# Models supported by emmeans

#### emmeans package, Version 1.11.2.8

Here we document what model objects may be used with emmeans, and some special features of some of them that may be accessed by passing additional arguments through `ref_grid` or `emmeans()`.

Certain objects are affected by optional arguments to functions that construct `emmGrid` objects, including `ref_grid()`, `emmeans()`, `emtrends()`, and `emmip()`. When “arguments” are mentioned in the subsequent quick reference and object-by-object documentation, we are talking about arguments in these constructors.

Some options cause transformations and links to be resolved at the time the reference grid is created, thus performing a kind of implied re-gridding at the very start. Such options are marked in the table with a “twiddle” (e.g., `"prob"~`). For those options, no links or transformations are passed along.

If a model type is not included here, users may be able to obtain usable results via the `qdrg()` function; see its help page. Package developers may support their models by writing appropriate `recover_data` and `emm_basis` methods. See the package documentation for `extending-emmeans` and `vignette("xtending")` for details.

Index of all vignette topics

## Quick reference for supported objects and options

Here is an alphabetical list of model classes that are supported, and the arguments that apply. Detailed documentation follows, with objects grouped by the code in the “Group” column. Scroll down or follow the links to those groups for more information.

Object.class | Package | Group | Arguments / notes (Suffix of `~` indicates re-gridding)
--- | --- | --- | ---
aov | stats | A |
aovList | stats | V | Best with balanced designs, orthogonal coding
averaging | MuMIn | I | `formula`, `subset` (see details)
betareg | betareg | B | `mode = c("link", "precision", "phi.link",`
|  |  | `"variance"~, "quantile"~)`
brmsfit | brms | P | Supported in brms package
carbayes | CARBayes | S | `data` is required
clm | ordinal | O | `mode = c("latent"~, "linear.predictor", "cum.prob"~,`
|  |  | `"exc.prob"~, "prob"~, "mean.class"~, "scale")`
clmm | ordinal | O | Like `clm` but no `"scale"` mode
coxme | coxme | G |
coxph | survival | G |
gam | mgcv | G | `freq = FALSE`, `unconditional = FALSE`,
|  |  | `what = c("location", "scale", "shape", "rate", "prob.gt.0")`
gamm | mgcv | G | `call = object$gam$call`
Gam | gam | G | `nboot = 800`
gamlss | gamlss | H | `what = c("mu", "sigma", "nu", "tau")`
gee | gee | E | `vcov.method = c("naive", "robust")`
geeglm | geepack | E | `vcov.method = c("vbeta", "vbeta.naiv", "vbeta.j1s",`
|  |  | `"vbeta.fij", "robust", "naive")` or a matrix
geese | geepack | E | Like `geeglm`
glm | stats | G |
glm.nb | MASS | G |
glmerMod | lme4 | G |
glmgee | glmtoolbox | E | `vcov.method = c("robust", "df-adjusted", "model",`
|  |  | `"bias-corrected", "jackknife")`
glmmadmb | glmmADMB |  | No longer supported
glmmPQL | MASS | G | inherits `lm` support
glmmTMB | glmmTMB | P | Supported in glmmTMB package
gls | nlme | K | `mode = c("auto", "df.error", "satterthwaite", "asymptotic")`
gnls | nlme | A | Supports `params` part. Requires `param = "<name>"`
hurdle | pscl | C | `mode = c("response", "count", "zero", "prob0"),`
|  |  | `lin.pred = c(FALSE~, TRUE)`
lavaan | lavaan | P | Supported by the semTools package (see `? lavaan2emmeans`)
lm | stats | A | Several other classes inherit from this and may be supported
lme | nlme | K | `sigmaAdjust = c(TRUE, FALSE),`
|  |  | `mode = c("auto", containment", "satterthwaite", "asymptotic"),`
|  |  | `extra.iter = 0`
lmerMod | lme4 | L | `lmer.df = c("kenward-roger", "satterthwaite", "asymptotic")`,
|  |  | `pbkrtest.limit = 3000`, `disable.pbkrtest = FALSE`.
|  |  | `emm_options(lmer.df =, pbkrtest.limit =, disable.pbkrtest =)`
logistf | glmmTMB | P | Supported in logistf package
lqm,lqmm | lqmm | Q | `tau = "0.5"` (must match an entry in `object$tau`)
|  |  | Optional: `method`, `R`, `seed`, `startQR` (must be fully spelled-out)
manova | stats | M | `mult.name`, `mult.levs`
maov | stats | M | `mult.name`, `mult.levs`
mblogit | mclogit | N | `mode = c("prob"~, "latent")`
|  |  | Always include response in specs for `emmeans()`
mcmc | mcmc | S | May require `formula`, `data`
MCMCglmm | MCMCglmm | S | (see also M) `mult.name`, `mult.levs`, `trait`,
|  |  | `mode = c("default", "multinomial")`; `data` is required
mira | mice | I | Optional arguments per class of `$analyses` elements
mixed | afex | P | Supported in afex package
mlm | stats | M | `mult.name`, `mult.levs`
mmer | sommer | G |
mmrm | mmrm | P | Supported in the mmrm package
multinom | nnet | N | `mode = c("prob"~, "latent")`
|  |  | Always include response in specs for `emmeans()`
nauf | nauf.xxx | P | Supported in nauf package
nlme | nlme | A | Supports fixed part. Requires `param = "<name>"`
polr | MASS | O | `mode = c("latent"~, "linear.predictor", "cum.prob"~,`
|  |  | `"exc.prob"~, "prob"~, "mean.class"~)`
rlm | MASS | A | inherits `lm` support
rms | rms | O | `mode = ("middle"~, "latent"~, "linear.predictor",`
|  |  | `"cum.prob"~, "exc.prob"~, "prob"~, "mean.class"~)`
rq,rqs | quantreg | Q | `tau = object$tau`
|  |  | Creates a pseudo-factor `tau` with levels `tau`
|  |  | Optional: `se`, `R`, `bsmethod`, etc.
rlmerMod | robustlmm | P | Supported in robustlmm package
rsm | rsm | P | Supported in rsm package
stanreg | rstanarm | S | Args for `stanreg_`xxx similar to those for xxx
survreg | survival | A |
svyglm | survey | A |
svyolr | survey | O | Piggybacks on `polr` support
zeroinfl | pscl | C | `mode = c("response", "count", "zero", "prob0")`,
|  |  | `lin.pred = c(FALSE~, TRUE)`

