// library(ascii)
// setwd("c://aaaWork//web//fishR//bookex//AIFFD//Box3_1")
// Sweave("Box3_1a.Rnw",driver=RweaveAsciidoc)








AIFFD Box 3-1  Vignette
=======================
:Author:    Derek H. Ogle
:Email:     dogle@northland.edu
:Date:      17-June-2009
:Revision:  2

== Required Packages
----
> library(NCStats)  # Summarize, sd.calc
> library(sciplot)  # se

----

== Preparing Data
The working directory is set, the link:box3_1.txt[] data file is read, and the structure of the data frame is observed with,
----
> setwd("c://aaaWork//web//fishR//bookex//AIFFD//Box3_1")
> d <- read.table("box3_1.txt", header = TRUE)
> str(d)
'data.frame':	15 obs. of  1 variable:
 $ catch: int  3 0 4 1 4 1 1 0 1 0 ... 
----

== Compute Summary Statistics
The mean and standard deviation for the quantitative variable can be computed with +*[red]#Summarize()#*+ from the +*NCStats*+ package.  The single required argument is the vector containing the quantitative data.  For example,
----
> Summarize(d$catch)
        n      Mean  St. Dev.      Min.   1st Qu.    Median   3rd Qu.      Max. 
15.000000  1.600000  1.404076  0.000000  0.500000  1.000000  2.500000  4.000000  
----

If you do not want to load the +*NCStats*+ package then the same calculations can be made with +*[red]#summary()#*+ and +*[red]#sd()#*+ of the base packages contained in R,
----
> summary(d$catch)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
    0.0     0.5     1.0     1.6     2.5     4.0  
> sd(d$catch)
[1] 1.404076 
----

Regardless of which way you summarize the results, you can compute the standard error with +*[red]#se()#*+ from the +*sciplot*+ package,
----
> se(d$catch)
[1] 0.3625308 
----

== Constructing Confidence Intervals for the Population Mean
The critical t value for computing a confidence interval can be found with +*[red]#qt()#*+.  This function requires an upper-tail probability (i.e., (1-C)/2 where C is the level of confidence), the degrees-of-freedom, and the +*[red]#lower.tail=FALSE#*+ argument.  For example, the t critical value in this situation is,
----
> df <- length(d$catch)-1          # compute df as n-1
> C <- 0.95                        # set confidence level
> a2 <- (1-C)/2                    # find upper tail probability
> ct <- qt(a2,df,lower.tail=FALSE) # find critical t-value
> ct
[1] 2.144787 
----

The two endpoints of the confidence interval can then be computed with,
----
> mn <- mean(d$catch)              # find (and save) the mean
> semn <- se(d$catch)              # find (and save) the SE of the mean
> LCI <- mn-ct*semn
> UCI <- mn+ct*semn
> c(LCI,UCI)                       # combine together for showing
[1] 0.8224488 2.3775512 
----

The confidence interval is computed from the raw data, along with a great deal of other information, with +*[red]#t.test()#*+.  In this simplest form this function only requires the vector of quantitative data as the first argument and an optional level of confidence in the +*[red]#conf.level=#*+ argument (it defaults to 95%).  For example,
----
> t.test(d$catch)
	One Sample t-test

data:  d$catch 
t = 4.4134, df = 14, p-value = 0.0005894
alternative hypothesis: true mean is not equal to 0 
95 percent confidence interval:
 0.8224488 2.3775512 
sample estimates:
mean of x 
      1.6  
----

== Demonstration of Standard Deviation Calculation
The intermediate steps in computing a standard deviation, as shown at the beginning of Box 3.1, can be demonstrated with +*[red]#sd.calc()#*+
----
> sd.calc(d$catch)
Demonstration of parts of a std. dev. calculation.

     x diffs diffs.sq
1    3   1.4     1.96
2    0  -1.6     2.56
3    4   2.4     5.76
4    1  -0.6     0.36
5    4   2.4     5.76
6    1  -0.6     0.36
7    1  -0.6     0.36
8    0  -1.6     2.56
9    1  -0.6     0.36
10   0  -1.6     2.56
11   2   0.4     0.16
12   2   0.4     0.16
13   3   1.4     1.96
14   2   0.4     0.16
15   0  -1.6     2.56
sum 24   0.0    27.60

Mean = x-bar = 24 / 15 = 1.6 

Variance = s^2 = 27.6 / 14 = 1.9714286 

Std. Dev = s = sqrt(1.9714286) = 1.4040757 
----