====== Contingency Tables ======

A contingency is obtained when by crossing two qualitatives nominal variables, typically artifacts type and archaeological assemblage.

Let's assume we have a data.frame object named ''MyData''. Each line corresponds to an artifact. The first column contains a label, the second the assemblage and the last the artifact type :


<file>label		assemblage	type
CLXIV-001	CL XIV 5+6	t
CLXIV-002	CL XIV 3+4	plh
...		...		... </file>

**''R''** provides two functions for computing contingency table, here assemblage against type:

<code C>MyCrossTable <- table(MyData[,2], MyData[,3])
</code>

an alternative :
<code C>MyCrossTable2 <- xtabs(~., NMB_2006[,c(1,6)] )
</code>

The latter can be used as argument to the ''corresp()'' function from the MASS package which computes correspondence analysis, a factorial data reduction method suitable for contingency tables.

==== Frequency Tables ====

The ''prop.table()'' function calculates the frequency table (percentages). Its first argument is an objet of class table. The second is the margin : 1 for row, 2 for columns.

<code C>MyCrosFreq <- prop.table(MyCrossTable, 1)
</code>

Here is an example :
<file>
     TYPE_REC_
PHASE     type 1    type 2    type 3    type 4
    1 0.29242348 0.2506487 0.1904153 0.2665125
    2 0.44952914 0.2752219 0.1045417 0.1707073
    3 0.00000000 0.5199755 0.1823170 0.2977075
    4 0.13439854 0.5759938 0.1875329 0.1020748
    5 0.07930212 0.1942093 0.1844235 0.5420651
    6 0.14209591 0.2609925 0.1652278 0.4316838
</file>
==== Plotting  ====

Now we can plot our frequency table. A graphical representation allows us to have a feel of the trends, even if there are many artifacts types and assemblages.

A common and popular way to represent a frequency table is Ford's Battleship diagram. Is is derived from the barplot.

Here is a code that implements it (Jammet-Reynal, 2006):
<code C>ford <- function(x, cex.row.labels=1) {
#################################################
##  FORD'S "BATTLESHIP" DIAGRAM                ##
##  Loic JAMMET-REYNAL, may 2006               ##
##  Departement d'Anthropologie et d'Ecologie  ##
##  University of Geneva                       ##
##  jammetr1[at]etu.unige.ch                   ##
#################################################

    dim(x)[2] -> jmax # colonnes j
    dim(x)[1] -> imax # lignes i
    
    set.up <- function(xlim, ylim) {
        # setting up coord. system
        plot(    xlim,    # x
                ylim,     # y
                type="n", # no plotting
                axes = FALSE,
                asp = NA,
                xlab = "",
                ylab = "")
    }
    
    ## initialisation du device
    ## on divise par le nombre de colonnes + 1
    ## 1ere colonne : labels
    op <- par(mfrow=c(1, jmax+1), mar=c(5,0,2,0))
    
    # labels des lignes (colonne 1)
    set.up(c(0,1),             # x
           c(0.9, imax+1.10) ) # y
    
    for (i in 1:imax) {
        text(0.5,
               i+0.5,
              row.names(x)[i],
              font = 2, # boldface
              cex = cex.row.labels)
    }

    for (j in 1:jmax) { # colonnes j
        set.up(xlim = c(-60,60)*max(x),   # x
               ylim = c(0.9, imax+1.10) ) # y
        
        title(sub=colnames(x)[j],
              font.sub=2, # boldface
              cex.sub = 1.5)
        
        for (i in 1: imax) { # lignes i
            # le plus important. boite multipliee
            # par les parametres
            X <- c(-50,+50,+50,-50,-50)*x[i,j]
            Y <- c(i,i,i+1,i+1,i)
            polygon(X,
                    Y,
                    xpd=FALSE, 
                    col="black",
                    mar=c(0,0,0,0) )
        }
    }
}
</code>

You first have to run the above code. A new function called ''ford()'' will be available. Its argument is a frequency table.

In order to represent a chronological hypothesis, you have to rearrange the order of rows and columns. A way to do it is giving two a vector of indices between brackets right after the frequency table object:
<code C>ford(MyCrosFreq[c(1,2,4,3), c(2,4,5,3,1,6)])
</code>

You can obtain optimal ordering by use of [[seriation]] techniques.

This is an output example :

{{ford.png?600|Ford diagram}}

==== Reference ====
**Jammet-Reynal, L. (2006).-** //La céramique de Clairvaux VII (Jura, France) : typologie, étude quantitative et sériation.// Genève : Département d'anthropologie et d'écologie de l'Université. Unpublished Master thesis.