Linear Mixed-effects Models

Population effects, clustered dependence, and partial pooling in Gaussian outcomes

Statistics

Mixed Models

Multilevel Models

Longitudinal Data

Linear mixed-model specification, marginal covariance, ML and REML estimation, inference, diagnostics, and bioinformatics design.

Author

Jiahong Sun

Published

October 29, 2026

Modified

October 29, 2026

1 A Regression Model for Clustered Gaussian Data

Linear mixed-effects models extend linear regression by adding latent effects shared by observations in the same group. They are useful for repeated measures, longitudinal data, multi-site experiments, blocked designs, pedigrees, and nested sampling structures. Their central task is not merely to “correct” standard errors. They jointly describe the conditional mean and the covariance created by the sampling hierarchy.

For all observations stacked into a vector,

\[ \boxed{ \mathbf y=\mathbf X\boldsymbol\beta +\mathbf Z\mathbf b+\boldsymbol\varepsilon, \qquad \mathbf b\sim N(\mathbf0,\mathbf G), \qquad \boldsymbol\varepsilon\sim N(\mathbf0,\mathbf R). } \]

$\mathbf X\boldsymbol\beta$ is the fixed-effect mean structure, $\mathbf Z\mathbf b$ contains random effects, and $\mathbf R$ describes residual variation. The usual model assumes $\mathbf b$ and $\boldsymbol\varepsilon$ are independent. Integrating out $\mathbf b$ gives the marginal model

\[ \boxed{ \mathbf y\sim N(\mathbf X\boldsymbol\beta,\mathbf V), \qquad \mathbf V=\mathbf Z\mathbf G\mathbf Z^\mathsf{T}+\mathbf R. } \]

This identity connects hierarchical interpretation to generalized least squares. Random effects are one structured way to build $\mathbf V$; they are not the only possible covariance model.

2 Start with the Scientific Target and Experimental Unit

Before writing a formula, define the target population, independent sampling unit, outcome, predictor levels, and desired generalization.

Goal	Example estimand	Relevant level
Population mean comparison	mean treatment contrast $\mathbf c^\mathsf{T}\boldsymbol\beta$	target population of biological units
Heterogeneity	donor-level variance component in $\mathbf G$	variation among donors
Existing-unit prediction	conditional mean given estimated $\mathbf b_i$	a sampled donor or site
New-unit prediction	outcome integrated over a new $\mathbf b_{\mathrm{new}}$	a future donor or site

The estimand is the population quantity in the scientific question. ML or REML fitting supplies an estimator, a random function of the data. Its observed coefficient or variance component is an estimate. A conditional prediction for a sampled donor is not the same target as a population-average fixed effect or a prediction for a new donor.

Fixed and random are properties of the model and inferential target, not innate properties of variables. A treatment is often represented by fixed coefficients because its specific contrast is the target. A donor effect may be random when donors are viewed as sampled from a population and their distribution is used for pooling. The number of levels alone does not decide the choice.

3 Conditional and Marginal Interpretation

Conditional on $\mathbf b$, the mean is

\[ E(\mathbf y\mid\mathbf b)=\mathbf X\boldsymbol\beta+ \mathbf Z\mathbf b. \]

Marginal over the random effects,

\[ E(\mathbf y)=\mathbf X\boldsymbol\beta. \]

For an identity-link Gaussian model, the fixed-effect coefficient has the same linear scale in both expressions, although predictions differ according to whether random effects are included. This convenient equality generally fails after a nonlinear link, which is one reason generalized linear mixed-model coefficients require special care.

Consider a donor random-intercept model,

\[ Y_{ij}=\beta_0+\beta_1T_{ij}+b_i+\varepsilon_{ij}, \qquad b_i\sim N(0,\tau^2). \]

$\beta_1$ is a conditional mean contrast at the modeled covariate values. A causal interpretation additionally needs treatment assignment, consistency, positivity, and control of confounding. The random effect accounts for modeled dependence; it does not by itself control unmeasured time-varying confounding.

