--- title: "Basics of estimated marginal means" author: "emmeans package, Version `r packageVersion('emmeans')`" output: emmeans::.emm_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(fig.width = 4.5, class.output = "ro") ``` ## Contents {#contents} 1. [Motivating example](#motivation) 2. [EMMs defined](#EMMdef) a. [Reference grids](#ref_grid) b. [Estimated marginal means](#emmeans) c. [Altering the reference grid](#altering) d. [Derived covariates](#depcovs) d. [Non-predictor variables](#params) e. [Graphical displays](#plots) e. [Formatting results](#formatting) f. [Weighting](#weights) g. [Multivariate models](#multiv) 3. [Objects, structures, and methods](#emmobj) 4. [P values, "significance", and recommendations](#pvalues) 5. [Summary](#summary) 6. [Further reading](#more) [Index of all vignette topics](vignette-topics.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, echo = FALSE} par(mar = .1 + c(4, 4, 1, 1)) # reduce head space ``` ```{r} 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 {#eqwts} 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.[^1] 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) ``` (*Note:* Many of the calculations that follow are meant to illustrate what is inside this reference-grid object; You don't need to do such calculations yourself in routine analysis; just use the `emmeans()` (or possibly `ref_grid()`) function as we do later.) In this model, 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 ``` [^1]: In newer versions of **emmeans**, however, covariates having only two distinct values are by default treated as two-level factors, though there is an option to reduce them to their mean. [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.keep = "percent") ``` Using `cov.keep = "percent"` specifies that, instead of using the mean, the reference grid should use all the unique values of `each covariate`"percent"`. Another option is to specify a `cov.reduce` function that is used in place of the mean; e.g., ```{r} ref_grid(pigs.lm2, cov.reduce = range) ``` Another option is to use the `at` argument. Consider this model for the built-in `mtcars` 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, cov.keep = 3, at = list(disp = c(100, 200, 300))) mtcars.rg ``` This illustrates two things: a new use of `cov.keep` and the `at` argument. `cov.keep = "3"` specifies that any covariates having 3 or fewer unique values is treated like a factor (the system default is `cov.keep = "2"`). The `at` specification gives three values of `disp`, overriding the default behavior to use the mean of `disp`. 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`, `cov.keep`, or the default methods. Also, covariate values in `at` need not be values that actually occur in the data, whereas `cov.keep` will use only values that are achieved. [Back to Contents](#contents) ### Derived covariates {#depcovs} You need to be careful when one covariate depends on the value of another. To illustrate in the `mtcars` example, suppose we want to use `cyl` as a factor and include a quadratic term for `disp`: ```{r} mtcars.1 <- lm(mpg ~ factor(cyl) + disp + I(disp^2), data = mtcars) emmeans(mtcars.1, "cyl") ``` Some users may not like function calls in the model formula, so they instead do something like this: ```{r} mtcars <- transform(mtcars, Cyl = factor(cyl), dispsq = disp^2) mtcars.2 <- lm(mpg ~ Cyl + disp + dispsq, data = mtcars) emmeans(mtcars.2, "Cyl") ``` Wow! Those are really different results -- even though the models are equivalent. Why is this? To understand, look at the reference grids: ```{r} ref_grid(mtcars.1) ref_grid(mtcars.2) ``` For both models, the reference grid uses the `disp` mean of 230.72. But for `mtcars.2`, we also set `dispsq` to its mean of 68113. This is not right, because `dispsq` should be the square of `disp` (about 53232, not 68113) in order to be consistent. If we use that value of `dispsq`, we get the same results (modulus rounding error) as for `mtcars.1`: ```{r} emmeans(mtcars.2, "Cyl", at = list(dispsq = 230.72^2)) ``` In summary, for polynomial models and others where some covariates depend on others in nonlinear ways, include that dependence in the model formula (as in `mtcars.1`) using `I()` or `poly()` expressions, or alter the reference grid so that the dependency among covariates is correct. ### Non-predictor variables {#params} 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: ```{r, eval = FALSE} 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., ```{r, eval = FALSE} emmeans(mod, ~ treat | x, at = list(x = 1:3), params = "deg") ``` [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. ###### {#plot.emmGrid} 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) ``` ###### {#ggplot} 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 = percent, y = log(conc)), data = pigs, pch = 2, color = "blue") ``` ### Formatting results {#formatting} 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: ```{r, eval = FALSE} ci <- confint(mtcars.rg_d.c, level = 0.90, adjust = "scheffe") xport <- print(ci, export = TRUE) cat("\n") knitr::kable(xport$summary, align = "r") for (a in xport$annotations) cat(paste(a, "
