Basics of estimated marginal means

Emphasis on experimental data

To start off with, we should emphasize that the underpinnings of estimated marginal means – and much of what the emmeans package offers – relate more to experimental data than to observational data. In observational data, we sample from some population, and the goal of statistical analysis is to characterize that population in some way. In contrast, with experimental data, the experimenter controls the environment under which test runs are conducted, and in which responses are observed and recorded. Thus with experimentation, the population is an abstract entity consisting of potential outcomes of test runs made under conditions we enforce, rather than a physical entity that we observe without changing it.

We say this because the default behavior of the emmeans() function is to average groups together with equal weights; this is common in analysis of experiments, but not common in analysis of observational data; and I think that misunderstandings about this underlie some criticisms such as are found here and here.

Consider, for example, a classic Latin square experimental design. RA Fisher and others expounded on such designs. Suppose we want to compare four treatments, say fertilizers, in an agricultural experiment. A Latin square plan would involve dividing a parcel of land into four rows and four columns, defining 16 plots. Then we apply one of the fertilizers to each plot in such a way that each fertilizer appears once in each row and once in each column (and thus, each row and each column contains all four fertilizers). This scheme, to some extent, controls for possible spatial effects within the land parcel. To compare the fertilizer, we average together the response values (say, yield of a crop) observed on the four plots where each fertilizer was used. It seems right to average these together with equal weight, because each experimental condition seems equally valid and there is no reason to give one more weight than another. In this illustration, the fertilizer means are not marginal means of some physical population; they are simply the means obtained under the four test conditions defined by the experiment.

Emphasis on models

The emmeans package requires you to fit a model to your data. All the results obtained in emmeans rely on this model. So, really, the analysis obtained is really an analysis of the model, not the data. This analysis does depend on the data, but only insofar as the fitted model depends on the data. We use predictions from this model to compute estimated marginal means (EMMs), which will be defined more explicitly below. For now, there are two things to know:

1. If you change the model, that changes the EMMs
2. If the model fits poorly, the EMMs represent the data poorly (the garbage in, garbage out principle)

So to use this package to analyze your data, the most important first step is to fit a good model.

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Illustration: pigs experiment

Consider the pigs dataset provided with the package (help("pigs") provides details). These data come from an experiment where pigs are given different percentages of protein (percent) from different sources (source) in their diet, and later we measured the concentration (conc) of leucine. The percent values are quantitative, but we chose those particular values deliberately, and (at least initially) we want separate estimates at each percent level; that is, we want to view percent as a factor, not a quantitative predictor.

As discussed, our first task is to come up with a good model. Doing so requires a lot of skill, and we don’t want to labor too much over the details; you really need other references to deal with this aspect adequately. But we will briefly discuss five models and settle on one of them:

mod1 <- lm(conc ~ source * factor(percent), data = pigs)
mod2 <- update(mod1, . ~ source + factor(percent))   # no interaction

These models have \(R^2\) values of 0.808 and 0.700, and adjusted \(R^2\) values of 0.684 and 0.634. mod1 is preferable to mod2, suggesting we need the interaction term. However, a residual-vs-predicted plot of mod2 has a classic “horn” shape (curving and fanning out), indicating a situation where a response transformation might help better than including the interaction.

It turns out that an inverse transformation, (1/conc) really serves us well. (Perhaps this isn’t too surprising, as concentrations are typically determined by titration, in which the actual measurements are volumes; and these are reciprocally related to concentrations, i.e., amounts per unit volume.)

So here are three more models:

mod3 <- update(mod1, inverse(conc) ~ .)
mod4 <- update(mod2, inverse(conc) ~ .)     # no interaction
mod5 <- update(mod4, . ~ source + percent)  # linear term for percent

(Note: We could have used 1/conc as the response variable, but emmeans provides an equivalent inverse() function that will prove more advantageous later.) The residual plots for these models look a lot more like a random scatter of points (and that is good). The \(R^2\) values for these models are 0.818, 0.787, and 0.749, respectively; and the adjusted \(R^2\)s are 0.700, 0.740, and 0.719. mod4 has the best adjusted \(R^2\) and will be our choice.

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Estimated marginal means

Now that we have a good model, let’s use the emmeans() function to obtain estimated marginal means (EMMs). We’ll explain them later.

