Chi-Square Goodness-of-Fit (Genetics) mechanism

Model scope

The core method is standard, but the specification is incomplete because α, degrees-of-freedom derivation, upper-tail p-value convention, ratio normalization, and minimum expected-count handling are not explicitly defined. Add these rules before implementation.

• The core statistic χ²=Σ(O−E)²/E is standard for a goodness-of-fit test, but the specification does not define the significance level α required for a reject/fail-to-reject decision.
• Degrees of freedom are not generally an arbitrary input: with k categories and no estimated parameters, df=k−1; parameters estimated from the data require further subtraction from the degrees of freedom.
• The p-value must be explicitly defined as the upper-tail probability P(Χ²_df≥χ²), not a lower-tail CDF.
• Expected counts must be computed by normalizing the supplied expected ratio to the observed total. The specification does not state this normalization step.
• The test requires strictly positive expected counts. Categories with expected count zero make the statistic undefined and must be rejected or handled explicitly.
• The specification does not state assumptions such as independent observations and sufficiently large expected counts; common rules of thumb require expected counts approximately at least 5, or an explicitly documented alternative.

Units and conventions

• Observed values are nonnegative integer counts and are dimensionless.
• Expected ratios are nonnegative relative proportions; they are normalized so expected counts sum to the observed total.
• The observed categories are mutually exclusive and collectively exhaustive.
• The significance level α is dimensionless and must be supplied; the vectors below use α=0.05.
• The p-value is the upper-tail probability from the chi-square distribution with the stated degrees of freedom.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "observed_counts": [
      75,
      25
    ],
    "expected_ratio": [
      3,
      1
    ],
    "df": 1,
    "alpha": 0.05
  },
  "expected_outputs": {
    "expected_counts": [
      75,
      25
    ],
    "chi_square": 0,
    "p_value": 1,
    "critical_value_at_alpha": 3.841459,
    "decision": "fail to reject"
  },
  "working": "Observed total = 75+25 = 100. Ratio total = 3+1 = 4. Expected counts = 100×[3/4,1/4] = [75,25]. χ² = (75−75)²/75 + (25−25)²/25 = 0/75 + 0/25 = 0. For df=1, p=P(Χ²₁≥0)=1. Since 0 < 3.841459, fail to reject at α=0.05."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

An expected ratio containing a zero produces an expected count E = 0 and a 0/0 χ² term when the corresponding observed count is also zero. The instrument must reject this input or explicitly define handling; otherwise the statistic and p-value are undefined.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche6/tranche6-results.json
• BOUNDARY_HAZARD_CATALOG.json