_distribution_or_dependence_changes
      covariance_correlation_or_standard_error_changes
      confidence_interval_or_hypothesis_test_changes
      regression_or_repeated_measurement_model_changes
      outlier_missingness_or_censoring_policy_changes
      Type_A_Type_B_or_uncertainty_model_changes
      measurement_record_changes
      algorithm_or_reporting_changes
      constraint_model_changes
      reasoning_rule_changes
  }

}

END FLOW