4 Likelihood and Generalized Least Squares

For known covariance parameters in $\mathbf V$, maximizing the Gaussian likelihood over $\boldsymbol\beta$ gives

\[ \widehat{\boldsymbol\beta}_{\mathrm{GLS}} =(\mathbf X^\mathsf{T}\mathbf V^{-1}\mathbf X)^{-1} \mathbf X^\mathsf{T}\mathbf V^{-1}\mathbf y, \]

with model-based covariance

\[ \operatorname{Var}(\widehat{\boldsymbol\beta}\mid\mathbf X) =(\mathbf X^\mathsf{T}\mathbf V^{-1}\mathbf X)^{-1}. \]

In practice, covariance parameters $\boldsymbol\theta$ inside $\mathbf G$ and $\mathbf R$ are unknown. The log-likelihood, apart from a constant, is

\[ \ell(\boldsymbol\beta,\boldsymbol\theta) =-\frac12\left[ \log|\mathbf V_{\boldsymbol\theta}| +(\mathbf y-\mathbf X\boldsymbol\beta)^\mathsf{T} \mathbf V_{\boldsymbol\theta}^{-1} (\mathbf y-\mathbf X\boldsymbol\beta) \right]. \]

Maximum likelihood estimates fixed effects and covariance parameters jointly. Restricted maximum likelihood (REML) instead maximizes a likelihood for error contrasts that remove the fixed-effect degrees of freedom. One form of the REML criterion adds

\[ \log|\mathbf X^\mathsf{T}\mathbf V^{-1}\mathbf X| \]

to the bracketed ML expression. REML often reduces small-sample bias in variance components, but it is not universally unbiased and does not rescue a misspecified covariance or mean model.

5 ML, REML, and Model Comparison

The estimation method must match the comparison.

Use ML, not REML, to compare nested models with different fixed-effect columns by a likelihood-ratio statistic, because REML criteria are based on different error-contrast spaces.
REML criteria can compare covariance structures when the fixed-effect design is identical and regularity conditions are appropriate.
Refit the selected fixed-effect structure with REML when REML variance estimates and associated summaries are the reporting target.
Information criteria are comparable only when based on the same response data, likelihood convention, and compatible estimation target.

Likelihood-ratio reference distributions can fail for testing a variance component equal to zero because the null lies on a boundary. Small numbers of groups also make large-sample chi-squared and normal approximations unreliable. Parametric bootstrap, restricted-likelihood methods with justified reference distributions, or carefully calibrated small-sample procedures may be preferable.

6 Fixed-effect Inference

For a prespecified contrast $\psi=\mathbf c^\mathsf{T}\boldsymbol\beta$, the estimate is $\widehat\psi=\mathbf c^\mathsf{T}\widehat{\boldsymbol\beta}$ with plug-in standard error

\[ \widehat{\operatorname{SE}}(\widehat\psi) =\sqrt{ \mathbf c^\mathsf{T} \widehat{\operatorname{Var}}(\widehat{\boldsymbol\beta}) \mathbf c}. \]

Wald intervals and tests often use a normal or approximate $t$ reference distribution. The denominator degrees of freedom are not uniquely defined after estimating variance components in an unbalanced design. Satterthwaite and Kenward–Roger methods approximate finite-sample uncertainty in different ways; software defaults should therefore be reported rather than hidden.

Profile-likelihood intervals can better represent asymmetric uncertainty. Parametric bootstrap can propagate covariance estimation and nonlinearity if the fitted data-generating model is credible and the complete fitting procedure is repeated. None of these methods corrects pseudoreplication, informative missing data, model selection, or an unsupported causal interpretation.

7 Random Effects and Prediction

Given covariance parameters, the conditional mean of the random effects is

\[ E(\mathbf b\mid\mathbf y) =\mathbf G\mathbf Z^\mathsf{T}\mathbf V^{-1} (\mathbf y-\mathbf X\boldsymbol\beta). \]

Plugging in estimated parameters produces empirical BLUPs. Their shrinkage is a strength for prediction but changes their sampling behavior. A conditional prediction interval for another observation from an existing unit differs from a marginal interval for a new unit: the latter must include new random-effect variation. State which population and level the prediction concerns.

