Creating and manipulating frequency tables
Michael Friendly
2022-04-21
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.
Ordered factors and reordered tables
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)
## 'table' num [1:4, 1:4, 1:2] 32 53 10 3 10 25 7 5 3 15 ...
## - attr(*, "dimnames")=List of 3
## ..$ Hair: chr [1:4] "Black" "Brown" "Red" "Blond"
## ..$ Eye : chr [1:4] "Brown" "Hazel" "Green" "Blue"
## ..$ Sex : chr [1:2] "Male" "Female"
This is also the order for both hair color and eye color shown in the result of a correspondence analysis () 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 () 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().
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.
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)
## Department A B C D E F
## Sex Admit?
## Male Yes 512 353 120 138 53 22
## No 313 207 205 279 138 351
## Female Yes 89 17 202 131 94 24
## No 19 8 391 244 299 317
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
## Eye Brown Hazel Green Blue
## Hair Sex
## Black Male 32 10 3 11
## Female 36 5 2 9
## Brown Male 53 25 15 50
## Female 66 29 14 34
## Red Male 10 7 7 10
## Female 16 7 7 7
## Blond Male 3 5 8 30
## Female 4 5 8 64
structable(Hair+Sex ~ Eye, HairEyeColor) # specify col ~ row variables
## Hair Black Brown Red Blond
## Sex Male Female Male Female Male Female Male Female
## Eye
## Brown 32 36 53 66 10 16 3 4
## Hazel 10 5 25 29 7 7 5 5
## Green 3 2 15 14 7 7 8 8
## Blue 11 9 50 34 10 7 30 64
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
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)
## The following objects are masked _by_ .GlobalEnv:
##
## A, B, C
mytable <- table(A,B) # A will be rows, B will be columns
mytable # print table
## B
## A b1 b2
## a1 116 114
## a2 138 132
margin.table(mytable, 1) # A frequencies (summed over B)
## A
## a1 a2
## 230 270
margin.table(mytable, 2) # B frequencies (summed over A)
## B
## b1 b2
## 254 246
prop.table(mytable) # cell percentages
## B
## A b1 b2
## a1 0.232 0.228
## a2 0.276 0.264
prop.table(mytable, 1) # row percentages
## B
## A b1 b2
## a1 0.5043478 0.4956522
## a2 0.5111111 0.4888889
prop.table(mytable, 2) # column percentages
## B
## A b1 b2
## a1 0.4566929 0.4634146
## a2 0.5433071 0.5365854
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)
## C c1 c2
## A B
## a1 b1 45 71
## b2 59 55
## a2 b1 62 76
## b2 76 56
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 .
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
## C c1 c2
## A B
## a1 b1 45 71
## b2 59 55
## a2 b1 62 76
## b2 76 56
summary(mytable) # chi-square test of indepedence
## Call: xtabs(formula = ~A + B + C, data = mydata)
## Number of cases in table: 500
## Number of factors: 3
## Test for independence of all factors:
## Chisq = 9.888, df = 4, p-value = 0.04235
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))
## party
## sex dem indep rep
## female 279 73 225
## male 165 47 191
summary(GSStab)
## Call: xtabs(formula = count ~ sex + party, data = GSS)
## Number of cases in table: 980
## Number of factors: 2
## Test for independence of all factors:
## Chisq = 7.01, df = 2, p-value = 0.03005
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)
## 'data.frame': 32 obs. of 6 variables:
## $ cigarette: Factor w/ 2 levels "Yes","No": 1 2 1 2 1 2 1 2 1 2 ...
## $ alcohol : Factor w/ 2 levels "Yes","No": 1 1 2 2 1 1 2 2 1 1 ...
## $ marijuana: Factor w/ 2 levels "Yes","No": 1 1 1 1 2 2 2 2 1 1 ...
## $ sex : Factor w/ 2 levels "female","male": 1 1 1 1 1 1 1 1 2 2 ...
## $ race : Factor w/ 2 levels "white","other": 1 1 1 1 1 1 1 1 1 1 ...
## $ Freq : num 405 13 1 1 268 218 17 117 453 28 ...
head(DaytonSurvey)
## cigarette alcohol marijuana sex race Freq
## 1 Yes Yes Yes female white 405
## 2 No Yes Yes female white 13
## 3 Yes No Yes female white 1
## 4 No No Yes female white 1
## 5 Yes Yes No female white 268
## 6 No Yes No female white 218
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
## cigarette alcohol marijuana Freq
## 1 Yes Yes Yes 911
## 2 No Yes Yes 44
## 3 Yes No Yes 3
## 4 No No Yes 2
## 5 Yes Yes No 538
## 6 No Yes No 456
## 7 Yes No No 43
## 8 No No No 279
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)
## cigarette Yes No
## alcohol Yes No Yes No
## marijuana Yes No Yes No Yes No Yes No
## sex race
## female white 405 268 1 17 13 218 1 117
## other 23 23 0 1 2 19 0 12
## male white 453 228 1 17 28 201 1 133
## other 30 19 1 8 1 18 0 17
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)
## cigarette Yes No
## alcohol Yes No Yes No
## marijuana
## Yes 911 3 44 2
## No 538 43 456 279
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)
## age 10-19 20-29 30-39 40-49 50-59 60-69
## sex education
## Male low 98 105 104 90 90 101
## med 97 105 101 88 97 107
## high 99 101 109 88 99 96
## Female low 102 117 101 105 85 88
## med 106 84 92 116 110 96
## high 106 96 121 91 107 102
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)
## age 10-29 30-49 50-69
## sex education
## Male <high 405 383 395
## high 200 197 195
## Female <high 409 414 379
## high 202 212 209
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.
