# Extended Model Formulas in R: Multiple Parts and Multiple Responses

#### Achim Zeileis

#### Yves Croissant

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 Hothorn et al. (2008) or in models with multivariate responses (e.g., supported in the party package, Zeileis, Hothorn, et al. 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 Kleiber and Zeileis (2008), variables in two-part models such as hurdle models or zero-inflated regressions (e.g., in the pscl package, Zeileis, Kleiber, et al. 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, et al. 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 (`NA`s), 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.

## Summary

The Formula package provides tools for processing multi-response and multi-part formulas in the R system for statistical computing. The new class `"Formula"` inherits from the existing `"formula"` class, only adds a single new formula operator `|`, and enables the usage of formula operators on the left-hand side of formulas. The methods provided for `"Formula"` objects are as similar as possible to the classic methods, facilitating their usage in model-fitting functions that require support for multiple responses and/or multiple parts of variables.

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