In rvlenth/emmeans: Estimated Marginal Means, aka Least-Squares Means

require("emmeans")
require("ggplot2")
options(show.signif.stars = FALSE) 
knitr::opts_chunk$set(fig.width = 4.5, class.output = "ro") 

Contents {#contents}

1. Issues with observational data
2. Mediating covariates
3. Mediating factors and weights
4. Nuisance factors
5. Counterfactuals and G-Computation
6. Sub-models
7. Nested fixed effects a. Avoiding mis-identified nesting

Index of all vignette topics

Issues with observational data {#issues}

In experiments, we control the conditions under which observations are made. Ideally, this leads to balanced datasets and clear inferences about the effects of those experimental conditions. In observational data, factor levels are observed rather than controlled, and in the analysis we control for those factors and covariates. It is possible that some factors and covariates lie in the causal path for other predictors. Observational studies can be designed in ways to mitigate some of these issues; but often we are left with a mess. Using EMMs does not solve the inherent problems in messy, undesigned studies; but they do give us ways to compensate for imbalance in the data, and allow us to estimate meaningful effects after carefully considering the ways in which they can be confounded.

# {#nutrex}

As an illustration, consider the nutrition dataset provided with the package. These data are used as an example in Milliken and Johnson (1992), Analysis of Messy Data, and contain the results of an observational study on nutrition education. Low-income mothers are classified by race, age category, and whether or not they received food stamps (the group factor); and the response variable is a gain score (post minus pre scores) after completing a nutrition training program. First, let's fit a model than includes all main effects and 2-way interactions, and obtain its "type II" ANOVA:

nutr.lm <- lm(gain ~ (age + group + race)^2, data = nutrition) 
car::Anova(nutr.lm)

There is definitely a group effect and a hint of and interaction with race. Here are the EMMs for those two factors, along with their counts:

emmeans(nutr.lm, ~ group * race, calc = c(n = ".wgt."))

# {#nonestex}

Hmmmm. The EMMs when race is "Hispanic" are not given; instead they are flagged as non-estimable. What does that mean? Well, when using a model to make predictions, it is impossible to do that beyond the linear space of the data used to fit the model. And we have no data for three of the age groups in the Hispanic population:

with(nutrition, table(race, age))

We can't make predictions for all the cases we are averaging over in the above EMMs, and that is why some of them are non-estimable. The bottom line is that we simply cannot include Hispanics in the mix when comparing factor effects. That's a limitation of this study that cannot be overcome without collecting additional data. Our choices for further analysis are to focus only on Black and White populations; or to focus only on age group 3. For example (the latter):

emmeans(nutr.lm, pairwise ~ group | race, at = list(age = "3")) |>
    summary(by = NULL)

(We used trickery with providing a by variable, and then taking it away, to make the output more compact.) Evidently, the training program has been beneficial to the Black and White groups in that age category. There is no conclusion for the Hispanic group -- for which we have very little data.

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Mediating covariates {#mediators}

The framing data in the mediation package has the results of an experiment conducted by Brader et al. (2008) where subjects were given the opportunity to send a message to Congress regarding immigration. However, before being offered this, some subjects (treat = 1) were first shown a news story that portrays Latinos in a negative way. Besides the binary response (whether or not they elected to send a message), the experimenters also measured emo, the subjects' emotional state after the treatment was applied. There are various demographic variables as well. Let's a logistic regression model, after changing the labels for educ to shorter strings.

framing <- mediation::framing 
levels(framing$educ) <- c("NA","Ref","< HS", "HS", "> HS","Coll +") 
framing.glm <- glm(cong_mesg ~ age + income + educ + emo + gender * factor(treat), 
    family = binomial, data = framing)

The conventional way to handle covariates like emo is to set them at their means and use those means for purposes of predictions and EMMs. These adjusted means are shown in the following plot.

emmip(framing.glm, treat ~ educ | gender, type = "response") 

This plot gives the impression that the effect of treat is reversed between male and female subjects; and also that the effect of education is not monotone. Both of these are counter-intuitive.

