# Sophisticated models in emmeans

#### emmeans package, Version 2.0.1

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.

## Contents

1. Linear mixed models (lmer)
   1. System options for lmerMod models
   2. Bias adjustment with random slopes
2. Models with offsets
3. Ordinal models
4. Models fitted using MCMC methods

Index of all vignette topics

## Linear mixed models (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.

    library(lme4)
    Oats.lmer <- 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

    ##  nitro emmean   SE   df lower.CL upper.CL
    ##    0.0   78.9 7.29 7.78     62.0     95.8
    ##    0.2   97.0 7.14 7.19     80.3    113.8
    ##    0.4  114.2 7.14 7.19     97.4    131.0
    ##    0.6  124.1 7.07 6.95    107.3    140.8
    ##
    ## Results are averaged over the levels of: Variety
    ## Degrees-of-freedom method: kenward-roger
    ## Confidence level used: 0.95

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")

    ##  nitro emmean   SE   df lower.CL upper.CL
    ##    0.0   78.9 7.28 7.28     61.8       96
    ##    0.2   97.0 7.13 6.72     80.0      114
    ##    0.4  114.2 7.13 6.72     97.2      131
    ##    0.6  124.1 7.07 6.49    107.1      141
    ##
    ## Results are averaged over the levels of: Variety
    ## Degrees-of-freedom method: satterthwaite
    ## Confidence level used: 0.95

######

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")

    ##  nitro emmean   SE  df asymp.LCL asymp.UCL
    ##    0.0   78.9 7.28 Inf      64.6      93.2
    ##    0.2   97.0 7.13 Inf      83.1     111.0
    ##    0.4  114.2 7.13 Inf     100.2     128.2
    ##    0.6  124.1 7.07 Inf     110.2     137.9
    ##
    ## Results are averaged over the levels of: Variety
    ## Degrees-of-freedom method: asymptotic
    ## Confidence level used: 0.95

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")

    ##  contrast  estimate    SE   df t.ratio p.value
    ##  linear      152.69 15.60 43.2   9.802 <0.0001
    ##  quadratic    -8.27  6.95 44.2  -1.190  0.2402
    ##  cubic        -6.32 15.20 42.8  -0.415  0.6800
    ##
    ## Results are averaged over the levels of: Variety
    ## Degrees-of-freedom method: kenward-roger

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)

    ## $emmeans
    ##  Variety     emmean   SE   df lower.CL upper.CL
    ##  Golden Rain  105.2 7.53 8.46     88.0      122
    ##  Marvellous   108.5 7.48 8.28     91.3      126
    ##  Victory       96.9 7.64 8.81     79.6      114
    ##
    ## Results are averaged over the levels of: nitro
    ## Degrees-of-freedom method: kenward-roger
    ## Confidence level used: 0.95
    ##
    ## $contrasts
    ##  contrast                 estimate   SE   df t.ratio p.value
    ##  Golden Rain - Marvellous    -3.23 6.55 9.56  -0.493  0.8764
    ##  Golden Rain - Victory        8.31 6.71 9.80   1.238  0.4595
    ##  Marvellous - Victory        11.54 6.67 9.80   1.729  0.2431
    ##
    ## Results are averaged over the levels of: nitro
    ## Degrees-of-freedom method: kenward-roger
    ## P value adjustment: tukey method for comparing a family of 3 estimates

### System options for lmerMod models

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.

### Bias adjustment with random slopes

In the `cbpp` example, we saw an example where we applied a bias adjustment to the inverse transformation. To do that adjustment, we required an estimate of the total SD of the response. That computation was (relatively) simple because the model had only random intercepts. But what if we have random slopes as well? The short answer is “it gets more complicated;” but here is an example of how we can muddle through it.