## Group A – “Standard” or minimally supported models

Models in this group, such as `lm`, do not have unusual features that need special support; hence no extra arguments are needed. Some may require `data` in the call.

## Group B – Beta regression

The additional `mode` argument for `betareg` objects has possible values of `"response"`, `"link"`, `"precision"`, `"phi.link"`, `"variance"`, and `"quantile"`, which have the same meaning as the `type` argument in `predict.betareg` – with the addition that `"phi.link"` is like `"link"`, but for the precision portion of the model. When `mode = "quantile"` is specified, the additional argument `quantile` (a numeric scalar or vector) specifies which quantile(s) to compute; the default is 0.5 (the median). Also in `"quantile"` mode, an additional variable `quantile` is added to the reference grid, and its levels are the values supplied. Modes `"variance"` and `"quantile"` cause an implied re-grid.

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## Group C – Count models

Two optional arguments – `mode` and `lin.pred` – are provided. The `mode` argument has possible values `"response"` (the default), `"count"`, `"zero"`, or `"prob0"`. `lin.pred` is logical and defaults to `FALSE`.

With `lin.pred = FALSE`, the results are comparable to those returned by `predict(..., type = "response")`, `predict(..., type = "count")`, `predict(..., type = "zero")`, or `predict(..., type = "prob")[, 1]`. See the documentation for `predict.hurdle` and `predict.zeroinfl`. Note that specifying `lin.pred = FALSE` causes re-gridding to take place.

The option `lin.pred = TRUE` only applies to `mode = "count"` and `mode = "zero"`. The results returned are on the linear-predictor scale, with the same transformation as the link function in that part of the model. The predictions for a reference grid with `mode = "count"`, `lin.pred = TRUE`, and `type = "response"` will be the same as those obtained with `lin.pred = FALSE` and `mode = "count"`; however, any EMMs derived from these grids will be different, because the averaging is done on the log-count scale and the actual count scale, respectively – thereby producing geometric means versus arithmetic means of the predictions.

If the `vcov.` argument is used (see details in the documentation for `ref_grid`), it must yield a matrix of the same size as would be obtained using `vcov.hurdle` or `vcov.zeroinfl` with its `model` argument set to `("full", "count", "zero")` in respective correspondence with `mode` of `("mean", "count", "zero")`. If `vcov.` is a function, it must support the `model` argument.