(EMM.source <- emmeans(mod4, "source"))

##  source emmean       SE df lower.CL upper.CL
##  fish   0.0337 0.000926 23   0.0318   0.0356
##  soy    0.0257 0.000945 23   0.0237   0.0276
##  skim   0.0229 0.000994 23   0.0208   0.0249
## 
## Results are averaged over the levels of: percent 
## Results are given on the inverse (not the response) scale. 
## Confidence level used: 0.95

(EMM.percent <- emmeans(mod4, "percent"))

##  percent emmean       SE df lower.CL upper.CL
##        9 0.0322 0.001030 23   0.0301   0.0344
##       12 0.0270 0.000969 23   0.0250   0.0290
##       15 0.0263 0.001100 23   0.0240   0.0286
##       18 0.0241 0.001340 23   0.0213   0.0268
## 
## Results are averaged over the levels of: source 
## Results are given on the inverse (not the response) scale. 
## Confidence level used: 0.95

Let’s compare these with the ordinary marginal means (OMMs) on inverse(conc):

with(pigs, tapply(inverse(conc), source, mean))

##       fish        soy       skim 
## 0.03331687 0.02632333 0.02372024

with(pigs, tapply(inverse(conc), percent, mean))

##          9         12         15         18 
## 0.03146170 0.02700341 0.02602757 0.02659336

Both sets of OMMs are vaguely similar to the corresponding EMMs. However, please note that the EMMs for percent form a decreasing sequence, while the the OMMs decrease but then increase at the end.

The reference grid, and definition of EMMs

Estimated marginal means are defined as marginal means of model predictions over the grid comprising all factor combinations – called the reference grid. For the example at hand, the reference grid is

(RG <- expand.grid(source = levels(pigs$source), percent = unique(pigs$percent)))

##    source percent
## 1    fish       9
## 2     soy       9
## 3    skim       9
## 4    fish      12
## 5     soy      12
## 6    skim      12
## 7    fish      15
## 8     soy      15
## 9    skim      15
## 10   fish      18
## 11    soy      18
## 12   skim      18

To get the EMMs, we first need to obtain predictions on this grid:

(preds <- matrix(predict(mod4, newdata = RG), nrow = 3))

##            [,1]       [,2]       [,3]       [,4]
## [1,] 0.03853514 0.03329091 0.03256404 0.03036586
## [2,] 0.03050486 0.02526063 0.02453376 0.02233558
## [3,] 0.02770292 0.02245869 0.02173182 0.01953364

then obtain the marginal means of these predictions:

apply(preds, 1, mean)   # row means -- for source

## [1] 0.03368899 0.02565870 0.02285677

apply(preds, 2, mean)   # column means -- for percent

## [1] 0.03224764 0.02700341 0.02627654 0.02407836

These marginal averages match the EMMs obtained earlier via emmeans().

Now let’s go back to the comparison with the ordinary marginal means. The source levels are represented by the columns of pred; and note that each row of pred is a decreasing set of values. So it is no wonder that the marginal means – the EMMs for source – are decreasing. That the OMMs for percent do not behave this way is due to the imbalance in sample sizes:

with(pigs, table(source, percent))

##       percent
## source 9 12 15 18
##   fish 2  3  2  3
##   soy  3  3  3  1
##   skim 3  3  2  1

This shows that the OMMs of the last column give most of the weight (3/5) to the first source, which tends to have higher inverse(conc), making the OMM for 18 percent higher than that for 15 percent, even though the reverse is true with every level of source. This kind of disconnect is an example of Simpson’s paradox, in which a confounding factor can distort your findings. The EMMs are not subject to this paradox, but the OMMs are, when the sample sizes are correlated with the expected values.

In summary, we obtain a references grid of all factor combinations, obtain model predictions on that grid, and then the expected marginal means are estimated as equally-weighted marginal averages of those predictions. Those EMMs are not subject to confounding by other factors, such as might happen with ordinary marginal means of the data. Moreover, unlike OMMs, EMMs are based on a model that is fitted to the data.

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More on the reference grid

In the previous section, we discussed the reference grid as being the set of all factor combinations. It is slightly more complicated than that when we have numerical predictors (AKA covariates) in the model. By default, we use the average of each covariate – thus not enlarging the number of combinations comprising the grid. Using the covariate average(s) yields what are often called adjusted means. There is one exception, though: if a covariate has only two different values, we treat it as a factor having those two levels. For example, a model could include an indicator variable male that is 1 if the subject is male, and 0 otherwise. Then male would be viewed as a factor with levels 0 and 1. Note, again, that the reference grid is formulated from the model we are using.

We can see a snapshot of the reference grid via the ref_grid function; for example

(RG4 <- ref_grid(mod4))

## 'emmGrid' object with variables:
##     source = fish, soy, skim
##     percent =  9, 12, 15, 18
## Transformation: "inverse"

ref_grid(mod5)

## 'emmGrid' object with variables:
##     source = fish, soy, skim
##     percent = 12.931
## Transformation: "inverse"

The reference grid for mod5 is different from that for mod4 because in those models, percent is a factor in mod4 and a covariate in mod5.

