propint

propint is an R package that includes several functions to compute confidence intervals for the estimation of proportions based on dichotomous data; each observation is a 1 or a 0 depending on whether a feature is present or absent. propint enables you to:

1. Compute confidence intervals for simple proportions like the prevalence of a disease. Functionality is included to estimate the confidence interval for independent data points, or from clustered data; e.g. if the data was collected in different cities that may differ in the average prevalence of a disease.
2. Compute confidence intervals for the difference between proportions, e.g. to compare the efficacy of a treatment and a placebo when the outcome was measured as a dichotomous feature.
3. Compute a confidence interval for the difference between two differences of proportions, that is, to test for the presence of an interaction between proportions (e.g. to test whether a treatment efficacy in comparison to a placebo differed between two hospitals).

Installation

library("devtools") # if not available: install.packages("devtools")
install_github("m-Py/propint")

# load the package via 
library("propint")

Confidence intervals for a simple proportion

A simple proportion can be estimated either or the basis of independent observations, or by taking into account dependency that might be known to the researcher.

Independent observations

Confidence intervals for single proportions are computed based on »Wilson's score« (1927) as recommended in Newcombe (Method 3, 1998c).

Clustered observations

Controlling for existing dependency in observations is crucial when applying methods of statistical inference. The prevalence of a disease may be estimated on observations in different cities that differ in their average prevalence; hence observations from cities are not statistically independent, and this clustering should be considered when computing a confidence interval. propint implements a straightforward method to control for clustering described by Bennet, Woods Iiyanage and Smith (1991). Its usage is as follows:

## Compute 95% confidence interval based on clustered data:
ci.one.prop.cluster(95, c(23, 19, 44, 9), c(30, 25, 77, 12))

$p
[1] 0.6597222

$l
[1] 0.5340171

$u
[1] 0.7854273

$se.cluster
[1] 0.06413645

$design.effect
[1] 2.638624

$intraclass.cor
[1] 0.04681784

We also get some additional information like the design effect and the intraclass correlation coefficient.

We can compare the results to the confidence interval for independent observations. The point estimate of course is the same, but the width of the interval differs. The confidence interval will generally be larger when clustering is accounted for; when clustering is ignored we will have undue confidence in the precision of our estimate (note that some persons do not think that confidence in confidence intervals is appropriate; but for the sake of simplicity I will use such language).

## Compare to unclustered CI:
ci.one.prop(95, sum(c(23, 19, 44, 9)), sum(c(30, 25, 77, 12)))

$p
[1] 0.6597222

$l
[1] 0.5790851

$u
[1] 0.732059

propint can be used to compute confidence intervals for simple pairs of (independent or dependent) proportions, and even to compute confidence intervals for higher order interaction (for example corresponding to a 2 x 2 x 2 ANOVA). All functions rely on confidence intervals of single proportions that are converted into "higher-order" confidence intervals using recommendations and developments by Newcombe (1998a, 1998b, 2001). Confidence intervals for single proportions are computed based on »Wilson's score« (1927) as recommended in Newcombe (Method 3, 1998c).

Interaction of independent proportions

The function ci.indep.interaction() computes the confidence interval for an interaction of independent proportions. Specifically, a confidence interval is computed for the differences of the difference of two pairs proportions: (p1 - p2) - (p3 - p4). If the confidence interval of this difference of differences does not cover zero, we assume there is an interaction.

ci.indep.interaction() can be used in two different ways:

1. Observed frequencies are directly passed ("successes" and "total trials" for each of the four proportions).
2. Proportions and boundaries of a confidence interval (e.g. 95% confidence interval) are directly passed to the function, because they have already been computed.

The following example illustrates usage 1 (example from Newcombe, 2001):

The Bristol Antenatal Care Study sought to evaluate the effectiveness of routine antenatal visits by asking whether choice improves well-being. Women were randomized to either a traditional or a Lexible schedule of antenatal care. As one of the outcomes, 277 women who developed a problem were asked whether their problem could have been recognized earlier than it was. Among nulliparous women, the proportions answering yes in the two groups were 17/65 and 17/75. Among multiparae, the proportions were 18/72 and 16/65. While clearly none of the differences between these four proportions are conventionally statistically significant, it is of interest to assess the degree to which the effect of the intervention differs between these two subsets of the parturient population, which it was anticipated could be quite different.

