Extended Model Formulas in R: Multiple Parts and Multiple Responses

Abstract

This introduction to the R package Formula is a (slightly) modified version of Zeileis and Croissant (2010), published in the Journal of Statistical Software. Model formulas are the standard approach for specifying the variables in statistical models in the S language. Although being eminently useful in an extremely wide class of applications, they have certain limitations including being confined to single responses and not providing convenient support for processing formulas with multiple parts. The latter is relevant for models with two or more sets of variables, e.g., different equations for different model parameters (such as mean and dispersion), regressors and instruments in instrumental variable regressions, two-part models such as hurdle models, or alternative-specific and individual-specific variables in choice models among many others. The R package Formula addresses these two problems by providing a new class "Formula" (inheriting from "formula") that accepts an additional formula operator | separating multiple parts and by allowing all formula operators (including the new |) on the left-hand side to support multiple responses.

Introduction

Since publication of the seminal “white book” (Chambers and Hastie 1992) the standard approach for fitting statistical models in the S language is to apply some model-fitting function (such as lm() or glm()) to a "formula" description of the variables involved in the model and typically stored in a "data.frame". The semantics of formula processing are based on the ideas of the Wilkinson and Rogers (1973) notation which in turn was targeted at specification of analysis of variance models. Despite this emphasis on specification of terms in models with linear predictors, formula notation has always been used much more generally in S, e.g., for specifying variables in classification and regression trees, margins in contingency tables, or variables in graphical displays. In such applications, the precise meaning of a particular formula depends on the function that processes it. Typically, the standard formula processing approach would encompass extraction of the specified terms using terms(), preparation of a preprocessed data frame using model.frame(), and computation of a “design” or “regressor” matrix using model.matrix(). However, there are certain limitations to these standard formula processing tools in S that can be rather inconvenient in certain applications:

The formula notation can just be used on the right-hand side (RHS) of a formula (to the right of the ~) while it has its original arithmetic meaning on the left-hand side (LHS). This makes it difficult to specify multiple responses, especially if these are not numeric (e.g., factors). This feature would be useful for specifying multivariate outcomes of mixed types in independence tests (e.g., in the coin package, Hothorn et al. 2006, @Formula:Hothorn+Hornik+VanDeWiel:2008) or in models with multivariate responses (e.g., supported in the party package, Zeileis, Hothorn, and Hornik 2008).
There is no simple construct in standard formula notation that allows one to separate several groups of variables from which separate model matrices can be derived. This task occurs in many types of models, e.g., when processing separate sets of variables for mean and dispersion (e.g., in the betareg package, Cribari-Neto and Zeileis 2010), separate equations for location, scatter, and shape (e.g., in the gamlss package, Stasinopoulos and Rigby 2007), regressors and instruments in instrumental variable regressions (e.g., in the plm package, Croissant and Millo 2008, or the AER package, @Formula:Kleiber+Zeileis:2008), variables in two-part models such as hurdle models or zero-inflated regressions (e.g., in the pscl package, Zeileis, Kleiber, and Jackman 2008), alternative-specific and individual-specific variables in choice models (e.g., in the mlogit package, Croissant 2010), efficiency level variables in stochastic frontier analysis (e.g., in the frontier package, Coelli and Henningsen 2010), or modeling variables and partitioning variables in model-based recursive partitioning techniques (e.g., in the party package, Zeileis, Hothorn, and Hornik 2008).

In many of the aformentioned packages, standard "formula" objects are employed but their processing is generalized, e.g., by using multiple formulas, multiple terms, by adding new formula operators or even more elaborate solutions. However, in many situations it is not easy to reuse these generalizations outside the package/function they were designed for. Therefore, as we repeatedly needed such generalizations in our own packages and addressed this in the past by various different solutions, it seemed natural to provide a more general unified approach by means of our new Formula package. This has already been reused in some of our own packages (including AER, betareg, mlogit, and plm) but can also be easily employed by other package developers (e.g., as in the frontier package). More applications in our own and other packages will hopefully follow.

