Package 'emmeans'

Estimated marginal means (aka Least-squares means)

Description

This package provides methods for obtaining estimated marginal means (EMMs, also known as least-squares means) for factor combinations in a variety of models. Supported models include [generalized linear] models, models for counts, multivariate, multinomial and ordinal responses, survival models, GEEs, and Bayesian models. For the latter, posterior samples of EMMs are provided. The package can compute contrasts or linear combinations of these marginal means with various multiplicity adjustments. One can also estimate and contrast slopes of trend lines. Some graphical displays of these results are provided.

Overview

Vignettes

A number of vignettes are provided to help the user get acquainted with the emmeans package and see some examples.

Concept

Estimated marginal means (see Searle et al. 1980 are popular for summarizing linear models that include factors. For balanced experimental designs, they are just the marginal means. For unbalanced data, they in essence estimate the marginal means you would have observed that the data arisen from a balanced experiment. Earlier developments regarding these techniques were developed in a least-squares context and are sometimes referred to as “least-squares means”. Since its early development, the concept has expanded far beyond least-squares settings.

Reference grids

The implementation in emmeans relies on our own concept of a reference grid, which is an array of factor and predictor levels. Predictions are made on this grid, and estimated marginal means (or EMMs) are defined as averages of these predictions over zero or more dimensions of the grid. The function ref_grid explicitly creates a reference grid that can subsequently be used to obtain least-squares means. The object returned by ref_grid is of class "emmGrid", the same class as is used for estimated marginal means (see below).

Our reference-grid framework expands slightly upon Searle et al.'s definitions of EMMs, in that it is possible to include multiple levels of covariates in the grid.

Models supported

As is mentioned in the package description, many types of models are supported by the package. See vignette("models", "emmeans") for full details. Some models may require other packages be installed in order to access all of the available features. For models not explicitly supported, it may still be possible to do basic post hoc analyses of them via the qdrg function.

Estimated marginal means

The emmeans function computes EMMs given a fitted model (or a previously constructed emmGrid object), using a specification indicating what factors to include. The emtrends function creates the same sort of results for estimating and comparing slopes of fitted lines. Both return an emmGrid object.

Summaries and analysis

The summary.emmGrid method may be used to display an emmGrid object. Special-purpose summaries are available via confint.emmGrid and test.emmGrid, the latter of which can also do a joint test of several estimates. The user may specify by variables, multiplicity-adjustment methods, confidence levels, etc., and if a transformation or link function is involved, may reverse-transform the results to the response scale.

Contrasts and comparisons

The contrast method for emmGrid objects is used to obtain contrasts among the estimates; several standard contrast families are available such as deviations from the mean, polynomial contrasts, and comparisons with one or more controls. Another emmGrid object is returned, which can be summarized or further analyzed. For convenience, a pairs.emmGrid method is provided for the case of pairwise comparisons.

Graphs

The plot.emmGrid method will display side-by-side confidence intervals for the estimates, and/or “comparison arrows” whereby the *P* values of pairwise differences can be observed by how much the arrows overlap. The emmip function displays estimates like an interaction plot, multi-paneled if there are by variables. These graphics capabilities require the lattice package be installed.

MCMC support

When a model is fitted using MCMC methods, the posterior chains(s) of parameter estimates are retained and converted into posterior samples of EMMs or contrasts thereof. These may then be summarized or plotted like any other MCMC results, using tools in, say coda or bayesplot.

multcomp interface

The as.glht function and glht method for emmGrids provide an interface to the glht function in the multcomp package, thus providing for more exacting simultaneous estimation or testing. The package also provides an emm function that works as an alternative to mcp in a call to glht.

---

Convert to and from emmGrid objects

Description

These are useful utility functions for creating a compact version of an emmGrid object that may be saved and later reconstructed, or for converting old ref.grid or lsmobj objects into emmGrid objects.

Usage

## S3 method for class 'emmGrid'
as.list(x, model.info.slot = FALSE, ...)

as.emm_list(object, ...)

as.emmGrid(object, ...)

Arguments

x	An emmGrid object
model.info.slot	Logical value: Include the model.info slot? Set this to TRUE if you want to preserve the original call and information needed by the submodel option. If FALSE, only the nesting information (if any) is saved
...	In as.emmGrid, additional arguments passed to update.emmGrid before returning the object. This argument is ignored in as.list.emmGrid
object	Object to be converted to class emmGrid. It may be a list returned by as.list.emmGrid, or a ref.grid or lsmobj object created by emmeans's predecessor, the lsmeans package. An error is thrown if object cannot be converted.

Details

An emmGrid object is an S4 object, and as such cannot be saved in a text format or saved without a lot of overhead. By using as.list, the essential parts of the object are converted to a list format that can be easily and compactly saved for use, say, in another session or by another user. Providing this list as the arguments for emmobj allows the user to restore a working emmGrid object.

Value

as.list.emmGrid returns an object of class list.

as.emm_list returns an object of class emm_list.

as.emmGrid returns an object of class emmGrid. However, in fact, both as.emmGrid and as.emm_list check for an attribute in object to decide whether to return an emmGrid or emm_list) object.

See Also

emmobj

Examples

pigs.lm <- lm(log(conc) ~ source + factor(percent), data = pigs)
pigs.sav <- as.list(ref_grid(pigs.lm))

pigs.anew <- as.emmGrid(pigs.sav)
emmeans(pigs.anew, "source")

---

Support for MCMC-based estimation

Description

When a model is fitted using Markov chain Monte Carlo (MCMC) methods, its reference grid contains a post.beta slot. These functions transform those posterior samples to posterior samples of EMMs or related contrasts. They can then be summarized or plotted using, e.g., functions in the coda package.

Usage

## S3 method for class 'emmGrid'
as.mcmc(x, names = TRUE, sep.chains = TRUE, likelihood,
  NE.include = FALSE, ...)

## S3 method for class 'emm_list'
as.mcmc(x, which = 1, ...)

## S3 method for class 'emmGrid'
as.mcmc.list(x, names = TRUE, ...)

## S3 method for class 'emm_list'
as.mcmc.list(x, which = 1, ...)

Arguments

x	An object of class emmGrid
names	Logical scalar or vector specifying whether variable names are appended to levels in the column labels for the as.mcmc or as.mcmc.list result – e.g., column names of treat A and treat B versus just A and B. When there is more than one variable involved, the elements of names are used cyclically.
sep.chains	Logical value. If TRUE, and there is more than one MCMC chain available, an mcmc.list object is returned by as.mcmc, with separate EMMs posteriors in each chain.
likelihood	Character value or function. If given, simulations are made from the corresponding posterior predictive distribution. If not given, we obtain the posterior distribution of the parameters in object. See Prediction section below.
NE.include	Logical value. If TRUE, non-estimable columns are kept but returned as columns of NA values (this may create errors or warnings in subsequent analyses using, say, coda). If FALSE, non-estimable columns are dropped, and a warning is issued. (If all are non-estimable, an error is thrown.)
...	arguments passed to other methods
which	item in the emm_list to use

Value

An object of class mcmc or mcmc.list.

Details

When the object's post.beta slot is non-trivial, as.mcmc will return an mcmc or mcmc.list object that can be summarized or plotted using methods in the coda package. In these functions, post.beta is transformed by post-multiplying it by t(linfct), creating a sample from the posterior distribution of LS means. In as.mcmc, if sep.chains is TRUE and there is in fact more than one chain, an mcmc.list is returned with each chain's results. The as.mcmc.list method is guaranteed to return an mcmc.list, even if it comprises just one chain.

Prediction

When likelihood is specified, it is used to simulate values from the posterior predictive distribution corresponding to the given likelihood and the posterior distribution of parameter values. Denote the likelihood function as $f(y|\theta,\phi)$, where $y$ is a response, $\theta$ is the parameter estimated in object, and $\phi$ comprises zero or more additional parameters to be specified. If likelihood is a function, that function should take as its first argument a vector of $\theta$ values (each corresponding to one row of object@grid). Any $\phi$ values should be specified as additional named function arguments, and passed to likelihood via .... This function should simulate values of $y$.

A few standard likelihoods are available by specifying likelihood as a character value. They are:

"normal"

The normal distribution with mean $\theta$ and standard deviation specified by additional argument sigma

"binomial"

The binomial distribution with success probability $theta$, and number of trials specified by trials

"poisson"

The Poisson distribution with mean $theta$ (no additional parameters)

"gamma"

The gamma distribution with scale parameter $\theta$ and shape parameter specified by shape

Examples

if(requireNamespace("coda")) 
    emm_example("as.mcmc-coda")
    # Use emm_example("as.mcmc-coda", list = TRUE) # to see just the code

---

Auto Pollution Filter Noise

Description

Three-factor experiment comparing pollution-filter noise for two filters, three sizes of cars, and two sides of the car.

Usage

auto.noise

Format

A data frame with 36 observations on the following 4 variables.

noise

Noise level in decibels (but see note) - a numeric vector.

size

The size of the vehicle - an ordered factor with levels S, M, L.

type

Type of anti-pollution filter - a factor with levels Std and Octel

side

The side of the car where measurement was taken – a factor with levels L and R.

Details

The data are from a statement by Texaco, Inc., to the Air and Water Pollution Subcommittee of the Senate Public Works Committee on June 26, 1973. Mr. John McKinley, President of Texaco, cited an automobile filter developed by Associated Octel Company as effective in reducing pollution. However, questions had been raised about the effects of filters on vehicle performance, fuel consumption, exhaust gas back pressure, and silencing. On the last question, he referred to the data included here as evidence that the silencing properties of the Octel filter were at least equal to those of standard silencers.

Note

While the data source claims that noise is measured in decibels, the values are implausible. I believe that these measurements are actually in tenths of dB (centibels?). Looking at the values in the dataset, note that every measurement ends in 0 or 5, and it is reasonable to believe that measurements are accurate to the nearest half of a decibel.

Source

The dataset was obtained from the Data and Story Library (DASL) at Carnegie-Mellon University. Apparently it has since been removed. The original dataset was altered by assigning meaningful names to the factors and sorting the observations in random order as if this were the run order of the experiment.

Examples

# (Based on belief that noise/10 is in decibel units)
noise.lm <- lm(noise/10 ~ size * type * side, data = auto.noise)

# Interaction plot of predictions
emmip(noise.lm, type ~ size | side)

# Confidence intervals
plot(emmeans(noise.lm, ~ size | side*type))

---

Compact letter displays

Description

A method for multcomp::cld() is provided for users desiring to produce compact-letter displays (CLDs). This method uses the Piepho (2004) algorithm (as implemented in the multcompView package) to generate a compact letter display of all pairwise comparisons of estimated marginal means. The function obtains (possibly adjusted) P values for all pairwise comparisons of means, using the contrast function with method = "pairwise". When a P value exceeds alpha, then the two means have at least one letter in common.

Usage

## S3 method for class 'emmGrid'
cld(object, details = FALSE, sort = TRUE, by,
  alpha = 0.05, Letters = c("1234567890", LETTERS, letters),
  reversed = decreasing, decreasing = FALSE, signif.sets = FALSE,
  delta = 0, ...)

## S3 method for class 'emm_list'
cld(object, ..., which = 1)

Arguments

object	An object of class emmGrid
details	Logical value determining whether detailed information on tests of pairwise comparisons is displayed
sort	Logical value determining whether the EMMs are sorted before the comparisons are produced. When TRUE, the results are displayed according to reversed.
by	Character value giving the name or names of variables by which separate families of comparisons are tested. If NULL, all means are compared. If missing, the object's by.vars setting, if any, is used.
alpha	Numeric value giving the significance level for the comparisons
Letters	Character vector of letters to use in the display. Any strings of length greater than 1 are expanded into individual characters
reversed, decreasing	Logical value (passed to multcompView::multcompLetters.) If TRUE, the order of use of the letters is reversed. Either reversed or decreasing may be specified, thus providing compatibility with both multcompView::multcompLetters(..., reversed, ...) and multcomp::cld(..., decreasing, ...). In addition, if both sort and reversed are TRUE, the sort order of results is reversed.
signif.sets	Logical value. If FALSE (and delta = 0), a ‘traditional’ compact-letter display is constructed with groupings representing sets of estimates that are not statistically different. If TRUE, the criteria are reversed so that two estimates sharing the same symbol test as significantly different. See also delta.
delta	Numeric value passed to test.emmGrid. If this is positive, it is used as an equivalence threshold in the TOST procedure for two-sided equivalence testing. In the resulting compact letter display, two estimates share the same grouping letter only if they are found to be statistically equivalent – that is, groupings reflect actual findings of equivalence rather than failure to find a significant difference. When delta is nonzero, signif.sets is ignored.
...	Arguments passed to contrast (for example, an adjust method)
which	Which element of the emm_list object to process (If length exceeds one, only the first one is used)

Value

A summary_emm object showing the estimated marginal means plus an additional column labeled .group (when signif.sets = FALSE), .signif.set (when signif.sets = TRUE), or .equiv.set (when delta > 0).

Note

We warn that the default display encourages a poor practice in interpreting significance tests. Such CLDs are misleading because they visually group means with comparisons P > alpha as though they are equal, when in fact we have only failed to prove that they differ. A better alternative if one wants to show groupings is to specify an equivalence threshold delta; then groupings will be based on actual findings of equivalence. Another way to display actual findings is to set signif.sets = TRUE, so that estimates in the same group are those found to be statistically different. Obviously, these different options require different interpretations of the results; the annotations and the label given the final column help guide how to assess the results.

As further alternatives, consider pwpp (graphical display of P values) or pwpm (matrix display).

References

Piepho, Hans-Peter (2004) An algorithm for a letter-based representation of all pairwise comparisons, Journal of Computational and Graphical Statistics, 13(2), 456-466.

Examples

if(requireNamespace("multcomp"))
    emm_example("cld-multcomp")
    # Use emm_example("cld-multcomp", list = TRUE) # to just list the code

---

Manipulate factors in a reference grid

Description

These functions manipulate the levels of factors comprising a reference grid by combining factor levels, splitting a factor's levels into combinations of newly-defined factors, creating a grouping factor in which factor(s) levels are nested, or permuting the order of levels of a factor

Usage

comb_facs(object, facs, newname = paste(facs, collapse = "."),
  drop = FALSE, ...)

split_fac(object, fac, newfacs, ...)

add_grouping(object, newname, refname, newlevs, ...)

permute_levels(object, fac, pos)

Arguments

object	An object of class emmGrid
facs	Character vector. The names of the factors to combine
newname	Character name of grouping factor to add (different from any existing factor in the grid)
drop	Logical value. If TRUE, any levels of the new factor that are dropped if all occurrences in the newly reconstructed object have weight zero. If FALSE, all levels are retained. (This argument is ignored if there is no .wgt. column in object@grid.)
...	arguments passed to other methods
fac	The name of a factor that is part of the grid in object
newfacs	A named list with the names of new factors and their levels. The names must not already exist in the object, and the product of the lengths of the levels must equal the number of levels of fac.
refname	Character name(s) of the reference factor(s)
newlevs	Character vector or factor of the same length as that of the (combined) levels for refname. The grouping factor newname will have the unique values of newlevs as its levels. The order of levels in newlevs is the same as the order of the level combinations produced by expand.grid applied to the levels of refname – that is, the first factor's levels change the fastest and the last one's vary the slowest.
pos	Integer vector consisting of some permutation of the sequence 1:k, where k is the number of levels of fac. This determines which position each level of fac will occupy after the levels are permuted; thus, if the levels of fac are A,B,C,D, and pos = c(3,1,2,4), then the permuted levels will be B,C,A,D.

