Creating and manipulating frequency tables"

knitr::opts_chunk$set(
  collapse = TRUE,
  warning = FALSE,
  fig.height = 6,
  fig.width = 7,
  fig.path = "fig/tut01-",
  dev = "png",
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)

# save some typing
knitr::set_alias(w = "fig.width",
                 h = "fig.height",
                 cap = "fig.cap")

# Old Sweave options
# \SweaveOpts{engine=R,eps=TRUE,height=6,width=7,results=hide,fig=FALSE,echo=TRUE}
# \SweaveOpts{engine=R,height=6,width=7,results=hide,fig=FALSE,echo=TRUE}
# \SweaveOpts{prefix.string=fig/vcd-tut,eps=FALSE}
# \SweaveOpts{keep.source=TRUE}


# preload datasets ???
set.seed(1071)
library(vcd)
library(vcdExtra)
library(ggplot2)
data(HairEyeColor)
data(PreSex)
data(Arthritis, package="vcd")
art <- xtabs(~Treatment + Improved, data = Arthritis)
if(!file.exists("fig")) dir.create("fig")

R provides many methods for creating frequency and contingency tables. Several are described below. In the examples below, we use some real examples and some anonymous ones, where the variables A, B, and C represent categorical variables, and X represents an arbitrary R data object.

Forms of frequency data

The first thing you need to know is that categorical data can be represented in three different forms in R, and it is sometimes necessary to convert from one form to another, for carrying out statistical tests, fitting models or visualizing the results. Once a data object exists in R, you can examine its complete structure with the str() function, or view the names of its components with the names() function.

Case form

Categorical data in case form are simply data frames containing individual observations, with one or more factors, used as the classifying variables. In case form, there may also be numeric covariates. The total number of observations is nrow(X), and the number of variables is ncol(X).

Example:

The Arthritis data is available in case form in the vcd package. There are two explanatory factors: Treatment and Sex. Age is a numeric covariate, and Improved is the response--- an ordered factor, with levels r paste(levels(Arthritis$Improved),collapse=' < '). Excluding Age, this represents a $2 \times 2 \times 3$ contingency table for Treatment, Sex and Improved, but in case form.

names(Arthritis)      # show the variables

str(Arthritis)        # show the structure

head(Arthritis,5)     # first 5 observations, same as Arthritis[1:5,] 

Frequency form

Data in frequency form is also a data frame containing one or more factors, and a frequency variable, often called Freq or count. The total number of observations is: sum(X$Freq), sum(X[,"Freq"]) or some equivalent form.

The number of cells in the table is given by nrow(X).

Example: For small frequency tables, it is often convenient to enter them in frequency form using expand.grid() for the factors and c() to list the counts in a vector. The example below, from [@vcd:Agresti:2002] gives results for the 1991 General Social Survey, with respondents classified by sex and party identification.

# Agresti (2002), table 3.11, p. 106
GSS <- data.frame(
  expand.grid(sex = c("female", "male"), 
              party = c("dem", "indep", "rep")),
  count = c(279,165,73,47,225,191))

GSS
names(GSS)
str(GSS)

sum(GSS$count)

Table form

Table form data is represented by a matrix, array or table object, whose elements are the frequencies in an $n$-way table. The variable names (factors) and their levels are given by dimnames(X). The total number of observations is sum(X). The number of dimensions of the table is length(dimnames(X)), and the table sizes are given by sapply(dimnames(X), length).

Example: The HairEyeColor is stored in table form in vcd.

str(HairEyeColor)                      # show the structure

sum(HairEyeColor)                      # number of cases

sapply(dimnames(HairEyeColor), length) # table dimension sizes

Example: Enter frequencies in a matrix, and assign dimnames, giving the variable names and category labels. Note that, by default, matrix() uses the elements supplied by columns in the result, unless you specify byrow=TRUE.

