Factors and Categorical Data

R has a special representation for categorical variables: factors. A factor stores observed category labels together with a fixed set of possible levels. This is more than a display convenience. Statistical models need to know which variables are categorical so they can create indicator columns, choose reference groups, and report coefficients in terms of category differences.

The book discusses factors as part of non-numeric values because categories sit between text and numbers. The labels are human-readable strings, but internally R stores integer codes with a levels attribute. That internal coding is useful, but it also creates classic mistakes when people treat factor codes as the original data. Good factor work means controlling levels deliberately and converting carefully.

Definitions

A factor is an integer vector with a levels attribute and class "factor". It represents nominal categories such as "low", "medium", and "high" or "control" and "treated".

An ordered factor is a factor whose levels have a meaningful order. Shirt sizes, severity grades, and survey responses such as "strongly disagree" through "strongly agree" are typical examples.

The levels of a factor are the allowed categories. levels(x) returns them, and factor(x, levels = ...) sets them explicitly.

The reference level is the baseline category used by many model functions. In treatment contrasts for lm, coefficients for other factor levels are interpreted relative to the reference level. relevel() changes it.

The function cut() converts numeric values into categorical intervals. This is useful for grouping ages, measurement bands, or probability bins, but the interval boundaries must be chosen carefully.

Key results

Factors are not just strings. A character vector records labels; a factor records labels plus a declared category set. That matters when a category is possible but absent in a subset. For example, a plot or table can preserve an empty level if the factor levels are already declared.

Task	Function	Example
Create nominal factor	factor(x)	factor(c("A", "B", "A"))
Set level order	factor(x, levels = ...)	factor(x, levels = c("low", "med", "high"))
Create ordered factor	ordered(x, levels = ...)	ordered(rating, levels = 1:5)
Inspect levels	levels(x)	levels(group)
Count categories	table(x)	table(iris$Species)
Drop unused levels	droplevels(x)	droplevels(subset_group)
Change reference	relevel(x, ref = "control")	relevel(group, "control")
Bin numeric values	cut(x, breaks = ...)	cut(age, c(0, 18, 65, Inf))

The safest factor conversion rule is: if a factor must become numeric values represented by its labels, convert through character first:

as.numeric(as.character(factor_numbers))

Using as.numeric(factor_numbers) returns the internal integer codes, not the label values.

Factors influence model matrices. For a factor with $k$ levels, a standard regression model usually creates $k - 1$ indicator columns when an intercept is present. Each coefficient compares one non-reference level with the reference level, holding other predictors constant.

Visual

Labels:       control treated control high
Levels:       control treated high
Codes:        1       2       1       3

The codes are storage. The labels and levels are the meaning.

Worked example 1: Controlling level order in summaries

Problem: survey responses are recorded as "medium", "low", "high", "medium", "low". Create an ordered factor with levels low, medium, high, then produce a table in that order.

Method:

1. Store the raw character responses.
2. Create an ordered factor with explicit levels.
3. Check the levels.
4. Tabulate.
5. Verify the counts manually.

response <- c("medium", "low", "high", "medium", "low")
rating <- ordered(response, levels = c("low", "medium", "high"))

levels(rating)
# [1] "low"    "medium" "high"

table(rating)
# rating
#    low medium   high
#      2      2      1

Checked answer: "low" appears twice, "medium" appears twice, and "high" appears once. The table prints in the intended conceptual order, not alphabetical order or first-seen order.

This matters in plots and reports. An ordered factor lets axes, legends, and summaries follow the scale's meaning rather than an arbitrary ordering.

Worked example 2: Reference levels in a linear model

Problem: in a small experiment, compare plant growth across "control" and "fertilizer" groups. Fit a model where control is the reference group, and interpret the coefficient.