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## Group E – GEE models

These models all have more than one covariance estimate available, and it may be selected by supplying a string as the `vcov.method` argument. It is partially matched with the available choices shown in the quick reference. In `geese` and `geeglm`, the aliases `"robust"` (for `"vbeta"`) and `"naive"` (for `"vbeta.naiv"` are also accepted.

If a matrix or function is supplied as `vcov.method`, it is interpreted as a `vcov.` specification as described for `...` in the documentation for `ref_grid`.

## Group G – Generalized linear models and relatives

Most models in this group receive only standard support as in Group A, but typically the tests and confidence intervals are asymptotic. Thus the `df` column for tabular results will be `Inf`.

Some objects in this group may require that the original or reference dataset be provided when calling `ref_grid()` or `emmeans()`.

For `coxph` objects, the estimates we obtain are comparable to running `predict.coxph()` with `reference = "zero"`; that is, no covariate centering is done. The user may use `at` to specify adjusted covariate values. For example, the default reference grid sets each covariate to its mean, so estimates comparable to `predict.coxph(..., reference = "sample")` would be obtained by specifying `at = list(x = 0)`.

In the case of `mgcv::gam` objects, there are optional `freq` and `unconditional` arguments as is detailed in the documentation for `mgcv::vcov.gam()`. Both default to `FALSE`. The value of `unconditional` matters only if `freq = FALSE` and `object$Vc` is non-null.

For `mgcv::gamm` objects, `emmeans()` results are based on the `object$gam` part. Unfortunately, that is missing its `call` component, so the user must supply it in the `call` argument (e.g., `call = quote(gamm(y ~ s(x), data = dat))`) or give the dataset in the `data` argument. Alternatively (and recommended), you may first set `object$gam$call` to the quoted call ahead of time. The `what` arguments are used to select which model formula to use: `"location", "scale"` apply to `gaulss` and `gevlss` families, `"shape"` applies only to `gevlss`, and `"rate", "prob.gt.0"` apply to `ziplss`.

With `gam::Gam` objects, standard errors are estimated using a bootstrap method when there are any smoothers involved. Accordingly, there is an optional `nboot` argument that sets the number of bootstrap replications used to estimate the variances and covariances of the smoothing portions of the model. Generally, it is better to use models fitted via `mgcv::gam()` rather than `gam::gam()`.

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## Group H – `gamlss` models

The `what` argument has possible values of `"mu"` (default), `"sigma"`, `"nu"`, or `"tau"` depending on which part of the model you want results for. Currently, there is no support when the selected part of the model contains a smoothing method like `pb()`.

## Group I – Multiple models (via imputation or averaging)

These objects are the results of fitting several models with different predictor subsets or imputed values. The `bhat` and `V` slots are obtained via averaging and, in the case of multiple imputation, adding a multiple of the between-imputation covariance per Rubin’s rules, along with an associated degrees-of-freedom adjustment (Barnard & Rubin 1999). In the case of `mira` models with model classes not supported by emmeans, GitHub issue #446 includes a function `pool_estimates_for_qdrg()` that may be useful for obtaining results via `qdrg()`. Another useful link may be a page that shows how to pool several `emmeans` results; that is, instead of pooling the models and then running `emmeans()`, we do just the reverse.

Support for `MuMIn::averaging` objects may be somewhat dodgy, as it is not clear that all supported model classes will work. The object must have a `"modelList"` attribute (obtained by constructing the object explicitly from a model list or by including `fit = TRUE` in the call). And each model should be fitted with `data` as a named argument in the call; or else provide a `data` argument in the call to `emmeans()` or `ref_grid()`. Only ``full’’ averaging is supported; conditional averaging can result in non-positive-definite covariance matrices, and so cannot be considered. No estimability checking is done at present (not clear what we even mean by it).

Also, be aware that support for `averaging` objects does not pay attention to the class of the models being averaged (nor any `emmeans` options associated with that class, such as alternative modes or d.f. methods), and that you can only obtain direct results of linear predictions or back-transformations thereof (`type = "response"`). It does not take apart multivariate models, nor multiple-intercept models (e.g. ordinal ones). (But keep reading…)

Finally, note that special care is needed with models having multiple components (e.g. hurdle models), where there is essentially more than one set of coefficients. We can handle only one at a time. A `subset` argument must be provided to specify which coefficients to use; that can be a named vector of integers (where the names are the names of the actual model terms), or the special character values `"prefix:pfx"`, `"pfx"`, or `wrap:wrp` which picks out all coefficients whose names are prefixed by `pfx` (e.g., `pfxtrtB`) or wrapped by `wrp` (e.g., `wrp(trtB)`). A `formula` option is also available for specifying an appropriate formula other than `object$formula` when that is not suitable.

