statsmodels.stats.proportion.confint_proportions_paired#

statsmodels.stats.proportion.confint_proportions_paired(table, method='newcomb', alpha=0.05)[source]#

Confidence interval for the difference of two paired proportions

This assumes that each of n units is classified twice as success or failure, for example before and after a treatment, or by two raters, so that the two proportions are correlated. The confidence interval is for the difference p1 - p2 of the two marginal proportions.

Parameters:

tablearray_like

2x2 contingency table of counts, [[n11, n12], [n21, n22]]. Rows correspond to the first classification and columns to the second, with success in the first row and first column. n12 and n21 are the discordant counts. The first marginal proportion is p1 = (n11 + n12) / n and the second is p2 = (n11 + n21) / n, where n is the total count.

method{“newcomb”, “newcombe”, “wald”}, optional

Method for computing the confidence interval.

“newcomb” : Newcombe’s method 10, which combines the Wilson score intervals of the two marginal proportions with a correction for the correlation between them. This is the default. The spelling “newcombe” is accepted as an alias; “newcomb” matches the method name in confint_proportions_2indep.
“wald” : simple asymptotic (Wald) interval without continuity correction, Newcombe’s method 1. It degenerates to a zero-width interval when there are no discordant pairs.

alphafloat, optional

Significance level for the confidence interval, default is 0.05. The nominal coverage probability is 1 - alpha.

Returns:

lowfloat: Lower confidence limit for p1 - p2.
uppfloat: Upper confidence limit for p1 - p2.

See also

confint_proportions_2indep: Confidence interval for two independent proportions.
statsmodels.stats.contingency_tables.mcnemar: Hypothesis test for marginal homogeneity of paired binary data.

Notes

Newcombe’s method 10 is the paired analogue of the hybrid score interval recommended for independent proportions in [1]. Let (l1, u1) and (l2, u2) be the Wilson score limits of p1 and p2, and let phi be the corrected phi coefficient of the table, which is zero if any marginal count is zero. The limits are

p1 - p2 - sqrt((p1 - l1)**2 - 2 * phi * (p1 - l1) * (u2 - p2) + (u2 - p2)**2)

and

p1 - p2 + sqrt((u1 - p1)**2 - 2 * phi * (u1 - p1) * (p2 - l2) + (p2 - l2)**2).

The interval always lies within [-1, 1] and has positive width even when all pairs are concordant. The Wald interval is not bounded.

The results have been verified against the R packages ratesci (moverpairci with Wilson base and the phi correction) and PropCIs (diffpropci.Wald.mp).

References

[1]

Newcombe, Robert G. 1998. “Improved Confidence Intervals for the Difference between Binomial Proportions Based on Paired Data.” Statistics in Medicine 17 (22): 2635-50. https://doi.org/10.1002/(SICI)1097-0258(19981130)17:22<2635::AID- SIM954>3.0.CO;2-C.

Examples

Of 37 patients, 15 respond to treatment A and 12 to treatment B, with 10 responding to both.

>>> from statsmodels.stats.proportion import confint_proportions_paired
>>> table = [[10, 5], [2, 20]]
>>> confint_proportions_paired(table)
(-0.0643926050711..., 0.2211160178958...)
>>> confint_proportions_paired(table, method="wald")
(-0.0566130751996..., 0.2187752373618...)