# Chapter 4 Ordered and unordered factors

A factor is a vector object used to specify a discrete classification (grouping) of the components of other vectors of the same length. R provides both ordered and unordered factors. While the “real” application of factors is with model formulae (see Contrasts), we here look at a specific example.

### 4.1 A specific example

Suppose, for example, we have a sample of 30 tax accountants from all the states and territories of Australia14 and their individual state of origin is specified by a character vector of state mnemonics as

> state <- c("tas", "sa",  "qld", "nsw", "nsw", "nt",  "wa",  "wa",
"qld", "vic", "nsw", "vic", "qld", "qld", "sa",  "tas",
"sa",  "nt",  "wa",  "vic", "qld", "nsw", "nsw", "wa",
"sa",  "act", "nsw", "vic", "vic", "act")

Notice that in the case of a character vector, “sorted” means sorted in alphabetical order.

A factor is similarly created using the factor() function:

> statef <- factor(state)

The print() function handles factors slightly differently from other objects:

> statef
[1] tas sa  qld nsw nsw nt  wa  wa  qld vic nsw vic qld qld sa
[16] tas sa  nt  wa  vic qld nsw nsw wa  sa  act nsw vic vic act
Levels:  act nsw nt qld sa tas vic wa

To find out the levels of a factor the function levels() can be used.

> levels(statef)
[1] "act" "nsw" "nt"  "qld" "sa"  "tas" "vic" "wa"

### 4.2 The function tapply() and ragged arrays

To continue the previous example, suppose we have the incomes of the same tax accountants in another vector (in suitably large units of money)

> incomes <- c(60, 49, 40, 61, 64, 60, 59, 54, 62, 69, 70, 42, 56,
61, 61, 61, 58, 51, 48, 65, 49, 49, 41, 48, 52, 46,
59, 46, 58, 43)

To calculate the sample mean income for each state we can now use the special function tapply():

> incmeans <- tapply(incomes, statef, mean)

giving a means vector with the components labelled by the levels

   act    nsw     nt    qld     sa    tas    vic     wa
44.500 57.333 55.500 53.600 55.000 60.500 56.000 52.250

The function tapply() is used to apply a function, here mean(), to each group of components of the first argument, here incomes, defined by the levels of the second component, here statef15, as if they were separate vector structures. The result is a structure of the same length as the levels attribute of the factor containing the results. The reader should consult the help document for more details.

Suppose further we needed to calculate the standard errors of the state income means. To do this we need to write an R function to calculate the standard error for any given vector. Since there is an builtin function var() to calculate the sample variance, such a function is a very simple one liner, specified by the assignment:

> stdError <- function(x) sqrt(var(x)/length(x))

(Writing functions will be considered later in Writing your own functions. Note that R’s a builtin function sd() is something different.) After this assignment, the standard errors are calculated by

> incster <- tapply(incomes, statef, stdError)

and the values calculated are then

> incster
act    nsw  nt    qld     sa tas   vic     wa
1.5 4.3102 4.5 4.1061 2.7386 0.5 5.244 2.6575

As an exercise you may care to find the usual 95% confidence limits for the state mean incomes. To do this you could use tapply() once more with the length() function to find the sample sizes, and the qt() function to find the percentage points of the appropriate t-distributions. (You could also investigate R’s facilities for t-tests.)

The function tapply() can also be used to handle more complicated indexing of a vector by multiple categories. For example, we might wish to split the tax accountants by both state and sex. However in this simple instance (just one factor) what happens can be thought of as follows. The values in the vector are collected into groups corresponding to the distinct entries in the factor. The function is then applied to each of these groups individually. The value is a vector of function results, labelled by the levels attribute of the factor.

The combination of a vector and a labelling factor is an example of what is sometimes called a ragged array, since the subclass sizes are possibly irregular. When the subclass sizes are all the same the indexing may be done implicitly and much more efficiently, as we see in the next section.

### 4.3 Ordered factors

The levels of factors are stored in alphabetical order, or in the order they were specified to factor if they were specified explicitly.

Sometimes the levels will have a natural ordering that we want to record and want our statistical analysis to make use of. The ordered() function creates such ordered factors but is otherwise identical to factor. For most purposes the only difference between ordered and unordered factors is that the former are printed showing the ordering of the levels, but the contrasts generated for them in fitting linear models are different.