Relative Risk / Odds Ratio Calculator mechanism

Model scope

The core RR and OR point-estimate formulas are standard, but the instrument is not implementation-complete because the CI and p-value algorithms are underspecified and zero-cell behavior is absent. Keep the educational/not-medical-advice label, and explicitly state the cell orientation, count-only input requirement, CI formulas, z critical value, chi-square variant, and zero-cell policy before release.

• The RR formula is standard for a=exposed outcome, b=exposed no outcome, c=unexposed outcome, and d=unexposed no outcome, but the table orientation must be explicitly enforced.
• The OR formula is standard under the same cell definitions.
• The log-SE confidence-interval method is underspecified: it must state whether the interval is the Wald interval, the z critical value (normally 1.96 for 95%), and the exact SE formulas. Standard choices are SE(log RR)=sqrt(1/a-1/(a+b)+1/c-1/(c+d)) and SE(log OR)=sqrt(1/a+1/b+1/c+1/d), followed by exponentiation of the log-scale limits.
• The p-value method is underspecified. Chi-square must specify Pearson versus likelihood-ratio chi-square, continuity correction, degrees of freedom, and one-sided versus two-sided reporting. The vectors below use uncorrected two-sided Pearson chi-square with df=1.
• No zero-cell policy is specified. Without a continuity correction or exact method, log-SE confidence intervals are undefined when any required cell is zero.
• A normal-distribution z-table alone is insufficient to calculate a chi-square p-value unless the implementation explicitly uses the df=1 identity p=2(1-Phi(sqrt(chi-square)).

Units and conventions

• a, b, c, and d are nonnegative integer counts, not percentages, proportions, rates, or person-time measurements.
• The table is a mutually exclusive 2x2 table: a=exposed with outcome, b=exposed without outcome, c=unexposed with outcome, d=unexposed without outcome.
• RR and OR are dimensionless. Counts must refer to the same observation unit and time/risk period.
• For the nonzero-cell vectors, 95% CIs use natural logarithms, z=1.96, the standard Wald log-SE formulas, and exponentiation back to the original scale.
• The chi-square p-values use the uncorrected Pearson statistic with df=1 and a two-sided upper-tail probability.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "a": 30,
    "b": 70,
    "c": 10,
    "d": 90,
    "ci_method": "Wald log scale, 95%, z=1.96",
    "p_method": "uncorrected two-sided Pearson chi-square, df=1"
  },
  "expected_outputs": {
    "RR": 3,
    "RR_95_CI": [
      1.5508,
      5.8034
    ],
    "OR": 3.8571428571,
    "OR_95_CI": [
      1.7666,
      8.4216
    ],
    "pearson_chi_square": 12.5,
    "p_value_approximately": 0.000407
  },
  "working": "Risk exposed = a/(a+b)=30/(30+70)=30/100=0.30. Risk unexposed = c/(c+d)=10/(10+90)=10/100=0.10. RR=0.30/0.10=3.00. SE(log RR)=sqrt(1/30-1/100+1/10-1/100)=sqrt(0.0333333-0.0100000+0.1000000-0.0100000)=sqrt(0.1133333)=0.336650. log(RR)=log(3)=1.098612. Limits on log scale=1.098612 +/- 1.96(0.336650)=1.098612 +/- 0.659834, giving [0.438778,1.758447]. Exponentiating gives RR CI [exp(0.438778),exp(1.758447)]=[1.5508,5.8034]. OR=(a*d)/(b*c)=(30x90)/(70x10)=2700/700=27/7=3.8571428571. SE(log OR)=sqrt(1/30+1/70+1/10+1/90)=sqrt(0.1587302)=0.398410. log(OR)=log(27/7)=1.349927. Limits=1.349927 +/- 1.96(0.398410)=1.349927 +/- 0.780883, so [0.569044,2.130810]. Exponentiating gives [1.7666,8.4216]. Pearson chi-square=n(ad-bc)^2/[(a+b)(c+d)(a+c)(b+d)]=200(30x90-70x10)^2/(100x100x40x160)=200(2000)^2/64000000=12.5. With df=1, p=2[1-Phi(sqrt(12.5))]=2[1-Phi(3.53553)] approximately 0.000407."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

Zero cells expose undefined or infinite odds ratios and log-based confidence intervals. The specification does not state zero-cell handling, continuity corrections, table-orientation validation, or a literal formula branch. Several intermediate chi-square arithmetic checks also require careful denominator handling; an implementation should validate the Pearson formula and reject or explicitly label undefined results.

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