In the remainder of this paper we discuss how multiple responses and multiple parts (both on the LHS and RHS) are enabled in the Formula package written in the R system for statistical computing (R Core Team 2009) and available from the Comprehensive R Archive Network (CRAN) at https://CRAN.R-project.org/package=Formula. Built on top of basic "formula" objects, the "Formula" class just adds a thin additional layer which is based on a single additional operator, namely |, that can be used to separate different parts (or groups) of variables. Formula essentially just handles the different formula parts and leverages the existing methods for "formula" objects for all remaining operations. (Throughout the paper, no terminological distinction is made between classic "formula" objects as such and the way they are interpreted by standard processing tools (e.g., terms(), model.frame(), model.matrix()). The reasons for this are twofold: First, it reflects their use as described in Chambers and Hastie (1992) and, second, generalizations generally require extra effort from the user.) In the “Motivating examples” we show the main ideas implemented in the package and how easily they can be employed. The details of the "Formula" class and its associated methods are discussed in the “Implementation” section and the “Usage in model fitting functions” illustrates how Formula can be used in the development of new functions/packages. A short “Summary” concludes the paper.

Motivating examples

To illustrate the basic ideas of the Formula package, we first generate a small artificial data set (with both numeric and categorical variables) and subsequently illustrate its usage with a multi-part and a multi-response "Formula", respectively.

set.seed(1090)
dat <- as.data.frame(matrix(round(runif(21), digits = 2), ncol = 7))
colnames(dat) <- c("y1", "y2", "y3", "x1", "x2", "x3", "x4")
for(i in c(2, 6:7)) dat[[i]] <- factor(dat[[i]] < 0.5,
  labels = c("a", "b"))
dat$y2[1] <- NA
dat
##     y1   y2   y3   x1   x2 x3 x4
## 1 0.82 <NA> 0.27 0.09 0.22  a  b
## 2 0.70    a 0.17 0.26 0.46  b  b
## 3 0.65    b 0.28 0.03 0.37  a  a

Multiple parts

We start out with a simple formula log(y1) ~ x1 + x2 | I(x1^2) which has a single response log(y1) on the LHS and two parts on the RHS, separated by |. The first part contains x1 and x2, the second contains I(x1^2), i.e., the squared values of x1. The initial "formula" can be transformed to a "Formula" using the constructor function Formula():

F1 <- Formula(log(y1) ~ x1 + x2 | I(x1^2))
length(F1)
## [1] 1 2

The length() method indicates that there is one part on the LHS and two parts on the RHS. The first step of processing data using a formula is typically the construction of a so-called model frame containing only the variables required by the formula. As usual, this can be obtained with the model.frame() method.

mf1 <- model.frame(F1, data = dat)
mf1
##      log(y1)   x1   x2 I(x1^2)
## 1 -0.1984509 0.09 0.22  0.0081
## 2 -0.3566749 0.26 0.46  0.0676
## 3 -0.4307829 0.03 0.37   9e-04

As this model just has a single response (as in base "formula" objects), the extractor function model.response() can be employed:

model.response(mf1)
##          1          2          3 
## -0.1984509 -0.3566749 -0.4307829

For constructing separate model matrices for the two parts on the RHS, the model.matrix() can be employed and additionally specifying the argument rhs:

model.matrix(F1, data = mf1, rhs = 1)
##   (Intercept)   x1   x2
## 1           1 0.09 0.22
## 2           1 0.26 0.46
## 3           1 0.03 0.37
## attr(,"assign")
## [1] 0 1 2
model.matrix(F1, data = mf1, rhs = 2)
##   (Intercept) I(x1^2)
## 1           1  0.0081
## 2           1  0.0676
## 3           1  0.0009
## attr(,"assign")
## [1] 0 1

Multiple responses

To accommodate multiple responses, all formula operators can be employed on the LHS in "Formula" objects (whereas they would have their original arithmetic meaning in "formula" objects). This also includes the new | operator for separating different parts. Thus, one could specify a two-part response via y1 | y2 ~ x3 or a single part with two variables via y1 + y2 ~ x3. We do the latter in the following illustration.