Value

A modified object of class emmGrid

The comb_facs function

comb_facs combines the levels of factors into a single factor in the reference grid (similar to interaction). This new factor replaces the factors that comprise it.

Additional note: The choice of whether to drop levels or not can make a profound difference. If the goal is to combine factors for use in joint_tests, we advise against drop = TRUE because that might change the weights used in deriving marginal means. If combining factors in a nested structure, dropping unused cases can considerably reduce the storage required.

The split_fac function

The levels in newfacs are expanded via expand.grid into combinations of levels, and the factor fac is replaced by those factor combinations. Unlike add_grouping, this creates a crossed, rather than a nested structure. Note that the order of factor combinations is systematic with the levels of first factor in newfacs varying the fastest; and those factor combinations are assigned respectively to the levels of fac as displayed in str(object).

The add_grouping function

This function adds a grouping factor to an existing reference grid or other emmGrid object, such that the levels of one or more existing factors (call them the reference factors) are mapped to a smaller number of levels of the new grouping factor. The reference factors are then nested in a new grouping factor named newname, and a new nesting structure refname %in% newname. This facilitates obtaining marginal means of the grouping factor, and contrasts thereof.

Additional notes: By default, the levels of newname will be ordered alphabetically. To dictate a different ordering of levels, supply newlevs as a factor having its levels in the desired order.

When refname specifies more than one factor, this can fundamentally (and permanently) change what is meant by the levels of those individual factors. For instance, in the gwrg example below, there are two levels of wool nested in each prod; and that implies that we now regard these as four different kinds of wool. Similarly, there are five different tensions (L, M, H in prod 1, and L, M in prod 2).

The permute_levels function

This function permutes the levels of fac. The returned object has the same factors, same by variables, but with the levels of fac permuted. The order of the columns in object@grid may be altered.

NOTE: fac must not be nested in another factor. permute_levels throws an error when fac is nested.

NOTE: Permuting the levels of a numeric predictor is tricky. For example, if you want to display the new ordering of levels in emmip(), you must add the arguments style = "factor" and nesting.order = TRUE.

Examples

mtcars.lm <- lm(mpg ~ factor(vs)+factor(cyl)*factor(gear), data = mtcars)
(v.c.g <- ref_grid(mtcars.lm))
(v.cg <- comb_facs(v.c.g, c("cyl", "gear")))
  
# One use is obtaining a single test for the joint contributions of two factors:
joint_tests(v.c.g)

joint_tests(v.cg)

# undo the 'comb_facs' operation:
split_fac(v.cg, "cyl.gear", list(cyl = c(4, 6, 8), gear = 3:5))

IS.glm <- glm(count ~ spray, data = InsectSprays, family = poisson)
IS.emm <- emmeans(IS.glm, "spray")
IS.new <- split_fac(IS.emm, "spray", list(A = 1:2, B = c("low", "med", "hi")))
str(IS.new)

fiber.lm <- lm(strength ~ diameter + machine, data = fiber)
( frg <- ref_grid(fiber.lm) )

# Suppose the machines are two different brands
brands <- factor(c("FiberPro", "FiberPro", "Acme"), levels = c("FiberPro", "Acme"))
( gfrg <- add_grouping(frg, "brand", "machine", brands) )

emmeans(gfrg, "machine")

emmeans(gfrg, "brand")

### More than one reference factor
warp.lm <- lm(breaks ~ wool * tension, data = warpbreaks)
gwrg <- add_grouping(ref_grid(warp.lm), 
    "prod",  c("tension", "wool"),  c(2, 1, 1,  1, 2, 1))
        # level combinations:         LA MA HA  LB MB HB

emmeans(gwrg, ~ wool * tension)   # some NAs due to impossible combinations

emmeans(gwrg, "prod")

str(v.c.g)
str(permute_levels(v.c.g, "cyl", c(2,3,1)))

---

Contrasts and linear functions of EMMs

Description

These methods provide for follow-up analyses of emmGrid objects: Contrasts, pairwise comparisons, tests, and confidence intervals. They may also be used to compute arbitrary linear functions of predictions or EMMs.

Usage

contrast(object, ...)

## S3 method for class 'emmGrid'
contrast(object, method = "eff", interaction = FALSE, by,
  offset = NULL, scale = NULL, name = "contrast",
  options = get_emm_option("contrast"), type, adjust, simple,
  combine = FALSE, ratios = TRUE, parens, enhance.levels = TRUE, wts,
  ...)

## S3 method for class 'emmGrid'
pairs(x, reverse = FALSE, ...)

## S3 method for class 'emmGrid'
coef(object, ...)

## S3 method for class 'emmGrid'
weights(object, ...)

Arguments

object	An object of class emmGrid
...	Additional arguments passed to other methods
method	Character value giving the root name of a contrast method (e.g. "pairwise" – see emmc-functions). Alternatively, a function of the same form, or a named list of coefficients (for a contrast or linear function) that must each conform to the number of results in each by group. In a multi-factor situation, the factor levels are combined and treated like a single factor.
interaction	Character vector, logical value, or list. If this is specified, method is ignored. See the “Interaction contrasts” section below for details.
by	Character names of variable(s) to be used for “by” groups. The contrasts or joint tests will be evaluated separately for each combination of these variables. If object was created with by groups, those are used unless overridden. Use by = NULL to use no by groups at all.
offset, scale	Numeric vectors of the same length as each by group. The scale values, if supplied, multiply their respective linear estimates, and any offset values are added. Scalar values are also allowed. (These arguments are ignored when interaction is specified.)
name	Character name to use to override the default label for contrasts used in table headings or subsequent contrasts of the returned object.
options	If non-NULL, a named list of arguments to pass to update.emmGrid, just after the object is constructed.
type	Character: prediction type (e.g., "response") – added to options
adjust	Character: adjustment method (e.g., "bonferroni") – added to options
simple	Character vector or list: Specify the factor(s) not in by, or a list thereof. See the section below on simple contrasts.
combine	Logical value that determines what is returned when simple is a list. See the section on simple contrasts.
ratios	Logical value determining how log and logit transforms are handled. These transformations are exceptional cases in that there is a valid way to back-transform contrasts: differences of logs are logs of ratios, and differences of logits are odds ratios. If ratios = TRUE and summarized with type = "response", contrast results are back-transformed to ratios whenever we have true contrasts (coefficients sum to zero). For other transformations, there is no natural way to back-transform contrasts, so even when summarized with type = "response", contrasts are computed and displayed on the linear-predictor scale. Similarly, if ratios = FALSE, log and logit transforms are treated in the same way as any other transformation.
parens	character or NULL. If a character value, the labels for levels being contrasted are parenthesized if they match the regular expression in parens[1] (via grep). The default is emm_option("parens"). Optionally, parens may contain second and third elements specifying what to use for left and right parentheses (default "(" and ")"). Specify parens = NULL or parens = "a^" (which won't match anything) to disable all parenthesization.
enhance.levels	character or logical. If character, the levels of the named factors that are contrasted are enhanced by appending the name of the factor; e.g., if a factor named "trt" has levels A and B, a trt comparison is labeled trtA - trtB. If enhance.levels is logical, then if TRUE (the default), only factors with numeric levels are enhanced; and of course if FALSE, no levels are enhanced. The levels of by variables are not enhanced, and any names of factors that don't exist are silently ignored. To enhance the labels beyond what is done here, change them directly via levels<-.
wts	The wts argument for some contrast methods. You should omit this argument unless you want unequal wts. Otherwise we recommend specifying wts = NA which instructs that wts be obtained from object, separately for each by group. If numerical wts are specified, they must conform to the number of levels in each by group, and those same weights are used in each group.
x	An emmGrid object
reverse	Logical value - determines whether to use "pairwise" (if TRUE) or "revpairwise" (if FALSE).

Value

contrast and pairs return an object of class emmGrid. Its grid will correspond to the levels of the contrasts and any by variables. The exception is that an emm_list object is returned if simple is a list and combine is FALSE.

coef returns a data.frame containing the "parent" object's grid, along with columns named c.1, c.2, ... containing the contrast coefficients used to produce the linear functions embodied in the object. coef() only returns coefficients if object is the result of a call to contrast(), and the parent object is the object that was handed to contrast. This is most useful for understanding interaction contrasts.

weights returns the weights stored for each row of object, or a vector of 1s if no weights are saved.

Pairs method

The call pairs(object) is equivalent to contrast(object, method = "pairwise"); and pairs(object, reverse = TRUE) is the same as contrast(object, method = "revpairwise").

Interaction contrasts

When interaction is specified, interaction contrasts are computed. Specifically contrasts are generated for each factor separately, one at a time; and these contrasts are applied to the object (the first time around) or to the previous result (subsequently). (Any factors specified in by are skipped.) The final result comprises contrasts of contrasts, or, equivalently, products of contrasts for the factors involved. Any named elements of interaction are assigned to contrast methods; others are assigned in order of appearance in object@levels. The contrast factors in the resulting emmGrid object are ordered the same as in interaction.

interaction may be a character vector or list of valid contrast methods (as documented for the method argument). If the vector or list is shorter than the number needed, it is recycled. Alternatively, if the user specifies contrast = TRUE, the contrast specified in method is used for all factors involved.

Simple contrasts

simple is essentially the complement of by: When simple is a character vector, by is set to all the factors in the grid except those in simple. If simple is a list, each element is used in turn as simple, and assembled in an "emm_list". To generate all simple main effects, use simple = "each" (this works unless there actually is a factor named "each"). Note that a non-missing simple will cause by to be ignored.

Ordinarily, when simple is a list or "each", the return value is an emm_list object with each entry in correspondence with the entries of simple. However, with combine = TRUE, the elements are all combined into one family of contrasts in a single emmGrid object using rbind.emmGrid.. In that case, the adjust argument sets the adjustment method for the combined set of contrasts.

Note

When object has a nesting structure (this can be seen via str(object)), then any grouping factors involved are forced into service as by variables, and the contrasts are thus computed separately in each nest. This in turn may lead to an irregular grid in the returned emmGrid object, which may not be valid for subsequent emmeans calls.

Examples

warp.lm <- lm(breaks ~ wool*tension, data = warpbreaks)
(warp.emm <- emmeans(warp.lm, ~ tension | wool))

contrast(warp.emm, "poly")    # inherits 'by = "wool"' from warp.emm

### Custom contrast coefs (we already have wool as 'by' thus 3 means to contrast)
contrast(warp.emm, list(mid.vs.ends = c(-1,2,-1)/2, lo.vs.hi = c(1,0,-1)))

pairs(warp.emm)

# Effects (dev from mean) of the 6 factor combs, with enhanced levels:
contrast(warp.emm, "eff", by = NULL, 
    enhance.levels = c("wool", "tension"))  
    
pairs(warp.emm, simple = "wool") # same as pairs(warp.emm, by = "tension")

# Do all "simple" comparisons, combined into one family
pairs(warp.emm, simple = "each", combine = TRUE)

## Not run: 

## Note that the following are NOT the same:
contrast(warp.emm, simple = c("wool", "tension"))
contrast(warp.emm, simple = list("wool", "tension"))
## The first generates contrasts for combinations of wool and tension
##   (same as by = NULL)
## The second generates contrasts for wool by tension, and for 
##   tension by wool, respectively.

## End(Not run)

# An interaction contrast for tension:wool
tw.emm <- contrast(warp.emm, interaction = c(tension = "poly", wool = "consec"), 
                   by = NULL)
tw.emm          # see the estimates
coef(tw.emm)    # see the contrast coefficients

# Use of scale and offset
#   an unusual use of the famous stack-loss data...
mod <- lm(Water.Temp ~ poly(stack.loss, degree = 2), data = stackloss)
(emm <- emmeans(mod, "stack.loss", at = list(stack.loss = 10 * (1:4))))
# Convert results from Celsius to Fahrenheit:
confint(contrast(emm, "identity", scale = 9/5, offset = 32))

---

Contrast families

Description

Functions with an extension of .emmc provide for named contrast families. One of the standard ones documented here may be used, or the user may write such a function.

Usage

pairwise.emmc(levs, exclude = integer(0), include, ...)

revpairwise.emmc(levs, exclude = integer(0), include, ...)

tukey.emmc(levs, reverse = FALSE, ...)

poly.emmc(levs, max.degree = min(6, k - 1), ...)

trt.vs.ctrl.emmc(levs, ref = 1, reverse = FALSE, exclude = integer(0),
  include, ...)

trt.vs.ctrl1.emmc(levs, ref = 1, ...)

trt.vs.ctrlk.emmc(levs, ref = length(levs), ...)

dunnett.emmc(levs, ref = 1, ...)

eff.emmc(levs, exclude = integer(0), include, wts = rep(1, length(levs)),
  ...)

del.eff.emmc(levs, exclude = integer(0), include, wts = rep(1,
  length(levs)), ...)

consec.emmc(levs, reverse = FALSE, exclude = integer(0), include, ...)

mean_chg.emmc(levs, reverse = FALSE, exclude = integer(0), include, ...)

wtcon.emmc(levs, wts, cmtype = "GrandMean", ...)

identity.emmc(levs, exclude = integer(0), include, ...)

Arguments

levs	Vector of factor levels
exclude	integer vector of indices, or character vector of levels to exclude from consideration. These levels will receive weight 0 in all contrasts. Character levels must exactly match elements of levs.
include	integer or character vector of levels to include (the complement of exclude). An error will result if the user specifies both exclude and include.
...	Additional arguments, passed to related methods as appropriate
reverse	Logical value to determine the direction of comparisons
max.degree	Integer specifying the maximum degree of polynomial contrasts
ref	Integer(s) or character(s) specifying which level(s) to use as the reference. Character values must exactly match elements of levs (including any enhancements – see examples)
wts	Optional weights to use with eff.emmc and del.eff.emmc contrasts. These default to equal weights. If exclude or include are specified, wts may be either the same length as levs or the length of the included levels. In the former case, weights for any excluded levels are set to zero. wts has no impact on the results unless there are at least three levels included in the contrast.
cmtype	the type argument passed to contrMat

Details

Each standard contrast family has a default multiple-testing adjustment as noted below. These adjustments are often only approximate; for a more exacting adjustment, use the interfaces provided to glht in the multcomp package.

pairwise.emmc, revpairwise.emmc, and tukey.emmc generate contrasts for all pairwise comparisons among estimated marginal means at the levels in levs. The distinction is in which direction they are subtracted. For factor levels A, B, C, D, pairwise.emmc generates the comparisons A-B, A-C, A-D, B-C, B-D, and C-D, whereas revpairwise.emmc generates B-A, C-A, C-B, D-A, D-B, and D-C. tukey.emmc invokes pairwise.emmc or revpairwise.emmc depending on reverse. The default multiplicity adjustment method is "tukey", which is only approximate when the standard errors differ.

poly.emmc generates orthogonal polynomial contrasts, assuming equally-spaced factor levels. These are derived from the poly function, but an ad hoc algorithm is used to scale them to integer coefficients that are (usually) the same as in published tables of orthogonal polynomial contrasts. The default multiplicity adjustment method is "none".

trt.vs.ctrl.emmc and its relatives generate contrasts for comparing one level (or the average over specified levels) with each of the other levels. The argument ref should be the index(es) (not the labels) of the reference level(s). trt.vs.ctrl1.emmc is the same as trt.vs.ctrl.emmc with a reference value of 1, and trt.vs.ctrlk.emmc is the same as trt.vs.ctrl with a reference value of length(levs). dunnett.emmc is the same as trt.vs.ctrl. The default multiplicity adjustment method is "dunnettx", a close approximation to the Dunnett adjustment. Note in all of these functions, it is illegal to have any overlap between the ref levels and the exclude levels. If any is found, an error is thrown.

consec.emmc and mean_chg.emmc are useful for contrasting treatments that occur in sequence. For a factor with levels A, B, C, D, E, consec.emmc generates the comparisons B-A, C-B, and D-C, while mean_chg.emmc generates the contrasts (B+C+D)/3 - A, (C+D)/2 - (A+B)/2, and D - (A+B+C)/3. With reverse = TRUE, these differences go in the opposite direction.

eff.emmc and del.eff.emmc generate contrasts that compare each level with the average over all levels (in eff.emmc) or over all other levels (in del.eff.emmc). These differ only in how they are scaled. For a set of k EMMs, del.eff.emmc gives weight 1 to one EMM and weight -1/(k-1) to the others, while eff.emmc gives weights (k-1)/k and -1/k respectively, as in subtracting the overall EMM from each EMM. The default multiplicity adjustment method is "fdr". This is a Bonferroni-based method and is slightly conservative; see p.adjust.

wtcon.emmc generates weighted contrasts based on the function contrMat function in the multcomp package, using the provided type as documented there. If the user provides wts, they have to conform to the length of levs; however, if wts is not specified, contrast will fill-in what is required, and usually this is safer (especially when by != NULL which usually means that the weights are different in each by group).

identity.emmc simply returns the identity matrix (as a data frame), minus any columns specified in exclude. It is potentially useful in cases where a contrast function must be specified, but none is desired.

