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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 hierarchical Bayesian model, replication conditional on fitted observation-level random effects checks the sampling layer. Replication with fresh random effects drawn from their population law also checks the population mean and variation. To assess a dose-response curve, sample the global parameters from their posterior distribution, draw fresh effects for new plates at each existing dose, and compare replicated means, spreads and trend discrepancies with the observed values. Reusing the fitted plate effects can conceal a misspecified mean curve by reproducing deviations those effects already absorbed.

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.

Drift-aware homogeneity analysis estimates material variation while accounting for time-dependent changes in the measurement system. Its purpose is to prevent instrumental drift from being assigned to differences between containers in a homogeneity study with replicated test portions. The recorded order is part of the experimental data.

A simple measurement model iswith container contribution $b_i$, linear drift rate $\gamma$ and residual $e_{ij}$. A linear regression can treat container differences as fixed effects; a random effect formulation instead models a between-container variance. Either approach must distinguish the time term from the container terms. Homogeneity-study time-term identifiability depends on the actual sampling schedule.

An original noiseless example uses A at times 0 and 5 with readings 100 and 105, B at 1 and 4 with readings 103 and 106, and C at 2 and 3 with readings 101 and 102. Each within-container difference divided by its time separation gives drift rate one. Subtracting that rate times observation time yields A readings 100, B readings 102 and C readings 99. The adjustment retains the true stipulated container differences instead of forcing all means to coincide.

With noise present, the rate is uncertain. Substituting a fitted rate and then analyzing the adjusted observations as if that rate were known omits its uncertainty and induced covariance. For example, six observations with an intercept, two independent container indicators and one time coefficient leave only two residual statistical degrees of freedom when the design has full rank. A more elaborate drift curve consumes information and may explain away genuine material differences.

Randomized order and control observations help diagnose the model. The IUPAC protocol requires randomized analysis of duplicate portions and checking trends or other suspect data features.^[ref] Separately, NIST's dimensional-comparison designs deliberately balance some drift contributions through observation order.^[ref]

Plot residuals against time and container, retain independent controls, and state how the time dependence was chosen. A fitted linear term does not establish that all drift is linear. If time and container effects cannot be distinguished, the study needs additional information or a better design rather than an unjustifiably precise homogeneity estimate.

A fixed effect represents an unknown nonrandom coefficient in the specified mean model. In a Gaussian linear mixed model, $X\beta$ is the population mean, while random effects are latent draws from a distribution whose parameters are estimated. Treating a group coefficient as fixed or random concerns the modelling and inferential target, not whether its estimate changes across samples.

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 generalized additive mixed model supplements smooth population effects with latent random effects. Given these effects, responses follow a specified response family; integrating them out induces dependence between related observations.

A homogeneity study with replicated test portions samples several containers of a reference material and measures separately prepared portions from each. The replication separates variation between containers from variation remaining within a container and the analytical procedure. Repeated detector readings of one prepared portion do not supply the same information.

The 2006 IUPAC protocol recommends randomly selecting at least ten distribution containers, homogenizing each selected container separately, and analyzing two portions from each in randomized order under a specified repeatability condition of measurement. It also requires adequate analytical precision and checks for trends, excessive rounding and discordant duplicate pairs.^[ref]

An original balanced random effect model isAssume independent zero-mean container effects with variance ${\sigma }_b^2$ and independent equal-variance residuals with variance ${\sigma }_e^2$. Residuals include analytical scatter and whatever within-container variation remains at the chosen portion size. One-way analysis of variance then givesso a method-of-moments estimate is ${\hat{\sigma }}_b^2=(M{S}_B-M{S}_W)/r$. A negative estimate can occur through sampling variation; it is not negative physical variance or proof of perfect reference material homogeneity.

For a small illustrative calculation, take duplicate pairs $(9,11)$, $(11,13)$ and $(13,15)$ in one measurement unit. The container means are 10, 12 and 14. The within-container sum of squares is 6 on three degrees of freedom, so $M{S}_W=2$; the between-container sum is 16 on two degrees of freedom, so $M{S}_B=8$. Thus ${\hat{\sigma }}_b^2=(8-2)/2=3$. These three invented containers explain the arithmetic; they do not meet the protocol's recommended ten-container study.

Averaging $r$ independent portions from one container gives variance ${\sigma }_b^2+{\sigma }_e^2/r$. More replication reduces the residual term without removing the common container effect. Randomizing the measurement order helps prevent instrumental drift from masquerading as between-container variation.

Fitness for purpose requires comparing material variation with an appropriate tolerance, not merely failing to reject zero variation. The protocol explicitly distinguishes sufficient homogeneity from the stricter hypothesis that sampling variance is exactly zero.^[ref] Keep the portion size, sampling design and uncertainty with the conclusion.