# A 4 x 4 table  Agresti (2002, Table 2.8, p. 57) Job Satisfaction
JobSat <- matrix(c( 1, 2, 1, 0, 
                    3, 3, 6, 1, 
                   10,10,14, 9, 
                    6, 7,12,11), 4, 4)

dimnames(JobSat) = list(
  income = c("< 15k", "15-25k", "25-40k", "> 40k"),
  satisfaction = c("VeryD", "LittleD", "ModerateS", "VeryS")
  )

JobSat

JobSat is a matrix, not an object of class("table"), and some functions are happier with tables than matrices. You can coerce it to a table with as.table(),

JobSat <- as.table(JobSat)
str(JobSat)

Ordered factors and reordered tables {#sec:ordered-factors}

In table form, the values of the table factors are ordered by their position in the table. Thus in the JobSat data, both income and satisfaction represent ordered factors, and the positions of the values in the rows and columns reflects their ordered nature.

Yet, for analysis, there are time when you need numeric values for the levels of ordered factors in a table, e.g., to treat a factor as a quantitative variable. In such cases, you can simply re-assign the dimnames attribute of the table variables. For example, here, we assign numeric values to income as the middle of their ranges, and treat satisfaction as equally spaced with integer scores.

dimnames(JobSat)$income <- c(7.5,20,32.5,60)
dimnames(JobSat)$satisfaction <- 1:4

For the HairEyeColor data, hair color and eye color are ordered arbitrarily. For visualizing the data using mosaic plots and other methods described below, it turns out to be more useful to assure that both hair color and eye color are ordered from dark to light. Hair colors are actually ordered this way already, and it is easiest to re-order eye colors by indexing. Again str() is your friend.

HairEyeColor <- HairEyeColor[, c(1,3,4,2), ]
str(HairEyeColor)

This is also the order for both hair color and eye color shown in the result of a correspondence analysis (\figref{fig:ca-haireye}) below.

With data in case form or frequency form, when you have ordered factors represented with character values, you must ensure that they are treated as ordered in R.

Imagine that the Arthritis data was read from a text file.
By default the Improved will be ordered alphabetically: Marked, None, Some --- not what we want. In this case, the function ordered() (and others) can be useful.

Arthritis <- read.csv("arthritis.txt",header=TRUE)
Arthritis$Improved <- ordered(Arthritis$Improved, 
                              levels=c("None", "Some", "Marked")
                              )

The dataset Arthritis in the vcd package is a data.frame in this form With this order of Improved, the response in this data, a mosaic display of Treatment and Improved (\figref{fig:arthritis}) shows a clearly interpretable pattern.

The original version of mosaic in the vcd package required the input to be a contingency table in array form, so we convert using xtabs().

#| Arthritis,
#| fig.height = 6,
#| fig.width = 6,
#| fig.cap = "Mosaic plot for the `Arthritis` data, showing the marginal model of independence for Treatment and Improved.  Age, a covariate, and Sex are ignored here."
data(Arthritis, package="vcd")
art <- xtabs(~Treatment + Improved, data = Arthritis)
mosaic(art, gp = shading_max, split_vertical = TRUE, main="Arthritis: [Treatment] [Improved]")

Finally, there are situations where, particularly for display purposes, you want to re-order the dimensions of an $n$-way table, or change the labels for the variables or levels. This is easy when the data are in table form: aperm() permutes the dimensions, and assigning to names and dimnames changes variable names and level labels respectively. We will use the following version of UCBAdmissions in \@(sec:mantel) below. ^[Changing Admit to Admit? might be useful for display purposes, but is dangerous--- because it is then difficult to use that variable name in a model formula. See \@(sec:tips) for options labeling_args and set_labelsto change variable and level names for displays in the strucplot framework.]

UCB <- aperm(UCBAdmissions, c(2, 1, 3))
dimnames(UCB)[[2]] <- c("Yes", "No")
names(dimnames(UCB)) <- c("Sex", "Admit?", "Department")

# display as a flattened table
stats::ftable(UCB)

structable() {#sec:structable}

For 3-way and larger tables the structable() function in vcd provides a convenient and flexible tabular display. The variables assigned to the rows and columns of a two-way display can be specified by a model formula.

structable(HairEyeColor)                   # show the table: default

structable(Hair+Sex ~ Eye, HairEyeColor)   # specify col ~ row variables

It also returns an object of class "structable" which may be plotted with mosaic() (not shown here).

