Sophisticated models in emmeans" In emmeans: Estimated Marginal Means, aka Least-Squares Means

```require("emmeans")
options(show.signif.stars = FALSE, width = 100)
knitr::opts_chunk\$set(fig.width = 4.5, class.output = "ro")
```

This vignette gives a few examples of the use of the emmeans package to analyze other than the basic types of models provided by the stats package. Emphasis here is placed on accessing the optional capabilities that are typically not needed for the more basic models. A reference for all supported models is provided in the "models" vignette.

Linear mixed models (lmer) {#lmer}

Linear mixed models are really important in statistics. Emphasis here is placed on those fitted using `lme4::lmer()`, but emmeans also supports other mixed-model packages such as nlme.

To illustrate, consider the `Oats` dataset in the nlme package. It has the results of a balanced split-plot experiment: experimental blocks are divided into plots that are randomly assigned to oat varieties, and the plots are subdivided into subplots that are randomly assigned to amounts of nitrogen within each plot. We will consider a linear mixed model for these data, excluding interaction (which is justified in this case). For sake of illustration, we will exclude a few observations.

```Oats.lmer <- lme4::lmer(yield ~ Variety + factor(nitro) + (1|Block/Variety),
data = nlme::Oats, subset = -c(1,2,3,5,8,13,21,34,55))
```

Let's look at the EMMs for `nitro`:

```Oats.emm.n <- emmeans(Oats.lmer, "nitro")
Oats.emm.n
```

You will notice that the degrees of freedom are fractional: that is due to the fact that whole-plot and subplot variations are combined when standard errors are estimated. Different degrees-of-freedom methods are available. By default, the Kenward-Roger method is used, and that's why you see a message about the pbkrtest package being loaded, as it implements that method. We may specify a different degrees-of-freedom method via the optional argument `lmer.df`:

```emmeans(Oats.lmer, "nitro", lmer.df = "satterthwaite")
```
{#dfoptions}

This latest result uses the Satterthwaite method, which is implemented in the lmerTest package. Note that, with this method, not only are the degrees of freedom slightly different, but so are the standard errors. That is because the Kenward-Roger method also entails making a bias adjustment to the covariance matrix of the fixed effects; that is the principal difference between the methods. A third possibility is `"asymptotic"`:

```emmeans(Oats.lmer, "nitro", lmer.df = "asymptotic")
```

This just sets all the degrees of freedom to `Inf` -- that's emmeans's way of using z statistics rather than t statistics. The asymptotic methods tend to make confidence intervals a bit too narrow and P values a bit too low; but they involve much, much less computation. Note that the SEs are the same as obtained using the Satterthwaite method.

Comparisons and contrasts are pretty much the same as with other models. As `nitro` has quantitative levels, we might want to test polynomial contrasts:

```contrast(Oats.emm.n, "poly")
```

The interesting thing here is that the degrees of freedom are much larger than they are for the EMMs. The reason is because `nitro` within-plot factor, so inter-plot variations have little role in estimating contrasts among `nitro` levels. On the other hand, `Variety` is a whole-plot factor, and there is not much of a bump in degrees of freedom for comparisons:

```emmeans(Oats.lmer, pairwise ~ Variety)
```

System options for lmerMod models {#lmerOpts}

The computation required to compute the adjusted covariance matrix and degrees of freedom may become cumbersome. Some user options (i.e., `emm_options()` calls) make it possible to streamline these computations through default methods and limitations on them. First, the option `lmer.df`, which may have values of `"kenward-roger"`, `"satterthwaite"`, or `"asymptotic"` (partial matches are OK!) specifies the default degrees-of-freedom method.

The options `disable.pbkrtest` and `disable.lmerTest` may be `TRUE` or `FALSE`, and comprise another way of controlling which method is used (e.g., the Kenward-Roger method will not be used if ```get_emm_option("disable.pbkrtest") == TRUE```). Finally, the options `pbkrtest.limit` and `lmerTest.limit`, which should be set to numeric values, enable the given package conditionally on whether the number of data rows does not exceed the given limit. The factory default is 3000 for both limits.