Random-effect estimates can be useful for checking the model or predicting sampled units, but naive histograms and normal Q–Q plots of heavily shrunken BLUPs do not display the latent distribution without distortion. Likewise, regressing an outcome on estimated BLUPs in a second stage ignores their uncertainty and can induce bias.

8 Missing and Unbalanced Data

Mixed models naturally accommodate unequal observation counts and irregular measurement times when the likelihood and covariance model are correctly specified. This is not the same as being immune to missingness.

Likelihood-based inference is generally justified when missingness is ignorable under the stated model—for example, missing at random given observed outcomes and covariates, together with distinct missingness parameters. Dropout depending on an unobserved current outcome or latent trajectory can make standard mixed model estimates biased. Record reasons for missingness, include relevant observed predictors, and use sensitivity models for plausible missing-not-at- random mechanisms.

9 Model Specification and Diagnostics

A useful workflow separates design, fitting, and criticism:

Identify biological units, repeated measurements, crossed or nested factors, and the target generalization.
Specify a scientifically interpretable fixed-effect mean, including required interactions and nonlinear time terms.
Encode design-supported random effects and any residual correlation or heterogeneous variance.
Check rank, group counts, within-group predictor variation, scaling, and optimizer convergence.
Examine marginal and conditional residuals, unit trajectories, fitted versus observed patterns, influence, and predictive performance at the correct unit.
Repeat key conclusions across plausible mean, covariance, and missingness specifications.

Warning	Possible mechanism	Useful response
Nonconvergence or large gradients	poor scaling, weak identification, unsuitable optimizer, overcomplex model	rescale, inspect design support, try validated optimizers, simplify transparently
Singular covariance estimate	zero or weakly identified component, too few informative groups	inspect within-group variation and report sensitivity
Residual trend or curvature	wrong mean trajectory or interaction	revise fixed effects before interpreting covariance
Residual spread varies	heteroskedasticity or wrong transformation	model variance by group/time or choose an outcome-appropriate model
Remaining serial pattern	random effects do not capture short-range residual dependence	add justified residual correlation and compare predictions
One group dominates	leverage, sparse treatment-by-group support, data error	verify data and perform leave-group-out influence analysis

Gaussian random effects and residuals are distributional assumptions. Fixed- effect estimates can sometimes be fairly robust to moderate shape deviations, but tail probabilities, variance components, BLUPs, and predictions may not be. Simulation from the fitted model and cluster-level resampling can reveal whether the intended inferential procedure is calibrated for the actual group count and imbalance.

10 Worked Bioinformatics Example

Suppose 60 patients contribute tumor biopsies before treatment and at two follow-up times. RNA abundance for a prespecified pathway score is measured from each biopsy, sometimes in duplicate libraries. Patients are recruited at four sites and processing batches overlap imperfectly with time.

A defensible model could include fixed effects for treatment, time, their interaction, baseline clinical covariates, and processing batch; patient random intercepts and time slopes; and a site effect if the sampling target and number of sites support it. Duplicate libraries are technical measurements. They may be summarized using measurement precision or represented by an explicit technical level, but they do not double the patient sample size.

The treatment-by-time contrast estimates a mean difference in pathway-score change for the modeled patient population. The patient slope variance describes unexplained trajectory heterogeneity. A conditional fitted curve describes a sampled patient; a prediction for a new patient includes between-patient variation. If dropout is related to unobserved disease progression, a standard MAR likelihood analysis needs an MNAR sensitivity analysis.

Statistically significant mean change is not necessarily biologically large or mechanistically attributable to treatment. Report the score scale, uncertainty, patient-level trajectories, heterogeneity, missingness, and whether the site and batch design supports transport beyond the observed centers.

11 Connections and Take-home Messages

Random intercepts and random slopes explains the most common covariance-building components.
Multiple linear regression provides the fixed-effect regression foundation.
Likelihood and log-likelihood and maximum-likelihood estimation develop the estimation principle.
Likelihood-based model comparison gives the general comparison framework and its nonregular limits.
Regression uncertainty and coefficient interpretation separates contrasts, mean responses, and prediction intervals.
Empirical Bayes explains the plug-in logic behind empirical BLUPs.
Sensitivity analysis and robustness organizes conclusions across credible alternatives.