{#med.covred}

However, note that the covariate emo is measured post-treatment. That suggests that in fact treat (and perhaps other factors) could affect the value of emo; and if that is true (as is in fact established by mediation analysis techniques), we should not pretend that emo can be set independently of treat as was done to obtain the EMMs shown above. Instead, let emo depend on treat and the other predictors -- easily done using cov.reduce -- and we obtain an entirely different impression:

emmip(framing.glm, treat ~ educ | gender, type = "response", 
    cov.reduce = emo ~ treat*gender + age + educ + income)

The reference grid underlying this plot has different emo values for each factor combination. The plot suggests that, after taking emotional response into account, male (but not female) subjects exposed to the negative news story are more likely to send the message than are females or those not seeing the negative news story. Also, the effect of educ is now nearly monotone.

{#adjcov}

By the way, the results in this plot are the same is what you would obtain by refitting the model with an adjusted covariate

emo.adj <- resid(lm(emo ~ treat*gender + age + educ + income, data = framing))

... and then using ordinary covariate-adjusted means at the means of emo.adj. This is a technique that is often recommended.

If there is more than one mediating covariate, their settings may be defined in sequence; for example, if x1, x2, and x3 are all mediating covariates, we might use

emmeans(..., cov.reduce = list(x1 ~ trt, x2 ~ trt + x1, x3 ~ trt + x1 + x2))

(or possibly with some interactions included as well).

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Mediating factors and weights {#weights}

A mediating covariate is one that is in the causal path; likewise, it is possible to have a mediating factor. For mediating factors, the moral equivalent of the cov.reduce technique described above is to use weighted averages in lieu of equally-weighted ones in computing EMMs. The weights used in these averages should depend on the frequencies of mediating factor(s). Usually, the "cells" weighting scheme described later in this section is the right approach. In complex situations, it may be necessary to compute EMMs in stages.

As described in the "basics" vignette, EMMs are usually defined as equally-weighted means of reference-grid predictions. However, there are several built-in alternative weighting schemes that are available by specifying a character value for weights in a call to emmeans() or related function. The options are "equal" (the default), "proportional", "outer", "cells", and "flat".

The "proportional" (or "prop" for short) method weights proportionally to the frequencies (or model weights) of each factor combination that is averaged over. The "outer" method uses the outer product of the marginal frequencies of each factor that is being averaged over. To explain the distinction, suppose the EMMs for A involve averaging over two factors B and C. With "prop", we use the frequencies for each combination of B and C; whereas for "outer", first obtain the marginal frequencies for B and for C and weight proportionally to the product of these for each combination of B and C. The latter weights are like the "expected" counts used in a chi-square test for independence. Put another way, outer weighting is the same as proportional weighting applied one factor at a time; the following two would yield the same results:

emmeans(model, "A", weights = "outer") 
emmeans(model, c("A", "B"), weights = "prop") |>  emmeans(weights = "prop") 

Using "cells" weights gives each prediction the same weight as occurs in the model; applied to a reference grid for a model with all interactions, "cells"-weighted EMMs are the same as the ordinary marginal means of the data. With "flat" weights, equal weights are used, except zero weight is applied to any factor combination having no data. Usually, "cells" or "flat" weighting will not produce non-estimable results, because we exclude empty cells. (That said, if covariates are linearly dependent with factors, we may still encounter non-estimable cases.)

Here is a comparison of predictions for nutr.lm defined above, using different weighting schemes:

sapply(c("equal", "prop", "outer", "cells", "flat"), \(w)
    emmeans(nutr.lm, ~ race, weights = w) |> predict())

In the other hand, if we do group * race EMMs, only one factor (age) is averaged over; thus, the results for "prop" and "outer" weights will be identical in that case.

