Required Packages

> library(NCStats)  # bin.ci

Preparing Data

The working directory is set, the box3_2.txt data file is read, and the structure of the data frame is observed with,

> setwd("c://aaaWork//web//fishR//bookex//AIFFD//Box3_2")
> d <- read.table("box3_2.txt", header = TRUE)
> d
  age  n
1   0 55
2   1 22
3   2 10
4   3 18
5   4  6
6   5  3
7   6  1
8   7  1

Compute Proportions in Each Age Group

The proportion of fish in each age group can be computed by dividing the catch in each age group by the total catch; i.e.,

> ttl <- sum(d$n)    # compute (and save) total catch
> p <- d$n/ttl       # find proportion of "successes"
> p
[1] 0.47413793 0.18965517 0.08620690 0.15517241 0.05172414 0.02586207 0.00862069 0.00862069

The standard errors are then computed with,

> q <- 1-p           # find proportion of "failures"
> seps <- sqrt(p*q/(ttl-1))

Confidence Intervals for Population Proportions I

The critical value from the F distribution can be found with qf(). This function requires an upper-tail probability as the first argument, numerator df in the second argument, denominator df in the third argument, and lower.tail=FALSE argument. For example, the two critical F values used for the first age-group in Box 3.2 can be found with,

> alpha <- 0.05
> n <- ttl
> a <- 55
> f.lwr <- qf(alpha, 2 * (n - a + 1), 2 * a, lower.tail = FALSE)
> f.lwr
[1] 1.360090
> f.upr <- qf(alpha, 2 * a + 2, 2 * (n - a + 1) - 2, lower.tail = FALSE)
> f.upr
[1] 1.355947

The CIs for the first age-group can then be computed with,

> LCI <- a/(a + (n - a + 1) * f.lwr)
> UCI <- ((a + 1) * f.upr)/(n - a + (a + 1) * f.upr)
> c(LCI, UCI)
[1] 0.3947589 0.5545268

The CIs for all of the age-groups

> alpha <- 0.05
> LCIs <- d$n/(d$n + (ttl - d$n + 1) * qf(alpha, 2 * (ttl - d$n + 1), 2 * d$n, lower.tail = FALSE))
> UCIs <- ((d$n + 1) * qf(alpha, 2 * d$n + 2, 2 * (ttl - d$n + 1) - 2, lower.tail = FALSE))/(ttl - d$n + (d$n + 1) *
+     qf(alpha, 2 * d$n + 2, 2 * (ttl - d$n + 1) - 2, lower.tail = FALSE))
> data.frame(age = d$age, p, seps, LCIs, UCIs)
  age          p       seps         LCIs       UCIs
1   0 0.47413793 0.04656283 0.3947588631 0.55452678
2   1 0.18965517 0.03655682 0.1320279845 0.25960996
3   2 0.08620690 0.02617255 0.0475164835 0.14183827
4   3 0.15517241 0.03376311 0.1027583277 0.22137272
5   4 0.05172414 0.02065214 0.0227626672 0.09953204
6   5 0.02586207 0.01480106 0.0070853559 0.06548328
7   6 0.00862069 0.00862069 0.0004420858 0.04024133
8   7 0.00862069 0.00862069 0.0004420858 0.04024133

These results match what is shown in Box 3.2

Confidence Intervals for Population Proportions II

Many authors suggest methods for computing confidence intervals for proportions other than using the F distribution as shown in the text. One such method is the so-called "Wilson" method which is the default method in bin.ci of the NCStats package. This function requires the "number of success" as the first argument, the "number of trials" as the second argument, and optional level of confidence in conf.level=. The default 95% confidence intervals using the Wilson method can be computed with,

> bin.ci(d$n, ttl)
      95% LCI    95% UCI
 0.3855640505 0.56436980
 0.1287138187 0.27049243
 0.0474993548 0.15144231
 0.1004660814 0.23198530
 0.0239186535 0.10826815
 0.0088338879 0.07328676
 0.0004421836 0.04721983
 0.0004421836 0.04721983