Confidence Intervals for differences in proportions

The BIS.Net Team BIS.Net Team

Although the following is directed at proportions, it also applies to ppm

In medical research differences in proportions are an important effect measure for randomized controlled trial (RCT) and cohort studies. In manufacturing the production manager may need to compare the proportion of defectives produced by two different processes. A market researcher may need to compare the percent of satisfied customers prior and after embarking on a customer satisfaction program. A politician needs to study the effect of a popularity enhancement campaign by comparing the proportion of voters who would vote for him if an election is held now with if one was held after the campaign.

Estimates based on a sample require confidence intervals to determine how ‘accurate’ the estimate is.

The traditional Wald Interval has limits based on the asymptotic normal distribution calculated through the following expression.

LL=p1-p2 -z(alpha/2)*sqrt(p1*(1-p1)/n1+p2*(1-p2)/n2)

UL=p1-p2 +z(alpha/2)*sqrt(p1*(1-p1)/n1+p2*(1-p2)/n2)

Where p1 is the sample proportion of population 1 and n1 the corresponding sample size and p2 is the sample proportion of population 2 and n2 the corresponding sample size.

Most software and textbooks still use the Wald method directly, or adjusted with a continuity factor, to compute confidence intervals. However, the Wald interval is very liberal with coverage probabilities considerably less than the nominal.

Many research papers have been written on this subject to find superior methods for calculating the confidence interval of the difference between two proportion with coverage closely equal to the nominal value.

Solutions have been offered by Agresti–Caffo, Newcombe, Miettinen–Nurminen and others. Agresti–Min proposed a method which involves inverting a one two-sided score test. This method appears to provide the closest coverage to the nominated level. To compute the confidence interval requires a complex machine powered algorithm which is used in the BIS.Net Inferences APP.

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