Value

A data.frame, each column containing contrast coefficients for levs. The "desc" attribute is used to label the results in emmeans, and the "adjust" attribute gives the default adjustment method for multiplicity.

Note

Caution is needed in cases where the user alters the ordering of results (e.g., using the the "[...]" operator), because the contrasts generated depend on the order of the levels provided. For example, suppose trt.vs.ctrl1 contrasts are applied to two by groups with levels ordered (Ctrl, T1, T2) and (T1, T2, Ctrl) respectively, then the contrasts generated will be for (T1 - Ctrl, T2 - Ctrl) in the first group and (T2 - T1, Ctrl - T1) in the second group, because the first level in each group is used as the reference level.

Examples

warp.lm <- lm(breaks ~ wool*tension, data = warpbreaks)
warp.emm <- emmeans(warp.lm, ~ tension | wool)
contrast(warp.emm, "poly")
contrast(warp.emm, "trt.vs.ctrl", ref = "M")
## Not run: 
## Same when enhanced labeling is used:
contrast(warp.emm, "trt.vs.ctrl", 
         enhance.levels = "tension", ref = "tensionM")
## End(Not run)

# Comparisons with grand mean
contrast(warp.emm, "eff")
# Comparisons with a weighted grand mean
contrast(warp.emm, "eff", wts = c(2, 5, 3))

# Compare only low and high tensions
# Note pairs(emm, ...) calls contrast(emm, "pairwise", ...)
pairs(warp.emm, exclude = 2)
# (same results using exclude = "M" or include = c("L","H") or include = c(1,3))

### Setting up a custom contrast function
helmert.emmc <- function(levs, ...) {
    M <- as.data.frame(contr.helmert(levs))
    names(M) <- paste(levs[-1],"vs earlier")
    attr(M, "desc") <- "Helmert contrasts"
    M
}
contrast(warp.emm, "helmert")
## Not run: 
# See what is used for polynomial contrasts with 6 levels
emmeans:::poly.emmc(1:6)

## End(Not run)

---

Calculate effect sizes and confidence bounds thereof

Description

Standardized effect sizes are typically calculated using pairwise differences of estimates, divided by the SD of the population providing the context for those effects. This function calculates effect sizes from an emmGrid object, and confidence intervals for them, accounting for uncertainty in both the estimated effects and the population SD.

Usage

eff_size(object, sigma, edf, method = "pairwise", ...)

Arguments

object	an emmGrid object, typically one defining the EMMs to be contrasted. If instead, class(object) == "emm_list", such as is produced by emmeans(model, pairwise ~ treatment), a message is displayed; the contrasts already therein are used; and method is replaced by "identity".
sigma	numeric scalar, value of the population SD.
edf	numeric scalar that specifies the equivalent degrees of freedom for the sigma. This is a way of specifying the uncertainty in sigma, in that we regard our estimate of sigma^2 as being proportional to a chi-square random variable with edf degrees of freedom. (edf should not be confused with the df argument that may be passed via ... to specify the degrees of freedom to use in $t$ statistics and confidence intervals.)
method	the contrast method to use to define the effects. This is passed to contrast after the elements of object are scaled.
...	Additional arguments passed to contrast

Details

Any by variables specified in object will remain in force in the returned effects, unless overridden in the optional arguments.

For models having a single random effect, such as those fitted using lm; in that case, the stats::sigma and stats::df.residual functions may be useful for specifying sigma and edf. For models with more than one random effect, sigma may be based on some combination of the random-effect variances.

Specifying edf can be rather unintuitive but is also relatively uncritical; but the smaller the value, the wider the confidence intervals for effect size. The value of sqrt(2/edf) can be interpreted as the relative accuracy of sigma; for example, with edf = 50, $\sqrt(2/50) = 0.2$, meaning that sigma is accurate to plus or minus 20 percent. Note in an example below, we tried two different edf values as kind of a bracketing/sensitivity-analysis strategy. A value of Inf is allowable, in which case you are assuming that sigma is known exactly. Obviously, this narrows the confidence intervals for the effect sizes – unrealistically if in fact sigma is unknown.

Value

an emmGrid object containing the effect sizes

Computation

This function uses calls to regrid to put the estimated marginal means (EMMs) on the log scale. Then an extra element is added to this grid for the log of sigma and its standard error (where we assume that sigma is uncorrelated with the log EMMs). Then a call to contrast subtracts log{sigma} from each of the log EMMs, yielding values of log(EMM/sigma). Finally, the results are re-gridded back to the original scale and the desired contrasts are computed using method. In the log-scaling part, we actually rescale the absolute values and keep track of the signs.

Note

The effects are always computed on the scale of the linear-predictor; any response transformation or link function is completely ignored. If you wish to base the effect sizes on the response scale, it is not enough to replace object with regrid(object), because this back-transformation changes the SD required to compute effect sizes.

Paired data: Be careful with paired-data situations, where Cohen's d is typically referenced to the SD of the paired differences rather than the residual SD. You may need to enlarge sigma by a factor of sqrt(2) to obtain comparable results with other software.

Disclaimer: There is substantial disagreement among practitioners on what is the appropriate sigma to use in computing effect sizes; or, indeed, whether any effect-size measure is appropriate for some situations. The user is completely responsible for specifying appropriate parameters (or for failing to do so).

The examples here illustrate a sobering message that effect sizes are often not nearly as accurate as you may think.

Examples

fiber.lm <- lm(strength ~ diameter + machine, data = fiber)

emm <- emmeans(fiber.lm, "machine")
eff_size(emm, sigma = sigma(fiber.lm), edf = df.residual(fiber.lm))

# or equivalently:
eff_size(pairs(emm), sigma(fiber.lm), df.residual(fiber.lm), method = "identity")


### Mixed model example:
if (require(nlme)) withAutoprint({
  Oats.lme <- lme(yield ~ Variety + factor(nitro), 
                  random = ~ 1 | Block / Variety,
                  data = Oats)
  # Combine variance estimates
  VarCorr(Oats.lme)
  (totSD <- sqrt(214.4724 + 109.6931 + 162.5590))
  # I figure edf is somewhere between 5 (Blocks df) and 51 (Resid df)
  emmV <- emmeans(Oats.lme, ~ Variety)
  eff_size(emmV, sigma = totSD, edf = 5)
  eff_size(emmV, sigma = totSD, edf = 51)
}, spaced = TRUE)

# Multivariate model for the same data:
 MOats.lm <- lm(yield ~ Variety, data = MOats)
 eff_size(emmeans(MOats.lm, "Variety"), 
          sigma = sqrt(mean(sigma(MOats.lm)^2)),   # RMS of sigma()
          edf = df.residual(MOats.lm))

---

Support for multcomp::glht

Description

These functions and methods provide an interface between emmeans and the multcomp::glht function for simultaneous inference provided by the multcomp package.

Usage

emm(...)

as.glht(object, ...)

## S3 method for class 'emmGrid'
as.glht(object, ...)

Arguments

...	In emm, the specs, by, and contr arguments you would normally supply to emmeans. Only specs is required. Otherwise, arguments are passed to other methods. You may also include a which argument; see Details.
object	An object of class emmGrid or emm_list

Value

emm returns an object of an intermediate class for which there is a multcomp::glht method.

as.glht returns an object of class glht or glht_list according to whether object is of class emmGrid or emm_list. See Details below for more on glht_lists.

Details for emm

emm is meant to be called only from "glht" as its second (linfct) argument. It works similarly to multcomp::mcp, except with specs (and optionally by and contr arguments) provided as in a call to emmeans.

If the specifications in ... would result in a list (i.e., an emm_list object), then by default, only the last element of that list is passed to glht. However, if ... contains a which argument consisting of integer values, the list elements with those indexes are selected and combined and passed on to glht. No checking is done on whether the indexes are valid, and the keyword which must be spelled-out.

Details for as.glht

When no by variable is in force, we obtain a glht object; otherwise it is a glht_list. The latter is defined in emmeans, not multcomp, and is simply a list of glht objects. Appropriate convenience methods coef, confint, plot, summary, and vcov are provided, which simply apply the corresponding glht methods to each member.

Note

The multivariate-$t$ routines used by glht require that all estimates in the family have the same integer degrees of freedom. In cases where that is not true, a message is displayed that shows what df is used. The user may override this via the df argument.

Examples

if(require(multcomp, quietly = TRUE)) 
    emm_example("glht-multcomp") 
    # Use emm_example("glht-multcomp", list = TRUE) # to see just the code

---

Run or list additional examples

Description

This function exists so as to provide cleaner-looking examples in help files when it must be run conditionally on another package. Typically we want to run the code (run = TRUE is the default), or otherwise just list it on the console (list = TRUE).

Usage

emm_example(name, run = !list, list = FALSE, ...)

Arguments

name	Character name of file to run. We look for a file with this name (with ".R" appended) in the system files provided with emmeans.
run	Logical choosing whether or not to run the example code
list	Logical choosing whether or not to list the example code
...	Used only by the developer

Examples

# List an example
emm_example("qdrg-biglm", list = TRUE)

# Run an example
if (require(biglm))
    emm_example("qdrg-biglm")

---

The emm_list class

Description

An emm_list object is simply a list of emmGrid objects. Such a list is returned, for example, by emmeans with a two-sided formula or a list as its specs argument. Several methods for this class are provided, as detailed below. Typically, these methods just quietly do the same thing as their emmGrid methods, using the first element of the list. You can specify which to select a different element, or just run the corresponding emmGrid method on object[[k]].

Usage

## S3 method for class 'emm_list'
contrast(object, ..., which = 1)

## S3 method for class 'emm_list'
pairs(x, ..., which = 1)

## S3 method for class 'emm_list'
test(object, ..., which = seq_along(object))

## S3 method for class 'emm_list'
confint(object, ..., which = seq_along(object))

## S3 method for class 'emm_list'
plot(x, ..., which = 1)

## S3 method for class 'emm_list'
coef(object, ..., which = 2)

## S3 method for class 'emm_list'
str(object, ...)

## S3 method for class 'emm_list'
summary(object, ..., which = seq_along(object))

## S3 method for class 'emm_list'
print(x, ...)

## S3 method for class 'emm_list'
as.data.frame(x, ...)

## S3 method for class 'summary_eml'
as.data.frame(x, row.names = NULL, optional = FALSE,
  which, ...)

Arguments

object, x	an object of class emm_list
...	additional arguments passed to corresponding emmGrid method
which	integer vector specifying which elements to select.
row.names, optional	Required arguments of as.data.frame, ignored

Value

a list of objects returned by the corresponding emmGrid method (thus, often, another emm_list object). However, if which has length 1, the one result is not wrapped in a list.

summary.emm_list returns an object of class summary_eml, which is a list of summary_emm objects.

The as.data.frame methods return a single data frame via as.data.frame(rbind(x)). See also rbind.emm_list and as.data.frame.emmGrid

Note

The plot method uses only the first element of which; the others are ignored.

No export option is provided for printing an emm_list (see print.emmGrid). If you wish to export these objects, you must do so separately for each element in the list.

---

Set or change emmeans options

Description

Use emm_options to set or change various options that are used in the emmeans package. These options are set separately for different contexts in which emmGrid objects are created, in a named list of option lists.

Usage

emm_options(..., disable)

get_emm_option(x, default = emm_defaults[[x]])

emm_defaults

Arguments

...	Option names and values (see Details)
disable	If non-missing, this will reset all options to their defaults if disable tests TRUE (but first save them for possible later restoration). Otherwise, all previously saved options are restored. This is important for bug reporting; please see the section below on reproducible bugs. When disable is specified, the other arguments are ignored.
x	Character value - the name of an option to be queried
default	Value to return if x is not found

Format

An object of class list of length 21.

Details

emmeans's options are stored as a list in the system option "emmeans". Thus, emm_options(foo = bar) is the same as options(emmeans = list(..., foo = bar)) where ... represents any previously existing options. The list emm_defaults contains the default values in case the corresponding element of system option emmeans is NULL.

Currently, the following main list entries are supported:

ref_grid

A named list of defaults for objects created by ref_grid. This could affect other objects as well. For example, if emmeans is called with a fitted model object, it calls ref_grid and this option will affect the resulting emmGrid object.

emmeans

A named list of defaults for objects created by emmeans or emtrends.

contrast

A named list of defaults for objects created by contrast.emmGrid or pairs.emmGrid.

summary

A named list of defaults used by the methods summary.emmGrid, predict.emmGrid, test.emmGrid, confint.emmGrid, and emmip. The only option that can affect the latter four is "predict.method".

sep

A character value to use as a separator in labeling factor combinations. Such labels are potentially used in several places such as contrast and plot.emmGrid when combinations of factors are compared or plotted. The default is " ".

parens

Character vector that determines which labels are parenthesized when they are contrasted. The first element is a regular expression, and the second and third elements are used as left and right parentheses. See details for the parens argument in contrast. The default will parenthesize labels containing the four arithmetic operators, using round parentheses.

cov.keep

The default value of cov.keep in ref_grid. Defaults to "2", i.e., two-level covariates are treated like factors.

graphics.engine

A character value matching c("ggplot", "lattice"), setting the default engine to use in emmip and plot.emmGrid. Defaults to "ggplot".

msg.interaction

A logical value controlling whether or not a message is displayed when emmeans averages over a factor involved in an interaction. It is probably not appropriate to do this, unless the interaction is weak. Defaults to TRUE.

msg.nesting

A logical value controlling whether or not to display a message when a nesting structure is auto-detected. The existence of such a structure affects computations of EMMs. Sometimes, a nesting structure is falsely detected – namely when a user has omitted some main effects but included them in interactions. This does not change the model fit, but it produces a different parameterization that is picked up when the reference grid is constructed. Defaults to TRUE.

rg.limit

An integer value setting a limit on the number of rows in a newly constructed reference grid. This is checked based on the number of levels of the factors involved; but it excludes the levels of any multivariate responses because those are not yet known. The reference grid consists of all possible combinations of the predictors, and this can become huge if there are several factors. An error is thrown if this limit is exceeded. One can use the nuisance argument of ref_grid to collapse on nuisance factors, thus making the grid smaller. Defaults to 10,000.

simplify.names

A logical value controlling whether to simplify (when possible) names in the model formula that refer to datasets – for example, should we simplify a predictor name like “data$trt” to just “trt”? Defaults to TRUE.

opt.digits

A logical value controlling the precision with which summaries are printed. If TRUE (default), the number of digits displayed is just enough to reasonably distinguish estimates from the ends of their confidence intervals; but always at least 3 digits. If FALSE, the system value getOption("digits") is used.

back.bias.adj

A logical value controlling whether we try to adjust bias when back-transforming. If FALSE, we use naive back transformation. If TRUE and sigma is available and valid, a second-order adjustment is applied to estimate the mean on the response scale. A warning is issued if no valid sigma is available

enable.submodel

A logical value. If TRUE, enables support for selected model classes to implement the submodel option. If FALSE, this support is disabled. Setting this option to FALSE could save excess memory consumption.