To test for this interaction we use:

# (the notation of a, m, b and n is taken from Newcombe (1998b))
> ci.indep.interaction(ci=95, a1=17, m1=65, b1=17, n1=75, 
>                      a2=18, m2=72, b2=16, n2=65)

$d
[1] 0.03102564

$l
[1] -0.1686752

$u
[1] 0.2343009

We obtain the difference between the differences d = (p1 - p2) - (p3 - p4) = 0.031 and the lower bound (l = -0.169) and the upper bound (u = 0.234) for the 95% confidence interval of this difference. Because the interval contains 0, there is no evidence for an interaction.

An alternative usage of ci.indep.interaction() is to compute confidence intervals for each proportion first, and then call ci.indep.interaction():

> # Alternative usage:
> p1 <- ci.one.prop(95, 17, 65)
> p2 <- ci.one.prop(95, 17, 75)
> p3 <- ci.one.prop(95, 18, 72)
> p4 <- ci.one.prop(95, 16, 65)

> ci.indep.interaction(p1=p1$p, p2=p2$p, p3=p3$p, p4=p4$p, l1=p1$l, l2=p2$l, 
>                      l3=p3$l, l4=p4$l, u1=p1$u, u2=p2$u, u3=p3$u, u4=p4$u)

$d 
[1] 0.03102564

$l
[1] -0.1686752

$u
[1] 0.2343009

As the code shows, it is possible to pass the lower and upper bounds of the four confidence intervals and the proportions directly to ci.indep.interaction(). In the illustrated case above, ci.one.prop() is used to determine the boundaries of the 95% confidence interval. ci.one.prop() is also part of the propint package and is used as the default computation of confidence intervals of single proportions (see Newcombe, 1998c). Consequently, you can pass confidence intervals to ci.indep.interaction() that you computed using some other method. Also note that we do not pass the confidence level in this case; ci.indep.interaction() computes the same confidence level that is implied in the lower and upper boundaries of the four proportions.

Interaction of dependent proportions ("mixed interaction")

To be written

Pairwise comparison of independent proportions

propint can be used for a classical test of the difference between two proportions. The function ci.two.indep.props() computes a confidence interval of the difference of two independent proportions (relying on method 10 in Newcombe, 1998b). It can be used in two ways analogue to ci.indep.interaction():

> # Usage 1:
> ci.two.indep.props(ci=95, a=56, m=70, b=48, n=80)

$d
[1] 0.2

$l
[1] 0.05243147

$u
[1] 0.3338727

> # Usage 2:
> p1 <- ci.one.prop(ci=95, r=56, n=70)
> p2 <- ci.one.prop(ci=95, r=48, n=80)
> ci.two.indep.props(p1=p1$p, p2=p2$p, l1=p1$l, l2=p2$l, u1=p1$u, u2=p2$u)

$d
[1] 0.2

$l
[1] 0.05243147

$u
[1] 0.3338727

We thus obtain the difference between the two proportions and the lower and upper boundary for the difference of the proportions.

Pairwise comparison of dependent proportions

To be written

More help

Use the help functions for more help on propint's functions in your R console:

> library("propint")
> ?ci.one.prop
> ?ci.two.indep.props
> ?ci.two.dep.props
> ?ci.indep.interaction
> ?ci.mixed.interaction

Questions and suggestions

If you have any questions or suggestions (which are greatly appreciated), just open an issue at Github or contact me via martin.papenberg at hhu.de.

References

Newcombe, R. G. (1998a). Improved confidence intervals for the difference between binomial proportions based on paired data. Statistics in medicine, 17(22), 2635-2650.

Newcombe, R. G. (1998b). Interval estimation for the difference between independent proportions: comparison of eleven methods. Statistics in medicine, 17(8), 873-890.

Newcombe, R. G. (1998c). Two-sided confidence intervals for the single proportion: comparison of seven methods. Statistics in medicine, 17(8), 857-872.

Newcombe, R. G. (2001). Estimating the difference between differences: measurement of additive scale interaction for proportions. Statistics in medicine, 20(19), 2885-2893.

Wilson, E. B. (1927). Probable inference, the law of succession, and statistical inference. Journal of the American Statistical Association, 22(158), 209-212.