Method:

1. Build a data frame with group and growth.
2. Convert group to a factor with "control" first.
3. Fit lm(growth ~ group, data = plants).
4. Compare model coefficients with manual group means.
5. Interpret the treatment coefficient.

plants <- data.frame(
  group = c("control", "control", "fertilizer", "fertilizer"),
  growth = c(4.8, 5.1, 6.0, 6.4)
)

plants$group \lt - factor(plants$group, levels = c("control", "fertilizer"))
fit <- lm(growth ~ group, data = plants)
coef(fit)
#      (Intercept) groupfertilizer
#             4.95            1.25

tapply(plants$growth, plants$group, mean)
#    control fertilizer
#       4.95       6.20

Checked answer: the control mean is (4.8 + 5.1) / 2 = 4.95. The fertilizer mean is (6.0 + 6.4) / 2 = 6.20. The difference is 6.20 - 4.95 = 1.25, matching the groupfertilizer coefficient. The intercept is the reference group mean.

The factor level order controls the interpretation. If "fertilizer" were the reference, the intercept and sign of the group coefficient would change, even though fitted values would remain the same.

Code

# Factor workflow with iris: order species by average petal length.

species_mean <- tapply(iris$Petal.Length, iris$Species, mean)
ordered_species <- names(sort(species_mean))

iris2 <- iris
iris2$Species \lt - factor(iris2$Species, levels = ordered_species)

print(species_mean)
print(levels(iris2$Species))
print(table(iris2$Species))

boxplot(
  Petal.Length ~ Species,
  data = iris2,
  xlab = "Species ordered by mean petal length",
  ylab = "Petal length",
  main = "Iris petal length by species"
)

This example orders a factor by a statistic rather than by spelling. That is often the right choice for exploratory plots because the visual order now carries information: species with smaller average petal length appear first, and species with larger average petal length appear later. The data values did not change; only the level order changed. This distinction matters because reordering a factor affects tables, model reference levels, legends, and axis order, but it does not alter the underlying observations.

When categories enter a model, document the reference level before interpreting coefficients. In a treatment comparison, the reference level is often the control group. In an ordered severity scale, the lowest severity is often the baseline. If the reference is arbitrary, use relevel() or explicit levels = ... to make the choice visible. A coefficient sign can flip when the reference changes even though fitted values and group means are unchanged.

Factors are also useful for preserving empty categories. If a subset contains no observations from one level, a factor can still remember that the level exists. That is helpful for consistent plots across subsets, but it can surprise summaries. Use droplevels() when the subset should forget unused categories, and avoid it when consistent category structure is needed.

A good factor audit has three lines: class(x), levels(x), and table(x, useNA = "ifany"). The class confirms that R will treat the variable categorically. The levels show order and reference candidates. The table shows observed counts and missingness. If any of those three checks is surprising, fix the factor before plotting or modeling.

For ordered factors, be precise about whether arithmetic is meaningful. The levels "low", "medium", and "high" have order, but the distance from low to medium may not equal the distance from medium to high. R can compare ordered factors, but that does not automatically make them interval-scale measurements. Choose summaries and models that match the scientific meaning of the scale.

When a factor appears in several related analyses, define its levels once near the cleaning step rather than redefining them inside each plot or model. That keeps tables, plots, and coefficient interpretations consistent. It also makes intentional changes, such as switching the reference group, easy to find in the script.

For review, compare a character vector and a factor side by side with str(). The printed labels can look similar, but the factor carries levels and class information. That extra structure is what modeling and plotting functions use to treat the variable as categorical.

If the order of categories would change the story in a table or plot, set it explicitly rather than accepting a default.

Common pitfalls

• Treating as.numeric(factor_x) as if it returns numeric labels. It returns internal codes.
• Letting R choose alphabetical levels when a conceptual order is needed.
• Forgetting to drop unused levels after subsetting, which can leave empty categories in tables and plots.
• Changing the reference level without updating model interpretation.
• Converting every character column to factor automatically. Modern R no longer does this by default in data.frame, and many text identifiers should remain character.
• Cutting continuous variables into categories without a reason. Binning can throw away information and create arbitrary thresholds.

Connections

• Vectors, arithmetic, and comparison
• Lists and data frames
• Statistical inference
• Linear and generalized models