F2 <- Formula(y1 + y2 ~ x3)
length(F2)
## [1] 1 1

As usual, the model frame can be derived by

mf2 <- model.frame(F2, data = dat)
mf2
##     y1 y2 x3
## 2 0.70  a  b
## 3 0.65  b  a

However, there is an important difference to the model frame mf1 derived in the previous example. As the (non-generic) model.response() function would only be able to extract a single response column from a model frame, multi-response model frames in Formula are implemented to have no response at all in the model.response() sense:

model.response(mf2)
## NULL

As model.response() cannot be extended (without overloading the base R function), Formula provides a new generic function model.part() which can be used to extract all variables from a model frame pertaining to specific parts. This can also be used to extract multiple responses. Its syntax is modeled after model.matrix() taking a "Formula" as the first argument. For further details see the “Implementation” section. Its application is straightforward and all LHS variables (in the first and only part of the LHS) can be extracted via

model.part(F2, data = mf2, lhs = 1)
##     y1 y2
## 2 0.70  a
## 3 0.65  b

The same method also works for single response models as in the previous example:

model.part(F1, data = mf1, lhs = 1, drop = TRUE)
##          1          2          3 
## -0.1984509 -0.3566749 -0.4307829

Implementation

Below we discuss the ideas for the design of the "Formula" class and methods. As all tools are built on top of the "formula" class and its associated methods whose most important feature are briefly outlined as well.

Working with classic `"formula"` objects

Classic "formula" objects (Chambers and Hastie 1992) are constructed by using ~ to separate LHS and RHS, typically (but not necessarily) interpreted as “dependent” and “explanatory” variables. For describing relationships between variables on the RHS, several operators can be used: +, -, *, /, :, %in%, ^. Thus, these do not have their original meaning in the top-level RHS while they keep their original arithmetic meaning on higher levels of the RHS (thus, within function calls such as I(x1^2)) and on the LHS in general. A first step in using "formula" objects is often to compute the associated "terms" using the function terms(). Based on the formula or the associated terms and a suitable set of variables (typically either in a data frame or in the global environment) model.frame() can build a so-called model frame that contains all variables in the formula/terms. This might include processing of missing values (NAs), carrying out variable transformations (such as logs, squares, or other functions of one or more variables) and providing further variables like weights, offset, etc. A model frame is simply a "data.frame" with additional attributes (including "terms") but without a specific class. From this preprocessed model frame several components can be extracted using model.extract() or model.response(), model.weights(), and model.offset(), all of which are non-generic. Last not least, model.matrix() (which is generic) can compute “design” or “regressor” matrices based on the formula/terms and the associated model frame.

Constructing `"Formula"` objects

To accomplish the main objectives of the "Formula" class, the following design principles have been used: reuse of "formula" objects, introduction of a single new operator |, and support of all formula operators on the LHS. Thus, | loses its original meaning (logical “or”) on the first level of formulas but can still be used with its original meaning on higher levels, e.g., factor(x1 > 0.5 | x3 == "a") still works as before. For assigning a new class to formulas containing |, the constructor function Formula() is used:

F3 <- Formula(y1 + y2 | log(y3) ~ x1 + I(x2^2) | 0 + log(x1) | x3 / x4)
F3
## y1 + y2 | log(y3) ~ x1 + I(x2^2) | 0 + log(x1) | x3/x4
length(F3)
## [1] 2 3

In this example, F3 is an artificially complex formula with two parts on the LHS and three parts on the RHS, both containing multiple terms, transformations or other formula operators. Apart from assigning the new class "Formula" (in addition to the old "formula" class), Formula() also splits up the formula into LHS and RHS parts which are stored as list attributes "lhs" and "rhs", respectively, e.g.,

attr(F3, "lhs")
## [[1]]
## y1 + y2
## 
## [[2]]
## log(y3)

and analogously attr(F3, "rhs"). The user never has to compute on these attributes directly, but many methods for "Formula" objects take lhs and/or rhs arguments. These always refer to index vectors for the two respective lists.

It would have been conceivable to generalize not only the notion of formulas but also of terms or model frames. However, there are few generics with methods for "terms" objects and there is no particular class for model frames at all. Hence, computing with generalized versions of these concepts would have required much more overhead for users of Formula. Hence, it was decided not to do so and keep the package interface as simple as possible.

Extracting `"formula"` and `"terms"` objects

As subsequent computations typically require a "formula" or a "terms" object, Formula provides suitable formula() and terms() extractors for "Formula" objects. For the former, the idea is to be able to switch back and forth between the "Formula" and "formula" representation, e.g., formula(Formula(...)) should recover the original input formula. For the latter, the objective is somewhat different: terms() should always return a "terms" object that can be processed by model.frame() and similar functions. Thus, the terms must not contain multiple responses and/or the new | operator.

