--- title: "1. Creating and manipulating frequency tables" author: "Michael Friendly" date: "`r Sys.Date()`" package: vcdExtra output: rmarkdown::html_vignette: fig_caption: yes bibliography: ["vcd.bib", "vcdExtra.bib"] csl: apa.csl vignette: > %\VignetteIndexEntry{1. Creating and manipulating frequency tables} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- ```{r setup, include = FALSE} knitr::opts_chunk$set( collapse = TRUE, message = FALSE, warning = FALSE, fig.height = 6, fig.width = 7, # fig.path = "fig/tut01-", dev = "png", comment = "##" ) # 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. ```{r, 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$Freq)`, `sum(X[,"Freq"])` or some equivalent form. The number of cells in the table is given by `nrow(X)`. ***Example***: For small frequency tables, it is often convenient to enter them in frequency form using `expand.grid()` for the factors and `c()` to list the counts in a vector. The example below, from [@vcd:Agresti:2002] gives results for the 1991 General Social Survey, with respondents classified by sex and party identification. ```{r, frequency-form} # Agresti (2002), table 3.11, p. 106 GSS <- data.frame( expand.grid(sex = c("female", "male"), party = c("dem", "indep", "rep")), count = c(279,165,73,47,225,191)) GSS names(GSS) str(GSS) sum(GSS$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`. ```{r, table-form1} 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`. ```{r, table-form2} # 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()`, ```{r, table-form3} 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 times 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. ```{r, relevel, eval=FALSE} 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. ```{r, reorder1} 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 (@ref(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. ```{r, reorder2, echo=TRUE, eval=FALSE} 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` (@ref(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()`. ```{r} #| 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]") ``` Several data sets in the package illustrate the salutary effects of reordering factor levels in mosaic displays and other analyses. See: * `help(AirCrash)` * `help(Glass)` * `help(HouseTasks)` The [seriate](https://CRAN.R-project.org/package=seriation) package now contains a general method to permute the row and column variables in a table according to the result of a correspondence analysis, using scores on the first CA dimension. ### Re-ordering dimensions 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 \@ref(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 \@ref(sec:tips) for options `labeling_args` and `set_labels`to change variable and level names for displays in the `strucplot` framework.] ```{r, reorder3} 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. ```{r, structable} 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). ```{r, structable1,eval=FALSE} 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: ```{r, table-setup} 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: ```{r, table-ex1} # 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. ```{r, table-ex2} # 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. ```{r, xtabs-ex1} # 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). ```{r, xtabs-ex2} (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. ```{r, dayton1} data("DaytonSurvey", package="vcdExtra") 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`. ```{r, dayton2} # 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`. ```{r, dayton3} # 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. ```{r, dayton4} # 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`]( https://CRAN.R-project.org/package=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. ```{r, dayton5} 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. ```{r, collapse1} # 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. ```{r, collapse2} # 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(" mutate(sibspF = case_match(sibsp, 0 ~ "0", 1 ~ "1", 2:max(sibsp) ~ "2+")) |> mutate(sibspF = ordered(sibspF)) |> mutate(parchF = case_match(parch, 0 ~ "0", 1 ~ "1", 2:max(parch) ~ "2+")) |> mutate(parchF = ordered(parchF)) table(Titanicp$sibspF, Titanicp$parchF) ``` `car::recode()` is a similar function, but with a less convenient interface. The [`forcats`]( https://CRAN.R-project.org/package=forcats) package provides a collection of functions for reordering the levels of a factor or grouping categories according to their frequency: * `forcats::fct_reorder()`: Reorder a factor by another variable. * `forcats::fct_infreq()`: Reorder a factor by the frequency of values. * `forcats::fct_relevel()`: Change the order of a factor by hand. * `forcats::fct_lump()`: Collapse the least/most frequent values of a factor into “other”. * `forcats::fct_collapse()`: Collapse factor levels into manually defined groups. * `forcats::fct_recode()`: Change factor levels by hand. ## 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. ```{r, convert-ex1} 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 \@ref(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.] ```{r, convert-ex2} 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`. ```{r, convert-ex3} 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: ```{r, tv1} 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: ```{r, tv2,eval=FALSE} 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 Neilsen 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 (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.] ```{r, tv3} 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()`: ```{r, tv3a,eval=FALSE} 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.] ```{r, tv4} 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. ```{r, tv5} 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. ```{r tv-mosaic1, fig.height=6, fig.width=7} 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