Gaussian linear mixed model ($Y=X\beta +Zb+\epsilon$)

A Gaussian linear mixed model combines fixed effects, random effects with a normal distribution, and independent Gaussian noise. For $b\sim N(0,D)$ and $\epsilon \sim N(0,R)$ independently, its marginal distribution is $Y\sim N(X\beta ,V)$ with $V=ZD{Z}^T+R$. The random effects induce dependence among observations sharing their design columns. A random-intercept linear mixed model and a random-slope linear mixed model are special cases.

A variance component is a nonnegative parameter multiplying a specified covariance contribution. In a Gaussian linear mixed model with independent standardized group effects, ${\tau }_k^2{Z}_k{Z}_k^T$ describes the covariance induced by one set of random effects. Testing whether a component is zero is a variance-component likelihood-ratio test at a boundary; ordinary regular chi-squared likelihood-ratio calibration need not apply.

With known covariance parameters, best linear unbiased prediction minimizes prediction-error variance over linear predictors that are unbiased over both observation errors and random effects, for every fixed-effect value. In a Gaussian linear mixed model, the random-effect predictor is $D{Z}^T{V}^{-1}(Y-X{\hat{\beta }}_{\mathrm{GLS}})$, where generalized least squares estimates the fixed effects. Plugging in covariance estimates gives an empirical predictor; its uncertainty must also account for estimating those covariance parameters. It differs from a best linear unbiased estimator of an unknown fixed coefficient.

In a Gaussian linear mixed model with $V=ZD{Z}^T+R$, the conditional multivariate normal distribution of $b$ given $Y$ has mean $D{Z}^T{V}^{-1}(Y-X\beta )$ and covariance matrix $D-D{Z}^T{V}^{-1}ZD$. When nonsingular, its mean is also its mode. Estimated parameters yield empirical conditional modes and shrink group deviations toward zero. At a zero variance component the associated effect is degenerate at zero; the covariance formula still applies, whereas formulas involving $D^{-1}$ require limits.

For $\kappa >0$, this error correlation decreases exponentially with elapsed time and permits irregular observation times. It is the stationary correlation of an Ornstein-Uhlenbeck process. Within a Gaussian linear mixed model, apply it to the errors conditional on random effects; the marginal correlation also includes those effects. nlme::corCAR1 parametrizes the same correlation by $\rho =e^{-\kappa }\in (0,1)$ and permits separate grouped time series.

Independent Gaussian random intercepts $u_j$ and random slopes $v_j$ give marginal within-group covariance ${\tau }_0^2+{\tau }_1^2{t}_{ij}{t}_{kj}+{\sigma }^2{1}_{\{i=k\}}$. In lme4, separate terms (1 | group) and (0 + x | group) impose zero intercept-slope covariance; (1 + x | group) estimates it. This independence restriction depends on the predictor origin: replacing $t$ by $t-c$ transforms the intercept effect to $u_j+c{v}_j$, generally correlated with $v_j$.