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Nuisance factors {#nuisance}

Consider a situation where we have a model with 15 factors, each at 5 levels. Regardless of how simple or complex the model is, the reference grid consists of all combinations of these factors -- and there are $5^{15}$ of these, or over 30 billion. If there are, say, 100 regression coefficients in the model, then just the linfct slot in the reference grid requires $100\times5^{15}\times8$ bytes of storage, or almost 23,000 gigabytes. Suppose in addition the model has a multivariate response with 5 levels. That multiplies both the rows and columns in linfct, increasing the storage requirements by a factor of 25. Either way, your computer can't store that much -- so this definitely qualifies as a messy situation!

The ref_grid() function now provides some relief, in the way of specifying some of the factors as "nuisance" factors. The reference grid is then constructed with those factors already averaged-out. So, for example with the same scenario, if only three of those 15 factors are of primary interest, and we specify the other 12 as nuisance factors to be averaged, that leaves us with only $3^5=125$ rows in the reference grid, and hence $125\times100\times8=10,000$ bytes of storage required for linfct. If there is a 5-level multivariate response, we'll have 625 rows in the reference grid and $25\times1000=250,000$ bytes in linfct. Suddenly a horribly unmanageable situation becomes quite manageable!

But of course, there is a restriction: nuisance factors must not interact with any other factors -- not even other nuisance factors. And a multivariate response (or an implied multivariate response, e.g., in an ordinal model) can never be a nuisance factor. Under that condition, the average effects of a nuisance factor are the same regardless of the levels of other factors, making it possible to pre-average them by considering just one case.

We specify nuisance factors by listing their names in a nuisance argument to ref_grid() (in emmeans(), this argument is passed to ref_grid)). Often, it is much more convenient to give the factors that are not nuisance factors, via a non.nuisance argument. If you do specify a nuisance factor that does interact with others, or doesn't exist, it is quietly excluded from the nuisance list.

{#nuis.example}

Time for an example. Consider the mtcars dataset standard in R, and the model

mtcars.lm <- lm(mpg ~ factor(cyl)*am + disp + hp + drat + log(wt) + vs + 
                  factor(gear) + factor(carb), data = mtcars)

And let's construct two different reference grids:

rg.usual <- ref_grid(mtcars.lm)
rg.usual
nrow(rg.usual@linfct)
rg.nuis = ref_grid(mtcars.lm, non.nuisance = "cyl")
rg.nuis
nrow(rg.nuis@linfct)

Notice that we left am out of non.nuisance and hence included it in nuisance. However, it interacts with cyl, so it was not allowed as a nuisance factor. But rg.nuis requires 1/36 as much storage. There's really nothing else to show, other than to demonstrate that we get the same EMMs either way, with slightly different annotations:

emmeans(rg.usual, ~ cyl * am)
emmeans(rg.nuis, ~ cyl * am)

By default, the pre-averaging is done with equal weights. If we specify wt.nuis as anything other than "equal", they are averaged proportionally. As described above, this really amounts to "outer" weights since they are averaged separately. Let's try it to see how the estimates differ:

predict(emmeans(mtcars.lm, ~ cyl * am, non.nuis = c("cyl", "am"), 
                wt.nuis = "prop"))
predict(emmeans(mtcars.lm, ~ cyl * am, weights = "outer"))

These are the same as each other, but different from the equally-weighted EMMs we obtained before. By the way, to help make things consistent, if weights is character, emmeans() passes wt.nuis = weights to ref_grid (if it is called), unless wt.nuis is also specified.

There is a trick to get emmeans to use the smallest possible reference grid: Pass the specs argument to ref_grid() as non.nuisance. But we have to quote it to delay evaluation, and also use all.vars() if (and only if) specs is a formula:

emmeans(mtcars.lm, ~ gear | am, non.nuis = quote(all.vars(specs)))

Observe that cyl was passed over as a nuisance factor because it interacts with another factor.