Our illustration is a model fitted to the `ChickWeight` data in the R datasets package. The data comprise weight determinations, over time (in days since birth), of chicks who are randomized to different diets. Our model fits a trend in the square root of weight for each diet, with random intercepts and slopes for each chick (this is likely not the best model, but it’s not totally stupid and it serves an an illustration).

    cw.lmer <- lmer(sqrt(weight) ~ Time*Diet + (1+Time|Chick), data = ChickWeight)

Our goal is to use this model to estimate the mean `weight` at times 5, 10, 15, and 20, for each diet. Accordingly, let’s get the estimates needed:

    cw.emm <- emmeans(cw.lmer, ~ Time|Diet, at = list(Time = c(5, 10, 15, 20)))

If we just summarize this with `type = “response”, we will under-estimate the mean weights. We need to apply a bias adjustment; but that involves providing an estimate of the SD of each transformed response. The problem is that since random slopes are involved, that SD depends on time. In particular, the model states that at time \(t\), \[ \sqrt Y_t = \mu_t + E + C + S\times t \] where \(\mu_t\) is the mean at time \(t\), \(E\) is the residual error, \(C\) is the random intercept for chicks, and \(S\) is the random slope for chicks. For purposes of bias correction, we need an estimate of \(SD(E + C + S\times t)\) for each \(t\).

The first step is to obtain an estimated covariance matrix for \((E, C, S)\):

    V <- matrix(0, nrow = 3, ncol = 3, dimnames = list(c("E","C","S"), c("E","C","S")))
    V[1, 1] <- sigma(cw.lmer)^2              # Var(E)
    V[2:3, 2:3] <- VarCorr(cw.lmer)$Chick    # Cov(C, S)
    V

    ##           E           C           S
    ## E 0.1867732  0.00000000  0.00000000
    ## C 0.0000000  0.15918129 -0.03977984
    ## S 0.0000000 -0.03977984  0.01513793

Now, using the matrix expression \(Var(a'X) = a'Va\) for \(X=c(E,C,S)\) and a given vector \(a\), we can obtain the needed SDs:

    sig <- sapply(c(5, 10, 15, 20), function(t) {
        a <- c(1, 1, t)
        sqrt(sum(a * V %*% a))
    })
    sig

    ## [1] 0.5714931 1.0315769 1.5995606 2.1931561

As expected, these values increase with \(t\). Finally, we obtain the bias-adjusted estimated weights. We can use `sigma = sig` as-is since the values follow the same ordering as `cw.emm@grid`.

    confint(cw.emm, type = "response", bias.adj = TRUE, sigma = sig)

    ## Diet = 1:
    ##  Time response    SE   df lower.CL upper.CL
    ##     5     63.2  1.49 45.5     60.2     66.2
    ##    10     91.1  4.11 45.8     83.0     99.5
    ##    15    124.5  7.81 46.1    109.3    140.8
    ##    20    163.6 12.40 46.2    139.6    189.7
    ##
    ## Diet = 2:
    ##  Time response    SE   df lower.CL upper.CL
    ##     5     69.3  2.15 45.0     65.1     73.7
    ##    10    106.6  6.14 44.9     94.6    119.3
    ##    15    152.3 12.00 44.9    129.1    177.4
    ##    20    206.5 19.40 44.9    169.3    247.5
    ##
    ## Diet = 3:
    ##  Time response    SE   df lower.CL upper.CL
    ##     5     72.8  2.20 45.0     68.4     77.3
    ##    10    121.3  6.56 44.9    108.5    134.9
    ##    15    182.7 13.10 44.9    157.2    210.1
    ##    20    256.9 21.70 44.9    215.1    302.4
    ##
    ## Diet = 4:
    ##  Time response    SE   df lower.CL upper.CL
    ##     5     76.4  2.26 45.0     71.9     81.0
    ##    10    119.4  6.50 45.0    106.6    132.8
    ##    15    172.4 12.80 45.0    147.7    199.1
    ##    20    235.5 20.80 45.0    195.5    279.2
    ##
    ## Degrees-of-freedom method: kenward-roger
    ## Confidence level used: 0.95
    ## Intervals are back-transformed from the sqrt scale
    ## Bias adjustment applied based on sigma = (various values)

This example illustrates that it is possible to deal with random slopes in bias corrections. However it does require some fairly careful attention to technical details and familiarity with matrix calculations; so if you don’t have a comfort level with those, it is best to get outside help.

