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Under fit.2, observations in each period are independent Poisson random variables with a common mean, so each sample variance should fluctuate around its sample mean. The plot instead has variances roughly 130–300 for means roughly 80–120, well above the line variance equals mean. This is overdispersion.

Day-to-day traffic variation can produce a Poisson mixture: a random daily intensity increases the marginal variance and can correlate observations from the same day. A useful replacement is a negative binomial regression, with variance $\mu +{\mu }^2/\kappa$, or a Poisson generalized linear mixed model with a day random intercept. A day-factor effect can also be included if inference is restricted to the observed days. A Quasi-Poisson regression estimates a dispersion multiplier for uncertainty, but does not itself explain the shared-day dependence and has no ordinary likelihood-based AIC. The means can retain the period-factor or selected jump structure.

Write $r=1,\dots ,16$ for rats and $j$ for repeated observations. With diet 1 as the reference level in a regression factor, the random-intercept linear mixed model iswith the two families of normal distributions independent and the covariates treated as fixed. The reported estimates areThe corresponding estimated standard deviations are $36.577$ and $8.176$ grams. Individual realized random intercepts are not parameters in this marginal model, and their predictions are not supplied by this summary. For distinct measurements on the same rat, $\mathrm{Cov}(Y_{rj},Y_{rk})={\tau }^2$, while the marginal variance is ${\tau }^2+{\sigma }^2$; different rats are independent.

The estimated mean weight rises by $0.58568$ grams per day, adjusting for diet and the rat's random intercept. This common slope is imposed across all diets: there is no time-by-diet interaction term, so these estimates do not compare diet-specific growth slopes. The intercept refers to time zero for diet 1 and a zero rat effect.

Let $Z_{ir}=1_{\{\text{observation }i\text{ belongs to rat }r\}}$. After integrating out the random intercepts,The fixed-effect design matrix has four independent columns, so the given $A$ has dimensions $176\times 172$, with $A^{T}A=I_{172}$ and $A^{T}X=0$. Consequently the error contrasts $U=A^{T}Y$ have a multivariate normal distribution with mean zero and covariance matrix $W=A^{T}VA$.

The restricted likelihood from orthogonal error contrasts is their density. Equivalently, the estimates maximizeThis is restricted maximum likelihood, not maximization over individual rat effects. The unknown $\beta$ disappears because $A^{T}X=0$. Replacing $A$ by $AQ$ for an orthogonal $Q$ rotates $U$ and conjugates $W$, preserving the determinant and quadratic form; hence the objective is independent of the chosen orthonormal basis. The software's integrated REML log-likelihood can differ by an additive constant depending only on $X$, which has no effect on these variance estimates.

Treatment was assigned by school, so observations within one school are clustered data and plausibly correlated. A random intercept models shared unobserved school characteristics and prevents the effective amount of treatment-level information from being exaggerated by treating all students as independent.

The fitted school standard deviation is $0.209$, so the estimated between-school variation is nonzero. Accounting for it raises the TV and SC standard errors from about $0.065$ in "lm1" to about $0.105$ in "lme1", reflecting that only 28 independently randomized schools identify those effects.

For distinct sibling supernovae $s$ and $s^{\prime }$ in the same galaxy, the shared fluctuation is the only common random term. Independence and linearity of covariance therefore giveTheir correlation coefficient isThis is the intraclass correlation coefficient of the galaxy-level random intercept model.

The generalized linear mixed model tries to explain overdispersion by replacing the fixed minority coefficient with a Gaussian random intercept. Conditional on the group effect $b_m$,

This is a poor use of a random effect because minority has only two levels. Two realized intercepts contain almost no information about a random-effects distribution or its variance, and the two levels are substantively fixed categories rather than a sample from a population of groups. The model also has worse Akaike information criterion than model 1, $1132.8>1122.3$, and its fit does not establish that the original overdispersion has disappeared.

For response $j$ from subject $s$, model2 is the generalized linear mixed modelwith conditional independence given the random intercepts. The fitted values are ${\hat{\beta }}_0=8.9966$, ${\hat{\beta }}_1=-1.2317$, ${\hat{\tau }}^2=1.7533$, and fitted residual dispersion $0.7591$.

The random intercept models persistent between-subject differences and the resulting within-subject dependence among repeated measurements. That is the main feature absent from model1.

Put $p_B={\mathrm{logit}}^{-1}(\eta +B)$ and $M(\eta )=\mathbb{E}(p_B)$. Differentiation gives $M^{\prime }=\mathbb{E}[p_B(1-p_B)]=M(1-M)-\mathrm{Var}(p_B)$. Hence the displayed derivative lies between zero and one, strictly below one for a nondegenerate random intercept. A marginal logit is therefore generally nonlinear and its slope differs from the conditional slope even without confounding. This is a precise instance of noncollapsibility of the odds ratio.

A random slope multiplies a predictor by a group-specific latent coefficient. With $b_j\sim N(0,{\tau }^2)$, a contribution $b_j{t}_i$ induces covariance ${\tau }^2{t}_i{t}_k$ for two observations in group $j$. A random slope does not necessarily include a random intercept.

Persistent unmeasured differences between individuals can generate outcome dependence after adjusting for observed covariates. A random intercept is one representation. Conditional independence given that latent variable need not persist after it is integrated out; apparent history dependence can therefore occur without true state dependence.