Limiting the size of the reference grid {#rg.limit}

We have just seen how easily the size of a reference grid can get out of hand. The rg.limit option (set via emm_options() or as an optional argument in ref_grid() or emmeans()) serves to guard against excessive memory demands. It specifies the number of allowed rows in the reference grid. But because of the way ref_grid() works, this check is made before any multivariate-response levels are taken into account. If the limit is exceeded, an error is thrown:

ref_grid(mtcars.lm, rg.limit = 200)

The default rg.limit is 10,000. With this limit, and if we have 1,000 columns in the model matrix, then the size of linfct is limited to about 80MB. If in addition, there is a 5-level multivariate response, the limit is 2GB -- darn big, but perhaps manageable. Even so, I suspect that the 10000-row default may be to loose to guard against some users getting into a tight situation.

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Counterfactuals and G-Computation {#counterfact}

G-computation is a method for model-based causal inference originated by JM Robins (Mathematical Modelling, 1986), and we want to remove confounding of treatment effects due to time-varying covariates and such. The idea is that, under certain assumptions, we can use the model to predict every subject's response to each treatment -- not just the treatment they received. To do this, we make several copies of the whole dataset, substituting the actual treatment(s) with each of the possible treatment levels; these provide us with counterfactual predictions. We then average those predictions over each copy of the dataset. Typically, this averaging is done on the response scale; that is the interesting case because on the link scale, everything is linear and we can obtain basically the same results using ordinary emmeans() computations with proportional weights.

An additional consideration is that when we average each of the counterfactual datasets, we are trying to represent the entire covariate distribution, rather than conditioning on the cases in the dataset. So it is a good idea to broaden the covariance estimate using, say, a sandwich estimate.

This kind of computation has just a little bit in common with nuisance variables, in that the net result is that we can sweep several predictors out of the reference grid just by averaging them away. For this to make sense, the predictors averaged-away will have been observed before treatment so that their effects are separate from the treatment effects.

The implementation of this in emmeans is via the counterfactuals argument in ref_grid() (but usually passed from emmeans()). We simply specify the factor(s) we want to keep. This creates an index variable .obs.no. to keep track of the observations in the dataset, and then the reference grid (before averaging) consists of every observation of the dataset in combination with the counterfactuals combinations.

{#neuralgia}

As an example, consider the neuralgia data, where we have a binary response, pain, a treatment of interest (two active treatments and placebo), and pre-treatment predictors of sex, age, and duration of the condition. We will include the vcovHC() covariance estimate in the sandwich package.

neuralgia.glm <- glm(Pain ~ Sex + Age + Duration + Treatment,
                     data = neuralgia, family = binomial)
emmeans(neuralgia.glm, "Treatment", counterfactuals = "Treatment",
        vcov. = sandwich::vcovHC)

Note that the results are already on the response (probability) scale, which is the default. Let's compare this with what we get without using counterfactuals (i.e., predicting at each covariate average):

emmeans(neuralgia.glm, "Treatment", weights = "prop", type = "response")

These results are markedly different; the counterfactual method produces smaller differences between each of the active treatments and placebo.

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Sub-models {#submodels}

We have just seen that we can assign different weights to the levels of containing factors. Another option is to constrain the effects of those containing factors to zero. In essence, that means fitting a different model without those containing effects; however, for certain models (not all), an emmGrid may be updated with a submodel specification so as to impose such a constraint. For illustration, return again to the nutrition example, and consider the analysis of group and race as before, after removing interactions involving age:

emmeans(nutr.lm, pairwise ~ group | race, submodel = ~ age + group*race) |> 
        summary(by = NULL)

If you like, you may confirm that we would obtain exactly the same estimates if we had fitted that sub-model to the data, except we continue to use the residual variance from the full model in tests and confidence intervals. Without the interactions with age, all of the marginal means become estimable. The results are somewhat different from those obtained earlier where we narrowed the scope to just age 3. These new estimates include all ages, averaging over them equally, but with constraints that the interaction effects involving age are all zero.

{#type2submodel}

There are two special character values that may be used with submodel. Specifying "minimal" creates a submodel with only the active factors:

emmeans(nutr.lm, ~ group * race, submodel = "minimal")

This submodel constrains all effects involving age to be zero. Another interesting option is "type2", whereby we in essence analyze the residuals of the model with all contained or overlapping effects, then constrain the containing effects to be zero. So what is left if only the interaction effects of the factors involved. This is most useful with joint_tests():

joint_tests(nutr.lm, submodel = "type2")

These results are identical to the type II anova obtained at the beginning of this example.