Back to Contents

Models with offsets {#offsets}

If a model is fitted and its formula includes an `offset()` term, then by default, the offset is computed and included in the reference grid. To illustrate, consider a hypothetical dataset on insurance claims (used as an example in SAS's documentation). There are classes of cars of varying counts (`n`), sizes (`size`), and age (`age`), and we record the number of insurance claims (`claims`). We fit a Poisson model to `claims` as a function of `size` and `age`. An offset of `log(n)` is included so that `n` functions as an "exposure" variable.

```ins <- data.frame(
n = c(500, 1200, 100, 400, 500, 300),
size = factor(rep(1:3,2), labels = c("S","M","L")),
age = factor(rep(1:2, each = 3)),
claims = c(42, 37, 1, 101, 73, 14))
ins.glm <- glm(claims ~ size + age + offset(log(n)),
data = ins, family = "poisson")
```

First, let's look at the reference grid obtained by default:

```ref_grid(ins.glm)
```

Note that `n` is included in the reference grid and that its average value of 500 is used for all predictions. Thus, if we obtain EMMs for, say, `age`, these are results are based on a pool of 500 cars:

```emmeans(ins.glm, "size", type = "response")
```

However, many users would like to ignore the offset for this kind of model, because then the estimates we obtain are rates per unit value of the (logged) offset. This may be accomplished by specifying an `offset` parameter in the call:

```emmeans(ins.glm, "size", type = "response", offset = 0)
```

You may verify that the above estimates are 1/500th of the previous ones. You may also verify that the above results are identical to those obtained by setting `n` equal to 1:

```emmeans(ins.glm, "size", type = "response", at = list(n = 1))
```

However, those who use these types of models will be more comfortable directly setting the offset to zero.

By the way, you may set some other reference value for the rates. For example, if you want estimates of claims per 100 cars, simply use (results not shown):

```emmeans(ins.glm, "size", type = "response", offset = log(100))
```

Back to Contents

Ordinal models {#ordinal}

Ordinal-response models comprise an example where several options are available for obtaining EMMs. To illustrate, consider the `wine` data in the ordinal package. The response is a rating of bitterness on a five-point scale. we will consider a probit model in two factors during fermentation: `temp` (temperature) and `contact` (contact with grape skins), with the judge making the rating as a scale predictor:

```require("ordinal")
wine.clm <- clm(rating ~ temp + contact, scale = ~ judge,
data = wine, link = "probit")
```

(in earlier modeling, we found little interaction between the factors.) Here are the EMMs for each factor using default options:

```emmeans(wine.clm, list(pairwise ~ temp, pairwise ~ contact))
```

These results are on the "latent" scale; the idea is that there is a continuous random variable (in this case normal, due to the probit link) having a mean that depends on the predictors; and that the ratings are a discretization of the latent variable based on a fixed set of cut points (which are estimated). In this particular example, we also have a scale model that says that the variance of the latent variable depends on the judges. The latent results are quite a bit like those for measurement data, making them easy to interpret. The only catch is that they are not uniquely defined: we could apply a linear transformation to them, and the same linear transformation to the cut points, and the results would be the same.

{#ordlp}

The `clm` function actually fits the model using an ordinary probit model but with different intercepts for each cut point. We can get detailed information for this model by specifying `mode = "linear.predictor"`:

```tmp <- ref_grid(wine.clm, mode = "lin")
tmp
```

Note that this reference grid involves an additional constructed predictor named `cut` that accounts for the different intercepts in the model. Let's obtain EMMs for `temp` on the linear-predictor scale:

```emmeans(tmp, "temp")
```

These are just the negatives of the latent results obtained earlier (the sign is changed to make the comparisons go the right direction). Closely related to this is `mode = "cum.prob"` and `mode = "exc.prob"`, which simply transform the linear predictor to cumulative probabilities and exceedance (1 - cumulative) probabilities. These modes give us access to the details of the fitted model but are cumbersome to use for describing results. When they can become useful is when you want to work in terms of a particular cut point. Let's look at `temp` again in terms of the probability that the rating will be at least 4:

```emmeans(wine.clm, ~ temp, mode = "exc.prob", at = list(cut = "3|4"))
```
{#ordprob}

There are yet more modes! With `mode = "prob"`, we obtain estimates of the probability distribution of each rating. Its reference grid includes a factor with the same name as the model response -- in this case `rating`. We usually want to use that as the primary factor, and the factors of interest as `by` variables:

```emmeans(wine.clm, ~ rating | temp, mode = "prob")
```

Using `mode = "mean.class"` obtains the average of these probability distributions as probabilities of the integers 1--5:

```emmeans(wine.clm, "temp", mode = "mean.class")
```

And there is a mode for the scale model too. In this example, the scale model involves only judges, and that is the only factor in the grid:

```summary(ref_grid(wine.clm, mode = "scale"), type = "response")
```

Judge 8's ratings don't vary much, relative to the others. The scale model is in terms of log(SD). Again, these are not uniquely identifiable, and the first level's estimate is set to log(1) = 0. so, actually, each estimate shown is a comparison with judge 1.

