---
title: "Cross-validating mixed-effects models"
author: "John Fox and Georges Monette"
date: "`r Sys.Date()`"
package: cv
output: 
  rmarkdown::html_vignette:
  fig_caption: yes
bibliography: ["cv.bib"]
csl: apa.csl
vignette: >
  %\VignetteIndexEntry{Cross-validating mixed-effects models}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  message = TRUE,
  warning = TRUE,
  fig.align = "center",
  fig.height = 6,
  fig.width = 7,
  fig.path = "fig/",
  dev = "png",
  comment = "#>" #,
)
library(cv)
library(lme4)
# save some typing
knitr::set_alias(w = "fig.width",
                 h = "fig.height",
                 cap = "fig.cap")

# colorize text: use inline as `r colorize(text, color)`
colorize <- function(x, color) {
  if (knitr::is_latex_output()) {
    sprintf("\\textcolor{%s}{%s}", color, x)
  } else if (knitr::is_html_output()) {
    sprintf("<span style='color: %s;'>%s</span>", color,
      x)
  } else x
}

.opts <- options(digits = 5)
CACHE <- FALSE

```

The fundamental analogy for cross-validation is to the collection of new data. That is, predicting the response in each fold from the model fit to data in the other folds is like using the model fit to all of the data to predict the response for new cases from the values of the predictors for those new cases. As we explained in the introductory vignette on cross-validating regression models, the application of this idea to independently sampled cases is straightforward---simply partition the data into random folds of equal size and leave each fold out in turn, or, in the case of LOO CV, simply omit each case in turn.

In contrast, mixed-effects models are fit to *dependent* data, in which cases as clustered, such as hierarchical data, where the clusters comprise higher-level units (e.g., students clustered in schools), or longitudinal data, where the clusters are individuals and the cases repeated observations on the individuals over time.[^crossed-effects] 

[^crossed-effects]: There are, however, more complex situations that give rise to so-called *crossed* (rather than *nested*) random effects. For example, consider students within classes within schools. In primary schools, students typically are in a single class, and so classes are nested within schools. In secondary schools, however, students typically take several classes and students who are together in a particular class may not be together in other classes; consequently, random effects based on classes within schools are crossed. The `lmer()` function in the **lme4** package is capable of modeling both nested and crossed random effects, and the `cv()` methods for mixed models in the **cv** package pertain to both nested and crossed random effects. We present an example of the latter later in the vignette.

We can think of two approaches to applying cross-validation to clustered data:[^cv-faq]

[^cv-faq]: We subsequently discovered that @Vehtari:2023 [Section 8] makes similar points.

1. Treat CV as analogous to predicting the response for one or more cases in a *newly observed cluster*. In this instance, the folds comprise one or more whole clusters; we refit the model with all of the cases in clusters in the current fold removed; and then we predict the response for the cases in clusters in the current fold. These predictions are based only on fixed effects because the random effects for the omitted clusters are presumably unknown, as they would be for data on cases in newly observed clusters.

2. Treat CV as analogous to predicting the response for a newly observed case in an *existing cluster*. In this instance, the folds comprise one or more individual cases, and the predictions can use both the fixed and random effects.

## Example: The High-School and Beyond data

Following their use by @RaudenbushBryk:2002, data from the 1982 *High School and Beyond* (HSB) survey have become a staple of the literature on mixed-effects models. The HSB data are used by @FoxWeisberg:2019 [Sec. 7.2.2] to illustrate the application of linear mixed models to hierarchical data, and we'll closely follow their example here.

The HSB data are included in the `MathAchieve` and `MathAchSchool` data sets in the **nlme** package  [@PinheiroBates:2000]. `MathAchieve` includes individual-level data on 7185  students in 160 high schools, and `MathAchSchool` includes school-level data:
```{r HSB-data}
data("MathAchieve", package = "nlme")
dim(MathAchieve)
head(MathAchieve, 3)
tail(MathAchieve, 3)