HSE < - structable(Hair+Sex ~ Eye, HairEyeColor)   # save structable object
mosaic(HSE)                                        # plot it

table() and friends {#sec:table}

You can generate frequency tables from factor variables using the table() function, tables of proportions using the prop.table() function, and marginal frequencies using margin.table().

For these examples, create some categorical vectors:

 n=500
 A <- factor(sample(c("a1","a2"), n, rep=TRUE))
 B <- factor(sample(c("b1","b2"), n, rep=TRUE))
 C <- factor(sample(c("c1","c2"), n, rep=TRUE))
 mydata <- data.frame(A,B,C)

These lines illustrate table-related functions:

# 2-Way Frequency Table
attach(mydata)
mytable <- table(A,B)   # A will be rows, B will be columns
mytable                 # print table

margin.table(mytable, 1) # A frequencies (summed over B)
margin.table(mytable, 2) # B frequencies (summed over A)

prop.table(mytable)    # cell percentages
prop.table(mytable, 1) # row percentages
prop.table(mytable, 2) # column percentages

table() can also generate multidimensional tables based on 3 or more categorical variables. In this case, you can use the ftable() or structable() function to print the results more attractively.

# 3-Way Frequency Table
mytable <- table(A, B, C)
ftable(mytable)

table() ignores missing values by default. To include NA as a category in counts, include the table option exclude=NULL if the variable is a vector. If the variable is a factor you have to create a new factor using \code{newfactor <- factor(oldfactor, exclude=NULL)}.

xtabs() {#sec:xtabs}

The xtabs() function allows you to create cross-tabulations of data using formula style input. This typically works with case-form data supplied in a data frame or a matrix. The result is a contingency table in array format, whose dimensions are determined by the terms on the right side of the formula.

# 3-Way Frequency Table
mytable <- xtabs(~A+B+C, data=mydata)

ftable(mytable)    # print table

summary(mytable)   # chi-square test of indepedence

If a variable is included on the left side of the formula, it is assumed to be a vector of frequencies (useful if the data have already been tabulated in frequency form).

(GSStab <- xtabs(count ~ sex + party, data=GSS))

summary(GSStab)

Collapsing over table factors: aggregate(), margin.table() and apply()}

It sometimes happens that we have a data set with more variables or factors than we want to analyse, or else, having done some initial analyses, we decide that certain factors are not important, and so should be excluded from graphic displays by collapsing (summing) over them. For example, mosaic plots and fourfold displays are often simpler to construct from versions of the data collapsed over the factors which are not shown in the plots.

The appropriate tools to use again depend on the form in which the data are represented--- a case-form data frame, a frequency-form data frame (aggregate()), or a table-form array or table object (margin.table() or apply()).

When the data are in frequency form, and we want to produce another frequency data frame, aggregate() is a handy tool, using the argument FUN=sum to sum the frequency variable over the factors not mentioned in the formula.

Example: The data frame DaytonSurvey in the vcdExtra package represents a $2^5$ table giving the frequencies of reported use (``ever used?'') of alcohol, cigarettes and marijuana in a sample of high school seniors, also classified by sex and race.

str(DaytonSurvey)
head(DaytonSurvey)

To focus on the associations among the substances, we want to collapse over sex and race. The right-hand side of the formula used in the call to aggregate() gives the factors to be retained in the new frequency data frame, Dayton.ACM.df.

# data in frequency form
# collapse over sex and race
Dayton.ACM.df <- aggregate(Freq ~ cigarette+alcohol+marijuana, 
                           data=DaytonSurvey, 
                           FUN=sum)
Dayton.ACM.df

When the data are in table form, and we want to produce another table, apply() with FUN=sum can be used in a similar way to sum the table over dimensions not mentioned in the MARGIN argument. margin.table() is just a wrapper for apply() using the sum() function.

Example: To illustrate, we first convert the DaytonSurvey to a 5-way table using xtabs(), giving Dayton.tab.

# in table form
Dayton.tab <- xtabs(Freq ~ cigarette+alcohol+marijuana+sex+race, 
                    data=DaytonSurvey)
structable(cigarette+alcohol+marijuana ~ sex+race, 
           data=Dayton.tab)

Then, use apply() on Dayton.tab to give the 3-way table Dayton.ACM.tab summed over sex and race. The elements in this new table are the column sums for Dayton.tab shown by structable() just above.

