 ## Summary and Analysis of Extension Program Evaluation in R

Salvatore S. Mangiafico

# Confidence Intervals for Proportions

A binomial proportion has counts for two levels of a nominal variable.  An example would be counts of students of only two sexes, male and female.  If there are 20 students in a class, and 12 are female, then the proportion of females are 12/20, or 0. 6, and the proportion of males are 8/20 or 0.4.  This is a binomial proportion.

Sex      Count   Proportion
Female   12      0.60
Male      8      0.40
------   --      -----------
Total    20      1.00

A multinomial proportion has counts for more than two levels of a nominal variable.  For example, we might have the following levels and counts for sex for students in a class:

Sex                Count   Proportion
Female                12   0.60
Male                   6   0.30
Other                  1   0.05
Prefer not to answer   1   0.05
--------------------  --   -----------
Total                 20   1.00

Confidence intervals can be produced for either binomial or multinomial proportions.

The binom.test function in the native stats package will provide the Clopper-Pearson confidence interval for a binomial proportion.  Other methods for a binomial proportion are provided by the BinomCI function in the DescTools package, as well as by various functions in the PropCIs package.

Confidence intervals for multinomial proportions can be produced with the MultinomCI function in the DescTools package.

### Packages used in this chapter

The packages used in this chapter include:

•  DescTools

•  PropCIs

The following commands will install these packages if they are not already installed:

if(!require(DescTools)){install.packages("DescTools")}
if(!require(PropCIs)){install.packages("PropCIs")}

### Example of confidence intervals for a binomial proportion

As part of a demographic survey of her scrapbooking 4-H course, Seras Victoria asks students if they have ever done scrapbooking before.  The following are the data from her course:

Experience   Count
Yes           7
No           14
----------   --
Total        21

Note that when calculating confidence intervals for a binomial variable, one level of the nominal variable is chosen to be the “success” level.  This is an arbitrary decision, but you should be cautious to remember that the confidence interval is reported for the proportion of “success” responses.  The BinomCI function in the DescTools package can produce the confidence interval for both “success” and “failure” in one step.

The binom.test function output includes a confidence interval for the proportion, and the proportion of “success” as a decimal number.  The binom.test function uses the Clopper–Pearson method for confidence intervals.

binom.test(7, 21,
0.5,
alternative="two.sided",
conf.level=0.95)

95 percent confidence interval:
0.1458769 0.5696755

sample estimates:
probability of success
0.3333333

The BinomCI function in the DescTools package has several methods for calculating confidence intervals for a binomial proportion.

library(DescTools)

BinomCI(7, 21,
conf.level = 0.95,
method = "clopper-pearson")

###  Methods: "wilson", "wald", "agresti-coull", "jeffreys",
###  "modified wilson", "modified jeffreys",
###  "clopper-pearson", "arcsine", "logit", "witting", "pratt"

est    lwr.ci    upr.ci
[1,] 0.3333333 0.1458769 0.5696755

The BinomCI function in the DescTools package can also produce the confidence intervals for “success” and “failure” in one step.

library(DescTools)

observed = c(7, 14)

total = sum(observed)

BinomCI(observed, total,
conf.level = 0.95,
method = "clopper-pearson")

###  Methods: "wilson", "wald", "agresti-coull", "jeffreys",
###  "modified wilson", "modified jeffreys",
###  "clopper-pearson", "arcsine", "logit", "witting", "pratt"

est    lwr.ci    upr.ci
[1,] 0.3333333 0.1458769 0.5696755
[2,] 0.6666667 0.4303245 0.8541231

The PropCIs package has functions for calculating confidence intervals for a binomial proportion.

The exactci function uses the Clopper–Pearson exact method.

7/21

 0.3333333

library(PropCIs)

exactci(7, 21,
conf.level=0.95)

95 percent confidence interval:
0.1458769 0.5696755

The blakerci function uses the Blaker exact method.

7/21

 0.3333333

library(PropCIs)

blakerci(7, 21,
conf.level=0.95)

95 percent confidence interval:
0.1523669 0.5510455

### Example of confidence intervals for a multinomial proportion

As part of a demographic survey of her scrapbooking 4-H course, Seras Victoria asks students to report their sex.  The following are the data from her course:

Sex          Count
Female       10
Male          9
Other         1
----------   -----
Total        21

library(DescTools)

observed = c(10, 9, 1, 1)

MultinomCI(observed,
conf.level=0.95,
method="sisonglaz")

### Methods: "sisonglaz", "cplus1", "goodman"

est    lwr.ci    upr.ci
[1,] 0.47619048 0.2857143 0.7009460
[2,] 0.42857143 0.2380952 0.6533270
[3,] 0.04761905 0.0000000 0.2723746
[4,] 0.04761905 0.0000000 0.2723746

### Optional analysis: confidence intervals for a difference in proportions

As part of a demographic survey of their scrapbooking 4-H courses, Seras Victoria and Integra Hellsing ask students if they have experience in scrapbooking.  They want to determine the difference of proportions of students having experience in each class, and calculate a confidence interval for that difference.  The following are the data:

Seras Victoria               Integra Hellsing

Experience   Count           Experience   Count
Yes           7              Yes          13
No           14              No            4
----------   -----           ----------   -----
Total        21              Total        17

Two functions in the PropCIs package can determine a confidence interval for a difference for in independent proportions.

7/21

 0.3333333

13/17

 0.7647059

(7/21) - (13/17)

 -0.4313725

library(PropCIs)

diffscoreci(7, 21, 13, 17,
conf.level=0.95)

95 percent confidence interval:
-0.6685985 -0.1103804

wald2ci(7, 21, 13, 17,
conf.level=0.95,

95 percent confidence interval:
-0.7165199 -0.1462252

sample estimates:
 -0.4313725

### References

“Confidence Limits” in Mangiafico, S.S. 2015a. An R Companion for the Handbook of Biological Statistics, version 1.09. rcompanion.org/rcompanion/c_04.html.