Back to Contents

Models fitted using MCMC methods {#mcmc}

To illustrate emmeans's support for models fitted using MCMC methods, consider the `example_model` available in the rstanarm package. The example concerns CBPP, a serious disease of cattle in Ethiopia. A generalized linear mixed model was fitted to the data using the code below. (This is a Bayesian equivalent of the frequentist model we considered in the "Transformations" vignette.) We subsequently obtain its reference grid in the usual way.

```cbpp <- transform(lme4::cbpp, unit = 1:56)
cbpp.rstan <- rstanarm::stan_glmer(
cbind(incidence, size - incidence) ~ period + (1|herd) + (1|unit),
data = cbpp, family = binomial,
chains = 2, cores = 1, seed = 12345, iter = 500)
cbpp.rg <- ref_grid(cbpp.rstan)
```
```cbpp.rg <- do.call(emmobj,
cbpp.sigma <- readRDS(system.file("extdata", "cbppsigma", package = "emmeans"))
```

Here is the structure of the reference grid:

```cbpp.rg
```

And, again in the usual way, we can obtain EMMs:

```period.emm <- emmeans(cbpp.rg, "period")
period.emm
```

The summary for EMMs of Bayesian models shows the median of the posterior distribution of each estimate, along with highest posterior density (HPD) intervals. Under the hood, the posterior sample of parameter estimates is used to compute a corresponding sample of posterior EMMs, and it is those that are summarized. (Technical note: the summary is actually rerouted to the `hpd.summary()` function.

{#bayesxtra}

We can access the posterior EMMs via the `as.mcmc` method for `emmGrid` objects. This gives us an object of class `mcmc` (defined in the coda package), which can be summarized and explored as we please.

```require("coda")
summary(as.mcmc(period.emm))
```

Note that `as.mcmc` will actually produce an `mcmc.list` when there is more than one chain present, as in this example. The 2.5th and 97.5th quantiles are similar, but not identical, to the 95% confidence intervals in the frequentist summary.

Next, let us consider the back-transformed results. As is discussed with the frequentist model, there are random effects present, and if wee want to think in terms of marginal probabilities across all herds and units, we should correct for bias; and to do that, we need the standard deviations of the random effects. The model object has MCMC results for the random effects of each herd and each unit, but after those, there are also summary results for the posterior SDs of the two random effects. (I used the `colnames` function to find that they are in the 78th and 79th columns.)

```cbpp.sigma = as.matrix(cbpp.rstan\$stanfit)[, 78:79]
```

Here are the first few:

```head(cbpp.sigma)
```

So to obtain bias-adjusted marginal probabilities, obtain the resultant SD and regrid with bias correction:

```totSD <- sqrt(apply(cbpp.sigma^2, 1, sum))
cbpp.rgrd <- regrid(cbpp.rg, bias.adjust = TRUE, sigma = totSD)
summary(cbpp.rgrd)
```

Here is a plot of the posterior incidence probabilities, back-transformed:

```bayesplot::mcmc_areas(as.mcmc(cbpp.rgrd))
```

... and here are intervals for each period compared with its neighbor:

```contrast(cbpp.rgrd, "consec", reverse = TRUE)
```

The only interval that excludes zero is the one that compares periods 1 and 2.

Bayesian prediction {#predict-mcmc}

To predict from an MCMC model, just specify the `likelihood` argument in `as.mcmc`. Doing so causes the function to simulate data from the posterior predictive distribution. For example, if we want to predict the CBPP incidence in future herds of 25 cattle, we can do:

```set.seed(2019.0605)
cbpp.preds <- as.mcmc(cbpp.rgrd, likelihood = "binomial", trials = 25)
bayesplot::mcmc_hist(cbpp.preds, binwidth = 1)
```

Back to Contents

Index of all vignette topics

Try the emmeans package in your browser

Any scripts or data that you put into this service are public.

emmeans documentation built on Dec. 1, 2019, 1:15 a.m.