---
title: "Basics of estimated marginal means"
author: "Russ Lenth"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Basics of EMMs}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---
```{r, echo = FALSE, results = "hide", message = FALSE}
require("emmeans")
knitr::opts_chunk$set(collapse = TRUE, fig.width = 4.5)
```

## Contents {#contents}

  1. [Motivating example](#motivation)
  2. [EMMs defined](#EMMdef)
      a. [Reference grids](#ref_grid)
      b. [Estimated marginal means](#emmeans)
      b. [Altering the reference grid](#altering)
      b. [Graphical displays](#plots)
      c. [Weighting](#weights)
      d. [Multivariate models](#multiv)
  3. [Summary](#summary)
  4. [Further reading](#more)
  
[Vignette index](index.html)


## Why we need EMMs {#motivation}
Consider the `pigs` dataset provided with the package (`help("pigs")` provides
details). These data come from an unbalanced experiment where pigs are given
different percentages of protein (`percent`) from different sources (`source`)
in their diet, and later we measure the concentration (`conc`) of leucine.
Here's an interaction plot showing the mean `conc` at each combination of 
the other factors.
```{r}
par(mar = .1 + c(4, 4, 1, 1))   # reduce head space
with(pigs, interaction.plot(percent, source, conc))
```

This plot suggests that with each `source`, `conc` tends to go up with 
`percent`, but that the mean differs with each `source`.

Now, suppose that we want to assess, numerically, the marginal results for
`percent`.  The natural thing to do is to obtain the marginal means:
```{r}
with(pigs, tapply(conc, percent, mean))
```
Looking at the plot, it seems a bit surprising that the last three means
are all about the same, with the one for 15 percent being the largest.

Hmmmm, so let's try another approach -- actually averaging together the values
we see in the plot. First, we need the means that are shown there:
```{r}
cell.means <- matrix(with(pigs, 
    tapply(conc, interaction(source, percent), mean)), 
    nrow = 3)
cell.means
```
Confirm that the rows of this matrix match the plotted values for fish,
soy, and skim, respectively. Now, average each column:
```{r}
apply(cell.means, 2, mean)
```
These results are decidedly different from the ordinary marginal means we 
obtained earlier. What's going on? The answer is that some observations were
lost, making the data unbalanced:
```{r}
with(pigs, table(source, percent))
```
We can reproduce the marginal means by weighting the cell means with these
frequencies. For example, in the last column:
```{r}
sum(c(3, 1, 1) * cell.means[, 4]) / 5
```
The big discrepancy between the ordinary mean for `percent = 18` and the marginal
mean from `cell.means` is due to the fact that the lowest value receives 3 times
the weight as the other two values.

### The point
The point is that the marginal means of `cell.means` give *equal weight* to each
cell. In many situations (especially with experimental data), that is a much
fairer way to compute marginal means, in that they are not biased by imbalances
in the data. we are, in a sense, estimating what the marginal means *would* be,
had the experiment been balanced. Estimated marginal means (EMMs) serve that
need.

All this said, there are certainly situations where equal weighting is *not* 
appropriate. Suppose, for example, we have data on sales of a product given 
different packaging and features. The data could be unbalanced because customers
are more attracted to some combinations than others. If our goal is to 
understand scientifically what packaging and features are inherently more 
profitable, then equally weighted EMMs may be appropriate; but if our goal is to
predict or maximize profit, the ordinary marginal means provide better estimates
of what we can expect in the marketplace.

[Back to Contents](#contents)

## What exactly are EMMs? {#EMMdef}

### Model and reference grid {#ref_grid}
Estimated marginal means are based on a *model* -- not directly on data. 
The basis for them is what we call the *reference grid* for a given model.
To obtain the reference grid, consider all the predictors in the model.
Here are the default rules for constructing the reference grid

  * For each predictor that is a *factor*, use its levels (dropping unused ones)
  * For each numeric predictor (covariate), use its average
  
The reference grid is then a regular grid of all combinations of these
reference levels.

As a simple example, consider again the `pigs` dataset (see `help("fiber")` for
details). Examination of residual plots from preliminary models suggests that it
is a good idea to work in terms of log concentration.

If we treat the predictor `percent` as a factor, we might fit the 
following model:
```{r}
pigs.lm1 <- lm(log(conc) ~ source + factor(percent), data = pigs)
```
The reference grid for this model can be found via the `ref_grid` function:
```{r}
ref_grid(pigs.lm1)
```
Both predictors are factors, and the reference grid consists of the 
$3\times4 = 12$ combinations of these factor levels. It can be seen explicitly
by looking at the `grid` slot of this object:
```{r}
ref_grid(pigs.lm1) @ grid
``` 
Note that other information is retained in the reference grid, e.g., the
transformation used on the response, and the cell counts as the `.wgt.` column.

