// library(ascii)
// setwd("c://aaaWork//web//fishR//bookex//AIFFD//Box3_2")
// Sweave("Box3_2a.Rnw",driver=RweaveAsciidoc)








AIFFD Box 3-2  Vignette
=======================
:Author:    Derek H. Ogle
:Email:     dogle@northland.edu
:Date:      17-June-2009
:Revision:  2

== Required Packages
----
> library(NCStats)  # bin.ci

----

== Preparing Data
The working directory is set, the link: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 +*[red]#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 +*[red]#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 +*[red]#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 +*[red]#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 
----