data("MathAchSchool", package = "nlme")
dim(MathAchSchool)
head(MathAchSchool, 2)
tail(MathAchSchool, 2)
```
The first few students are in school number 1224 and the last few in school 9586. 

We'll use only the `School`, `SES` (students' socioeconomic status), and `MathAch` (their score on a standardized math-achievement test) variables in the `MathAchieve` data set, and `Sector` (`"Catholic"` or `"Public"`) in the `MathAchSchool` data set.

Some data-management is required before fitting a mixed-effects model to the HSB data, for which we use the **dplyr** package [@WickhamEtAl:2023]:
```{r HSB-data-management, cache=CACHE}
library("dplyr")
MathAchieve %>% group_by(School) %>%
  summarize(mean.ses = mean(SES)) -> Temp
Temp <- merge(MathAchSchool, Temp, by = "School")
HSB <- merge(Temp[, c("School", "Sector", "mean.ses")],
             MathAchieve[, c("School", "SES", "MathAch")], by = "School")
names(HSB) <- tolower(names(HSB))

HSB$cses <- with(HSB, ses - mean.ses)
```
In the process, we created two new school-level variables: `meanses`, which is the average SES for students in each school; and `cses`, which is school-average SES centered at its mean. For details, see @FoxWeisberg:2019 [Sec. 7.2.2].

Still following Fox and Weisberg, we proceed to use the `lmer()` function in the **lme4** package [@BatesEtAl:2015] to fit a mixed model for math achievement to the HSB data:
```{r HSB-lmer, cache=CACHE}
library("lme4")
hsb.lmer <- lmer(mathach ~ mean.ses * cses + sector * cses
                 + (cses | school), data = HSB)
summary(hsb.lmer, correlation = FALSE)
```

We can then cross-validate at the cluster (i.e., school) level,
```{r HSB-lmer-CV-cluster, cache=CACHE}
library("cv")

summary(cv(hsb.lmer,
   k = 10,
   clusterVariables = "school",
   seed = 5240))
```
or at the case (i.e., student) level,
```{r HSB-lmer-CV-case, cache=CACHE}
summary(cv(hsb.lmer, seed = 1575))
```
For cluster-level CV, the `clusterVariables` argument tells `cv()` how the clusters are defined. Were there more than one clustering variable, say classes within schools, these would be provided as a character vector of variable names: `clusterVariables = c("school", "class")`. For cluster-level CV, the default is `k = "loo"`, that is, leave one cluster out at a time; we instead specify `k = 10` folds of clusters, each fold therefore comprising $160/10 = 16$ schools. 

If the `clusterVariables` argument is omitted, then case-level CV is employed, with `k = 10` folds as the default, here each with $7185/10 \approx 719$ students. Notice that one of the 10 models refit with a fold removed failed to converge. Convergence problems are common in mixed-effects modeling. The apparent issue here is that an estimated variance component is close to or equal to 0, which is at a boundary of the parameter space. That shouldn't disqualify the fitted model for the kind of prediction required for cross-validation.

There is also a `cv()` method for linear mixed models fit by the `lme()` function in the **nlme** package, and the arguments for `cv()` in this case are the same as for a model fit by `lmer()` or `glmer()`. We illustrate with the mixed model fit to the HSB data:
```{r hsb-lme, cache=CACHE}
library("nlme")
hsb.lme <- lme(
  mathach ~ mean.ses * cses + sector * cses,
  random = ~ cses | school,
  data = HSB,
  control = list(opt = "optim")
)
summary(hsb.lme)

summary(cv(hsb.lme,
   k = 10,
   clusterVariables = "school",
   seed = 5240))