In many cases (especially with multivariate or ordinal models), you are better off making a copy of one of the averaged models that has all required terms, and hacking that object by replacing the coefficients by `coef(object, full = TRUE)` and the covariance matrix by `vcov(object, full = TRUE)`. You need to be careful to match the names of the coefficients correctly, and to use the same indexes to permute the rows and columns of the covariance matrix. You then use the `emmeans` support, including any options available, for the class of the model object, rather than for class `averaging`. An example is given towards the end of Issue 442.

## Group K – `gls` and `lme` models

The `sigmaAdjust` argument is a logical value that defaults to `TRUE`. It is comparable to the `adjustSigma` option in `nlme::summary.lme` (the name-mangling is to avoid conflicts with the often-used `adjust` argument), and determines whether or not a degrees-of-freedom adjustment is performed with models fitted using the ML method.

The optional `mode` argument affects the degrees of freedom. The `mode = "satterthwaite"` option determines degrees of freedom via the Satterthwaite method: If `s^2` is the estimate of some variance, then its Satterthwaite d.f. is `2*s^4 / Var(s^2)`. In case our numerical methods for this fail, we also offer `mode = "appx-satterthwaite"` as a backup, by which quantities related to `Var(s^2)` are obtained by randomly perturbing the response values. Currently, only `"appx-satterthwaite"` is available for `lme` objects, and it is used if `"satterthwaite"` is requested. Because `appx-satterthwaite` is simulation-based, results may vary if the same analysis is repeated. An `extra.iter` argument may be added to request additional simulation runs (at [possibly considerable] cost of repeating the model-fitting that many more times). (Note: Previously, `"appx-satterthwaite"` was termed `"boot-satterthwaite"`; this is still supported for backward compatibility. The “boot” was abandoned because it is really an approximation method, not a bootstrap method in the sense as many statistical methods.)

An alternative method is `"df.error"` (for `gls`) and `"containment"` (for `lme`). `df.error` is just the error degrees of freedom for the model, minus the number of extra random effects estimated; it generally over-estimates the degrees of freedom. The `asymptotic` mode simply sets the degrees of freedom to infinity. `"containment"` mode (for `lme` models) determines the degrees of freedom for the coarsest grouping involved in the contrast or linear function involved, so it tends to under-estimate the degrees of freedom. The default is `mode = "auto"`, which uses Satterthwaite if there are estimated random effects and the non-Satterthwaite option otherwise.

User reports indicate that models with special terms like `poly()` are not adequately supported by `gls` in that the needed basis is not recoverable from its `terms` component. This is not a problem with `lme`.

The `extra.iter` argument is ignored unless the d.f. method is (or defaults to) `appx-satterthwaite`.

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## Group L – `lmerMod` models

There is an optional `lmer.df` argument that defaults to `get_EMM_option("lmer.df")` (which in turn defaults to `"kenward-roger"`). The possible values are `"kenward-roger"`, `"satterthwaite"`, and `"asymptotic"` (these are partially matched and case-insensitive). With `"kenward-roger"`, d.f. are obtained using code from the pbkrtest package, if installed. With `"satterthwaite"`, d.f. are obtained using code from the lmerTest package, if installed. With `"asymptotic"`, or if the needed package is not installed, d.f. are set to `Inf`. (For backward compatibility, the user may specify `mode` in lieu of `lmer.df`.)

A by-product of the Kenward-Roger method is that the covariance matrix is adjusted using `pbkrtest::vcovAdj()`. This can require considerable computation; so to avoid that overhead, the user should opt for the Satterthwaite or asymptotic method; or, for backward compatibility, may disable the use of pbkrtest via `emm_options(disable.pbkrtest = TRUE)` (this does not disable the pbkrtest package entirely, just its use in emmeans). The computation time required depends roughly on the number of observations, N, in the design matrix (because a major part of the computation involves inverting an N x N matrix). Thus, pbkrtest is automatically disabled if N exceeds the value of `get_emm_option("pbkrtest.limit")`, for which the factory default is 3000. (The user may also specify `pbkrtest.limit` or `disable.pbkrtest` as an argument in the call to `emmeans()` or `ref_grid()`)

Similarly to the above, the `disable.lmerTest` and `lmerTest.limit` options or arguments affect whether Satterthwaite methods can be implemented.