The formula() method is straightforward. When no additional arguments are supplied it recovers the original "formula". Furthermore, there are two optional additional arguments lhs and rhs. With these arguments subsets of formulas can be chosen, indexing the LHS and RHS parts. The default value for both is NULL, meaning that all parts are employed.

formula(F3)
## y1 + y2 | log(y3) ~ x1 + I(x2^2) | 0 + log(x1) | x3/x4
formula(F3, lhs = 2, rhs = -2)
## log(y3) ~ x1 + I(x2^2) | x3/x4
formula(F3, lhs = c(TRUE, FALSE), rhs = 0)
## y1 + y2 ~ 0

Similarly, terms() computes a "terms" object, by default using all parts in the formula, but lhs and rhs can be used as above. To remove the | operator, all parts are collapsed using the + operator. Furthermore, the LHS variables can only be kept on the LHS if they contain a single term. Otherwise, to stop subsequent computations from interpreting the formula operators as arithmetic operators, all LHS components are added on the RHS as well. Thus, for F3 we obtain

terms(F3)
## ~y1 + y2 + log(y3) + (x1 + I(x2^2)) + (0 + log(x1)) + x3/x4
## attr(,"variables")
## list(y1, y2, log(y3), x1, I(x2^2), log(x1), x3, x4)
## attr(,"factors")
##         y1 y2 log(y3) x1 I(x2^2) log(x1) x3 x3:x4
## y1       1  0       0  0       0       0  0     0
## y2       0  1       0  0       0       0  0     0
## log(y3)  0  0       1  0       0       0  0     0
## x1       0  0       0  1       0       0  0     0
## I(x2^2)  0  0       0  0       1       0  0     0
## log(x1)  0  0       0  0       0       1  0     0
## x3       0  0       0  0       0       0  1     2
## x4       0  0       0  0       0       0  0     1
## attr(,"term.labels")
## [1] "y1"      "y2"      "log(y3)" "x1"      "I(x2^2)" "log(x1)" "x3"     
## [8] "x3:x4"  
## attr(,"order")
## [1] 1 1 1 1 1 1 1 2
## attr(,"intercept")
## [1] 0
## attr(,"response")
## [1] 0
## attr(,".Environment")
## <environment: R_GlobalEnv>

Instead of using all parts, subsets can again be selected. We illustrate this below but only show the associated "formula" to save output space:

formula(terms(F3))
## ~y1 + y2 + log(y3) + (x1 + I(x2^2)) + (0 + log(x1)) + x3/x4
formula(terms(F3, lhs = 2, rhs = -2))
## log(y3) ~ x1 + I(x2^2) + x3/x4
formula(terms(F3, lhs = c(TRUE, FALSE), rhs = 0))
## ~y1 + y2

Computing model frames, matrices, and responses

Given that suitable "terms" can be extracted from "Formula" objects, it is straightforward to set up the corresponding model frame. The model.frame() method simply first calls the terms() method and then applies the default model.frame(). Hence, all further arguments are processed as usual, e.g.,

mf3 <- model.frame(F3, data = dat, subset = y1 < 0.75, weights = x1)
mf3
##     y1 y2   log(y3)   x1 I(x2^2)   log(x1) x3 x4 (weights)
## 2 0.70  a -1.771957 0.26  0.2116 -1.347074  b  b      0.26
## 3 0.65  b -1.272966 0.03  0.1369 -3.506558  a  a      0.03

All subsequent computations are then based on this preprocessed model frame (and possibly the original "Formula"). Thus, the model matrices for each RHS part can be easily computed, again setting the rhs argument:

model.matrix(F3, data = mf3, rhs = 2)
##     log(x1)
## 2 -1.347074
## 3 -3.506558
## attr(,"assign")
## [1] 1

Typically, just a single RHS will be selected and hence rhs = 1 and not rhs = NULL is the default in this method. However, multiple RHS parts are also supported. Also, there is a lhs argument available (defaulting to NULL) which might seem unnecessary at first sight but it is important in case the selected RHS part(s) contain(s) a “.” that needs to be resolved (see ?model.matrix.Formula for an example).