")) cat("\n") ``` ```{r, results = "asis", echo = FALSE} ci <- confint(mtcars.rg_d.c, level = 0.90, adjust = "scheffe") xport <- print(ci, export = TRUE) cat("\n") knitr::kable(xport$summary, align = "r") for (a in xport$annotations) cat(paste(a, "
")) cat("\n") ``` [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 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`. [Back to Contents](#contents) ## Objects, structures, and methods {#emmobj} The `ref_grid()` and `emmeans()` functions are introduced previously. These functions, and a few related ones, return an object of class `emmGrid`: ```{r} pigs.rg <- ref_grid(pigs.lm1) class(pigs.rg) pigs.emm.s <- emmeans(pigs.rg, "source") class(pigs.emm.s) ``` If you simply show these objects, you get different-looking results: ```{r} pigs.rg pigs.emm.s ``` This is based on guessing what users most need to see when displaying the object. You can override these defaults; for example to just see a quick summary of what is there, do ```{r} str(pigs.emm.s) ``` The most important method for `emmGrid` objects is `summary()`. It is used as the print method for displaying an `emmeans()` result. For this reason, arguments for `summary()` may also be specified within most functions that produce `these kinds of results.`emmGrid` objects. For example: ```{r} # equivalent to summary(emmeans(pigs.lm1, "percent"), level = 0.90, infer = TRUE)) emmeans(pigs.lm1, "percent", level = 0.90, infer = TRUE) ``` This `summary()` method for `emmGrid` objects) actually produces a `data.frame`, but with extra bells and whistles: ```{r} class(summary(pigs.emm.s)) ``` This can be useful to know because if you want to actually *use* `emmeans()` results in other computations, you should save its summary, and then you can access those results just like you would access data in a data frame. The `emmGrid` object itself is not so accessible. There is a `print.summary_emm()` function that is what actually produces the output you see above -- a data frame with extra annotations. [Back to Contents](#contents) ## P values, "significance", and recommendations {#pvalues} There is some debate among statisticians and researchers about the appropriateness of *P* values, and that the term "statistical significance" can be misleading. If you have a small *P* value, it *only* means that the effect being tested is unlikely to be explained by chance variation alone, in the context of the current study and the current statistical model underlying the test. If you have a large *P* value, it *only* means that the observed effect could plausibly be due to chance alone: it is *wrong* to conclude that there is no effect. The American Statistical Association has for some time been advocating very cautious use of *P* values (Wasserman *et al.* 2014) because it is too often misinterpreted, and too often used carelessly. Wasserman *et al.* (2019) even goes so far as to advise against *ever* using the term "statistically significant". The 43 articles it accompanies in the same issue of *TAS*, recommend a number of alternatives. I do not agree with all that is said in the main article, and there are portions that are too cutesy or wander off-topic. Further, it is quite dizzying to try to digest all the accompanying articles, and to reconcile their disagreeing viewpoints. For some time I included a summary of Wasserman *et al.*'s recommendations and their *ATOM* paradigm (Acceptance of uncertainty, Thoughtfulness, Openness, Modesty). But in the meantime, I have handled a large number of user questions, and many of those have made it clear to me that there are more important fish to fry in a vignette section like this. It is just a fact that *P* values are used, and are useful. So I have my own set of recommendations regarding them. #### A set of comparisons or well-chosen contrasts is more useful and interpretable than an omnibus *F* test {#recs1} *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 {#recs2} ... 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 {#recs3} Statistics is hard. It is a lot more than just running programs and copying output. 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. [Back to Contents](#contents) ## Summary of main points {#summary} * EMMs are derived from 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). * Be cautious with the terms "significant" and "nonsignificant", and don't ever interpret a "nonsignificant" result as saying that there is no effect. * Follow good practices such as getting the model right first, and using adjusted *P* values for appropriately chosen families of comparisons or contrasts. [Back to Contents](#contents) ### References Wasserman 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 Wasserman 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 ## 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) * Making predictions from your model: ["predictions"](predictions.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) * Adding **emmeans** support to your package: ["xtending"](xtending.html) [Back to Contents](#contents) [Index of all vignette topics](vignette-topics.html)