More details on how submodel works may be found in vignette("xplanations")

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Nested fixed effects {#nesting}

A factor A is nested in another factor B if the levels of A have a different meaning in one level of B than in another. Often, nested factors are random effects---for example, subjects in an experiment may be randomly assigned to treatments, in which case subjects are nested in treatments---and if we model them as random effects, these random nested effects are not among the fixed effects and are not an issue to emmeans. But sometimes we have fixed nested factors.

{#cows}

Here is an example of a fictional study of five fictional treatments for some disease in cows. Two of the treatments are administered by injection, and the other three are administered orally. There are varying numbers of observations for each drug. The data and model follow:

cows <- data.frame (
    route = factor(rep(c("injection", "oral"), c(5, 9))),
    drug = factor(rep(c("Bovineumab", "Charloisazepam", 
              "Angustatin", "Herefordmycin", "Mollycoddle"), c(3,2,  4,2,3))),
    resp = c(34, 35, 34,   44, 43,      36, 33, 36, 32,   26, 25,   25, 24, 24)
)
cows.lm <- lm(resp ~ route + drug, data = cows)

The ref_grid function finds a nested structure in this model:

cows.rg <- ref_grid(cows.lm)
cows.rg

When there is nesting, emmeans computes averages separately in each group\ldots

route.emm <- emmeans(cows.rg, "route")
route.emm

... and insists on carrying along any grouping factors that a factor is nested in:

drug.emm <- emmeans(cows.rg, "drug")
drug.emm

Here are the associated pairwise comparisons:

pairs(route.emm, reverse = TRUE)

pairs(drug.emm, by = "route", reverse = TRUE)

In the latter result, the contrast itself becomes a nested factor in the returned emmGrid object. That would not be the case if there had been no by variable.

Graphs with nesting

It can be very helpful to take advantage of special features of ggplot2 when graphing results with nested factors. For example, the default plot for the cows example is not ideal:

emmip(cows.rg, ~ drug | route)

We can instead remove route from the call and instead handle it with ggplot2 code to use separate x scales:

require(ggplot2)
emmip(cows.rg, ~ drug) + facet_wrap(~ route, scales = "free_x")

Similarly with plot.emmGrid():

plot(drug.emm, PIs = TRUE) + 
    facet_wrap(~ route, nrow = 2, scales = "free_y")

Auto-identification of nested factors -- avoid being trapped! {#nest-trap}

ref_grid() and emmeans() tries to discover and accommodate nested structures in the fixed effects. It does this in two ways: first, by identifying factors whose levels appear in combination with only one level of another factor; and second, by examining the terms attribute of the fixed effects. In the latter approach, if an interaction A:B appears in the model but A is not present as a main effect, then A is deemed to be nested in B. Note that this can create a trap: some users take shortcuts by omitting some fixed effects, knowing that this won't affect the fitted values. But such shortcuts do affect the interpretation of model parameters, ANOVA tables, etc., and I advise against ever taking such shortcuts. Here are some ways you may notice mistakenly-identified nesting:

• A message is displayed when nesting is detected
• A str() listing of the emmGrid object shows a nesting component
• An emmeans() summary unexpectedly includes one or more factors that you didn't specify
• EMMs obtained using by factors don't seem to behave right, or give the same results with different specifications

To override the auto-detection of nested effects, use the nesting argument in ref_grid() or emmeans(). Specifying nesting = NULL will ignore all nesting. Incorrectly-discovered nesting can be overcome by specifying something akin to nesting = "A %in% B, C %in% (A * B)" or, equivalently, nesting = list(A = "B", C = c("A", "B")).

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Index of all vignette topics

rvlenth/emmeans documentation built on April 9, 2024, 5:19 a.m.