Some other options have more specific purposes:

estble.tol

Tolerance for determining estimability in rank-deficient cases. If absent, the value in emm_defaults$estble.tol) is used.

save.ref_grid

Logical value of TRUE if you wish the latest reference grid created to be saved in .Last.ref_grid. The default is FALSE.

Options for lme4::lmerMod models

Options lmer.df, disable.pbkrtest, pbkrtest.limit, disable.lmerTest, and lmerTest.limit options affect how degrees of freedom are computed for lmerMod objects produced by the lme4 package). See that section of the "models" vignette for details.

Value

emm_options returns the current options (same as the result of ‘⁠getOption("emmeans")⁠’) – invisibly, unless called with no arguments.

get_emm_option returns the currently stored option for x, or its default value if not found.

Reproducible bugs

Most options set display attributes and such that are not likely to be associated with bugs in the code. However, some other options (e.g., cov.keep) are essentially configuration settings that may affect how/whether the code runs, and the settings for these options may cause subtle effects that may be hard to reproduce. Therefore, when sending a bug report, please create a reproducible example and make sure the bug occurs with all options set at their defaults. This is done by preceding it with emm_options(disable = TRUE).

By the way, disable works like a stack (LIFO buffer), in that disable = TRUE is equivalent to emm_options(saved.opts = emm_options()) and emm_options(disable = FALSE) is equivalent to options(emmeans = get_emm_option("saved.opts")). To completely erase all options, use options(emmeans = NULL)

See Also

update.emmGrid

Examples

## Not run: 
emm_options(ref_grid = list(level = .90),
            contrast = list(infer = c(TRUE,FALSE)),
            estble.tol = 1e-6)
# Sets default confidence level to .90 for objects created by ref.grid
# AS WELL AS emmeans called with a model object (since it creates a 
# reference grid). In addition, when we call 'contrast', 'pairs', etc.,
# confidence intervals rather than tests are displayed by default.

## End(Not run)

## Not run: 
emm_options(disable.pbkrtest = TRUE)
# This forces use of asymptotic methods for lmerMod objects.
# Set to FALSE or NULL to re-enable using pbkrtest.

## End(Not run)

# See tolerance being used for determining estimability
get_emm_option("estble.tol")

## Not run: 
# Set all options to their defaults
emm_options(disable = TRUE)
# ... and perhaps follow with code for a minimal reproducible bug,
#     which may include emm_options() clls if they are pertinent ...

# restore options that had existed previously
emm_options(disable = FALSE)

## End(Not run)

---

Estimated marginal means (Least-squares means)

Description

Compute estimated marginal means (EMMs) for specified factors or factor combinations in a linear model; and optionally, comparisons or contrasts among them. EMMs are also known as least-squares means.

Usage

emmeans(object, specs, by = NULL, fac.reduce = function(coefs) apply(coefs,
  2, mean), contr, options = get_emm_option("emmeans"), weights, offset, ...,
  tran)

Arguments

object	An object of class emmGrid; or a fitted model object that is supported, such as the result of a call to lm or lmer. Many fitted-model objects are supported; see vignette("models", "emmeans") for details.
specs	A character vector specifying the names of the predictors over which EMMs are desired. specs may also be a formula or a list (optionally named) of valid specs. Use of formulas is described in the Overview section below. Note: We recommend against using two-sided formulas; see the note below for contr.
by	A character vector specifying the names of predictors to condition on.
fac.reduce	A function that combines the rows of a matrix into a single vector. This implements the “marginal averaging” aspect of EMMs. The default is the mean of the rows. Typically if it is overridden, it would be some kind of weighted mean of the rows. If fac.reduce is nonlinear, bizarre results are likely, and EMMs will not be interpretable. NOTE: If the weights argument is non-missing, fac.reduce is ignored.
contr	A character value or list specifying contrasts to be added. See contrast. Note: contr is ignored when specs is a formula. Note 2:: We recommend against using this argument; obtaining means and obtaining contrasts are two different things, and it is best to do them in separate steps, using the contrast function for the contrasts.
options	If non-NULL, a named list of arguments to pass to update.emmGrid, just after the object is constructed. (Options may also be included in ...; see the ‘options’ section below.)
weights	Character value, numeric vector, or numeric matrix specifying weights to use in averaging predictions. See “Weights” section below. Also, if object is not already a reference grid, weights (if it is character) is passed to ref_grid as wt.nuis in case nuisance factors are specified. We can override this by specifying wt.nuis explicitly. This more-or-less makes the weighting of nuisance factors consistent with that of primary factors.
offset	Numeric vector or scalar. If specified, this adds an offset to the predictions, or overrides any offset in the model or its reference grid. If a vector of length differing from the number of rows in the result, it is subsetted or cyclically recycled.
...	When object is not already a "emmGrid" object, these arguments are passed to ref_grid. Common examples are at, cov.reduce, data, type, regrid, df, nesting, and vcov.. Model-type-specific options (see vignette("models", "emmeans")), commonly mode, may be used here as well. In addition, if the model formula contains references to variables that are not predictors, you must provide a params argument with a list of their names. These arguments may also be used in lieu of options. See the ‘Options’ section below.
tran	Placeholder to prevent it from being included in .... If non-missing, it is added to 'options'. See the ‘Options’ section.

Details

Users should also consult the documentation for ref_grid, because many important options for EMMs are implemented there, via the ... argument.

Value

When specs is a character vector or one-sided formula, an object of class "emmGrid". A number of methods are provided for further analysis, including summary.emmGrid, confint.emmGrid, test.emmGrid, contrast.emmGrid, and pairs.emmGrid. When specs is a list or a formula having a left-hand side, the return value is an emm_list object, which is simply a list of emmGrid objects.

Overview

Estimated marginal means or EMMs (sometimes called least-squares means) are predictions from a linear model over a reference grid; or marginal averages thereof. The ref_grid function identifies/creates the reference grid upon which emmeans is based.

For those who prefer the terms “least-squares means” or “predicted marginal means”, functions lsmeans and pmmeans are provided as wrappers. See wrappers.

If specs is a formula, it should be of the form ~ specs, ~ specs | by, contr ~ specs, or contr ~ specs | by. The formula is parsed and the variables therein are used as the arguments specs, by, and contr as indicated. The left-hand side is optional (and we don't recommend it), but if specified it should be the name of a contrast family (e.g., pairwise). Operators like * or : are needed in the formula to delineate names, but otherwise are ignored.

In the special case where the mean (or weighted mean) of all the predictions is desired, specify specs as ~ 1 or "1".

A number of standard contrast families are provided. They can be identified as functions having names ending in .emmc – see the documentation for emmc-functions for details – including how to write your own .emmc function for custom contrasts.

Weights

If weights is a vector, its length must equal the number of predictions to be averaged to obtain each EMM. If a matrix, each row of the matrix is used in turn, wrapping back to the first row as needed. When in doubt about what is being averaged (or how many), first call emmeans with weights = "show.levels".

If weights is a string, it should partially match one of the following:

"equal"

Use an equally weighted average.

"proportional"

Weight in proportion to the frequencies (in the original data) of the factor combinations that are averaged over.

"outer"

Weight in proportion to each individual factor's marginal frequencies. Thus, the weights for a combination of factors are the outer product of the one-factor margins

"cells"

Weight according to the frequencies of the cells being averaged.

"flat"

Give equal weight to all cells with data, and ignore empty cells.

"show.levels"

This is a convenience feature for understanding what is being averaged over. Instead of a table of EMMs, this causes the function to return a table showing the levels that are averaged over, in the order that they appear.

Outer weights are like the 'expected' counts in a chi-square test of independence, and will yield the same results as those obtained by proportional averaging with one factor at a time. All except "cells" uses the same set of weights for each mean. In a model where the predicted values are the cell means, cell weights will yield the raw averages of the data for the factors involved. Using "flat" is similar to "cells", except nonempty cells are weighted equally and empty cells are ignored.

Offsets

Unlike in ref_grid, an offset need not be scalar. If not enough values are supplied, they are cyclically recycled. For a vector of offsets, it is important to understand that the ordering of results goes with the first name in specs varying fastest. If there are any by factors, those vary slower than all the primary ones, but the first by variable varies the fastest within that hierarchy. See the examples.

Options and ...

Arguments that could go in options may instead be included in ..., typically, arguments such as type, infer, etc. that in essence are passed to summary.emmGrid. Arguments in both places are overridden by the ones in ....

There is a danger that ... arguments could partially match those used by both ref_grid and update.emmGrid, creating a conflict. If these occur, usually they can be resolved by providing complete (or at least longer) argument names; or by isolating non-ref_grid arguments in options; or by calling ref_grid separately and passing the result as object. See a not-run example below.

Also, when specs is a two-sided formula, or contr is specified, there is potential confusion concerning which ... arguments apply to the means, and which to the contrasts. When such confusion is possible, we suggest doing things separately (a call to emmeans with no contrasts, followed by a call to contrast). We treat adjust as a special case: it is applied to the emmeans results only if there are no contrasts specified, otherwise it is passed only to contrast.

See Also

ref_grid, contrast, vignette("models", "emmeans")

Examples

warp.lm <- lm(breaks ~ wool * tension, data = warpbreaks)
emmeans (warp.lm,  ~ wool | tension)
# or equivalently emmeans(warp.lm, "wool", by = "tension")

# 'adjust' argument ignored in emmeans, passed to contrast part...
emmeans (warp.lm, poly ~ tension | wool, adjust = "sidak")

# 'adjust' argument NOT ignored ...
emmeans (warp.lm, ~ tension | wool, adjust = "sidak")


## Not run: 
  ### Offsets: Consider a silly example:
  emmeans(warp.lm, ~ tension | wool, offset = c(17, 23, 47)) @ grid
  # note that offsets are recycled so that each level of tension receives
  # the same offset for each wool.
  # But using the same offsets with ~ wool | tension will probably not
  # be what you want because the ordering of combinations is different.

## End(Not run)

---

The emmGrid class

Description

The emmGrid class encapsulates linear functions of regression parameters, defined over a grid of predictors. This includes reference grids and grids of marginal means thereof (aka estimated marginal means). Objects of class 'emmGrid' may be used independently of the underlying model object. Instances are created primarily by ref_grid and emmeans, and several related functions.

Slots

model.info

list. Contains the elements call (the call that produced the model), terms (its terms object), and xlev (factor-level information)

roles

list. Contains at least the elements predictors, responses, and multresp. Each is a character vector of names of these variables.

grid

data.frame. Contains the combinations of the variables that define the reference grid. In addition, there is an auxiliary column named ".wgt." holding the observed frequencies or weights for each factor combination (excluding covariates). If the model has one or more offset() calls, there is an another auxiliary column named ".offset.". Auxiliary columns are not considered part of the reference grid. (However, any variables included in offset calls are in the reference grid.)

levels

list. Each entry is a character vector with the distinct levels of each variable in the reference grid. Note that grid is obtained by applying the function expand.grid to this list

matlevs

list. Like levels but has the levels of any matrices in the original dataset. Matrix columns are always concatenated and treated as a single variable for purposes of the reference grid

linfct

matrix. Each row consists of the linear function of the regression coefficients for predicting its corresponding element of the reference grid. The rows of this matrix go in one-to-one correspondence with the rows of grid, and the columns with elements of bhat.

bhat

numeric. The regression coefficients. If there is a multivariate response, the matrix of coefficients is flattened to a single vector, and linfct and V redefined appropriately. Important: bhat must include any NA values produced as a result of collinearity in the predictors. These are taken care of later in the estimability check.

nbasis

matrix. The basis for the non-estimable functions of the regression coefficients. Every EMM will correspond to a linear combination of rows of linfct, and that result must be orthogonal to all the columns of nbasis in order to be estimable. If everything is estimable, nbasis should be a 1 x 1 matrix of NA.

V

matrix. The symmetric variance-covariance matrix of bhat

dffun

function having two arguments. dffun(k, dfargs) should return the degrees of freedom for the linear function sum(k*bhat), or NA if unavailable

dfargs

list. Used to hold any additional information needed by dffun.

misc

list. Additional information used by methods. These include at least the following: estName (the label for the estimates of linear functions), and the default values of infer, level, and adjust to be used in the summary.emmGrid method. Elements in this slot may be modified if desired using the update.emmGrid method.

post.beta

matrix. A sample from the posterior distribution of the regression coefficients, if MCMC methods were used; or a 1 x 1 matrix of NA otherwise. When it is non-trivial, the as.mcmc.emmGrid method returns post.beta %*% t(linfct), which is a sample from the posterior distribution of the EMMs.

Methods

All methods for these objects are S3 methods except for show. They include [.emmGrid, as.glht.emmGrid, as.mcmc.emmGrid, as.mcmc.list.emmGrid (see coda), cld.emmGrid (see multcomp), coef.emmGrid, confint.emmGrid, contrast.emmGrid, pairs.emmGrid, plot.emmGrid, predict.emmGrid, print.emmGrid, rbind.emmGrid, show.emmGrid, str.emmGrid, summary.emmGrid, test.emmGrid, update.emmGrid, vcov.emmGrid, and xtable.emmGrid

---

Interaction-style plots for estimated marginal means

Description

Creates an interaction plot of EMMs based on a fitted model and a simple formula specification.

Usage

emmip(object, formula, ...)

## Default S3 method:
emmip(object, formula, type, CIs = FALSE, PIs = FALSE,
  style, engine = get_emm_option("graphics.engine"), plotit = TRUE,
  nesting.order = FALSE, ...)

emmip_ggplot(emms, style = "factor", dodge = 0.1, xlab = labs$xlab,
  ylab = labs$ylab, tlab = labs$tlab, facetlab = "label_context", scale,
  dotarg = list(shape = "circle"), linearg = list(linetype = "solid"),
  CIarg = list(lwd = 2, alpha = 0.5), PIarg = list(lwd = 1.25, alpha =
  0.33), col, ...)

emmip_lattice(emms, style = "factor", xlab = labs$xlab, ylab = labs$ylab,
  tlab = labs$tlab, pch = c(1, 2, 6, 7, 9, 10, 15:20), lty = 1,
  col = NULL, ...)