#### Adding variables not in fixed model {addl.vars}

Consider a model like

    mod <- lmer(log(response) ~ treatment + (1 + x | subject), data = mydata)

Ordinarily, the reference grid will not include the variable `x` because it is not part of the fixed-effects formula. However, you can include it via the `addl.vars` argument:

    EMM <- emmeans(mod, ~ x|treatment, addl.vars = "x", at = list(x = -1:1))

We will then obtain EMMs for combinations of `treatment` and `x`. (For a given `treatment`, all those means will be equal for every `x`.) But the bias adjustments will depend on `x`.

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## Models with 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)

    ## 'emmGrid' object with variables:
    ##     size = S, M, L
    ##     age = 1, 2
    ##     n = 500
    ## Transformation: "log"

Note that `n` is included in the reference grid and that its average value of 500 is displayed. But let’s look at the EMMs:

    (EMM <- emmeans(ins.glm, "size", type = "response"))

    ##  size rate   SE  df asymp.LCL asymp.UCL
    ##  S    69.3 6.25 Inf     58.03      82.7
    ##  M    34.6 3.34 Inf     28.67      41.9
    ##  L    11.9 3.14 Inf      7.07      19.9
    ##
    ## Results are averaged over the levels of: age
    ## Confidence level used: 0.95
    ## Intervals are back-transformed from the log scale

We can see more explicitly what is happening by examining the internal structure of `EMM`:

    EMM@grid

    ##   size .offset. .wgt.
    ## 1    S 6.214608     2
    ## 2    M 6.214608     2
    ## 3    L 6.214608     2

and note that \(\log(500) \approx 6.215\) is used as the offset value in calculating these estimates.

All this said, many users would like to ignore that average offset for this kind of model, and instead use one corresponding to `n = 1`, because then the estimates we obtain are estimated rates per unit `n`. This may be accomplished by specifying an `offset` parameter in the call:

    emmeans(ins.glm, "size", type = "response", offset = log(1))

    ##  size   rate      SE  df asymp.LCL asymp.UCL
    ##  S    0.1385 0.01250 Inf    0.1161    0.1653
    ##  M    0.0693 0.00669 Inf    0.0573    0.0837
    ##  L    0.0237 0.00627 Inf    0.0141    0.0398
    ##
    ## Results are averaged over the levels of: age
    ## Confidence level used: 0.95
    ## Intervals are back-transformed from the log scale

An alternative way to achieve the same results is to set `n` equal to 1 in the reference grid itself (output not shown, because it is identical):

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

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))

For more details on how offsets are handled, and how and why an `offset()` model term is treated differently than an `offset` argument in model fitting, see the “xplanations” vignette.

An additional complication may come up in models zero-inflated or hurdle models. In those cases, it is somewhat ambiguous what one might mean by a “rate”, but one interpretation would be to just go with the above techniques with estimates for just the Poisson component of the model. Another approach would be to estimate the response mean with the zero-inflation included, and divide by the appropriate offset. This can be done, but it is messy. An example is given on the Cross-Validated site.

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## Ordinal models

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")

    ## Loading required package: 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))

    ## $`emmeans of temp`
    ##  temp emmean    SE  df asymp.LCL asymp.UCL
    ##  cold -0.884 0.290 Inf    -1.452    -0.316
    ##  warm  0.601 0.225 Inf     0.161     1.041
    ##
    ## Results are averaged over the levels of: contact, judge
    ## Confidence level used: 0.95
    ##
    ## $`pairwise differences of temp`
    ##  1           estimate    SE  df z.ratio p.value
    ##  cold - warm    -1.07 0.422 Inf  -2.547  0.0109
    ##
    ## Results are averaged over the levels of: contact, judge
    ##
    ## $`emmeans of contact`
    ##  contact emmean    SE  df asymp.LCL asymp.UCL
    ##  no      -0.614 0.298 Inf   -1.1990   -0.0297
    ##  yes      0.332 0.201 Inf   -0.0632    0.7264
    ##
    ## Results are averaged over the levels of: temp, judge
    ## Confidence level used: 0.95
    ##
    ## $`pairwise differences of contact`
    ##  1        estimate    SE  df z.ratio p.value
    ##  no - yes   -0.684 0.304 Inf  -2.251  0.0244
    ##
    ## Results are averaged over the levels of: temp, judge

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.