summary(cv(hsb.lme, seed = 1575))
```
We used the same random-number generator seeds as in the previous example cross-validating the model fit by `lmer()`, and so the same folds are employed in both cases.[^optimizer] The estimated covariance components and fixed effects in the summary output differ slightly between the `lmer()` and `lme()` solutions, although both functions seek to maximize the REML criterion. This is, of course, to be expected when different algorithms are used for numerical optimization.  To the precision reported, the cluster-level CV results for the `lmer()` and `lme()` models are identical, while the case-level CV results are very similar but not identical.

[^optimizer]: The observant reader will notice that we set the argument `control=list(opt="optim")` in the call to `lme()`, changing the optimizer employed from the default `"nlminb"`. We did this because with the default optimizer, `lme()` encountered the same convergence issue as `lmer()`, but rather than issuing a warning, `lme()` failed, reporting an error. As it turns out, setting the optimizer to `"optim"` avoids this problem.

## Example: Contrasting cluster-based and case-based CV

We introduce four artificial data sets that exemplify aspects of cross-validation particular to hierarchical models. Using these data sets, we show that model comparisons employing cluster-based and those employing case-based cross-validation may not agree on a "best" model. Furthermore, commonly used measures of fit, such as mean-squared error, do not necessarily become smaller as models become larger, even when the models are nested, and even when the measure of fit is computed for the whole data set. 

The four datasets differ in the relative magnitude of between-cluster variance compared with within-cluster variance. They serve to illustrate how fitting mixed models, and, consequently, the cross-validation of mixed models, is sensitive to relative variance, which determines the degree of shrinkage of within-cluster estimates of effects towards between-cluster estimates.           

For these analyses, we will use the `glmmTMB()` function in the **glmmTMB** package [@BrooksEtAl:2017] because, in our experience, it is more likely to converge than functions in the **nlme** and the **lme4** packages for models with low between-cluster variance.

Consider a researcher studying the effect of the dosage of a drug on the severity of symptoms for a hypothetical disease. The researcher has longitudinal data on 20 patients, each of whom was observed on five occasions in which patients received different dosages of the drug.  The data are observational, with dosages prescribed by the patients' physicians, so that patients who were more severely affected by the disease received higher dosages of the drug.

Our four contrived data sets (see below) illustrate possible results for data obtained in such a scenario. The relative configuration of dosages `x` and symptoms `y` are identical within patients in each of the four datasets. Within patients, higher dosages are generally associated with a reduction in symptoms. 

Between patients, however, higher dosages are associated with higher levels of symptoms. A plausible mechanism is a reversal of causality: Within patients, higher dosages alleviate symptoms, but between patients higher morbidity causes the prescription of higher dosages.

The four data sets differ in the between-patient variance of patient centroids from a common between-patients regression line. The data sets exhibit a progression from low to high variance around the common regression line.  

We start by generating a data set to serve as a common template for the four sample data sets.  We then apply different multipliers to the between-patient variation using parameters consistent with the description above to highlight the issues that arise in cross-validating mixed-effects models:[^data-template]

[^data-template]: We invite the interested reader to experiment with varying the parameters of our example. 

```{r include=FALSE, echo=FALSE}
library("glmmTMB") # necessary for some reason to knit vignette in RStudio, harmless otherwise
```
```{r parameters}

# Parameters:

Nb <- 20     # number of patients
Nw <- 5      # number of occasions for each patient

Bb <- 1.0    # between-patient regression coefficient on patient means
Bw <- -0.5   # within-patient effect of x

SD_between <- c(0, 5, 6, 8)               # SD between patients
SD_within <- rep(2.5, length(SD_between)) # SD within patients

Nv <- length(SD_within)       # number of variance profiles
SD_ratio <- paste0('SD ratio = ', SD_between,' / ',SD_within)
SD_ratio <- factor(SD_ratio, levels = SD_ratio)

set.seed(833885) 

Data_template <- expand.grid(patient = 1:Nb, obs = 1:Nw) |>
  within({
    xw <- seq(-2, 2, length.out = Nw)[obs]
    x <- patient + xw
    xm  <- ave(x, patient)   # within-patient mean

    # Scaled random error within each SD_ratio_i group

    re_std <- scale(resid(lm(rnorm(Nb*Nw) ~ x)))
    re_between <- ave(re_std, patient)
    re_within <- re_std - re_between
    re_between <- scale(re_between)/sqrt(Nw)
    re_within <- scale(re_within)
  })

Data <- do.call(
  rbind,
  lapply(
    1:Nv,
    function(i) {
      cbind(Data_template, SD_ratio_i = i)
    }
  )
)

Data <- within(
  Data,
  {
    SD_within_ <- SD_within[SD_ratio_i]
    SD_between_ <- SD_between[SD_ratio_i]
    SD_ratio <- SD_ratio[SD_ratio_i]
    y <- 10 +
      Bb * xm +                  # contextual effect
      Bw * (x - xm) +            # within-patient effect
      SD_within_ * re_within +   # within patient random effect
      SD_between_ * re_between   # adjustment to between patient random effect
  }
)
```
Here is a scatterplot of the data sets showing estimated 50% concentration ellipses for each cluster:[^lattice]

[^lattice]: We find it convenient to use the **lattice** [@Sarkar:2008] and **latticeExtra** [@SarkarAndrews:2022] packages for this and other graphs in this section.