Arguments

object	An object of class emmGrid, or a fitted model of a class supported by the emmeans package
formula	Formula of the form trace.factors ~ x.factors | by.factors. The EMMs are plotted against x.factor for each level of trace.factors. by.factors is optional, but if present, it determines separate panels. Each element of this formula may be a single factor in the model, or a combination of factors using the * operator.
...	Additional arguments passed to emmeans (when object is not already an emmGrid object), predict.emmGrid, emmip_ggplot, or emmip_lattice.
type	As in predict.emmGrid, this determines whether we want to inverse-transform the predictions (type = "response") or not (any other choice). The default is "link", unless the "predict.type" option is in force; see emm_options. In addition, the user may specify type = "scale" to create a transformed scale for the vertical axis based on object's response transformation or link function.
CIs	Logical value. If TRUE, confidence intervals (or HPD intervals for Bayesian models) are added to the plot (works only with engine = "ggplot").
PIs	Logical value. If TRUE, prediction intervals are added to the plot (works only with engine = "ggplot"). This is allowed only if the underlying model family is "gaussian". If both CIs and PIs are TRUE, the prediction intervals will be somewhat longer, lighter, and thinner than the confidence intervals. Additional parameters to predict.emmGrid (e.g., sigma) may be passed via .... For Bayesian models, PIs require frequentist = TRUE and a value for sigma.
style	Optional character value. This has an effect only when the horizontal variable is a single numeric variable. If style is unspecified or "numeric", the horizontal scale will be numeric and curves are plotted using lines (and no symbols). With style = "factor", the horizontal variable is treated as the levels of a factor (equally spaced along the horizontal scale), and curves are plotted using lines and symbols. When the horizontal variable is character or factor, or a combination of more than one predictor, "factor" style is always used.
engine	Character value matching "ggplot" (default), "lattice", or "none". The graphics engine to be used to produce the plot. These require, respectively, the ggplot2 or lattice package to be installed. Specifying "none" is equivalent to setting plotit = FALSE.
plotit	Logical value. If TRUE, a graphical object is returned; if FALSE, a data.frame is returned containing all the values used to construct the plot.
nesting.order	Logical value. If TRUE, factors that are nested are presented in order according to their nesting factors, even if those nesting factors are not present in formula. If FALSE, only the variables in formula are used to order the variables.
emms	A data.frame created by calling emmip with plotit = FALSE. Certain variables and attributes are expected to exist in this data frame; see the section detailing the rendering functions.
dodge	Numerical amount passed to ggplot2::position_dodge by which points and intervals are offset so they do not collide.
xlab, ylab, tlab	Character labels for the horizontal axis, vertical axis, and traces (the different curves), respectively. The emmip function generates these automatically and provides therm via the labs attribute, but the user may override these if desired.
facetlab	Labeller for facets (when by variables are in play). Use "label_value" to show just the factor levels, or "label_both" to show both the factor names and factor levels. The default of "label_context" decides which based on how many by factors there are. See the documentation for ggplot2::label_context.
scale	If not missing, an object of class scales::trans specifying a (usually) nonlinear scaling for the vertical axis. For example, scales = scales::log_trans() specifies a logarithmic scale. For fine-tuning purposes, additional arguments to ggplot2::scale_y_continuous may be included in ... .
dotarg	list of arguments passed to geom_point to customize appearance of points
linearg	list of arguments passed to geom_line to customize appearance of lines
CIarg, PIarg	lists of arguments passed to geom_linerange to customize appearance of intervals. (Note: the linetype aesthetic defaults to "solid" under the hood)
col	With emmip_ggplot, this adds color = col (not colour) to all of the *arg lists. This is intended for setting a common color for everything, such as a black-and-white plot. With emmip_lattice, col specifies the colors to use for each group, recycled as needed. If not specified, the default trellis colors are used.
pch	(Lattice only) The plotting characters to use for each group (i.e., levels of trace.factors). They are recycled as needed.
lty	(Lattice only) The line types to use for each group. Recycled as needed.

Value

If plotit = FALSE, a data.frame (actually, a summary_emm object) with the table of EMMs that would be plotted. The variables plotted are named xvar and yvar, and the trace factor is named tvar. This data frame has an added "labs" attribute containing the labels xlab, ylab, and tlab for these respective variables. The confidence limits are also included, renamed LCL and UCL.

If plotit = TRUE, the function returns an object of class "ggplot" or a "trellis", depending on engine.

Details

If object is a fitted model, emmeans is called with an appropriate specification to obtain estimated marginal means for each combination of the factors present in formula (in addition, any arguments in ... that match at, trend, cov.reduce, or fac.reduce are passed to emmeans). Otherwise, if object is an emmGrid object, its first element is used, and it must contain one estimate for each combination of the factors present in formula.

Rendering functions

The functions emmip_ggplot and emmip_lattice are called when plotit == TRUE to render the plots; but they may also be called later on an object saved via plotit = FALSE (or engine = "none"). The functions require that emms contains variables xvar, yvar, and tvar, and attributes "labs" and "vars". Confidence intervals are plotted if variables LCL and UCL exist; and prediction intervals are plotted if LPL and UPL exist. Finally, it must contain the variables named in attr(emms, "vars").

In emmip_ggplot, colors, linetypes, and shapes are all assigned to groups (according to tvar) unless overridden. So, for example, one may have different symbols for each group by simply specifying dotarg = list().

Note

Conceptually, this function is equivalent to interaction.plot where the summarization function is thought to return the EMMs.

See Also

emmeans, interaction.plot.

Examples

#--- Three-factor example
noise.lm = lm(noise ~ size * type * side, data = auto.noise)

# Separate interaction plots of size by type, for each side
emmip(noise.lm, type ~ size | side)

# One interaction plot, using combinations of size and side as the x factor
# ... with added confidence intervals and some formatting changes
emmip(noise.lm, type ~ side * size, CIs = TRUE,
    CIarg = list(lwd = 1, alpha = 1, color = "cyan"),
    dotarg = list(color = "black"))

# Create a black-and-white version of above with different linetypes
# (Let the linetypes and symbols default to the palette)
emmip(noise.lm, type ~ side * size, CIs = TRUE, col = "black",
      linearg = list(), dotarg = list(size = 2), CIarg = list(alpha = 1)) +
    ggplot2::theme_bw()

# One interaction plot using combinations of type and side as the trace factor
emmip(noise.lm, type * side ~ size)

# Individual traces in panels
emmip(noise.lm, ~ size | type * side)

# Example for the 'style' argument
fib.lm = lm(strength ~ machine * sqrt(diameter), data = fiber)
fib.rg = ref_grid(fib.lm, at = list(diameter = c(3.5, 4, 4.5, 5, 5.5, 6)^2))
emmip(fib.rg, machine ~ diameter)   # curves (because diameter is numeric)
emmip(fib.rg, machine ~ diameter, style = "factor")  # points and lines

# For an example using extra ggplot2 code, see 'vignette("messy-data")',
# in the section on nested models.

### Options with transformations or link functions
neuralgia.glm <- glm(Pain ~ Treatment * Sex + Age, family = binomial(), 
                     data = neuralgia) 

# On link scale:
emmip(neuralgia.glm, Treatment ~ Sex)

# On response scale:
emmip(neuralgia.glm, Treatment ~ Sex, type = "response")

# With transformed axis scale and custom scale divisions
emmip(neuralgia.glm, Treatment ~ Sex, type = "scale",
    breaks = seq(0.10, 0.90, by = 0.10))

---

Construct an emmGrid object from scratch

Description

This allows the user to incorporate results obtained by some analysis into an emmGrid object, enabling the use of emmGrid methods to perform related follow-up analyses.

Usage

emmobj(bhat, V, levels, linfct = diag(length(bhat)), df = NA, dffun,
  dfargs = list(), post.beta = matrix(NA), nesting = NULL, ...)

Arguments

bhat	Numeric. Vector of regression coefficients
V	Square matrix. Covariance matrix of bhat
levels	Named list or vector. Levels of factor(s) that define the estimates defined by linfct. If not a list, we assume one factor named "level"
linfct	Matrix. Linear functions of bhat for each combination of levels.
df	Numeric value or function with arguments (x, dfargs). If a number, that is used for the degrees of freedom. If a function, it should return the degrees of freedom for sum(x*bhat), with any additional parameters in dfargs.
dffun	Overrides df if specified. This is a convenience to match the slot names of the returned object.
dfargs	List containing arguments for df. This is ignored if df is numeric.
post.beta	Matrix whose columns comprise a sample from the posterior distribution of the regression coefficients (so that typically, the column averages will be bhat). A 1 x 1 matrix of NA indicates that such a sample is unavailable.
nesting	Nesting specification as in ref_grid. This is ignored if model.info is supplied.
...	Arguments passed to update.emmGrid

Details

The arguments must be conformable. This includes that the length of bhat, the number of columns of linfct, and the number of columns of post.beta must all be equal. And that the product of lengths in levels must be equal to the number of rows of linfct. The grid slot of the returned object is generated by expand.grid using levels as its arguments. So the rows of linfct should be in corresponding order.

The functions qdrg and emmobj are close cousins, in that they both produce emmGrid objects. When starting with summary statistics for an existing grid, emmobj is more useful, while qdrg is more useful when starting from an unsupported fitted model.

Value

An emmGrid object

See Also

qdrg, an alternative that is useful when starting with a fitted model not supported in emmeans.

Examples

# Given summary statistics for 4 cells in a 2 x 2 layout, obtain 
# marginal means and comparisons thereof. Assume heteroscedasticity
# and use the Satterthwaite method
levels <- list(trt = c("A", "B"), dose = c("high", "low"))
ybar <- c(57.6, 43.2, 88.9, 69.8)
s <-    c(12.1, 19.5, 22.8, 43.2)
n <-    c(44,   11,   37,   24)
se2 = s^2 / n
Satt.df <- function(x, dfargs)
    sum(x * dfargs$v)^2 / sum((x * dfargs$v)^2 / (dfargs$n - 1))
    
expt.rg <- emmobj(bhat = ybar, V = diag(se2),
    levels = levels, linfct = diag(c(1, 1, 1, 1)),
    df = Satt.df, dfargs = list(v = se2, n = n), estName = "mean")
plot(expt.rg)

( trt.emm <- emmeans(expt.rg, "trt") )
( dose.emm <- emmeans(expt.rg, "dose") )

rbind(pairs(trt.emm), pairs(dose.emm), adjust = "mvt")

---

Estimated marginal means of linear trends

Description

The emtrends function is useful when a fitted model involves a numerical predictor $x$ interacting with another predictor a (typically a factor). Such models specify that $x$ has a different trend depending on $a$; thus, it may be of interest to estimate and compare those trends. Analogous to the emmeans setting, we construct a reference grid of these predicted trends, and then possibly average them over some of the predictors in the grid.

Usage

emtrends(object, specs, var, delta.var = 0.001 * rng, max.degree = 1, ...)

Arguments

object	A supported model object (not a reference grid)
specs	Specifications for what marginal trends are desired – as in emmeans. If specs is missing or NULL, emmeans is not run and the reference grid for specified trends is returned.
var	Character value giving the name of a variable with respect to which a difference quotient of the linear predictors is computed. In order for this to be useful, var should be a numeric predictor that interacts with at least one factor in specs. Then instead of computing EMMs, we compute and compare the slopes of the var trend over levels of the specified other predictor(s). As in EMMs, marginal averages are computed for the predictors in specs and by. See also the “Generalizations” section below.
delta.var	The value of h to use in forming the difference quotient $(f(x+h) - f(x))/h$. Changing it (especially changing its sign) may be necessary to avoid numerical problems such as logs of negative numbers. The default value is 1/1000 of the range of var over the dataset.
max.degree	Integer value. The maximum degree of trends to compute (this is capped at 5). If greater than 1, an additional factor degree is added to the grid, with corresponding numerical derivatives of orders 1, 2, ..., max.degree as the estimates.
...	Additional arguments passed to ref_grid or emmeans as appropriate. See Details.

Details

The function works by constructing reference grids for object with various values of var, and then calculating difference quotients of predictions from those reference grids. Finally, emmeans is called with the given specs, thus computing marginal averages as needed of the difference quotients. Any ... arguments are passed to the ref_grid and emmeans; examples of such optional arguments include optional arguments (often mode) that apply to specific models; ref_grid options such as data, at, cov.reduce, mult.names, nesting, or transform; and emmeans options such as weights (but please avoid trend or offset.

Value

An emmGrid or emm_list object, according to specs. See emmeans for more details on when a list is returned.

Generalizations

Instead of a single predictor, the user may specify some monotone function of one variable, e.g., var = "log(dose)". If so, the chain rule is applied. Note that, in this example, if object contains log(dose) as a predictor, we will be comparing the slopes estimated by that model, whereas specifying var = "dose" would perform a transformation of those slopes, making the predicted trends vary depending on dose.

Note

In earlier versions of emtrends, the first argument was named model rather than object. (The name was changed because of potential mis-matching with a mode argument, which is an option for several types of models.) For backward compatibility, model still works provided all arguments are named.

It is important to understand that trends computed by emtrends are not equivalent to polynomial contrasts in a parallel model where var is regarded as a factor. That is because the model object here is assumed to fit a smooth function of var, and the estimated trends reflect local behavior at particular value(s) of var; whereas when var is modeled as a factor and polynomial contrasts are computed, those contrasts represent the global pattern of changes over all levels of var.

See the pigs.poly and pigs.fact examples below for an illustration. The linear and quadratic trends depend on the value of percent, but the cubic trend is constant (because that is true of a cubic polynomial, which is the underlying model). The cubic contrast in the factorial model has the same P value as for the cubic trend, again because the cubic trend is the same everywhere.

See Also

emmeans, ref_grid

Examples

fiber.lm <- lm(strength ~ diameter*machine, data=fiber)
# Obtain slopes for each machine ...
( fiber.emt <- emtrends(fiber.lm, "machine", var = "diameter") )
# ... and pairwise comparisons thereof
pairs(fiber.emt)

# Suppose we want trends relative to sqrt(diameter)...
emtrends(fiber.lm, ~ machine | diameter, var = "sqrt(diameter)", 
         at = list(diameter = c(20, 30)))

# Obtaining a reference grid
mtcars.lm <- lm(mpg ~ poly(disp, degree = 2) * (factor(cyl) + factor(am)), data = mtcars)

# Center trends at mean disp for each no. of cylinders
mtcTrends.rg <- emtrends(mtcars.lm, var = "disp", 
                          cov.reduce = disp ~ factor(cyl))
summary(mtcTrends.rg)  # estimated trends at grid nodes
emmeans(mtcTrends.rg, "am", weights = "prop")


### Higher-degree trends ...

pigs.poly <- lm(conc ~ poly(percent, degree = 3), data = pigs)
emt <- emtrends(pigs.poly, ~ degree | percent, "percent", max.degree = 3,
                at = list(percent = c(9, 13.5, 18)))
       # note: 'degree' is an extra factor created by 'emtrends'
       
summary(emt, infer = c(TRUE, TRUE))

# Compare above results with poly contrasts when 'percent' is modeled as a factor ...
pigs.fact <- lm(conc ~ factor(percent), data = pigs)
emm <- emmeans(pigs.fact, "percent")

contrast(emm, "poly")
# Some P values are comparable, some aren't! See Note in documentation

---

Support functions for model extensions

Description

This documents some functions and methods that may be useful to package developers wishing to add support for emmeans for their model objects.A user or package developer may add emmeans support for a model class by writing recover_data and emm_basis methods for that class. (Users in need for a quick way to obtain results for a model that is not supported may be better served by the qdrg function.) There are several other exported functions that may be useful. See the "xtending" vignette for more details.