A linear mixed model combines a fixed-effect mean with covariance induced by random effects and residual structure.
Population contrasts, variance components, existing-unit predictions, and new-unit predictions are different estimands.
ML and REML serve different comparison and estimation roles; boundary and small-group problems invalidate automatic asymptotic tests.
Convergence is computational evidence, not proof that the mean, covariance, missingness, or causal model is adequate.
Biological experimental units determine independent information; technical measurements must remain nested in that design.

12 References

Laird, N. M., and Ware, J. H. (1982). Random-effects models for longitudinal data. Biometrics, 38(4), 963–974. https://doi.org/10.2307/2529876
Patterson, H. D., and Thompson, R. (1971). Recovery of inter-block information when block sizes are unequal. Biometrika, 58(3), 545–554. https://doi.org/10.1093/biomet/58.3.545
Pinheiro, J. C., and Bates, D. M. (2000). Mixed-Effects Models in S and S-PLUS. Springer. https://doi.org/10.1007/b98882
Verbeke, G., and Molenberghs, G. (2000). Linear Mixed Models for Longitudinal Data. Springer. https://doi.org/10.1007/978-1-4419-0300-6
Bates, D., Mächler, M., Bolker, B. M., and Walker, S. C. (2015). Fitting linear mixed-effects models using lme4. Journal of Statistical Software, 67(1), 1–48. https://doi.org/10.18637/jss.v067.i01
Kenward, M. G., and Roger, J. H. (1997). Small sample inference for fixed effects from restricted maximum likelihood. Biometrics, 53(3), 983–997. https://doi.org/10.2307/2533558