######

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

    ## 'emmGrid' object with variables:
    ##     temp = cold, warm
    ##     contact = no, yes
    ##     judge = 1, 2, 3, 4, 5, 6, 7, 8, 9
    ##     cut = multivariate response levels: 1|2, 2|3, 3|4, 4|5
    ## Transformation: "probit"

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")

    ##  temp emmean    SE  df asymp.LCL asymp.UCL
    ##  cold  0.884 0.290 Inf     0.316     1.452
    ##  warm -0.601 0.225 Inf    -1.041    -0.161
    ##
    ## Results are averaged over the levels of: contact, judge, cut
    ## Results are given on the probit (not the response) scale.
    ## Confidence level used: 0.95

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"))

    ##  temp exc.prob     SE  df asymp.LCL asymp.UCL
    ##  cold   0.0748 0.0318 Inf    0.0124     0.137
    ##  warm   0.4069 0.0706 Inf    0.2686     0.545
    ##
    ## Results are averaged over the levels of: contact, judge
    ## Confidence level used: 0.95

######

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")

    ## temp = cold:
    ##  rating   prob     SE  df asymp.LCL asymp.UCL
    ##  1      0.1292 0.0625 Inf   0.00667    0.2518
    ##  2      0.4877 0.0705 Inf   0.34948    0.6259
    ##  3      0.3083 0.0594 Inf   0.19186    0.4248
    ##  4      0.0577 0.0238 Inf   0.01104    0.1043
    ##  5      0.0171 0.0127 Inf  -0.00768    0.0419
    ##
    ## temp = warm:
    ##  rating   prob     SE  df asymp.LCL asymp.UCL
    ##  1      0.0156 0.0129 Inf  -0.00961    0.0408
    ##  2      0.1473 0.0448 Inf   0.05959    0.2350
    ##  3      0.4302 0.0627 Inf   0.30723    0.5532
    ##  4      0.2685 0.0625 Inf   0.14593    0.3910
    ##  5      0.1384 0.0506 Inf   0.03923    0.2376
    ##
    ## Results are averaged over the levels of: contact, judge
    ## Confidence level used: 0.95

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")

    ##  temp mean.class    SE  df asymp.LCL asymp.UCL
    ##  cold       2.35 0.144 Inf      2.06      2.63
    ##  warm       3.37 0.146 Inf      3.08      3.65
    ##
    ## Results are averaged over the levels of: contact, judge
    ## Confidence level used: 0.95

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 response    SE  df
    ##  1        1.000 0.000 Inf
    ##  2        1.043 0.570 Inf
    ##  3        1.053 0.481 Inf
    ##  4        0.710 0.336 Inf
    ##  5        0.663 0.301 Inf
    ##  6        0.758 0.341 Inf
    ##  7        1.071 0.586 Inf
    ##  8        0.241 0.179 Inf
    ##  9        0.533 0.311 Inf

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.

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## Models fitted using MCMC methods

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.) In fitting the model, we first set the contrast coding to `bayestestR::contr.bayes` because this equalizes the priors across different treatment levels (a correction from an earlier version of this vignette.) We subsequently obtain the reference grids for these models in the usual way. For later use, we also fit the same model with just the prior information.

    cbpp <- transform(lme4::cbpp, unit = 1:56)
    require("bayestestR")
    options(contrasts = c("contr.bayes", "contr.poly"))
    cbpp.rstan <- rstanarm::stan_glmer(
        cbind(incidence, size - incidence) ~ period + (1|herd) + (1|unit),
        data = cbpp, family = binomial,
        prior = student_t(df = 5, location = 0, scale = 2, autoscale = FALSE),
        chains = 2, cores = 1, seed = 2021.0120, iter = 1000)
    cbpp_prior.rstan <- update(cbpp.rstan, prior_PD = TRUE)
    cbpp.rg <- ref_grid(cbpp.rstan)
    cbpp_prior.rg <- ref_grid(cbpp_prior.rstan)

Here is the structure of the reference grid:

    cbpp.rg

    ## 'emmGrid' object with variables:
    ##     period = 1, 2, 3, 4
    ## Transformation: "logit"

So here are the EMMs (no averaging needed in this simple model):

    summary(cbpp.rg)

    ##  period prediction lower.HPD upper.HPD
    ##  1           -1.60     -2.26    -0.987
    ##  2           -2.77     -3.65    -1.974
    ##  3           -2.90     -3.77    -2.040
    ##  4           -3.32     -4.43    -2.385
    ##
    ## Point estimate displayed: median
    ## Results are given on the logit (not the response) scale.
    ## HPD interval probability: 0.95

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.