Usage

recover_data(object, ...)

## S3 method for class 'call'
recover_data(object, trms, na.action, data = NULL,
  params = "pi", frame, pwts, addl.vars, ...)

emm_basis(object, trms, xlev, grid, ...)

.recover_data(object, ...)

.emm_basis(object, trms, xlev, grid, ...)

.emm_register(classes, pkgname)

.std.link.labels(fam, misc)

.combine.terms(...)

.aovlist.dffun(k, dfargs)

.cmpMM(X, weights = rep(1, nrow(X)), assign = attr(X$qr, "assign"))

.get.excl(levs, exc, inc)

.get.offset(terms, grid)

.my.vcov(object, vcov. = .statsvcov, ...)

.all.vars(expr, retain = c("\\$", "\\[\\[", "\\]\\]", "'", "\""),
  ...)

.diag(x, nrow, ncol)

.num.key(levs, key)

.emm_vignette(css = system.file("css", "clean-simple.css", package =
  "emmeans"), highlight = NULL, ...)

.hurdle.support(cmu, cshape, cp0, cmean, zmu, zshape, zp0)

.zi.support(zmu, zshape, zp0)

Arguments

object	An object of the same class as is supported by a new method.
...	Additional parameters that may be supported by the method.
trms	The terms component of object (typically with the response deleted, e.g. via delete.response)
na.action	Integer vector of indices of observations to ignore; or NULL if none
data	Data frame. Usually, this is NULL. However, if non-null, this is used in place of the reconstructed dataset. It must have all of the predictors used in the model, and any factor levels must match those used in fitting the model.
params	Character vector giving the names of any variables in the model formula that are not predictors. For example, a spline model may involve a local variable knots that is not a predictor, but its value is needed to fit the model. Names of parameters not actually used are harmless, and the default value "pi" (the only numeric constant in base R) is provided in case the model involves it. An example involving splines may be found at https://github.com/rvlenth/emmeans/issues/180.
frame	Optional data.frame. Many model objects contain the model frame used when fitting the model. In cases where there are no predictor transformations, this model frame has all the original predictor values and so is usable for recovering the data. Thus, if frame is non-missing and data is NULL, a check is made on trms and if there are no function calls, we use data = frame. This can be helpful because it provides a modicum of security against the possibility that the original data used when fitting the model has been altered or removed.
pwts	Optional vector of prior weights. Typically, this may be obtained from the fitted model via weights(model). If this is provided, it is used to set weights as long as it is non-NULL and the same length as the number of rows of the data.
addl.vars	Character value or vector specifying additional predictors to include in the reference grid. These must be names of variables that exist, or you will get an error. This may be useful if you need to do additional computations later on that depend on these variables; e.g., bias adjustments for random slopes of variables not among the fixed predictors.
xlev	Named list of factor levels (excluding ones coerced to factors in the model formula)
grid	A data.frame (provided by ref_grid) containing the predictor settings needed in the reference grid
classes	Character names of one or more classes to be registered. The package must contain the functions recover_data.foo and emm_basis.foo for each class foo listed in classes.
pkgname	Character name of package providing the methods (usually should be the second argument of .onLoad)
fam	Result of call to family(object)
misc	A list intended for the @misc slot of an emmGrid object
k, dfargs	Arguments to .aovlist.dffun, which is made available as a convenience to developers providing support similar to that provided for aovlist objects
X, weights, assign	Arguments for .cmpMM, which compacts a model matrix X into a much smaller matrix that has the same row space. Specifically, it returns the R portion of its QR decomposition. If X is already of class qr, it is used directly. weights should be the weights used in the model fit, and assign is used for unravelling any pivoting done by qr.
levs, key	The .num.key function returns the numeric indices of the levels in levs to the set of all levels in key
exc, inc	Arguments for .get.excl which is useful in writing .emmc functions for generating contrast coefficients, and supports arguments exclude or include for excluding or specifying which levels to use.
terms	A terms component
vcov.	Function or matrix that returns a suitable covariance matrix. The default is .statsvcov which is stats::vcov. The .my.vcov function should be called in place of vcov, and it supports the user being able to specify a different matrix or function via the optional vcov. argument.
expr, retain	Arguments for .all.vars, which is an alternative to all.vars that has special provisions for retaining the special characters in retain, thus allowing model specifications like y ~ data$trt * df[["dose"]]
x, nrow, ncol	Arguments for .diag, which is an alternative to diag that lacks its idiosyncrasy of returning an identity matrix when x is of length 1.
css, package, highlight	Arguments for .emm_vignette, which is a clean and simple alternative to such as html_document for use as the output style of a Markdown file. All the vignettes in the emmeans package use this output style.
cmu, zmu	In .hurdle.support and .zi.support, these specify a vector of back-transformed estimates for the count and zero model, respectively
cshape, zshape	Shape parameter for the count and zero model, respectively
cp0, zp0	Function of (mu, shape) for computing Prob(Y = 0) for the count and zero model, respectively
cmean	Function of (mu, shape) for computing the mean of the count model. Typically, this just returns mu

Value

The recover_data method must return a data.frame containing all the variables that appear as predictors in the model, and attributes "call", "terms", "predictors", and "responses". (recover_data.call will provide these attributes.)

The emm_basis method should return a list with the following elements:

X

The matrix of linear functions over grid, having the same number of rows as grid and the number of columns equal to the length of bhat.

bhat

The vector of regression coefficients for fixed effects. This should include any NAs that result from rank deficiencies.

nbasis

A matrix whose columns form a basis for non-estimable functions of beta, or a 1x1 matrix of NA if there is no rank deficiency.

V

The estimated covariance matrix of bhat.

dffun

A function of (k, dfargs) that returns the degrees of freedom associated with sum(k * bhat).

dfargs

A list containing additional arguments needed for dffun

.

.recover_data and .emm_basis are hidden exported versions of recover_data and emm_basis, respectively. They run in emmeans's namespace, thus providing access to all existing methods.

.std.link.llabels returns a modified version of misc with the appropriate information included corresponding to the information in fam

combine.terms returns a terms object resulting from combining all the terms or formulas in ....

.get.offset returns the values, based on grid, of any offset component in terms

.hurdle.support returns a matrix with 3 rows containing the estimated mean responses and the differentials wrt cmu and zmu, resp.

.zi.support returns a matrix with 2 rows containing the estimated probabilities of 0 and the differentials wrt mu. See the section on hurdle and zero-inflated models.

Details

To create a reference grid, the ref_grid function needs to reconstruct the data used in fitting the model, and then obtain a matrix of linear functions of the regression coefficients for a given grid of predictor values. These tasks are performed by calls to recover_data and emm_basis respectively. A vignette giving details and examples is available via vignette("xtending", "emmeans")

To extend emmeans's support to additional model types, one need only write S3 methods for these two functions. The existing methods serve as helpful guidance for writing new ones. Most of the work for recover_data can be done by its method for class "call", providing the terms component and na.action data as additional arguments. Writing an emm_basis method is more involved, but the existing methods (e.g., emmeans:::emm_basis.lm) can serve as models. Certain recover_data and emm_basis methods are exported from emmeans. (To find out, do methods("recover_data").) If your object is based on another model-fitting object, it may be that all that is needed is to call one of these exported methods and perhaps make modifications to the results. Contact the developer if you need others of these exported.

If the model has a multivariate response, bhat needs to be “flattened” into a single vector, and X and V must be constructed consistently.

In models where a non-full-rank result is possible (often, you can tell by seeing if there is a singular.ok argument in the model-fitting function), summary.emmGrid and its relatives check the estimability of each prediction, using the nonest.basis function in the estimability package.

The models already supported are detailed in the "models" vignette. Some packages may provide additional emmeans support for its object classes.

Communication between methods

If the recover_data method generates information needed by emm_basis, that information may be incorporated by creating a "misc" attribute in the returned recovered data. That information is then passed as the misc argument when ref_grid calls emm_basis.

Optional hooks

Some models may need something other than standard linear estimates and standard errors. If so, custom functions may be pointed to via the items misc$estHook, misc$vcovHook and misc$postGridHook. If just the name of the hook function is provided as a character string, then it is retrieved using get.

The estHook function should have arguments ‘⁠(object, do.se, tol, ...)⁠’ where object is the emmGrid object, do.se is a logical flag for whether to return the standard error, and tol is the tolerance for assessing estimability. It should return a matrix with 3 columns: the estimates, standard errors (NA when do.se==FALSE), and degrees of freedom (NA for asymptotic). The number of rows should be the same as ‘⁠object@linfct⁠’. The vcovHook function should have arguments ‘⁠(object, tol, ...)⁠’ as described. It should return the covariance matrix for the estimates. Finally, postGridHook, if present, is called at the very end of ref_grid; it takes one argument, the constructed object, and should return a suitably modified emmGrid object.

Registering S3 methods for a model class

The .emm_register function is provided as a convenience to conditionally register your S3 methods for a model class, recover_data.foo and emm_basis.foo, where foo is the class name. Your package should implement an .onLoad function and call .emm_register if emmeans is installed. See the example.

Support for Hurdle and Zero-inflated models

The functions .hurdle.support and .zi.support help facilitate calculations needed to estimate the mean response (count model and zero model combined) of these models. .hurdle.support returns a matrix of three rows. The first is the estimated mean for a hurdle model, and the 2nd and 3rd rows are differentials for the count and zero models, which needed for delta-method calculations. To use these, regard the @linfct slot as comprising two sets of columns, for the count and zero models respectively. To do the delta method calculations, multiply the rows of the count part by its differentials times link$mu.eta evcaluated at that part of the linear predictor. Do the same for the zero part, using its differentials and mu.eta. If the resulting matrix is A, then the covariance of the mean response is AVA' where Vis the @V slot of the object.

The function zi.support works the same way, only it is much simpler, and is used to estimate the probability of 0 and its differential for either part of a zero-inflated model or hurdle model.

See the code for emm_basis.zeroinfl and emm_basis.hurdle for how these are used with models fitted by the pscl package.

Note

Without an explicit data argument, recover_data returns the current version of the dataset. If the dataset has changed since the model was fitted, then this will not be the data used to fit the model. It is especially important to know this in simulation studies where the data are randomly generated or permuted, and in cases where several datasets are processed in one step (e.g., using dplyr). In those cases, users should be careful to provide the actual data used to fit the model in the data argument.

See Also

Vignette on extending emmeans

Examples

## Not run: 
#--- If your package provides recover_data and emm_grid methods for class 'mymod',
#--- put something like this in your package code -- say in zzz.R:
  .onLoad <- function(libname, pkgname) {
    if (requireNamespace("emmeans", quietly = TRUE))
      emmeans::.emm_register("mymod", pkgname)
  }

## End(Not run)

---

Feedlot data

Description

This is an unbalanced analysis-of-covariance example, where one covariate is affected by a factor. Feeder calves from various herds enter a feedlot, where they are fed one of three diets. The weight of the animal at entry is the covariate, and the weight at slaughter is the response.

Usage

feedlot

Format

A data frame with 67 observations and 4 variables:

herd

a factor with levels 9 16 3 32 24 31 19 36 34 35 33, designating the herd that a feeder calf came from.

diet

a factor with levels Low Medium High: the energy level of the diet given the animal.

swt

a numeric vector: the weight of the animal at slaughter.

ewt

a numeric vector: the weight of the animal at entry to the feedlot.

Details

The data arise from a Western Regional Research Project conducted at New Mexico State University. Calves born in 1975 in commercial herds entered a feedlot as yearlings. Both diets and herds are of interest as factors. The covariate, ewt, is thought to be dependent on herd due to different genetic backgrounds, breeding history, etc. The levels of herd ordered to similarity of genetic background.

Note: There are some empty cells in the cross-classification of herd and diet.

Source

Urquhart NS (1982) Adjustment in covariates when one factor affects the covariate. Biometrics 38, 651-660.

Examples

feedlot.lm <- lm(swt ~ ewt + herd*diet, data = feedlot)

# Obtain EMMs with a separate reference value of ewt for each 
# herd. This reproduces the last part of Table 2 in the reference
emmeans(feedlot.lm,  ~ diet | herd,  cov.reduce = ewt ~ herd)

---

Fiber data

Description

Fiber data from Montgomery Design (8th ed.), p.656 (Table 15.10). Useful as a simple analysis-of-covariance example.

Usage

fiber

Format

A data frame with 15 observations and 3 variables:

machine

a factor with levels A B C. This is the primary factor of interest.

strength

a numeric vector. The response variable.

diameter

a numeric vector. A covariate.

Details

The goal of the experiment is to compare the mean breaking strength of fibers produced by the three machines. When testing this, the technician also measured the diameter of each fiber, and this measurement may be used as a concomitant variable to improve precision of the estimates.

Source

Montgomery, D. C. (2013) Design and Analysis of Experiments (8th ed.). John Wiley and Sons, ISBN 978-1-118-14692-7.

Examples

fiber.lm <- lm(strength ~ diameter + machine, data=fiber)
ref_grid(fiber.lm)

# Covariate-adjusted means and comparisons
emmeans(fiber.lm, pairwise ~ machine)

---

Summarize an emmGrid from a Bayesian model

Description

This function computes point estimates and HPD intervals for each factor combination in object@emmGrid. While this function may be called independently, it is called automatically by the S3 method summary.emmGrid when the object is based on a Bayesian model. (Note: the level argument, or its default, is passed as prob).

Usage

hpd.summary(object, prob, by, type, point.est = median, delta,
  bias.adjust = get_emm_option("back.bias.adj"), sigma, ...)

Arguments

object	an emmGrid object having a non-missing post.beta slot
prob	numeric probability content for HPD intervals (note: when not specified, the current level option is used; see emm_options)
by	factors to use as by variables
type	prediction type as in summary.emmGrid
point.est	function to use to compute the point estimates from the posterior sample for each grid point
delta	Numeric equivalence threshold (on the linear predictor scale regardless of type). See the section below on equivalence testing.
bias.adjust	Logical value for whether to adjust for bias in back-transforming (type = "response"). This requires a value of sigma to exist in the object or be specified.
sigma	Error SD assumed for bias correction (when type = "response". If not specified, object@misc$sigma is used, and a warning if it is not found or invalid. Note: sigma may be a vector, as long as it conforms to the number of observations in the posterior sample.
...	required but not used

Value

an object of class summary_emm

Equivalence testing note

If delta is positive, two columns labeled p.equiv and odds.eq are appended to the summary. p.equiv is the fraction of posterior estimates having absolute values less than delta. The odds.eq column is just p.equiv converted to an odds ratio; so it is the posterior odds of equivalence.

A high value of p.equiv is evidence in favor of equivalence. It can be used to obtain something equivalent (in spirit) to the frequentist Schuirmann (TOST) procedure, whereby we would conclude equivalence at significance level $\alpha$ if the $(1 - 2\alpha)$ confidence interval falls entirely in the interval $[-\delta, \delta]$. Similarly in the Bayesian context, an equally strong argument for equivalence is obtained if p.equiv exceeds $1 - 2\alpha$.

A closely related quantity is the ROPE (region of practical equivalence), obtainable via bayestestR::rope(object, range = c(-delta, delta)). Its value is approximately 100 * p.equiv / 0.95 if the default ci = 0.95 is used. See also bayestestR's issue #567.

Finally, a Bayes factor for equivalence is obtainable by dividing odds.eq by the prior odds of equivalence, assessed or elicited separately.