It is possible to modify the reference grid. In the context of the present example, it might be inetersting to compare EMMs based on mod4 and mod5, and we can put them on an equal footing by using the same percent values as reference levels:

(RG5 <- ref_grid(mod5, at = list(percent = c(9, 12, 15, 18))))

## 'emmGrid' object with variables:
##     source = fish, soy, skim
##     percent =  9, 12, 15, 18
## Transformation: "inverse"

We could also have done this using

(RG5 <- ref_grid(mod5, cov.reduce = FALSE)

… which tells ref_grid() to set covariate levels using unique values. It’s safer to use at because cov.reduce affects all covariates instead of specific ones. ###### {#emmip} The two models’ predictions can be compared using interaction-style plots via the emmip() function

emmip(RG4, source ~ percent, style = "factor")

emmip(RG5, source ~ percent, style = "factor")

Both plots show three parallel trends, because neither model includes an interaction term; but of course for mod5, those trends are straight lines.

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Passing arguments

Quite a few functions in the emmeans package, including emmeans() and emmip(), can take either a model object or a reference-grid object as their first argument. Thus we can obtain EMMs for mod5 directly from RG5, e.g.

emmeans(RG5, "source")

##  source emmean       SE df lower.CL upper.CL
##  fish   0.0336 0.000958 25   0.0316   0.0355
##  soy    0.0255 0.000971 25   0.0235   0.0275
##  skim   0.0227 0.001030 25   0.0206   0.0248
## 
## Results are averaged over the levels of: percent 
## Results are given on the inverse (not the response) scale. 
## Confidence level used: 0.95

These are slightly different results than we had earlier for mod4. In these functions where the model and the reference grid are interchangeable, the first thing the function does is to check which it is; and if it is a model object, it constructs the reference grid. When it does that, it passes its arguments to ref_grid() in case they are needed. For instance, the above EMMs could have been obtained using

emmeans(mod5, "source", at = list(percent = c(9, 12, 15, 18)))
## (same results as above)

It is a great convenience to be able to pass arguments to ref_grid(), but it also can confuse new users, because if we look at the help page for emmeans(), it does not list at as a possible argument. It is mentioned, though, if you look at the ... argument. So develop a habit of looking at documentation for other functions, especially ref_grid(), for other arguments that may affect your results.

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Transformations

In our running example with pigs, by now you are surely tired of seeing all the answers on the inverse(conc) scale. What about estimating things on the conc scale? You may have noticed that the inverse transformation has not been forgotten; it is mentioned in the annotations below the emmeans() output. [I’d also comment that having used inverse(conc) rather than 1/conc as the response variable in the model has made it easier to sort things out, because inverse() is a named transformation that emmeans() can work with.] We can back-transform the results by specifying type = "response" in any function call where it makes sense. For instance,

emmeans(RG4, "source", type = "response")

##  source response    SE df lower.CL upper.CL
##  fish       29.7 0.816 23     28.1     31.5
##  soy        39.0 1.440 23     36.2     42.2
##  skim       43.8 1.900 23     40.1     48.1
## 
## Results are averaged over the levels of: percent 
## Confidence level used: 0.95 
## Intervals are back-transformed from the inverse scale

emmip(RG4, source ~ percent, type = "response")

We are now on the conc scale, and that will likely be less confusing. Compared with the earlier plots in which the trends were decreasing and parallel, this plot has them increasing (because of the inverse relationship) and non-parallel. An interaction that occurs on the response scale is pretty well explained by a model with no interactions on the inverse scale.

Transformations have a lot of nuances, and we refer you to the vignette of transformations for more details.

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Derived covariates

You need to be careful when one covariate depends on the value of another. To illustrate using the datasets::mtcars data, suppose we want to predict mpg using cyl (number of cylinders) as a factor disp (displacement) as a covariate, and include a quadratic term for disp. Here are two equivalent models:

mcmod1 <- lm(mpg ~ factor(cyl) + disp + I(disp^2), data = mtcars)
mtcars <- transform(mtcars, 
                    dispsq = disp^2)
mcmod2 <- lm(mpg ~ factor(cyl) + disp + dispsq, data = mtcars)

These two models have exactly the same predicted values. But look at the EMMs:

emmeans(mcmod1, "cyl")