|
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(). 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)
## sex party Freq
## 1 female dem 279
## 2 male dem 165
## 3 female indep 73
## 4 male indep 47
## 5 female rep 225
## 6 male rep 191
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().%
Art.tab <- with(Arthritis, table(Treatment, Sex, Improved))
str(Art.tab)
## 'table' int [1:2, 1:2, 1:3] 19 6 10 7 7 5 0 2 6 16 ...
## - attr(*, "dimnames")=List of 3
## ..$ Treatment: chr [1:2] "Placebo" "Treated"
## ..$ Sex : chr [1:2] "Female" "Male"
## ..$ Improved : chr [1:3] "None" "Some" "Marked"
ftable(Art.tab)
## Improved None Some Marked
## Treatment Sex
## Placebo Female 19 7 6
## Male 10 0 1
## Treated Female 6 5 16
## Male 7 2 5
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)
## 'data.frame': 84 obs. of 3 variables:
## $ Treatment: chr "Placebo" "Placebo" "Placebo" "Placebo" ...
## $ Sex : chr "Female" "Female" "Female" "Female" ...
## $ Improved : chr "None" "None" "None" "None" ...
A complex example
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)
## V1 V2 V3 V4 V5
## 1 1 1 1 1 6
## 2 2 1 1 1 18
## 3 3 1 1 1 6
## 4 4 1 1 1 2
## 5 5 1 1 1 11
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 (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:
V1– values 1:5 correspond to the days Monday–Friday;V2– values 1:11 correspond to the quarter hour times 8:00PM through 10:30PM;V3– values 1:5 correspond to ABC, CBS, NBC, Fox, and non-network choices;V4– values 1:3 correspond to transition states: turn the television Off, Switch channels, or Persist in viewing the current channel.
We are interested just the cell frequencies, and rely on the facts that the
- the table is complete— there are no missing cells, so
nrow(tv.data) = 825;
- 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.
TV2 <- TV[,,1:3,] # keep only ABC, CBS, NBC
TV2 <- TV2[,,,3] # keep only Persist -- now a 3 way table
structable(TV2)
## Time 8:00 8:15 8:30 8:45 9:00 9:15 9:30 9:45 10:00 10:15 10:30
## Day Network
## Monday ABC 146 151 156 83 325 350 386 340 352 280 278
## CBS 337 293 304 233 311 251 241 164 252 265 272
## NBC 263 219 236 140 226 235 239 246 279 263 283
## Tuesday ABC 244 181 231 205 385 283 345 192 329 351 364
## CBS 173 180 184 109 218 235 256 250 274 263 261
## NBC 315 254 280 241 370 214 195 111 188 190 210
## Wednesday ABC 233 161 194 156 339 264 279 140 237 228 203
## CBS 158 126 207 59 98 103 122 86 109 105 110
## NBC 134 146 166 66 194 230 264 143 274 289 306
## Thursday ABC 174 183 197 181 187 198 211 86 110 122 117
## CBS 196 185 195 104 106 116 116 47 102 84 84
## NBC 515 463 472 477 590 473 446 349 649 705 747
## Friday ABC 294 281 305 239 278 246 245 138 246 232 233
## CBS 130 144 154 81 129 153 136 126 138 136 152
## NBC 195 220 248 160 172 164 169 85 183 198 204
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)
## Day Monday Tuesday Wednesday Thursday Friday
## Time Network
## 8:00 ABC 297 425 394 357 575
## CBS 630 353 284 381 274
## NBC 482 569 280 978 415
## 8:30 ABC 239 436 350 378 544
## CBS 537 293 266 299 235
## NBC 376 521 232 949 408
## 9:00 ABC 675 668 603 385 524
## CBS 562 453 201 222 282
## NBC 461 584 424 1063 336
## 9:30 ABC 726 537 419 297 383
## CBS 405 506 208 163 262
## NBC 485 306 407 795 254
## 10:00 ABC 632 680 465 232 478
## CBS 517 537 214 186 274
## NBC 542 378 563 1354 381
## 10:30 ABC 278 364 203 117 233
## CBS 272 261 110 84 152
## NBC 283 210 306 747 204
We’ve come this far, so we might as well show a mosaic display. This is analogous to that used by 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:
- Overall, there are about the same number of viewers on each weekday, with slightly more on Thursday.
- Looking at time slots, viewership is slightly greater from 9:00 - 10:00 overall and also 8:00 - 9:00 on Thursday and Friday
From the residual shading of the tiles:
- Monday: CBS dominates in all time slots.
- Tuesday” ABC and CBS dominate after 9:00
- Thursday: is a largely NBC day
- Friday: ABC dominates in the early evening
References
Agresti, A. (2002). Categorical data analysis (2nd ed.). Hoboken, New Jersey: John Wiley & Sons.
Hartigan, J. A., & Kleiner, B. (1984). A mosaic of television ratings. The American Statistician, 38, 32–35.