```{r plot1}
#| out.width = "100%",
#| fig.height = 4.5,
#| fig.cap = "Data sets showing identical within-patient configurations with increasing between-patient variance. The labels above each panel show the between-patient SD/within-patient SD. The ellipses are estimated 50% concentration ellipses for each patient. The population between-patient regression line, $E(y) = 10 + x$, is shown in each panel."
library("lattice")
library("latticeExtra")
plot <- xyplot(y ~ x | SD_ratio, data = Data, group = patient,
           layout = c(Nv, 1),
           par.settings = list(superpose.symbol = list(pch = 1, cex = 0.7))) +
      layer(panel.ellipse(..., center.pch = 16, center.cex = 1.5,  
                          level = 0.5),
            panel.abline(a = 10, b = 1))
plot # display graph
```
The population from which these datasets is generated has a between-patient effect of dosage of 1 and a within-patient effect of $-0.5$. The researcher may attempt to obtain an estimate of the within-patient effect of dosage through the use of mixed models with a random intercept. We will illustrate how this approach results in estimates that are highly sensitive to the relative variance at the two levels of the mixed model by considering four models, all of which include a random intercept.  The first three models have sequentially nested fixed effects: (1)`~ 1`, intercept only; (2)`~ 1 + x`, intercept and effect of `x`; and (3) `~ 1 + x + xm`, intercept, effect of `x`, and a contextual variable, `xm`, consisting of the within-patient mean of `x`. The fourth model, `~ 1 + I(x - xm)`, uses an intercept and the centered-within-group variable `x - xm`. We thus fit four models to four datasets for a total of 16 models.

```{r model-fits}
model.formulas <- c(
  ' ~ 1'             =  y ~ 1 + (1 | patient),
  '~ 1 + x'          =  y ~ 1 + x + (1 | patient),
  '~ 1 + x + xm'     =  y ~ 1 + x + xm + (1 | patient),
  '~ 1 + I(x - xm)'  =  y ~ 1 + I(x - xm) + (1 | patient)
)

fits <- lapply(split(Data, ~ SD_ratio),
               function(d) {
                 lapply(model.formulas, function(form) {
                   glmmTMB(form, data = d)
                 })
               })
```
We proceed to obtain predictions from each model based on fixed effects alone, as would be used for cross-validation based on clusters (i.e., patients), and for fixed and random effects---so-called best linear unbiased predictions or "BLUPs"---as would be used for cross-validation based on cases (i.e., occasions within patients):
```{r predict}
# predicted fixed and random effects:
pred.BLUPs <- lapply(fits, lapply, predict)
# predicted fixed effects:
pred.fixed <- lapply(fits, lapply, predict, re.form = ~0)  
```
We then prepare the data for plotting:
```{r data-predictions}
Dataf <- lapply(split(Data, ~ SD_ratio),
                    function(d) {
                      lapply(names(model.formulas), 
                             function(form) cbind(d, formula = form))
                    }) |> 
             lapply(function(dlist) do.call(rbind, dlist)) |> 
             do.call(rbind, args = _)
    
Dataf <- within(
  Dataf,
  {
    pred.fixed <- unlist(pred.fixed)
    pred.BLUPs <- unlist(pred.BLUPs)
    panel <- factor(formula, levels = c(names(model.formulas), 'data'))
  }
)

Data$panel <- factor('data', levels = c(names(model.formulas), 'data'))
```

The fixed-effects predictions from these models are shown in the following graph:

```{r plot-fits-fixed}
#| out.width = "100%",
#| fig.height = 6,
#| fig.cap = "Fixed-effect predictions using each model applied to data sets with varying between- and within-patient variance ratios.  The top row shows summaries of the within-patient data using estimated 50% concentration ellipses for each patient."
{
  xyplot(y ~ x |SD_ratio * panel, Data,
         groups = patient, type = 'n', 
         par.strip.text = list(cex = 0.7),
         drop.unused.levels = FALSE) +
    glayer(panel.ellipse(..., center.pch = 16, center.cex = 0.5,  
                         level = 0.5),
           panel.abline(a = 10, b = 1)) +
    xyplot(pred.fixed  ~ x |SD_ratio * panel, Dataf, type = 'l', 
           groups = patient,
           drop.unused.levels = F,
           ylab = 'fixed-effect predictions') 
}|> 
  useOuterStrips() |> print()
```
The BLUPs from these models are shown in the following graph:
```{r plot-fits-blups}
#| out.width = "100%",
#| fig.height = 6,
#| fig.cap = "Fixed- and random-effect predictions (BLUPs) using each model applied to data sets with varying between- and within-patient variance ratios. The top row shows summaries of the within-patient data using estimated 50% concentration ellipses for each patient."
{
  xyplot(y ~ x | SD_ratio * panel, Data,
         groups = patient, type = 'n',
         drop.unused.levels = F, 
         par.strip.text = list(cex = 0.7),
         ylab = 'fixed- and random-effect predictions (BLUPS)') +
    glayer(panel.ellipse(..., center.pch = 16, center.cex = 0.5,  
                         level = 0.5),
           panel.abline(a = 10, b = 1)) +
    xyplot(pred.BLUPs  ~ x | SD_ratio * panel, Dataf, type = 'l', 
           groups = patient,
           drop.unused.levels = F) 
}|> 
  useOuterStrips() |> print()
```
Data sets with relatively low between-patient variance result in strong shrinkage of fixed-effects predictions and also of BLUPs towards the between-patient relationship between `y` and `x` in the model with an intercept and `x` as fixed-effect predictors, `~ 1 + x`. The inclusion of a contextual variable in the model corrects this problem.

Although the BLUPs fit the observed data more closely than predictions based on fixed effects alone, the slopes of within-patient BLUPs do not conform with the within-patient slopes for the `~ 1 + x` model in the two datasets with the smallest between-patient variances. 

For data with a small between-patient variance, fixed-effects predictions for the `~ 1 + x` model have a slope that is close to the between-patient slope but provide better overall predictions than the fixed-effect predictions for datasets with larger between-subject variance.  With data whose between-patient variance is relatively large, predictions based on the model with a common intercept and slope for all clusters, are very poor---indeed, much worse than the fixed-effects-only predictions based on the simpler random-intercept model.

We therefore anticipate (and show later in this section) that case-based cross-validation may prefer the intercept-only model, `~ 1`  to the larger `~ 1 + x` model when the between-cluster variance is relatively small, but that cluster-based cross-validation will prefer the latter to the former.

We will discover that case-based cross-validation prefers the `~ 1 + x` model to the `~ 1` model for the '5 / 2.5' dataset, but cluster-based cross-validation prefers the latter model to the former. The situation is entirely reversed with the '8 / 2.5' dataset. 

The third model, `~ 1 + x + xm`, includes a contextual effect of `x`---that is, the cluster mean `xm`---along with `x` and the intercept in the fixed-effect part of the model, along with a random intercept. This model is equivalent to fitting `y ~ I(x - xm) + xm + (1 | patient)`, which is the model that generated the data.  The fit of the mixed model `~ 1 + x + xm` is consequently similar to that of a fixed-effects only model with `x` and a categorical predictor for individual patients (i.e., `y ~ factor(patient) + x`, treating patients as a factor, and not shown here). 

We next carry out case-based cross-validation, which, as we have explained, is based on both fixed and predicted random effects (i.e., BLUPs), and cluster-based cross-validation, which is based on fixed effects only. In order to reduce between-model random variability in comparisons of models on the same dataset, we apply `cv()` to the list of models created by the `models()` function (introduced previously), performing cross-validation with the same folds for each model.