See Also

summary.emmGrid

Examples

if(require("coda")) 
    emm_example("hpd.summary-coda")
    # Use emm_example("hpd.summary-coda", list = TRUE) # to see just the code

---

Compute joint tests of the terms in a model

Description

This function produces an analysis-of-variance-like table based on linear functions of predictors in a model or emmGrid object. Specifically, the function constructs, for each combination of factors (or covariates reduced to two or more levels), a set of (interaction) contrasts via contrast, and then tests them using test with joint = TRUE. Optionally, one or more of the predictors may be used as by variable(s), so that separate tables of tests are produced for each combination of them.

Usage

joint_tests(object, by = NULL, show0df = FALSE, showconf = TRUE,
  cov.reduce = make.meanint(1), ...)

make.meanint(delta)

meanint(x)

make.symmint(ctr, delta)

symmint(ctr)

Arguments

object	a fitted model, emmGrid, or emm_list. If the latter, its first element is used.
by	character names of by variables. Separate sets of tests are run for each combination of these.
show0df	logical value; if TRUE, results with zero numerator degrees of freedom are displayed, if FALSE they are skipped
showconf	logical value. When we have models with estimability issues (e.g., missing cells), then with showconf = TRUE, we test any remaining effects that are not purely due to contrasts of a single term. If found, they are labeled (confounded). See vignette("xplanations") for more information.
cov.reduce	a function. If object is a fitted model, it is replaced by ref_grid(object, cov.reduce = cov.reduce, ...). For this purpose, the functions meanint and symmint are available for returning an interval around the mean or around zero, respectively. Se the section below on covariates.
...	additional arguments passed to ref_grid and emmeans
delta, ctr	arguments for make.meanint and make.symmint
x	argument for meanint and symmint

Details

In models with only factors, no covariates, these tests correspond to “type III” tests a la SAS, as long as equal-weighted averaging is used and there are no estimability issues. When covariates are present and they interact with factors, the results depend on how the covariate is handled in constructing the reference grid. See the section on covariates below. The point that one must always remember is that joint_tests always tests contrasts among EMMs, in the context of the reference grid, whereas type III tests are tests of model coefficients – which may or may not have anything to do with EMMs or contrasts.

Value

a summary_emm object (same as is produced by summary.emmGrid). All effects for which there are no estimable contrasts are omitted from the results. There may be an additional row named (confounded) which accounts for additional degrees of freedom for effects not accounted for in the preceding rows.

The returned object also includes an "est.fcns" attribute, which is a named list containing the linear functions associated with each joint test. No estimable functions are included for confounded effects.

make.meanint returns the function function(x) mean(x) + delta * c(-1, 1), and make.symmint(ctr, delta) returns the function function(x) ctr + delta * c(-1, 1) (which does not depend on x). The cases with delta = 1, meanint = make.meanint(1) and symmint(ctr) = make.symmint(ctr, 1) are retained for back-compatibility reasons. These functions are available primarily for use with cov.reduce.

Dealing with covariates

A covariate (or any other predictor) must have more than one value in the reference grid in order to test its effect and be included in the results. Therefore, when object is a model, we default to cov.reduce = meanint which sets each covariate at a symmetric interval about its mean. But when object is an existing reference grid, it often has only one value for covariates, in which case they are excluded from the joint tests.

Covariates present further complications in that their values in the reference grid can affect the joint tests of other effects. When covariates are centered around their means (the default), then the tests we obtain can be described as joint tests of covariate-adjusted means; and that is our intended use here. However, some software such as SAS and car::Anova adopt the convention of centering covariates around zero; and for that purpose, one can use cov.reduce = symmint(0) when calling with a model object (or in constructing a reference grid). However, adjusted means with covariates set at or around zero do not make much sense in the context of interpreting estimated marginal means, unless the covariate means really are zero.

See the examples below with the toy dataset.

See Also

test

Examples

pigs.lm <- lm(log(conc) ~ source * factor(percent), data = pigs)

(jt <- joint_tests(pigs.lm))             ## will be same as type III ANOVA

### Estimable functions associated with "percent"
attr(jt, "est.fcns") $ "percent"

joint_tests(pigs.lm, weights = "outer")  ## differently weighted

joint_tests(pigs.lm, by = "source")      ## separate joint tests of 'percent'

### Comparisons with type III tests in SAS
toy = data.frame(
    treat = rep(c("A", "B"), c(4, 6)),
    female = c(1, 0, 0, 1,   0, 0, 0, 1, 1, 0 ),
    resp = c(17, 12, 14, 19, 28, 26, 26, 34, 33, 27))
toy.fac = lm(resp ~ treat * factor(female), data = toy)
toy.cov = lm(resp ~ treat * female, data = toy)
# (These two models have identical fitted values and residuals)

# -- SAS output we'd get with toy.fac --
## Source          DF    Type III SS    Mean Square   F Value   Pr > F
## treat            1    488.8928571    488.8928571    404.60   <.0001
## female           1     78.8928571     78.8928571     65.29   0.0002
## treat*female     1      1.7500000      1.7500000      1.45   0.2741
# 
# -- SAS output we'd get with toy.cov --
## Source          DF    Type III SS    Mean Square   F Value   Pr > F
## treat            1    252.0833333    252.0833333    208.62   <.0001
## female           1     78.8928571     78.8928571     65.29   0.0002
## female*treat     1      1.7500000      1.7500000      1.45   0.2741

joint_tests(toy.fac)
joint_tests(toy.cov)   # female is regarded as a 2-level factor by default

## Treat 'female' as a numeric covariate (via cov.keep = 0)
## ... then tests depend on where we center things

# Center around the mean
joint_tests(toy.cov, cov.keep = 0, cov.reduce = make.meanint(delta = 1))
# Center around zero (like SAS's results for toy.cov)
joint_tests(toy.cov, cov.keep = 0, cov.reduce = make.symmint(ctr = 0, delta = 1))
# Center around 0.5 (like SAS's results for toy.fac)
joint_tests(toy.cov, cov.keep = 0, cov.reduce = range)

### Example with empty cells and confounded effects
low3 <- unlist(attr(ubds, "cells")[1:3]) 
ubds.lm <- lm(y ~ A*B*C, data = ubds, subset = -low3)

# Show overall joint tests by C:
ref_grid(ubds.lm, by = "C") |> contrast("consec") |> test(joint = TRUE)

# Break each of the above into smaller components:
joint_tests(ubds.lm, by = "C")

---

Wrappers for alternative naming of EMMs

Description

These are wrappers for emmeans and related functions to provide backward compatibility, or for users who may prefer to use other terminology than “estimated marginal means” – namely “least-squares means”. These functions also provide the functionality formerly provided by the lsmeans package, which is now just a front-end for emmeans.

Usage

lsmeans(...)

lstrends(...)

lsmip(...)

lsm(...)

lsmobj(...)

lsm.options(...)

get.lsm.option(x, default = emm_defaults[[x]])

Arguments

...	Arguments passed to the corresponding emxxxx function
x	Character name of desired option
default	default value to return if x not found

Details

For each function with lsxxxx in its name, the same function named emxxxx is called. Any estimator names or list items beginning with “em” are replaced with “ls” before the results are returned

Value

The result of the call to emxxxx, suitably modified.

get.lsm.option and lsm.options remap options from and to corresponding options in the emmeans options system.

See Also

emmeans, emtrends, emmip, emm, emmobj, emm_options, get_emm_option

Examples

pigs.lm <- lm(log(conc) ~ source + factor(percent), data = pigs)
lsmeans(pigs.lm, "source")

---

Response-transformation extensions

Description

The make.tran function creates the needed information to perform transformations of the response variable, including inverting the transformation and estimating variances of back-transformed predictions via the delta method. make.tran is similar to make.link, but it covers additional transformations. The result can be used as an environment in which the model is fitted, or as the tran argument in update.emmGrid (when the given transformation was already applied in an existing model).

Usage

make.tran(type = c("genlog", "power", "boxcox", "sympower", "asin.sqrt",
  "atanh", "bcnPower", "scale"), alpha = 1, beta = 0, param, y, inner, ...)

inverse(y)

Arguments

type	The name of a standard transformation supported by stat::make.link, or of a special transformation described under Details.
alpha, beta	Numeric parameters needed for special transformations.
param	If non-missing, this specifies either alpha or c(alpha, beta) (provided for backward compatibility). Also, for the same reason, if alpha is of length more than 1, it is taken as param.
y	A numeric response variable used (and required) with type = "scale", where scale(y) determines alpha and beta.
inner	another transformation. See the section on compound transformations
...	Additional arguments passed to other functions/methods

Value

A list having at least the same elements as those returned by make.link. The linkfun component is the transformation itself. Each of the functions is associated with an environment where any parameter values are defined.

inverse returns the reciprocal of its argument. It allows the "inverse" link to be auto-detected as a response transformation.

Details

The make.tran function returns a suitable list of functions for several popular transformations. Besides being usable with update, the user may use this list as an enclosing environment in fitting the model itself, in which case the transformation is auto-detected when the special name linkfun (the transformation itself) is used as the response transformation in the call. See the examples below.

The primary purpose of make.tran is to support transformations that require additional parameters, specified as alpha and beta; these are the onse shown in the argument-matching list. However, standard transformations supported by stats::make.link are also supported. In the following discussion of ones requiring parameters, we use $\alpha$ and $\beta$ to denote alpha and beta, and $y$ to denote the response variable. The type argument specifies the following transformations:

"genlog"

Generalized logarithmic transformation: $\log_\beta(y + \alpha)$, where $y > -\alpha$. When $\beta = 0$ (the default), we use $\log_e(y + \alpha)$

"power"

Power transformation: $(y-\beta)^\alpha$, where $y > \beta$. When $\alpha = 0$, $\log(y-\beta)$ is used instead.

"boxcox"

The Box-Cox transformation (unscaled by the geometric mean): $((y - \beta)^\alpha - 1) / \alpha$, where $y > \beta$. When $\alpha = 0$, $\log(y - \beta)$ is used.

"sympower"

A symmetrized power transformation on the whole real line: $|y - \beta|^\alpha\cdot sign(y - \beta)$. There are no restrictions on $y$, but we require $\alpha > 0$ in order for the transformation to be monotone and continuous.

"asin.sqrt"

Arcsin-square-root transformation: $\sin^{-1}(y/\alpha)^{1/2}$. Typically, alpha will be either 1 (default) or 100.

"atanh"

Arctanh transformation: $\tanh^{-1}(y/\alpha)$. Typically, alpha will be either 1 (default) or 100.

"bcnPower"

Box-Cox with negatives allowed, as described for the bcnPower function in the car package. It is defined as the Box-Cox transformation $(z^\alpha - 1) / \alpha$ of the variable $z = y + (y^2+\beta^2)^{1/2}$. Note that this requires both parameters and that beta > 0.

"scale"

This one is a little different than the others, in that alpha and beta are ignored; instead, they are determined by calling scale(y, ...). The user should give as y the response variable in the model to be fitted to its scaled version.

Note that with the "power", "boxcox", or "sympower" transformations, the argument beta specifies a location shift. In the "genpower" transformation, beta specifies the base of the logarithm – however, quirkily, the default of beta = 0 is taken to be the natural logarithm. For example, make.tran(0.5, 10) sets up the $\log_{10}(y + \frac12)$ transformation. In the "bcnPower" transformation, beta must be specified as a positive value.

For purposes of back-transformation, the ‘⁠sqrt(y) + sqrt(y+1)⁠’ transformation is treated exactly the same way as ‘⁠2*sqrt(y)⁠’, because both are regarded as estimates of $2\sqrt\mu$.

Cases where make.tran may not be needed

For standard transformations with no parameters, we usually don't need to use make.tran; just the name of the transformation is all that is needed. The functions emmeans, ref_grid, and related ones automatically detect response transformations that are recognized by examining the model formula. These are log, log2, log10, log1p, sqrt, logit, probit, cauchit, cloglog; as well as (for a response variable y) asin(sqrt(y)), asinh(sqrt(y)), atanh(y), and sqrt(y) + sqrt(y+1). In addition, any constant multiple of these (e.g., 2*sqrt(y)) is auto-detected and appropriately scaled (see also the tran.mult argument in update.emmGrid).

A few additional transformations may be specified as character strings and are auto-detected: "identity", "1/mu^2", "inverse", "reciprocal", "log10", "log2", "asin.sqrt", "asinh.sqrt", and "atanh".

Compound transformations

A transformation that is a function of another function can be created by specifying inner for the other function. For example, the transformation $1/\sqrt{y}$ can be created either by make.tran("inverse", inner = "sqrt") or by make.tran("power", -0.5). In principle, transformations can be compounded to any depth. Also, if type is "scale", y is replaced by inner$linkfun(y), because that will be the variable that is scaled.

Note

The genlog transformation is technically unneeded, because a response transformation of the form log(y + c) is now auto-detected by ref_grid.

We modify certain make.link results in transformations where there is a restriction on valid prediction values, so that reasonable inverse predictions are obtained, no matter what. For example, if a sqrt transformation was used but a predicted value is negative, the inverse transformation is zero rather than the square of the prediction. A side effect of this is that it is possible for one or both confidence limits, or even a standard error, to be zero.

Examples

# Fit a model using an oddball transformation:
bctran <- make.tran("boxcox", 0.368)
warp.bc <- with(bctran, 
    lm(linkfun(breaks) ~ wool * tension, data = warpbreaks))
# Obtain back-transformed LS means:    
emmeans(warp.bc, ~ tension | wool, type = "response")

### Using a scaled response...
# Case where it is auto-detected:
mod <- lm(scale(yield[, 1]) ~ Variety, data = MOats)
emmeans(mod, "Variety", type = "response")

# Case where scaling is not auto-detected -- and what to do about it:
copt <- options(contrasts = c("contr.sum", "contr.poly"))
mod.aov <- aov(scale(yield[, 1]) ~ Variety + Error(Block), data = MOats)
emm.aov <- suppressWarnings(emmeans(mod.aov, "Variety", type = "response"))

# Scaling was not retrieved, but we can do:
emm.aov <- update(emm.aov, tran = make.tran("scale", y = MOats$yield[, 1]))
emmeans(emm.aov, "Variety", type = "response")

### Compound transformations
# The following amount to the same thing:
t1 <- make.tran("inverse", inner = "sqrt")
t2 <- make.tran("power", -0.5)

options(copt)


## Not run: 
### An existing model 'mod' was fitted with a y^(2/3) transformation...
  ptran = make.tran("power", 2/3)
  emmeans(mod, "treatment", tran = ptran)

## End(Not run)

pigs.lm <- lm(inverse(conc) ~ source + factor(percent), data = pigs)
emmeans(pigs.lm, "source", type = "response")

---

Oats data in multivariate form

Description

This is the Oats dataset provided in the nlme package, but it is rearranged as one multivariate observation per plot.

Usage

MOats

Format

A data frame with 18 observations and 3 variables

Variety

a factor with levels Golden Rain, Marvellous, Victory

Block

an ordered factor with levels VI < V < III < IV < II < I

yield

a matrix with 4 columns, giving the yields with nitrogen concentrations of 0, .2, .4, and .6.

Details

These data arise from a split-plot experiment reported by Yates (1935) and used as an example in Pinheiro and Bates (2000) and other texts. Six blocks were divided into three whole plots, randomly assigned to the three varieties of oats. The whole plots were each divided into 4 split plots and randomized to the four concentrations of nitrogen.

Source

The dataset Oats in the nlme package.