# collapse over sex and race
Dayton.ACM.tab <- apply(Dayton.tab, MARGIN=1:3, FUN=sum)
Dayton.ACM.tab <- margin.table(Dayton.tab, 1:3)   # same result

structable(cigarette+alcohol ~ marijuana, data=Dayton.ACM.tab)

Many of these operations can be performed using the **ply() functions in the plyr package. For example, with the data in a frequency form data frame, use ddply() to collapse over unmentioned factors, and plyr::summarise() as the function to be applied to each piece.

library(plyr)
Dayton.ACM.df <- plyr::ddply(DaytonSurvey, 
                             .(cigarette, alcohol, marijuana), 
                             plyr::summarise, Freq=sum(Freq))

Dayton.ACM.df

Collapsing table levels: collapse.table()

A related problem arises when we have a table or array and for some purpose we want to reduce the number of levels of some factors by summing subsets of the frequencies. For example, we may have initially coded Age in 10-year intervals, and decide that, either for analysis or display purposes, we want to reduce Age to 20-year intervals. The collapse.table() function in vcdExtra was designed for this purpose.

Example: Create a 3-way table, and collapse Age from 10-year to 20-year intervals. First, we generate a $2 \times 6 \times 3$ table of random counts from a Poisson distribution with mean of 100.

# create some sample data in frequency form
sex <- c("Male", "Female")
age <- c("10-19", "20-29",  "30-39", "40-49", "50-59", "60-69")
education <- c("low", 'med', 'high')
data <- expand.grid(sex=sex, age=age, education=education)
counts <- rpois(36, 100)   # random Possion cell frequencies
data <- cbind(data, counts)

# make it into a 3-way table
t1 <- xtabs(counts ~ sex + age + education, data=data)
structable(t1)

Now collapse age to 20-year intervals, and education to 2 levels. In the arguments, levels of age and education given the same label are summed in the resulting smaller table.

# collapse age to 3 levels, education to 2 levels
t2 <- collapse.table(t1, 
         age=c("10-29", "10-29",  "30-49", "30-49", "50-69", "50-69"),
         education=c("<high", "<high", "high"))
structable(t2)

Converting among frequency tables and data frames

As we've seen, a given contingency table can be represented equivalently in different forms, but some R functions were designed for one particular representation.

The table below shows some handy tools for converting from one form to another.

| From this | | To this | | |:-----------------|:--------------------|:---------------------|-------------------| | | Case form | Frequency form | Table form | | Case form | noop | xtabs(~A+B) | table(A,B) | | Frequency form | expand.dft(X) | noop | xtabs(count~A+B)| | Table form | expand.dft(X) | as.data.frame(X) | noop |

For example, a contingency table in table form (an object of class(table)) can be converted to a data.frame with as.data.frame(). ^[Because R is object-oriented, this is actually a short-hand for the function as.data.frame.table().] The resulting data.frame contains columns representing the classifying factors and the table entries (as a column named by the responseName argument, defaulting to Freq. This is the inverse of xtabs().

Example: Convert the GSStab in table form to a data.frame in frequency form.

as.data.frame(GSStab)

Example: Convert the Arthritis data in case form to a 3-way table of Treatment $\times$ Sex $\times$ Improved. Note the use of with() to avoid having to use Arthritis\$Treatment etc. within the call to table().% ^[table() does not allow a data argument to provide an environment in which the table variables are to be found. In the examples in \@(sec:table) I used attach(mydata) for this purpose, but attach() leaves the variables in the global environment, while with() just evaluates the table() expression in a temporary environment of the data.]

Art.tab <- with(Arthritis, table(Treatment, Sex, Improved))
str(Art.tab)

ftable(Art.tab)

There may also be times that you will need an equivalent case form data.frame with factors representing the table variables rather than the frequency table. For example, the mca() function in package MASS only operates on data in this format. Marc Schwartz initially provided code for expand.dft() on the Rhelp mailing list for converting a table back into a case form data.frame. This function is included in vcdExtra.

Example: Convert the Arthritis data in table form (Art.tab) back to a data.frame in case form, with factors Treatment, Sex and Improved.