The `df` argument may be used to specify some other degrees of freedom. Note that if `df` and `method = "kenward-roger"` are both specified, the covariance matrix is adjusted but the K-R degrees of freedom are not used.

Finally, note that a user-specified covariance matrix (via the `vcov.` argument) will also disable the Kenward-Roger method; in that case, the Satterthwaite method is used in place of Kenward-Roger.

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## Group M – Multivariate models

When there is a multivariate response, the different responses are treated as if they were levels of a factor – named `rep.meas` by default. The `mult.name` argument may be used to change this name. The `mult.levs` argument may specify a named list of one or more sets of levels. If this has more than one element, then the multivariate levels are expressed as combinations of the named factor levels via the function `base::expand.grid`.

## N - Multinomial responses

The reference grid includes a pseudo-factor with the same name and levels as the multinomial response. (If the response is an expression, the name of that pseudo-factor will be the first name in the expression; e.g., if the model formula is `cbind(col1, col2) ~ trt`, the grid factors will be `cbind` and `trt` and the levels of `cbind` will be `1` and `2`.) (You can change the assigned name for the multinomial response via the `mult.resp` argument.)

There is an optional `mode` argument which should match `"prob"` or `"latent"`. With `mode = "prob"`, the reference-grid predictions consist of the estimated multinomial probabilities – and this implies a re-gridding so no link functions are passed on. The `"latent"` mode returns the linear predictor, re-centered so that it averages to zero over the levels of the response variable (similar to sum-to-zero contrasts). Thus each latent variable can be regarded as the log probability at that level minus the average log probability over all levels.

There are two optional arguments: `mode` and `rescale` (which defaults to `c(0, 1)`).

Please note that, because the probabilities sum to 1 (and the latent values sum to 0) over the multivariate-response levels, all sensible results from `emmeans()` must involve that response as one of the factors. For example, if `resp` is a response with k levels, `emmeans(model, ~ resp | trt)` will yield the estimated multinomial distribution for each `trt`; but `emmeans(model, ~ trt)` will just yield the average probability of 1/k for each `trt`.

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## Group O - Ordinal responses

The reference grid for ordinal models will include all variables that appear in the main model as well as those in the `scale` or `nominal` models (if provided). There are two optional arguments: `mode` (a character string) and `rescale` (which defaults to `c(0, 1)`). `mode` should match one of `"latent"` (the default), `"linear.predictor"`, `"cum.prob"`, `"exc.prob"`, `"prob"`, `"mean.class"`, or `"scale"` – see the quick reference and note which are supported. With the exception of `"linear.predictor"`, all of these modes do an implied regrid.

With `mode = "latent"`, the reference-grid predictions are made on the scale of the latent variable implied by the model. The scale and location of this latent variable are arbitrary, and may be altered via `rescale`. The predictions are multiplied by `rescale[2]`, then added to `rescale[1]`. Keep in mind that the scaling is related to the link function used in the model; for example, changing from a probit link to a logistic link will inflate the latent values by around \(\pi/\sqrt{3}\), all other things being equal. `rescale` has no effect for other values of `mode`. Even though the latent means comprise a re-scaling of the linear predictor, we regard this as a re-gridding, and the cumulative link function is not included in the reference grid.

With `mode = "linear.predictor"`, `mode = "cum.prob"`, and `mode = "exc.prob"`, the boundaries between categories (i.e., thresholds) in the ordinal response are included in the reference grid as a pseudo-factor named `cut`. The reference-grid predictions are then of the cumulative probabilities at each threshold (for `mode = "cum.prob"`), exceedance probabilities (one minus cumulative probabilities, for `mode = "exc.prob"`), or the link function thereof (for `mode = "linear.predictor"`).

With `mode = "prob"`, a pseudo-factor with the same name as the model’s response variable is created, and the grid predictions are of the probabilities of each class of the ordinal response. With `"mean.class"`, the returned results are means of the ordinal response, interpreted as a numeric value from 1 to the number of classes, using the `"prob"` results as the estimated probability distribution for each case.

With `mode = "scale"`, and the fitted object incorporates a scale model, EMMs are obtained for the factors in the scale model (with a log response) instead of the response model. The grid is constructed using only the factors in the scale model.