References

Pinheiro, J. C. and Bates D. M. (2000) Mixed-Effects Models in S and S-PLUS, Springer, New York. (Appendix A.15)

Yates, F. (1935) Complex experiments, Journal of the Royal Statistical Society Suppl. 2, 181-247

Examples

MOats.lm <- lm (yield ~ Block + Variety, data = MOats)
MOats.rg <- ref_grid (MOats.lm, mult.name = "nitro")
emmeans(MOats.rg, ~ nitro | Variety)

---

Models supported in emmeans

Description

Documentation for models has been moved to a vignette. To access it, use vignette("models", "emmeans").

---

Multivariate contrasts

Description

This function displays tests of multivariate comparisons or contrasts. The contrasts are constructed at each level of the variable in mult.name, and then we do a multivariate test that the vector of estimates is equal to null (zero by default). The F statistic and degrees of freedom are determined via the Hotelling distribution. that is, if there are $m$ error degrees of freedom and multivariate dimensionality $d$, then the resulting $F$ statistic has degrees of freedom $(d, m - d + 1)$ as shown in Hotelling (1931).

Usage

mvcontrast(object, method = "eff", mult.name = object@roles$multresp,
  null = 0, by = object@misc$by.vars, adjust = c("sidak",
  p.adjust.methods), show.ests = FALSE, ...)

Arguments

object	An object of class emmGrid
method	A contrast method, per contrast.emmGrid
mult.name	Character vector of nNames of the factors whose levels define the multivariate means to contrast. If the model itself has a multivariate response, that is what is used. Otherwise, mult.name must be specified.
null	Scalar or conformable vector of null-hypothesis values to test against
by	Any by variable(s). These should not include the primary variables to be contrasted. For convenience, the by variable is nulled-out if it would result in no primary factors being contrasted.
adjust	Character value of a multiplicity adjustment method ("none" for no adjustment). The available adjustment methods are more limited that in contrast, and any default adjustment returned via method is ignored.
show.ests	Logical flag determining whether the multivariate means are displayed
...	Additional arguments passed to contrast

Value

An object of class summary_emm containing the multivariate test results; or a list of the estimates and the tests if show.ests is TRUE. The test results include the Hotelling $T^2$ statistic, $F$ ratios, degrees of freedom, and $P$ values.

Note

If some interactions among the primary and mult.name factors are absent, the covariance of the multivariate means is singular; this situation is accommodated, but the result has reduced degrees of freedom and a message is displayed. If there are other abnormal conditions such as non-estimable results, estimates are shown as NA.

While designed primarily for testing contrasts, multivariate tests of the mean vector itself can be implemented via method = "identity") (see the examples).

References

Hotelling, Harold (1931) "The generalization of Student's ratio", Annals of Mathematical Statistics 2(3), 360–378. doi:10.1214/aoms/1177732979

Examples

MOats.lm <- lm(yield ~ Variety + Block, data = MOats)
MOats.emm <- emmeans(MOats.lm, ~ Variety | rep.meas)
mvcontrast(MOats.emm, "consec", show.ests = TRUE)  # mult.name defaults to rep.meas

# Test each mean against a specified null vector
mvcontrast(MOats.emm, "identity", name = "Variety", 
           null = c(80, 100, 120, 140), adjust = "none")
# (Note 'name' is passed to contrast() and overrides default name "contrast")

# 'mult.name' need not refer to a multivariate response
mvcontrast(MOats.emm, "trt.vs.ctrl1", mult.name = "Variety")

---

Neuralgia data

Description

These data arise from a study of analgesic effects of treatments of elderly patients who have neuralgia. Two treatments and a placebo are compared. The response variable is whether the patient reported pain or not. Researchers recorded the age and gender of 60 patients along with the duration of complaint before the treatment began.

Usage

neuralgia

Format

A data frame with 60 observations and 5 variables:

Treatment

Factor with 3 levels A, B, and P. The latter is placebo

Sex

Factor with two levels F and M

Age

Numeric covariate – patient's age in years

Duration

Numeric covariate – duration of the condition before beginning treatment

Pain

Binary response factor with levels No and Yes

Source

Cai, Weijie (2014) Making Comparisons Fair: How LS-Means Unify the Analysis of Linear Models, SAS Institute, Inc. Technical paper 142-2014, page 12, http://support.sas.com/resources/papers/proceedings14/SAS060-2014.pdf

Examples

# Model and analysis shown in the SAS report:
neuralgia.glm <- glm(Pain ~ Treatment * Sex + Age, family = binomial(),
   data = neuralgia) 
pairs(emmeans(neuralgia.glm, ~ Treatment, at = list(Sex = "F")), 
    reverse = TRUE, type = "response", adjust = "bonferroni")

---

Nutrition data

Description

This observational dataset involves three factors, but where several factor combinations are missing. It is used as a case study in Milliken and Johnson, Chapter 17, p.202. (You may also find it in the second edition, p.278.)

Usage

nutrition

Format

A data frame with 107 observations and 4 variables:

age

a factor with levels 1, 2, 3, 4. Mother's age group.

group

a factor with levels FoodStamps, NoAid. Whether or not the family receives food stamp assistance.

race

a factor with levels Black, Hispanic, White. Mother's race.

gain

a numeric vector (the response variable). Gain score (posttest minus pretest) on knowledge of nutrition.

Details

A survey was conducted by home economists “to study how much lower-socioeconomic-level mothers knew about nutrition and to judge the effect of a training program designed to increase their knowledge of nutrition.” This is a messy dataset with several empty cells.

Source

Milliken, G. A. and Johnson, D. E. (1984) Analysis of Messy Data – Volume I: Designed Experiments. Van Nostrand, ISBN 0-534-02713-7.

Examples

nutr.aov <- aov(gain ~ (group + age + race)^2, data = nutrition)

# Summarize predictions for age group 3
nutr.emm <- emmeans(nutr.aov, ~ race * group, at = list(age="3"))
                   
emmip(nutr.emm, race ~ group)

# Hispanics seem exceptional; but this doesn't test out due to very sparse data
pairs(nutr.emm, by = "group")
pairs(nutr.emm, by = "race")

---

Sales of oranges

Description

This example dataset on sales of oranges has two factors, two covariates, and two responses. There is one observation per factor combination.

Usage

oranges

Format

A data frame with 36 observations and 6 variables:

store

a factor with levels 1 2 3 4 5 6. The store that was observed.

day

a factor with levels 1 2 3 4 5 6. The day the observation was taken (same for each store).

price1

a numeric vector. Price of variety 1.

price2

a numeric vector. Price of variety 2.

sales1

a numeric vector. Sales (per customer) of variety 1.

sales2

a numeric vector. Sales (per customer) of variety 2.

Source

This is (or once was) available as a SAS sample dataset.

References

Littell, R., Stroup W., Freund, R. (2002) SAS For Linear Models (4th edition). SAS Institute. ISBN 1-59047-023-0.

Examples

# Example on p.244 of Littell et al.
oranges.lm <- lm(sales1 ~ price1*day, data = oranges)
emmeans(oranges.lm, "day")

# Example on p.246 of Littell et al.
emmeans(oranges.lm, "day", at = list(price1 = 0))

# A more sensible model to consider, IMHO (see vignette("interactions"))
org.mlm <- lm(cbind(sales1, sales2) ~ price1 * price2 + day + store, 
              data = oranges)

---

Effects of dietary protein on free plasma leucine concentration in pigs

Description

A two-factor experiment with some observations lost

Usage

pigs

Format

A data frame with 29 observations and 3 variables:

source

Source of protein in the diet (factor with 3 levels: fish meal, soybean meal, dried skim milk)

percent

Protein percentage in the diet (numeric with 4 values: 9, 12, 15, and 18)

conc

Concentration of free plasma leucine, in mcg/ml

Source

Windels HF (1964) PhD thesis, Univ. of Minnesota. (Reported as Problem 10.8 in Oehlert G (2000) A First Course in Design and Analysis of Experiments, licensed under Creative Commons, http://users.stat.umn.edu/~gary/Book.html.) Observations 7, 22, 23, 31, 33, and 35 have been omitted, creating a more notable imbalance.

Examples

pigs.lm <- lm(inverse(conc) ~ source + factor(percent), data = pigs)
  emmeans(pigs.lm, "source")

---

Plot an emmGrid or summary_emm object

Description

Methods are provided to plot EMMs as side-by-side CIs, and optionally to display “comparison arrows” for displaying pairwise comparisons.

Usage

## S3 method for class 'emmGrid'
plot(x, y, type, CIs = TRUE, PIs = FALSE,
  comparisons = FALSE, colors = c("black", "blue", "blue", "red"),
  alpha = 0.05, adjust = "tukey", int.adjust = "none", intervals, ...)

## S3 method for class 'summary_emm'
plot(x, y, horizontal = TRUE, CIs = TRUE, xlab, ylab,
  layout, scale = NULL, colors = c("black", "blue", "blue", "red"),
  intervals, plotit = TRUE, ...)

Arguments

x	Object of class emmGrid or summary_emm
y	(Required but ignored)
type	Character value specifying the type of prediction desired (matching "linear.predictor", "link", or "response"). See details under summary.emmGrid. In addition, the user may specify type = "scale", in which case a transformed scale (e.g., a log scale) is displayed based on the transformation or link function used. Additional customization of this scale is available through including arguments to ggplot2::scale_x_continuous in ... .
CIs	Logical value. If TRUE, confidence intervals are plotted for each estimate.
PIs	Logical value. If TRUE, prediction intervals are plotted for each estimate. If object is a Bayesian model, this requires the ... arguments to include frequentist = TRUE and sigma = (some value). Note that the PIs option is not available with summary_emm objects – only for emmGrid objects. Also, prediction intervals are not available with engine = "lattice".
comparisons	Logical value. If TRUE, “comparison arrows” are added to the plot, in such a way that the degree to which arrows overlap reflects as much as possible the significance of the comparison of the two estimates. (A warning is issued if this can't be done.) Note that comparison arrows are not available with 'summary_emm' objects.
colors	Character vector of color names to use for estimates, CIs, PIs, and comparison arrows, respectively. CIs and PIs are rendered with some transparency, and colors are recycled if the length is less than four; so all plot elements are visible even if a single color is specified.
alpha	The significance level to use in constructing comparison arrows
adjust	Character value: Multiplicity adjustment method for comparison arrows only.
int.adjust	Character value: Multiplicity adjustment method for the plotted confidence intervals only.
intervals	If specified, it is used to set CIs. This is the previous argument name for CIs and is provided for backward compatibility.
...	Additional arguments passed to update.emmGrid, summary.emmGrid, predict.emmGrid, or dotplot
horizontal	Logical value specifying whether the intervals should be plotted horizontally or vertically
xlab	Character label for horizontal axis
ylab	Character label for vertical axis
layout	Numeric value passed to dotplot when engine == "lattice".
scale	Object of class trans (in the scales package) to specify a nonlinear scale. This is used in lieu of type = "scale" when plotting a summary_emm object created with type = "response". This is ignored with other types of summaries.
plotit	Logical value. If TRUE, a graphical object is returned; if FALSE, a data.frame is returned containing all the values used to construct the plot.

Value

If plotit = TRUE, a graphical object is returned.

If plotit = FALSE, a data.frame with the table of EMMs that would be plotted. In the latter case, the estimate being plotted is named the.emmean, and any factors involved have the same names as in the object. Confidence limits are named lower.CL and upper.CL, prediction limits are named lpl and upl, and comparison-arrow limits are named lcmpl and ucmpl. There is also a variable named pri.fac which contains the factor combinations that are not among the by variables.

Details

If any by variables are in force, the plot is divided into separate panels. For "summary_emm" objects, the ... arguments in plot are passed only to dotplot, whereas for "emmGrid" objects, the object is updated using ... before summarizing and plotting.

In plots with comparisons = TRUE, the resulting arrows are only approximate, and in some cases may fail to accurately reflect the pairwise comparisons of the estimates – especially when estimates having large and small standard errors are intermingled in just the wrong way. Note that the maximum and minimum estimates have arrows only in one direction, since there is no need to compare them with anything higher or lower, respectively. See the vignette("xplanations", "emmeans") for details on how these are derived.

If adjust or int.adjust are not supplied, they default to the internal adjust setting saved in pairs(x) and x respectively (see update.emmGrid).

Note

In order to play nice with the plotting functions, any variable names that are not syntactically correct (e.g., contain spaces) are altered using make.names.

Examples

warp.lm <- lm(breaks ~ wool * tension, data = warpbreaks)
warp.emm <- emmeans(warp.lm, ~ tension | wool)
plot(warp.emm)
plot(warp.emm, by = NULL, comparisons = TRUE, adjust = "mvt", 
     horizontal = FALSE, colors = "darkgreen")

### Using a transformed scale
pigs.lm <- lm(log(conc + 2) ~ source * factor(percent), data = pigs)
pigs.emm <- emmeans(pigs.lm, ~ percent | source)
plot(pigs.emm, type = "scale", breaks = seq(20, 100, by = 10))

# Based on a summary. 
# To get a transformed axis, must specify 'scale'; but it does not necessarily
# have to be the same as the actual response transformation
pigs.ci <- confint(pigs.emm, type = "response")
plot(pigs.ci, scale = scales::log10_trans())

---

Pairwise P-value matrix (plus other statistics)

Description

This function presents results from emmeans and pairwise comparisons thereof in a compact way. It displays a matrix (or matrices) of estimates, pairwise differences, and P values. The user may opt to exclude any of these via arguments means, diffs, and pvals, respectively. To control the direction of the pairwise differences, use reverse; and to control what appears in the upper and lower triangle(s), use flip. Optional arguments are passed to contrast.emmGrid and/or summary.emmGrid, making it possible to control what estimates and tests are displayed.

Usage

pwpm(emm, by, reverse = FALSE, pvals = TRUE, means = TRUE,
  diffs = TRUE, flip = FALSE, digits, ...)

Arguments

emm	An emmGrid object
by	Character vector of variable(s) in the grid to condition on. These will create different matrices, one for each level or level-combination. If missing, by is set to emm@misc$by.vars. Grid factors not in by are the primary factors: whose levels or level combinations are compared pairwise.
reverse	Logical value passed to pairs.emmGrid. Thus, FALSE specifies "pairwise" comparisons (earlier vs. later), and TRUE specifies "revpairwise" comparisons (later vs. earlier).
pvals	Logical value. If TRUE, the pairwise differences of the EMMs are included in each matrix according to flip.
means	Logical value. If TRUE, the estimated marginal means (EMMs) from emm are included in the matrix diagonal(s).
diffs	Logical value. If TRUE, the pairwise differences of the EMMs are included in each matrix according to flip.
flip	Logical value that determines where P values and differences are placed. FALSE places the P values in the upper triangle and differences in the lower, and TRUE does just the opposite.
digits	Integer. Number of digits to display. If missing, an optimal number of digits is determined.
...	Additional arguments passed to contrast.emmGrid and summary.emmGrid. You should not include method here, because pairwise comparisons are always used.

Value

A matrix or 'list' of matrices, one for each 'by' level.

Note

If emm is the result of a Bayesian analysis, pwpm is based on a frequentist analysis

See Also

A graphical display of essentially the same results is available from pwpp

Examples

warp.lm <- lm(breaks ~ wool * tension, data = warpbreaks)
warp.emm <- emmeans(warp.lm, ~ tension | wool)

pwpm(warp.emm)

# use dot options to specify noninferiority tests
pwpm(warp.emm, by = NULL, side = ">", delta = 5, adjust = "none")

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