The LHS parts can be extracted using the method for the new model.part() generic, employing a syntax similar to model.matrix():

model.part(F3, data = mf3, lhs = 1)
##     y1 y2
## 2 0.70  a
## 3 0.65  b
model.part(F3, data = mf3, lhs = 2)
##     log(y3)
## 2 -1.771957
## 3 -1.272966

As argued above, introduction of a new generic is necessary for supporting multi-response formulas because model.response() is non-generic. For model frames derived from single-response formulas, model.response() can be used as usual. The remaining extractors work as usual:

model.weights(mf3)
## [1] 0.26 0.03

Further methods

To conclude the suite of methods available for the new "Formula" class, Formula provides an update() method and a new as.Formula() generic with suitable methods. The former updates the formula part by part, adding new parts if necessary:

update(F1, . ~ . - x1 | . + x1)
## log(y1) ~ x2 | I(x1^2) + x1
update(F1, . + y2 | y3 ~ .)
## log(y1) + y2 | y3 ~ x1 + x2 | I(x1^2)

The as.Formula() method coerces to "Formula", possibly also processing multiple arguments:

as.Formula(y1 ~ x1, y2 ~ x2, ~ x3)
## y1 | y2 ~ x1 | x2 | x3

Usage in model fitting functions

A typical application of Formula is to provide the workhorse for formula processing in model-fitting functions that require specification of multiple parts or multiple responses. To provide a very brief and simple example, we show how such a function can be set up. For illustration, we compute the coefficients in an instrumental variables regression using two-stage least squares.

The ivcoef() function below takes the usual arguments formula, data, subset, and na.action (and further arguments weights and offset could be included in the same way). The formula should be a two-part formula like y ~ x1 + x2 | z1 + z2 + z3. There is a single response on the LHS, one RHS part with the regressors and a second RHS part with the instruments. The function ivcoef() uses the typical workflow of model-fitting functions and processes its arguments in the following four steps: (1) process the call, (2) set up the model frame (using the "Formula" method), (3) extract response and regressors from the model frame, (4) estimate the model (by calling lm.fit() twice to compute the two-stage least squares coefficients).

ivcoef <- function(formula, data, subset, na.action, ...)
{
  mf <- match.call(expand.dots = FALSE)
  m <- match(c("formula", "data", "subset", "na.action"), names(mf), 0)
  mf <- mf[c(1, m)]
  
  f <- Formula(formula)
  mf[[1]] <- as.name("model.frame")
  mf$formula <- f
  mf <- eval(mf, parent.frame())
  
  y <- model.response(mf)
  x <- model.matrix(f, data = mf, rhs = 1)
  z <- model.matrix(f, data = mf, rhs = 2)

  xz <- as.matrix(lm.fit(z, x)$fitted.values)
  lm.fit(xz, y)$coefficients
}

The resulting function can then be applied easily (albeit not very meaningfully) to the dat data frame:

ivcoef(log(y1) ~ x1 | x2, data = dat)
## (Intercept)          x1 
##   -0.169027   -1.260073

The same coefficients can be derived along with all the usual inference using the ivreg() function from the AER package (Kleiber and Zeileis 2008), which also uses the Formula tools in its latest release. Apart from providing inference and many other details, ivreg() also supports weights, offsets etc. Finally, for backward compatibility, the function also allows separate formulas for regressors and instruments (i.e., formula = y ~ x1 + x2 and instruments = ~ z1 + z2 + z3) which can be easily incorporated using the Formula tools, e.g., replacing f <- Formula(formula) by

  f <- if(!is.null(instruments)) as.Formula(formula, instruments)
    else as.Formula(formula)
  stopifnot(isTRUE(all.equal(length(f), c(1, 2))))

In summary, the usage of Formula should reduce the overhead for the developers of model-fitting functions in R with multiple responses and/or multiple parts and make the resulting programs more intelligible. Further R packages employing the "Formula" approach can be obtained from CRAN, including betareg, frontier, mlogit, and plm.

Extended Model Formulas in R: Multiple Parts and Multiple Responses

Achim Zeileis

Yves Croissant

Introduction

Motivating examples

Multiple parts

Multiple responses

Implementation

Working with classic `"formula"` objects

Constructing `"Formula"` objects

Extracting `"formula"` and `"terms"` objects

Computing model frames, matrices, and responses

Further methods

Usage in model fitting functions

Summary

References

Extended Model Formulas in R: Multiple Parts and Multiple Responses

Achim Zeileis

Yves Croissant

Introduction

Motivating examples

Multiple parts

Multiple responses

Implementation

Working with classic "formula" objects

Constructing "Formula" objects

Extracting "formula" and "terms" objects

Computing model frames, matrices, and responses

Further methods

Usage in model fitting functions

Summary

References

Working with classic `"formula"` objects

Constructing `"Formula"` objects

Extracting `"formula"` and `"terms"` objects