##  cyl emmean   SE df lower.CL upper.CL
##    4   19.3 2.66 27     13.9     24.8
##    6   17.2 1.36 27     14.4     20.0
##    8   18.8 1.47 27     15.7     21.8
## 
## Confidence level used: 0.95

emmeans(mcmod2, "cyl")

##  cyl emmean   SE df lower.CL upper.CL
##    4   20.8 2.05 27     16.6     25.0
##    6   18.7 1.19 27     16.3     21.1
##    8   20.2 1.77 27     16.6     23.9
## 
## Confidence level used: 0.95

Wow! Those are really different results – even though the models are equivalent. Why is this – and which (if either) is right? To understand, look at the reference grids:

ref_grid(mcmod1)

## 'emmGrid' object with variables:
##     cyl = 4, 6, 8
##     disp = 230.72

ref_grid(mcmod2)

## 'emmGrid' object with variables:
##     cyl = 4, 6, 8
##     disp = 230.72
##     dispsq = 68113

For both models, the reference grid uses the disp mean of 230.72. But for mcmod2, dispsq is a separate covariate, and it is set to its mean of 68113. This is not right, because it is impossible to have disp equal to 230.72 and its square equal to 68113 at the same time! If we use consistent values of disp anddispsq, we get the same results as for mcmod1:

emmeans(mcmod2, "cyl", at = list(disp = 230.72, dispsq = 230.72^2))

##  cyl emmean   SE df lower.CL upper.CL
##    4   19.3 2.66 27     13.9     24.8
##    6   17.2 1.36 27     14.4     20.0
##    8   18.8 1.47 27     15.7     21.8
## 
## Confidence level used: 0.95

In summary, for polynomial models and others where some covariates depend on others in nonlinear ways, it is definitely best to include that dependence in the model formula (as in mcmod1) using I() or poly() expressions, or alter the reference grid so that the dependency among covariates is correct.

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Non-predictor variables

Reference grids are derived using the variables in the right-hand side of the model formula. But sometimes, these variables are not actually predictors. For example:

deg <- 2
mod <- lm(y ~ treat * poly(x, degree = deg), data = mydata)

If we call ref_grid() or emmeans() with this model, it will try to construct a grid of values of treat, x, and deg – causing an error because deg is not a predictor in this model. To get things to work correctly, you need to name deg in a params argument, e.g.,

emmeans(mod, ~ treat | x, at = list(x = 1:3), params = "deg")

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Graphical displays

The results of ref_grid() or emmeans() (these are objects of class emmGrid) may be plotted in two different ways. One we have already seen is an interaction-style plot, using emmip().

The formula specification we used in emmip(RG4, source ~ percent) sets the x variable to be the one on the right-hand side and the “trace” factor (what is used to define the different curves) on the left.

mcmod3 <- lm(mpg ~ disp * cyl, data = mtcars)

EMM3 <- emmeans(mcmod3, ~ cyl | disp, 
                at = list(cyl = c(4,6,8), disp = c(100,200,300)))
plot(EMM3)

mcrg <- ref_grid(mcmod3, at = list(cyl = c(4,6,8)),
                         cov.reduce = disp ~ cyl)
mcrg @ grid

##        disp cyl .wgt.
## 1  93.78673   4     1
## 2 218.98458   6     1
## 3 344.18243   8     1

plot(mcrg)

require("ggplot2")
emmip(mod4, ~ percent | source, CIs = TRUE, type = "response") +
    geom_point(aes(x = percent, y = conc), data = pigs, pch = 2, color = "blue")

Formatting results

If you want to include emmeans() results in a report, you might want to have it in a nicer format than just the printed output. We provide a little bit of help for this, especially if you are using RMarkdown or SWeave to prepare the report. There is an xtable method for exporting these results, which we do not illustrate here but it works similarly to xtable() in other contexts. Also, the export option the print() method allows the user to save exactly what is seen in the printed output as text, to be saved or formatted as the user likes (see the documentation for print.emmGrid for details). Here is an example using one of the objects above:

ci <- confint(mcrg, level = 0.90, adjust = "scheffe")
xport <- print(ci, export = TRUE)
cat("<font color = 'blue'>\n")
knitr::kable(xport$summary, align = "r")
for (a in xport$annotations) cat(paste(a, "<br>"))
cat("</font>\n")

Confidence level used: 0.9
Conf-level adjustment: scheffe method with rank 3

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Using weights

As we have mentioned, emmeans() uses equal weighting by default, based on its foundations in experimental situations. When you have observational data, you are more likely to use unequal weights that more accurately characterize the population. Accordingly, a weights argument is provided in emmeans(). For example, using weights = "cells" in the call will weight the predictions according to their cell frequencies (recall this information is retained in the reference grid). This produces results comparable to ordinary marginal means:

emmeans(mod4, "percent", weights = "cells")