######

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")

    ## Loading required package: coda

    summary(as.mcmc(cbpp.rg))

    ##
    ## Iterations = 1:500
    ## Thinning interval = 1
    ## Number of chains = 2
    ## Sample size per chain = 500
    ##
    ## 1. Empirical mean and standard deviation for each variable,
    ##    plus standard error of the mean:
    ##
    ##            Mean     SD Naive SE Time-series SE
    ## period 1 -1.595 0.3333  0.01054        0.01279
    ## period 2 -2.790 0.4327  0.01368        0.01367
    ## period 3 -2.916 0.4491  0.01420        0.01706
    ## period 4 -3.379 0.5384  0.01703        0.01845
    ##
    ## 2. Quantiles for each variable:
    ##
    ##            2.5%    25%    50%    75%   97.5%
    ## period 1 -2.283 -1.823 -1.597 -1.357 -0.9929
    ## period 2 -3.707 -3.038 -2.766 -2.511 -2.0197
    ## period 3 -3.834 -3.196 -2.899 -2.610 -2.0915
    ## period 4 -4.502 -3.725 -3.318 -3.022 -2.4079

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.

The bayestestR package provides `emmGrid` methods for most of its description and testing functions. For example:

    bayestestR::bayesfactor_parameters(pairs(cbpp.rg), prior = pairs(cbpp_prior.rg))

    ## Warning: Bayes factors might not be precise.
    ##   For precise Bayes factors, sampling at least 40,000 posterior samples is recommended.

    ## Bayes Factor (Savage-Dickey density ratio)
    ##
    ## contrast          |    BF
    ## -------------------------
    ## period1 - period2 |  2.91
    ## period1 - period3 |  5.13
    ## period1 - period4 | 14.85
    ## period2 - period3 | 0.173
    ## period2 - period4 | 0.268
    ## period3 - period4 | 0.221
    ##
    ## * Evidence Against The Null: 0

    bayestestR::p_rope(pairs(cbpp.rg), range = c(-0.25, 0.25))

    ## Proportion of samples inside the ROPE [-0.25, 0.25]
    ##
    ## contrast          | p (ROPE)
    ## ----------------------------
    ## period1 - period2 |    0.021
    ## period1 - period3 |    0.015
    ## period1 - period4 |    0.004
    ## period2 - period3 |    0.367
    ## period2 - period4 |    0.184
    ## period3 - period4 |    0.290

Both of these sets of results suggest that period 1 is different from the others. For more information on these methods, refer to the CRAN page for bayestestR and its vignettes, e.g., the one on Bayes factors.

### Bias-adjusted incidence probabilities

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)

    ##           parameters
    ## iterations Sigma[unit:(Intercept),(Intercept)] Sigma[herd:(Intercept),(Intercept)]
    ##       [1,]                            1.154694                         0.167807505
    ##       [2,]                            1.459379                         0.040318460
    ##       [3,]                            1.482619                         0.006198847
    ##       [4,]                            1.236694                         0.206057981
    ##       [5,]                            1.460472                         0.088491844
    ##       [6,]                            1.412277                         0.070334431

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)

    ##  period   prob lower.HPD upper.HPD
    ##  1      0.2199    0.1324     0.322
    ##  2      0.0864    0.0329     0.156
    ##  3      0.0767    0.0241     0.137
    ##  4      0.0524    0.0120     0.106
    ##
    ## Point estimate displayed: median
    ## HPD interval probability: 0.95

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)

    ##  contrast          estimate lower.HPD upper.HPD
    ##  period1 - period2  0.13283    0.0280     0.235
    ##  period2 - period3  0.00918   -0.0635     0.097
    ##  period3 - period4  0.02331   -0.0427     0.103
    ##
    ## Point estimate displayed: median
    ## HPD interval probability: 0.95

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

### Bayesian prediction

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)

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