Any grid point that is non-estimable by either the location or the scale model (if present) is set to `NA`, and any EMMs involving such a grid point will also be non-estimable. A consequence of this is that if there is a rank-deficient `scale` model, then all latent responses become non-estimable because the predictions are made using the average log-scale estimate.

`rms` models have an additional `mode`. With `mode = "middle"` (this is the default), the middle intercept is used, comparable to the default for `rms::Predict()`. This is quite similar in concept to `mode = "latent"`, where all intercepts are averaged together.

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## Group P – Other packages

Models in this group have their emmeans support provided by another package (usually the one that implements the model-fitting procedure). Users should refer to the package documentation for details on emmeans support. In some cases, a package’s models may have been supported here in emmeans; if so, the other package’s support overrides it.

## Group Q – Quantile regression

The elements of `tau` are included in the reference grid as a pseudo-factor named `tau`. In these models, the covariance matrix is obtained via the model’s `summary()` method with `covariance = TRUE`. The user may specify one or more of the other arguments for `summary` (e.g., `se = "boot"`) or to be passed to the `...` argument.

A caveat is that when there is more than one `tau` value, we do not have estimates of the covariances between regression coefficients associated with different `tau`s. Thus, a contrast involving different `tau`s can be estimated but its SE will be `NA`. Also, due to `NA`s in the covariance matrix, the `"mvt"` adjustment is unavailable.

Note: Older versions of this `rq` and `rqs` support required `tau`; now it is optional. Only `tau` values included in `object$tau` are allowed; others are silently ignored. It is more efficient (and less memory-greedy) to specify `tau` than to subset them using the `at` argument to `ref_grid`.

## Group S – Sampling (MCMC) methods

Models fitted using MCMC methods contain a sample from the posterior distribution of fixed-effect coefficients. In some cases (e.g., results of `MCMCpack::MCMCregress()` and `MCMCpack::MCMCpoisson()`), the object may include a `"call"` attribute that `emmeans()` can use to reconstruct the data and obtain a basis for the EMMs. If not, a `formula` and `data` argument are provided that may help produce the right results. In addition, the `contrasts` specifications are not necessarily recoverable from the object, so the system default must match what was actually used in fitting the model.

The `summary.emmGrid()` method provides credibility intervals (HPD intervals) of the results, and ignores the frequentist-oriented arguments (`infer`, `adjust`, etc.) An `as.mcmc()` method is provided that creates an `mcmc` object that can be summarized or plotted using the coda package (or others that support those objects). It provides a posterior sample of EMMs, or contrasts thereof, for the given reference grid, based on the posterior sample of the fixed effects from the model object.

In `MCMCglmm` objects, the `data` argument is required; however, if you save it as a member of the model object (e.g., `object$data = quote(mydata)`), that removes the necessity of specifying it in each call. The special keyword `trait` is used in some models. When the response is multivariate and numeric, `trait` is generated automatically as a factor in the reference grid, and the arguments `mult.levels` can be used to name its levels. In other models such as a multinomial model, use the `mode` argument to specify the type of model, and `trait = <factor name>` to specify the name of the data column that contains the levels of the factor response.

The brms package version 2.13 and later, has its own `emmeans` support. Refer to the documentation in that package.

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## Group V – `aovList` objects (also used with `afex_aov` objects)

Support for these objects is limited. To avoid strong biases in the predictions, it is strongly recommended that when fitting the model, the `contrasts` attribute of all factors should be of a type that sums to zero – for example, `"contr.sum"`, `"contr.poly"`, or `"contr.helmert"` but not `"contr.treatment"`. If that is found not to be the case, the model is re-fitted using sum-to-zero contrasts (thus requiring additional computation). Doing so does not remove all bias in the EMMs unless the design is perfectly balanced, and an annotation is added to warn of that. This bias cancels out when doing comparisons and contrasts.

Only intra-block estimates of covariances are used. That is, if a factor appears in more than one error stratum, only the covariance structure from its lowest stratum is used in estimating standard errors. Degrees of freedom are obtained using the Satterthwaite method. In general, `aovList` support is best with balanced designs, with due caution in the use of contrasts. If a `vcov.` argument is supplied, it must yield a single covariance matrix for the unique fixed effects (not a set of them for each error stratum). In that case, the degrees of freedom are set to `NA`.

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Index of all vignette topics