##  percent emmean       SE df lower.CL upper.CL
##        9 0.0315 0.001030 23   0.0293   0.0336
##       12 0.0270 0.000969 23   0.0250   0.0290
##       15 0.0260 0.001100 23   0.0238   0.0283
##       18 0.0266 0.001300 23   0.0239   0.0293
## 
## Results are averaged over the levels of: source 
## Results are given on the inverse (not the response) scale. 
## Confidence level used: 0.95

Note that, as in the ordinary marginal means we obtained long ago, the highest estimate is for percent = 15 rather than percent = 18. It is interesting to compare this with the results for a model that includes only percent as a predictor.

mod6 <- lm(inverse(conc) ~ factor(percent), data = pigs)
emmeans(mod6, "percent")

##  percent emmean      SE df lower.CL upper.CL
##        9 0.0315 0.00196 25   0.0274   0.0355
##       12 0.0270 0.00185 25   0.0232   0.0308
##       15 0.0260 0.00210 25   0.0217   0.0303
##       18 0.0266 0.00248 25   0.0215   0.0317
## 
## Results are given on the inverse (not the response) scale. 
## Confidence level used: 0.95

The EMMs in these two tables are identical, so in some sense, weights = "cells" amounts to throwing-out the uninvolved factors. However, note that these outputs show markedly different standard errors. That is because the model mod4 accounts for variations due to source while mod6 does not. The lesson here is that it is possible to obtain statistics comparable to ordinary marginal means, while still accounting for variations due to the factors that are being averaged over.

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Multivariate responses

The emmeans package supports various multivariate models. When there is a multivariate response, the dimensions of that response are treated as if they were levels of a factor. For example, the MOats dataset provided in the package has predictors Block and Variety, and a four-dimensional response yield giving yields observed with varying amounts of nitrogen added to the soil. Here is a model and reference grid:

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

## 'emmGrid' object with variables:
##     Block = VI, V, III, IV, II, I
##     Variety = Golden Rain, Marvellous, Victory
##     nitro = multivariate response levels: 0, 0.2, 0.4, 0.6

So, nitro is regarded as a factor having 4 levels corresponding to the 4 dimensions of yield. We can subsequently obtain EMMs for any of the factors Block, Variety, nitro, or combinations thereof. The argument mult.name = "nitro" is optional; if it had been excluded, the multivariate levels would have been named rep.meas.

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A set of comparisons or well-chosen contrasts is more useful and interpretable than an omnibus F test

F tests are useful for model selection, but don’t tell you anything specific about the nature of an effect. If F has a small P value, it suggests that there is some effect, somewhere. It doesn’t even necessarily imply that any two means differ statistically.

Use adjusted P values

When you run a bunch of tests, there is a risk of making too many type-I errors, and adjusted P values (e.g., the Tukey adjustment for pairwise comparisons) keep you from making too many mistakes. That said, it is possible to go overboard; and it’s usually reasonable to regard each “by” group as a separate family of tests for purposes of adjustment.

It is not necessary to have a significant F test as a prerequisite to doing comparisons or contrasts

… as long as an appropriate adjustment is used. There do exist rules such as the “protected LSD” by which one is given license to do unadjusted comparisons provided the \(F\) statistic is “significant.” However, this is a very weak form of protection for which the justification is, basically, “if \(F\) is significant, you can say absolutely anything you want.”

Get the model right first

Everything the emmeans package does is an interpretation of the model that you fitted to the data. If the model is bad, you will get bad results from emmeans() and other functions. Every single limitation of your model, be it presuming constant error variance, omitting interaction terms, etc., becomes a limitation of the results emmeans() produces. So do a responsible job of fitting the model. And if you don’t know what’s meant by that…

Consider seeking the advice of a statistical consultant

Statistics is hard. It is a lot more than just running programs and copying output. We began this vignette by emphasizing we need to start with a good model; that is an artful task, and certainly what is shown here only hints at what is required; you may need help with it. It is your research; is it important that it be done right? Many academic statistics and biostatistics departments can refer you to someone who can help.

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References

Wasserstein RL, Lazar NA (2016) “The ASA’s Statement on p-Values: Context, Process, and Purpose,” The American Statistician, 70, 129–133, https://doi.org/10.1080/00031305.2016.1154108

Wasserstein RL, Schirm AL, Lazar, NA (2019) “Moving to a World Beyond ‘p < 0.05’,” The American Statistician, 73, 1–19, https://doi.org/10.1080/00031305.2019.1583913