--- title: "Linear Mixed-effects Models" subtitle: "Population effects, clustered dependence, and partial pooling in Gaussian outcomes" description: "Linear mixed-model specification, marginal covariance, ML and REML estimation, inference, diagnostics, and bioinformatics design." author: "Jiahong Sun" date: "2026-10-29" date-modified: "2026-10-29" categories: [Statistics, Mixed Models, Multilevel Models, Longitudinal Data] toc: true number-sections: true code-fold: show draft: false --- # A Regression Model for Clustered Gaussian Data Linear mixed-effects models extend linear regression by adding latent effects shared by observations in the same group. They are useful for repeated measures, longitudinal data, multi-site experiments, blocked designs, pedigrees, and nested sampling structures. Their central task is not merely to “correct” standard errors. They jointly describe the conditional mean and the covariance created by the sampling hierarchy. For all observations stacked into a vector, $$ \boxed{ \mathbf y=\mathbf X\boldsymbol\beta +\mathbf Z\mathbf b+\boldsymbol\varepsilon, \qquad \mathbf b\sim N(\mathbf0,\mathbf G), \qquad \boldsymbol\varepsilon\sim N(\mathbf0,\mathbf R). } $$ $\mathbf X\boldsymbol\beta$ is the fixed-effect mean structure, $\mathbf Z\mathbf b$ contains random effects, and $\mathbf R$ describes residual variation. The usual model assumes $\mathbf b$ and $\boldsymbol\varepsilon$ are independent. Integrating out $\mathbf b$ gives the marginal model $$ \boxed{ \mathbf y\sim N(\mathbf X\boldsymbol\beta,\mathbf V), \qquad \mathbf V=\mathbf Z\mathbf G\mathbf Z^\mathsf{T}+\mathbf R. } $$ This identity connects hierarchical interpretation to generalized least squares. Random effects are one structured way to build $\mathbf V$; they are not the only possible covariance model. # Start with the Scientific Target and Experimental Unit Before writing a formula, define the target population, independent sampling unit, outcome, predictor levels, and desired generalization. | Goal | Example estimand | Relevant level | |---|---|---| | Population mean comparison | mean treatment contrast $\mathbf c^\mathsf{T}\boldsymbol\beta$ | target population of biological units | | Heterogeneity | donor-level variance component in $\mathbf G$ | variation among donors | | Existing-unit prediction | conditional mean given estimated $\mathbf b_i$ | a sampled donor or site | | New-unit prediction | outcome integrated over a new $\mathbf b_{\mathrm{new}}$ | a future donor or site | The **estimand** is the population quantity in the scientific question. ML or REML fitting supplies an **estimator**, a random function of the data. Its observed coefficient or variance component is an **estimate**. A conditional prediction for a sampled donor is not the same target as a population-average fixed effect or a prediction for a new donor. Fixed and random are properties of the model and inferential target, not innate properties of variables. A treatment is often represented by fixed coefficients because its specific contrast is the target. A donor effect may be random when donors are viewed as sampled from a population and their distribution is used for pooling. The number of levels alone does not decide the choice. # Conditional and Marginal Interpretation Conditional on $\mathbf b$, the mean is $$ E(\mathbf y\mid\mathbf b)=\mathbf X\boldsymbol\beta+ \mathbf Z\mathbf b. $$ Marginal over the random effects, $$ E(\mathbf y)=\mathbf X\boldsymbol\beta. $$ For an identity-link Gaussian model, the fixed-effect coefficient has the same linear scale in both expressions, although predictions differ according to whether random effects are included. This convenient equality generally fails after a nonlinear link, which is one reason generalized linear mixed-model coefficients require special care. Consider a donor random-intercept model, $$ Y_{ij}=\beta_0+\beta_1T_{ij}+b_i+\varepsilon_{ij}, \qquad b_i\sim N(0,\tau^2). $$ $\beta_1$ is a conditional mean contrast at the modeled covariate values. A causal interpretation additionally needs treatment assignment, consistency, positivity, and control of confounding. The random effect accounts for modeled dependence; it does not by itself control unmeasured time-varying confounding. # Likelihood and Generalized Least Squares For known covariance parameters in $\mathbf V$, maximizing the Gaussian likelihood over $\boldsymbol\beta$ gives $$ \widehat{\boldsymbol\beta}_{\mathrm{GLS}} =(\mathbf X^\mathsf{T}\mathbf V^{-1}\mathbf X)^{-1} \mathbf X^\mathsf{T}\mathbf V^{-1}\mathbf y, $$ with model-based covariance $$ \operatorname{Var}(\widehat{\boldsymbol\beta}\mid\mathbf X) =(\mathbf X^\mathsf{T}\mathbf V^{-1}\mathbf X)^{-1}. $$ In practice, covariance parameters $\boldsymbol\theta$ inside $\mathbf G$ and $\mathbf R$ are unknown. The log-likelihood, apart from a constant, is $$ \ell(\boldsymbol\beta,\boldsymbol\theta) =-\frac12\left[ \log|\mathbf V_{\boldsymbol\theta}| +(\mathbf y-\mathbf X\boldsymbol\beta)^\mathsf{T} \mathbf V_{\boldsymbol\theta}^{-1} (\mathbf y-\mathbf X\boldsymbol\beta) \right]. $$ Maximum likelihood estimates fixed effects and covariance parameters jointly. Restricted maximum likelihood (REML) instead maximizes a likelihood for error contrasts that remove the fixed-effect