Art.df <- expand.dft(Art.tab)
str(Art.df)

A complex example {#sec:complex}

If you've followed so far, you're ready for a more complicated example. The data file, tv.dat represents a 4-way table of size $5 \times 11 \times 5 \times 3$ where the table variables (unnamed in the file) are read as V1 -- V4, and the cell frequency is read as V5. The file, stored in the doc/extdata directory of vcdExtra, can be read as follows:

tv.data<-read.table(system.file("extdata","tv.dat", package="vcdExtra"))
head(tv.data,5)

For a local file, just use read.table() in this form:

tv.data<-read.table("C:/R/data/tv.dat")

The data tv.dat came from the initial implementation of mosaic displays in R by Jay Emerson. In turn, they came from the initial development of mosaic displays [@vcd:Hartigan+Kleiner:1984] that illustrated the method with data on a large sample of TV viewers whose behavior had been recorded for the Neilson ratings. This data set contains sample television audience data from Neilsen Media Research for the week starting November 6, 1995.

The table variables are:

We are interested just the cell frequencies, and rely on the facts that the

(a) the table is complete--- there are no missing cells, so nrow(tv.data) = r nrow(tv.data); (b) the observations are ordered so that V1 varies most rapidly and V4 most slowly. From this, we can just extract the frequency column and reshape it into an array. [That would be dangerous if any observations were out of order.]

TV <- array(tv.data[,5], dim=c(5,11,5,3))                                        
dimnames(TV) <- list(c("Monday","Tuesday","Wednesday","Thursday","Friday"), 
                     c("8:00","8:15","8:30","8:45","9:00","9:15","9:30",         
                       "9:45","10:00","10:15","10:30"),                            
                     c("ABC","CBS","NBC","Fox","Other"), 
                     c("Off","Switch","Persist"))

names(dimnames(TV))<-c("Day", "Time", "Network", "State")

More generally (even if there are missing cells), we can use xtabs() (or plyr::daply()) to do the cross-tabulation, using V5 as the frequency variable. Here's how to do this same operation with xtabs():

TV <- xtabs(V5 ~ ., data=tv.data)
dimnames(TV) <- list(Day = c("Monday","Tuesday","Wednesday","Thursday","Friday"), 
                     Time = c("8:00","8:15","8:30","8:45","9:00","9:15","9:30",         
                              "9:45","10:00","10:15","10:30"),                            
                     Network = c("ABC","CBS","NBC","Fox","Other"), 
                     State = c("Off","Switch","Persist"))

# table dimensions
dim(TV)

But this 4-way table is too large and awkward to work with. Among the networks, Fox and Other occur infrequently. We can also cut it down to a 3-way table by considering only viewers who persist with the current station. ^[This relies on the fact that that indexing an array drops dimensions of length 1 by default, using the argument drop=TRUE; the result is coerced to the lowest possible dimension.]

TV2 <- TV[,,1:3,]      # keep only ABC, CBS, NBC
TV2 <- TV2[,,,3]       # keep only Persist -- now a 3 way table
structable(TV2)

Finally, for some purposes, we might want to collapse the 11 times into a smaller number. Half-hour time slots make more sense. Here, we use as.data.frame.table() to convert the table back to a data frame, levels() to re-assign the values of Time, and finally, xtabs() to give a new, collapsed frequency table.

TV.df <- as.data.frame.table(TV2)
levels(TV.df$Time) <- c(rep("8:00", 2),
                        rep("8:30", 2),
                        rep("9:00", 2), 
                        rep("9:30", 2), 
                        rep("10:00",2),
                            "10:30"
                        )

TV3 <- xtabs(Freq ~ Day + Time + Network, TV.df)

structable(Day ~ Time+Network, TV3)

We've come this far, so we might as well show a mosaic display. This is analogous to that used by @vcd:Hartigan+Kleiner:1984.

mosaic(TV3, shade = TRUE,
       labeling = labeling_border(rot_labels = c(0, 0, 0, 90)))

This mosaic displays can be read at several levels, corresponding to the successive splits of the tiles and the residual shading. Several trends are clear for viewers who persist:

From the residual shading of the tiles:

References



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vcdExtra documentation built on April 21, 2022, 5:10 p.m.