Now, suppose instead that we treat `percent` as a numeric predictor. 
This leads to a different model -- and a different reference grid.
```{r}
pigs.lm2 <- lm(log(conc) ~ source + percent, data = pigs)
ref_grid(pigs.lm2)
```
This reference grid has the levels of `source`, but only one `percent` value, 
its average. Thus, the grid has only three elements:
```{r}
ref_grid(pigs.lm2) @ grid
```

[Back to Contents](#contents)

### Estimated marginal means {#emmeans}
Once the reference grid is established, we can consider using the model to
estimate the mean at each point in the reference grid. (Curiously, the
convention is to call this "prediction" rather than "estimation"). For
`pigs.lm1`, we have
```{r}
pigs.pred1 <- matrix(predict(ref_grid(pigs.lm1)), nrow = 3)
pigs.pred1
```
Estimated marginal means (EMMs) are defined as equally weighted means of these
predictions at specified margins:
```{r}
apply(pigs.pred1, 1, mean) ### EMMs for source

apply(pigs.pred1, 2, mean) ### EMMs for percent
``` 
For the other model, `pigs.lm2`, we have only one point in the reference
grid for each `source` level; so the EMMs for `source` are just the predictions
themselves:
```{r}
predict(ref_grid(pigs.lm2))
```
These are slightly different from the previous EMMs for `source`, emphasizing
the fact that EMMs are model-dependent. In models with covariates, EMMs are
often called *adjusted means*.

The `emmeans` function computes EMMs, accompanied by standard errors and 
confidence intervals. For example,
```{r}
emmeans(pigs.lm1, "percent")
```

In these examples, all the results are presented on the `log(conc)` scale
(and the annotations in the output warn of this).
It is possible to convert them back to the `conc` scale by back-transforming.
This topic is discussed in [the vignette on transformations](transformations.html).

An additional note: There is an exception to the definition of EMMs given
here. If the model has a nested structure in the fixed effects, then averaging
is performed separately in each nesting group. See the [section on nesting in the
"messy-data" vignette](messy-data.html#nesting) for an example.

[Back to Contents](#contents)

### Altering the reference grid {#altering}
It is possible to alter the reference grid. We might, for example, want to 
define a reference grid for `pigs.lm2` that is comparable to the one for 
`pigs.lm1`.
```{r}
ref_grid(pigs.lm2, cov.reduce = FALSE)
```
Using `cov.reduce = FALSE` specifies that, instead of using the mean, the
reference grid should use all the unique values of each covariate. Be careful
with this: it can create quite a mess if there is a covariate that was measured
rather than experimentally varied. Another option is to give a function; e.g.,
```{r}
ref_grid(pigs.lm2, cov.reduce = range)
```

Perhaps more common is to use the `at` argument. Consider this model for the
built-in `cars` dataset:
```{r}
mtcars.lm <- lm(mpg ~ disp * cyl, data = mtcars)
ref_grid(mtcars.lm)
```
Since both predictors are numeric, the default reference grid has only one 
point. For purposes of describing the fitted model, you might want to obtain
predictions at a grid of points, like this:
```{r}
mtcars.rg <- ref_grid(mtcars.lm, 
    at = list(disp = c(100, 200, 300), cyl = c(4, 6, 8)))
```
... which will create a 3 x 3 grid. (We'll look at this model again shortly.)
Another use of `at` is to focus on only some of the levels of a factor. Note that
`at` does not need to specify every predictor; those not mentioned in `at` are
handled by `cov.reduce` or the default methods.


[Back to Contents](#contents)

### Graphical displays {#plots}
The results of `ref_grid()` or `emmeans()` (these are objects of class `emmGrid`)
may be plotted in two different 
ways. One is an interaction-style plot, using `emmip()`. In the following, let's 
use it to compare the predictions from `pigs.lm1` and `pigs.lm2`:
```{r}
emmip(pigs.lm1, source ~ percent)
emmip(ref_grid(pigs.lm2, cov.reduce = FALSE), source ~ percent)
```

Notice that `emmip()` may also be used on a fitted model. The formula
specification needs the *x* variable on the right-hand side and the "trace"
factor (what is used to define the different curves) on the left.
This is a good time to yet again emphasize that EMMs are based on a *model*.
Neither of these plots is an interaction plot of the *data*; they are 
interaction plots of model predictions; and since both models do not include
an interaction, no interaction at all is evident in the plots.