degrees of freedom. One form of the REML criterion adds $$ \log|\mathbf X^\mathsf{T}\mathbf V^{-1}\mathbf X| $$ to the bracketed ML expression. REML often reduces small-sample bias in variance components, but it is not universally unbiased and does not rescue a misspecified covariance or mean model. # ML, REML, and Model Comparison The estimation method must match the comparison. - Use ML, not REML, to compare nested models with different fixed-effect columns by a likelihood-ratio statistic, because REML criteria are based on different error-contrast spaces. - REML criteria can compare covariance structures when the fixed-effect design is identical and regularity conditions are appropriate. - Refit the selected fixed-effect structure with REML when REML variance estimates and associated summaries are the reporting target. - Information criteria are comparable only when based on the same response data, likelihood convention, and compatible estimation target. Likelihood-ratio reference distributions can fail for testing a variance component equal to zero because the null lies on a boundary. Small numbers of groups also make large-sample chi-squared and normal approximations unreliable. Parametric bootstrap, restricted-likelihood methods with justified reference distributions, or carefully calibrated small-sample procedures may be preferable. # Fixed-effect Inference For a prespecified contrast $\psi=\mathbf c^\mathsf{T}\boldsymbol\beta$, the estimate is $\widehat\psi=\mathbf c^\mathsf{T}\widehat{\boldsymbol\beta}$ with plug-in standard error $$ \widehat{\operatorname{SE}}(\widehat\psi) =\sqrt{ \mathbf c^\mathsf{T} \widehat{\operatorname{Var}}(\widehat{\boldsymbol\beta}) \mathbf c}. $$ Wald intervals and tests often use a normal or approximate $t$ reference distribution. The denominator degrees of freedom are not uniquely defined after estimating variance components in an unbalanced design. Satterthwaite and Kenward--Roger methods approximate finite-sample uncertainty in different ways; software defaults should therefore be reported rather than hidden. Profile-likelihood intervals can better represent asymmetric uncertainty. Parametric bootstrap can propagate covariance estimation and nonlinearity if the fitted data-generating model is credible and the complete fitting procedure is repeated. None of these methods corrects pseudoreplication, informative missing data, model selection, or an unsupported causal interpretation. # Random Effects and Prediction Given covariance parameters, the conditional mean of the random effects is $$ E(\mathbf b\mid\mathbf y) =\mathbf G\mathbf Z^\mathsf{T}\mathbf V^{-1} (\mathbf y-\mathbf X\boldsymbol\beta). $$ Plugging in estimated parameters produces empirical BLUPs. Their shrinkage is a strength for prediction but changes their sampling behavior. A conditional prediction interval for another observation from an existing unit differs from a marginal interval for a new unit: the latter must include new random-effect variation. State which population and level the prediction concerns. Random-effect estimates can be useful for checking the model or predicting sampled units, but naive histograms and normal Q--Q plots of heavily shrunken BLUPs do not display the latent distribution without distortion. Likewise, regressing an outcome on estimated BLUPs in a second stage ignores their uncertainty and can induce bias. # Missing and Unbalanced Data Mixed models naturally accommodate unequal observation counts and irregular measurement times when the likelihood and covariance model are correctly specified. This is not the same as being immune to missingness. Likelihood-based inference is generally justified when missingness is ignorable under the stated model—for example, missing at random given observed outcomes and covariates, together with distinct missingness parameters. Dropout depending on an unobserved current outcome or latent trajectory can make standard mixed model estimates biased. Record reasons for missingness, include relevant observed predictors, and use sensitivity models for plausible missing-not-at- random mechanisms. # Model Specification and Diagnostics A useful workflow separates design, fitting, and criticism: 1. Identify biological units, repeated measurements, crossed or nested factors, and the target generalization. 2. Specify a scientifically interpretable fixed-effect mean, including required interactions and nonlinear time terms. 3. Encode design-supported random effects and any residual correlation or heterogeneous variance. 4. Check rank, group counts, within-group predictor variation, scaling, and optimizer convergence. 