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title: "Linear Mixed Model Methodology"
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```{r}
#| include: false
load(file.path("..", "data", "ex125.RData"))
```

## Model Form

The book presents linear mixed models in the form

$$
Y = X\beta + Zb + e,
$$

where $Y$ is the response vector, $X$ is the fixed effect design matrix,
$\beta$ is the vector of fixed effect parameters, $Z$ is the random effect
design matrix, $b$ is the random effect vector, and $e$ is the residual error.
The usual assumptions are

$$
b \sim N(0, G), \quad e \sim N(0, R), \quad \operatorname{Cov}(b, e) = 0.
$$

The marginal variance is therefore

$$
V = ZGZ^T + R.
$$

## Fixed Effects

When variance components are known or estimated, generalized least squares
estimates the fixed effects as

$$
\hat{\beta} = (X^T V^{-1} X)^{-} X^T V^{-1}Y.
$$

The generalized inverse is needed for overparameterized model descriptions,
which are common in the book because the SAS examples use reference-level
constraints.

## Variance Components

Example 2.4.2.2 uses the `ex125` split-plot data. The book reports maximum
likelihood and restricted maximum likelihood variance component estimates for
region, herd within region and drug, and residual error.

```{r}
if (requireNamespace("lme4", quietly = TRUE)) {
  fit_ml <- lme4::lmer(
    Pcv ~ dose * Drug + (1 | Region / Drug),
    data = ex125,
    REML = FALSE
  )
  if (requireNamespace("report", quietly = TRUE)) {
    report::report(fit_ml)
  }

  fit_reml <- lme4::lmer(
    Pcv ~ dose * Drug + (1 | Region / Drug),
    data = ex125,
    REML = TRUE
  )
  if (requireNamespace("report", quietly = TRUE)) {
    report::report(fit_reml)
  }

  data.frame(
    component = as.data.frame(lme4::VarCorr(fit_ml))$grp,
    ml = round(as.data.frame(lme4::VarCorr(fit_ml))$vcov, 3),
    reml = round(as.data.frame(lme4::VarCorr(fit_reml))$vcov, 3)
  )
}
```

These estimates match the readable printed table in the book to rounding:
`4.292`, `0.322`, and `1.747` for ML, and `5.151`, `0.387`, and `2.096` for
REML.

## Reporting Fitted Models

The package keeps `report` optional, but fitted mixed models can be interpreted
with `report_mixed_model()` when the easystats reporting package is available.
This reporting step does not refit the model or change the estimates; it only
turns the fitted model object into a narrative summary. The helper wraps the
same guarded call to `report::report()`.

```{r}
if (requireNamespace("report", quietly = TRUE) && exists("fit_reml")) {
  report::report(fit_reml)
} else {
  data.frame(
    workflow = "Optional model report",
    requirement = "Install report to generate easystats-style summaries"
  )
}
```

## Post Hoc Inference with emmeans

Model reports and marginal means answer different questions. A report gives a
narrative summary of the fitted model, while `emmeans` focuses on model-based
means and comparisons among scientifically meaningful factor levels. The package
helper `emmeans_mixed_model()` wraps this same guarded `emmeans` workflow for
users who want a package-level entry point.

```{r}
if (requireNamespace("emmeans", quietly = TRUE) && exists("fit_reml")) {
  dose_emm <- emmeans::emmeans(
    fit_reml,
    ~ dose | Drug,
    lmer.df = "asymptotic"
  )
  dose_pairs <- emmeans::contrast(dose_emm, method = "pairwise")

  as.data.frame(dose_emm)
} else {
  data.frame(
    workflow = "Optional marginal means",
    requirement = "Install emmeans to compute post hoc model summaries"
  )
}
```

```{r}
if (exists("dose_pairs")) {
  as.data.frame(dose_pairs)
}
```

The marginal means summarize model-predicted packed cell volume by dose within
drug. The pairwise contrasts then test the high-versus-low dose difference
within each drug group, which is a post hoc interpretation layer rather than a
replacement for the fitted mixed model.

## Inference

Inference for fixed effects in mixed models depends on variance component
estimation and denominator degrees of freedom. The package examples use
`lmerTest` for Satterthwaite-style tests where the book uses SAS `PROC MIXED`.
Small numerical differences are expected across software versions and methods,
especially for degrees of freedom and p-values.