The other graphics option offered is the `plot()` method for `emmGrid` objects. In
the following, we display the estimates and 95% confidence intervals for
`mtcars.rg` in separate panels for each `disp`.
```{r}
plot(mtcars.rg, by = "disp")
```

This plot illustrates, as much as anything else, how silly it is to try to
predict mileage for a 4-cylinder car having high displacement, or an 8-cylinder
car having low displacement. The widths of the intervals give us a clue that we
are extrapolating. A better idea is to acknowledge that displacement largely
depends on the number of cylinders. So here is yet another way to 
use `cov.reduce` to modify the reference grid:
```{r}
mtcars.rg_d.c <- ref_grid(mtcars.lm, at = list(cyl = c(4,6,8)),
                          cov.reduce = disp ~ cyl)
mtcars.rg_d.c @ grid
```
The `ref_grid` call specifies that `disp` depends on `cyl`; so a linear model 
is fitted with the given formula and its fitted values are used as the `disp`
values -- only one for each `cyl`. If we plot this grid, the results are 
sensible, reflecting what the model predicts for typical cars with each 
number of cylinders:
```{r fig.height = 1.5}
plot(mtcars.rg_d.c)
```

Wizards with the **ggplot2** package can further enhance these plots if 
they like. For example, we can add the data to an interaction plot -- this
time we opt to include confidence intervals and put the three sources 
in separate panels:
```{r}
require("ggplot2")
emmip(pigs.lm1, ~ percent | source, CIs = TRUE) +
    geom_point(aes(x = factor(percent), y = log(conc)), data = pigs, pch = 2, color = "blue")
```

[Back to Contents](#contents)

### Using weights {#weights}
It is possible to override the equal-weighting method for computing EMMs. 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:
```{r}
emmeans(pigs.lm1, "percent", weights = "cells")
```
Note that, as in the ordinary means in [the motivating example](#motivation),
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.
```{r}
pigs.lm3 <- lm(log(conc) ~ factor(percent), data = pigs)
emmeans(pigs.lm3, "percent")
```
The EMMs in these two tables are identical, but their standard errors are
considerably different. That is because the model `pigs.lm1` accounts for 
variations due to `source`. 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.

[Back to Contents](#contents)

### Multivariate responses {#multiv}
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:
```{r}
MOats.lm <- lm (yield ~ Block + Variety, data = MOats)
ref_grid (MOats.lm, mult.name = "nitro")
```
So, `nitro` is in essence 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`.

[Back to Contents](#contents)

## Summary of main points {#summary}
  * EMMs are based on a *model*. A different model for the same data may lead
    to different EMMs.
  * EMMs are based on a *reference grid* consisting of all combinations
    of factor levels, with each covariate set to its average (by default).
  * For purposes of defining the reference grid, dimensions of
    a multivariate response are treated as levels of a factor.
  * EMMs are then predictions on this reference grid, or marginal
    averages thereof (equally weighted by default).
  * Reference grids may be modified using `at` or `cov.reduce`;
    the latter may be logical, a function, or a formula.
  * Reference grids and `emmeans()` results may be plotted via `plot()`
    (for parallel confidence intervals) or `emmip()` (for an interaction-style
    plot).

## Further reading {#more}
The reader is referred to other vignettes for more details and advanced use.
The strings linked below are the names of the vignettes; i.e., they can
also be accessed via `vignette("`*name*`", "emmeans")`

  * Models that are supported in **emmeans** (there are lots of them)
    ["models"](models.html)
  * Confidence intervals and tests: 
    ["confidence-intervals"](confidence-intervals.html)
  * Often, users want to compare or contrast EMMs: ["comparisons"](comparisons.html)
  * Working with response transformations and link functions:
    ["transformations"](transformations.html)
  * Multi-factor models with interactions: ["interactions"](interactions.html)
  * Working with messy data and nested effects: ["messy-data"](messy-data.html)
  * Examples of more sophisticated models (e.g., mixed, ordinal, MCMC)
    ["sophisticated"](sophisticated.html)
  * Utilities for working with `emmGrid` objects: ["utilities"](utilities.html)
  * Frequently asked questions: ["FAQs"](FAQs.html)

[Back to Contents](#contents)

[Vignette index](index.html)