5. Examine marginal and conditional residuals, unit trajectories, fitted versus observed patterns, influence, and predictive performance at the correct unit. 6. Repeat key conclusions across plausible mean, covariance, and missingness specifications. | Warning | Possible mechanism | Useful response | |---|---|---| | Nonconvergence or large gradients | poor scaling, weak identification, unsuitable optimizer, overcomplex model | rescale, inspect design support, try validated optimizers, simplify transparently | | Singular covariance estimate | zero or weakly identified component, too few informative groups | inspect within-group variation and report sensitivity | | Residual trend or curvature | wrong mean trajectory or interaction | revise fixed effects before interpreting covariance | | Residual spread varies | heteroskedasticity or wrong transformation | model variance by group/time or choose an outcome-appropriate model | | Remaining serial pattern | random effects do not capture short-range residual dependence | add justified residual correlation and compare predictions | | One group dominates | leverage, sparse treatment-by-group support, data error | verify data and perform leave-group-out influence analysis | Gaussian random effects and residuals are distributional assumptions. Fixed- effect estimates can sometimes be fairly robust to moderate shape deviations, but tail probabilities, variance components, BLUPs, and predictions may not be. Simulation from the fitted model and cluster-level resampling can reveal whether the intended inferential procedure is calibrated for the actual group count and imbalance. # Worked Bioinformatics Example Suppose 60 patients contribute tumor biopsies before treatment and at two follow-up times. RNA abundance for a prespecified pathway score is measured from each biopsy, sometimes in duplicate libraries. Patients are recruited at four sites and processing batches overlap imperfectly with time. A defensible model could include fixed effects for treatment, time, their interaction, baseline clinical covariates, and processing batch; patient random intercepts and time slopes; and a site effect if the sampling target and number of sites support it. Duplicate libraries are technical measurements. They may be summarized using measurement precision or represented by an explicit technical level, but they do not double the patient sample size. The treatment-by-time contrast estimates a mean difference in pathway-score change for the modeled patient population. The patient slope variance describes unexplained trajectory heterogeneity. A conditional fitted curve describes a sampled patient; a prediction for a new patient includes between-patient variation. If dropout is related to unobserved disease progression, a standard MAR likelihood analysis needs an MNAR sensitivity analysis. Statistically significant mean change is not necessarily biologically large or mechanistically attributable to treatment. Report the score scale, uncertainty, patient-level trajectories, heterogeneity, missingness, and whether the site and batch design supports transport beyond the observed centers. # Connections and Take-home Messages - [Random intercepts and random slopes](random-intercepts-and-random-slopes.qmd) explains the most common covariance-building components. - [Multiple linear regression](../regression-models/multiple-linear-regression.qmd) provides the fixed-effect regression foundation. - [Likelihood and log-likelihood](../statistical-inference/likelihood-and-log-likelihood.qmd) and [maximum-likelihood estimation](../statistical-inference/maximum-likelihood-estimation.qmd) develop the estimation principle. - [Likelihood-based model comparison](../regression-models/likelihood-based-model-comparison.qmd) gives the general comparison framework and its nonregular limits. - [Regression uncertainty and coefficient interpretation](../regression-models/regression-uncertainty-and-coefficient-interpretation.qmd) separates contrasts, mean responses, and prediction intervals. - [Empirical Bayes](empirical-bayes.qmd) explains the plug-in logic behind empirical BLUPs. - [Sensitivity analysis and robustness](../statistical-inference/sensitivity-analysis-and-robustness.qmd) organizes conclusions across credible alternatives. 1. A linear mixed model combines a fixed-effect mean with covariance induced by random effects and residual structure. 2. Population contrasts, variance components, existing-unit predictions, and new-unit predictions are different estimands. 3. ML and REML serve different comparison and estimation roles; boundary and small-group problems invalidate automatic asymptotic tests. 4. Convergence is computational evidence, not proof that the mean, covariance, missingness, or causal model is adequate. 5. Biological experimental units determine independent information; technical measurements must remain nested in that design. # References - Laird, N. M., and Ware, J. H. (1982). Random-effects models for longitudinal data. *Biometrics*, 38(4), 963--974. <https://doi.org/10.2307/2529876> - Patterson, H. D., and Thompson, R. (1971). Recovery of inter-block information when block sizes are unequal. *Biometrika*, 58(3), 545--554. <https://doi.org/10.1093/biomet/58.3.545> - Pinheiro, J. C., and Bates, D. M. (2000). *Mixed-Effects Models in S and S-PLUS*. Springer. <https://doi.org/10.1007/b98882> - Verbeke, G., and Molenberghs, G. (2000). *Linear Mixed Models for Longitudinal Data*. Springer. <https://doi.org/10.1007/978-1-4419-0300-6> - Bates, D., Mächler, M., Bolker, B. M., and Walker, S. C. (2015). Fitting linear mixed-effects models using lme4. *Journal of Statistical Software*, 67(1), 1--48. <https://doi.org/10.18637/jss.v067.i01> - Kenward, M. G., and Roger, J. H. (1997). Small sample inference for fixed effects from restricted maximum likelihood. *Biometrics*, 53(3), 983--997. <https://doi.org/10.2307/2533558>