```{r echo=FALSE,include=FALSE}
library(cv) # unclear why it's necessary to reload cv
.opts <- options(warn = -2) 
```
```{r cross-validation,cache=CACHE}

model_lists <- lapply(fits, function(fitlist) do.call(models, fitlist))

cvs_cases <-
  lapply(1:Nv,
         function(i){
           cv(model_lists[[i]], k = 10, 
              data = split(Data, ~ SD_ratio)[[i]])
         })

cvs_clusters <-
  lapply(1:Nv,
         function(i){
           cv(model_lists[[i]],  k = 10, 
              data = split(Data, ~SD_ratio)[[i]], 
              clusterVariables = 'patient')
         })
```
For a given dataset, we can plot the results for a list of models using the plot method for `"cvList"` objects. For example
```{r plot-cv-example}
#| out.width = "60%",
#| fig.height = 6,
#| fig.cap = "10-fold cluster-based cross-validation comparing random intercept mixed models with varying fixed effects."
plot(cvs_clusters[[2]], main="Comparison of Fixed Effects")
```
We assemble the results for all datasets to show them in a common figure.
```{r cross-validation-data}
names(cvs_clusters) <- names(cvs_cases) <- SD_ratio 
dsummary <- expand.grid(SD_ratio_i = names(cvs_cases), model = names(cvs_cases[[1]]))

dsummary$cases <-
  sapply(1:nrow(dsummary), function(i){
    with(dsummary[i,], cvs_cases[[SD_ratio_i]][[model]][['CV crit']])
  })

