Statistical & Clinical Analysis
Miettinen Test-Based Confidence Interval Formula: OR^(1 ± z / SQRT(Chi-Square))
Reconstruct 95% confidence limits for any Odds Ratio, Risk Ratio, or Hazard Ratio published with only a point estimate and Mantel-Haenszel chi-square statistic using Olli S. Miettinen’s 1976 test-based formula OR^(1 ± z / SQRT(chi²)).
Guaranteeing exact p-value consistency between Mantel-Haenszel chi-square tests and 95% confidence limits!
In classic epidemiology and meta-analytic data extraction from older clinical trials, studies frequently report a pooled Mantel-Haenszel Odds Ratio OR and 1-df chi-square statistic chi^2 without printing standard errors—or worse, Woolf’s asymptotic interval crosses 1.00 even when the Mantel-Haenszel chi^2 > 3.841 is statistically significant! Olli S. Miettinen (1976, American Journal of Epidemiology) solved both problems by equating |LN(OR)| / SE_TB = SQRT(chi^2), yielding the test-based standard error SE_TB = |LN(OR)| / SQRT(chi^2) and elegant exponentiated limits OR^(1 - z / SQRT(chi^2)) and OR^(1 + z / SQRT(chi^2))!
Notice that whenever chi^2 = z^2 = 1.96^2 = 3.8416, the exponent (1 - 1.96 / 1.96) = 0, so the lower limit hits OR^0 = 1.0000 EXACTLY! For OR = 2.65 with Mantel-Haenszel chi^2 = 6.76 (SQRT = 2.60) and z = 1.96, the 95% test-based CI is [1.2711, 5.5247]—shifting to lower limit 0.9170 (< 1.00, non-significant) when chi^2 = 3.24 (< 3.8416)!
| Point Estimate & 1-df Chi-Square (z = 1.96) | Lower 95% Limit OR^(1 - z/SQRT(chi²)) | Upper 95% Limit OR^(1 + z/SQRT(chi²)) | Hypothesis Test Alignment |
|---|---|---|---|
| OR = 2.65, chi² = 6.76 (p < 0.01) | 1.2711 (> 1.00) | 5.5247 | Excludes 1.00 iff chi² > z² |
| OR = 2.65, chi² = 3.24 (p = 0.072) | 0.9170 (< 1.00) | 7.6584 | Crosses 1.00 cleanly when p > 0.05 |
| Null Estimate OR = 1.00 or chi² <= 0 | Undefined SE_TB | "Requires Positive Non-Null OR & Chi-Square" | Singularity Guard |
Miettinen Test-Based Lower Confidence Limit = MIN(OR^(1 - z/SQRT(chi²)), OR^(1 + z/SQRT(chi²)))
Uses MIN across both signs so protective ratios (OR < 1) and harmful ratios (OR > 1) both order [Lower, Upper] properly.
=IF(OR(B2 <= 0, B2 = 1, B3 <= 0, B4 <= 0), "Requires Positive Non-Null OR & Chi-Square", ROUND(MIN(B2 ^ (1 - B4 / SQRT(B3)), B2 ^ (1 + B4 / SQRT(B3))), 4))Miettinen Test-Based Upper Confidence Limit = MAX(OR^(1 - z/SQRT(chi²)), OR^(1 + z/SQRT(chi²)))
Returns the upper test-based confidence bound for point ratio B2, 1-df chi-square B3, and normal quantile z B4.
=IF(OR(B2 <= 0, B2 = 1, B3 <= 0, B4 <= 0), "Requires Positive Non-Null OR & Chi-Square", ROUND(MAX(B2 ^ (1 - B4 / SQRT(B3)), B2 ^ (1 + B4 / SQRT(B3))), 4))Try the example
Use the original values or change them to test your understanding.