dsummary$clusters <-
  sapply(1:nrow(dsummary), function(i){
    with(dsummary[i,], cvs_clusters[[SD_ratio_i]][[model]][['CV crit']])
  })
```
```{r cross-validation-data-plot}
#| out.width = "100%",
#| fig.height = 4,
#| fig.cap = "10-fold cluster- and case-based cross-validation comparing mixed models with random intercepts and various fixed effects."
xyplot(cases + clusters ~ model|SD_ratio_i, dsummary,
       auto.key = list(space = 'top', reverse.rows = T, columns = 2), type = 'b',
       xlab = "Fixed Effects",
       ylab = 'CV criterion (MSE)',
       layout= c(Nv,1),
       par.settings =
         list(superpose.line=list(lty = c(2, 3), lwd = 3),
              superpose.symbol=list(pch = 15:16, cex = 1.5)),
       scales = list(y = list(log = TRUE), x = list(alternating = F, rot = 60))) |> print()
```
In summary, when between-cluster variance is relatively large, the model `~ 1 + x`, with `x` alone and without the contextual mean of `x`, is assessed as fitting very poorly by cluster-based CV, but relatively much better by case-based CV. In all our examples, the model `~ 1 + x + xm`, which includes both `x` and its contextual mean, produces better results using both cluster-based and case-based CV.  These conclusions are consistent with our observations based on graphing predictions from the various models, and they illustrate the desirability of assessing mixed-effect models at different hierarchical levels.

## Example: Crossed random effects

Crossed random effects arise when the structure of the data aren't strictly hierarchical. Nevertheless, crossed and nested random effects can be handled in much the same manner, by refitting the mixed-effects model to the data with a fold of clusters or cases removed and using the refitted model to predict the response in the removed fold.

We'll illustrate with data on pig growth, introduced by @DiggleLiangZeger:1994 [Table 3.1]. The data are in the `Pigs` data frame in the **cv** package:
```{r pigs}
head(Pigs, 9)
head(xtabs( ~ id + week, data = Pigs), 3)
tail(xtabs( ~ id + week, data = Pigs), 3)
```
Each of 48 pigs is observed weekly over a period of 9 weeks, with the weight of the pig recorded in kg. The data are in "long" format, as is appropriate for use with the `lmer()` function in the **lme4** package. The data are very regular, with no missing cases.

The following graph, showing the growth trajectories of the pigs, is similar to Figure 3.1 in @DiggleLiangZeger:1994; we add an overall least-squares line and a loess smooth, which are nearly indistinguishable:
```{r pigs-graph}
#| out.width = "60%",
#| fig.height = 6,
#| fig.cap = "Growth trajectories for 48 pigs, with overall least-squares line (sold blue) and loess line (broken magenta)."
plot(weight ~ week, data = Pigs, type = "n")
for (i in unique(Pigs$id)) {
  with(Pigs, lines(
    x = 1:9,
    y = Pigs[id == i, "weight"],
    col = "gray"
  ))
}
abline(lm(weight ~ week, data = Pigs),
       col = "blue",
       lwd = 2)
lines(
  with(Pigs, loess.smooth(week, weight, span = 0.5)),
  col = "magenta",
  lty = 2,
  lwd = 2
)
```
The individual "growth curves" and the overall trend are generally linear, with some tendency for variability of pig weight to increase over weeks (a feature of the data that we ignore in the mixed model that we fit to the data below). 

The **Stata** mixed-effects models manual proposes a model with crossed random effects for the `Pigs` data [@Stata:2023 page 37]:

> [S]uppose that we wish to fit
$$
\mathrm{weight}_{ij} = \beta_0 + \beta_1 \mathrm{week}_{ij} + u_i + v_j + \varepsilon_{ij} 
$$
for the $i = 1, \ldots, 9$ weeks and $j = 1, \dots, 48$ pigs and
$$
u_i \sim N(0, \sigma^2_u); v_j \sim N(0, \sigma^2_v ); \varepsilon_{ij} \sim N(0, \sigma^2_\varepsilon)
$$
all independently. That is, we assume an overall population-average growth curve $\beta_0 + \beta_1 \mathrm{week}$ and a random pig-specific shift. In other words, the effect due to week, $u_i$, is systematic to that week and  common to all pigs. The rationale behind [this model] could be that, assuming that the pigs were measured contemporaneously, we might be concerned that week-specific random factors such as weather and feeding patterns had significant systematic effects on all pigs.

Although we might prefer an alternative model,[^pig-alternative] we think that this is a reasonable specification.

[^pig-alternative]: These are repeated-measures data, which would be more conventionally modeled with autocorrelated errors within pigs. The `lme()` function in the **nlme** package, for example, is capable of fitting a mixed-model of this form.

The **Stata** manual fits the mixed model by maximum likelihood (rather than REML), and we duplicate the results reported there using `lmer()`:
```{r pigs-lmer}
m.p <- lmer(
  weight ~ week + (1 | id) + (1 | week),
  data = Pigs,
  REML = FALSE, # i.e., ML
  control = lmerControl(optimizer = "bobyqa")
)
summary(m.p)
```
We opt for the non-default `"bobyqa"` optimizer because it provides more numerically stable results for subsequent cross-validation in this example.

We can then cross-validate the model by omitting folds composed of pigs, folds composed of weeks, or folds composed of pig-weeks (which in the `Pigs` data set correspond to individual cases, using only the fixed effects):
```{r pigs-cv}
summary(cv(m.p, clusterVariables = "id"))

summary(cv(m.p, clusterVariables = "week"))

summary(cv(
  m.p,
  clusterVariables = c("id", "week"),
  k = 10,
  seed = 8469))
```
We can also cross-validate the individual cases taking account of the random effects (employing the same 10 folds):
```{r pigs-cv-cases}
summary(cv(m.p, k = 10, seed = 8469))
```
Because these predictions are based on BLUPs, they are more accurate than the predictions based only on fixed effects.[^crossed-vs-nested] As well, the difference between the MSE computed for the model fit to the full data and the CV estimates of the MSE is greater here than for cluster-based predictions.

[^crossed-vs-nested]: Even though there is only one observation per combination of pigs and weeks, we can use the BLUP for the omitted case because of the crossed structure of the random effects; that is each pig-week has a pig random effect and a week random effect. Although it probably isn't sensible, we can imagine a mixed model for the pig data that employs nested random effects, which would be specified by `lmer(weight ~ week + (1 | id/week), data=Pigs)`---that is, a random intercept that varies by combinations of `id` (pig) and `week`. This model can't be fit, however: With only one case per combination of `id` and `week`, the nested random-effect variance is indistinguishable from the case-level variance.

```{r coda, include = FALSE}
options(.opts)
```

## References