Differences between supplied proficiency-test items can contribute to a proficiency-test z-score even when the laboratory's own measurement procedure has zero systematic error. Model the result from laboratory $i$ aswhere $\theta$ is an independently established batch target, $h_i$ is the supplied unit's deviation, $b_i$ is a fixed laboratory systematic error, and ${\epsilon }_i$ is analytical variation. Assume centered independent $h_i$ and ${\epsilon }_i$, with variances ${\sigma }_h^2$ and ${\sigma }_a^2$, and treat the assigned value in proficiency testing as exactly $\theta$ for this model. Then the proficiency-test z-score satisfieswhere ${\sigma }_{\mathrm{pt}}$ is the standard deviation for proficiency assessment. A large deviation cannot be attributed to $b_i$ from the score alone. The primary guide requires item reference material homogeneity and reference material stability to remain appropriate when a stricter assessment scale is selected.^[ref]

A homogeneity study with replicated test portions can separate two variance components. Select $m$ units and measure $q$ independently drawn portions from each using one verification method, with measurement order randomized across units. Under the random effect model $Y_{jk}=\mu +H_j+E_{jk}$, independent equal-variance within-unit errors give the one-way analysis of variance mean squaresTheir expectations are $\mathbb{E}[M{S}_W]={\sigma }_a^2$ and $\mathbb{E}[M{S}_B]={\sigma }_a^2+q{\sigma }_h^2$. A method-of-moments variance component estimate is therefore $(M{S}_B-M{S}_W)/q$. Here the within-unit component is treated as analytical variation; variation between portions of the same unit would also enter it unless separated by further measurements. Clipping a negative estimate to zero enforces a nonnegative variance component; it introduces boundary bias and does not establish perfect material uniformity.

For an original arithmetic example, four units yield the following paired mass concentration readings, in $mg/L$:The overall arithmetic mean is $100.05$. These chosen data give $M{S}_W=0.02$, $M{S}_B=0.50$ and ${\hat{\sigma }}_h^2=0.24$, with squared concentration units. Suppose a participant also has analytical variance $0.02$ and the assessment scale is $0.20mg/L$. The stipulated unbiased score then has standard deviation $\sqrt{0.26}/0.20\approx 2.55$, despite the absence of a laboratory systematic error in the model.

Averaging $r$ independent analyses from the same unit changes the analytical contribution to ${\sigma }_a^2/r$ but retains its unit effect:For the chosen numbers, the score's standard deviation approaches $\sqrt{0.24}/0.20\approx 2.45$ as $r$ increases. Repeatedly reading one aliquot may reduce an even narrower instrumental contribution. Sampling independently supplied units addresses a different source of variation.

This is an explanatory measurement model, not a prescribed proficiency-testing acceptance rule. Its variance components require evidence about portion size, preparation and measurement conditions. Instrumental drift confounded with unit order, changing material, correlated errors or an uncertain consensus benchmark require additional terms and an appropriate design. The provider's homogeneity evidence is essential to interpreting the score; a smaller numerical scoring denominator does not improve the items.

A statistical model combining fixed effects with random effects. The grouping structure determines the correlations represented by the model.

Nan Laird develops statistical methods for incomplete observations, repeated measurements and the combination of evidence across studies. Her Harvard biography identifies contributions to the expectation-maximization algorithm, meta-analysis, longitudinal studies and statistical genetics.^[ref]

In a longitudinal study, repeated outcomes from one person are usually dependent. A model with random effects can represent persistent differences between people while estimating population relationships. Treating those observations as independent can distort uncertainty. Laird's work on longitudinal analysis addresses this connection between a scientifically plausible dependence structure and practical inference; the same concern arises when results from heterogeneous studies are combined by random-effects meta-analysis.

Homepage: hsph.harvard.edu/profile/nan-m-laird/

Nested measurements are responses grouped inside larger units, with each measured object belonging to just one such unit. Three leaves within each pot give nested measurements; repeatedly reading one leaf gives another level of repetition. Either can yield clustered data. A common random effect can describe the shared influence of the containing experimental unit.

For the constructed pot example, label treatments by $g\in \{A,B\}$, pots within each group by $i=1,\dots ,n$, and leaves within each pot by $j=1,\dots ,m$. Consider the statistical modelHere $Y_{gij}$ is a leaf length, $\mu +{\alpha }_g$ is its group expected value, $U_{gi}$ is a pot random effect, and ${\epsilon }_{gij}$ describes remaining variation among leaves and measurement error. Assume that all pot random effects are mutually independent, all remaining errors are mutually independent, and the two collections are independent of one another and of the randomization. These are explicit modelling assumptions, conditional on the assigned treatments; no normal distribution assumption is needed for the following variance calculation.

Two distinct leaf measurements in one pot share $U_{gi}$. Their covariance is ${\tau }^2$, their individual variance is ${\tau }^2+{\sigma }^2$, and their intraclass correlation coefficient isMeasurements from different pots have zero covariance and are independent under the stated statistical model. Statistical independence therefore concerns the whole response structure, not just the fact that the measured leaves are different objects.

A nested wringing-repeatability study separates variation between independently prepared gauge block wringings from readout scatter within one unchanged wring. Repeating a detector reading many times does not sample the contact variability introduced by separating and rejoining the surfaces.

For $m$ wrings with $r$ readings each, an original balanced random effect model iswhere independent wring effects have variance ${\tau }^2$ and independent readout errors have variance ${\sigma }^2$. Assume stable conditions and independence between those levels. One-way analysis of variance givesThus $M{S}_W$ estimates readout variance and $(M{S}_B-M{S}_W)/r$ estimates the between-wring variance in this model. A negative variance estimate reflects sampling variability or model limitations, not a physically negative contact variance.

For a small invented data set, duplicate readings in nm relative to an arbitrary reference are $(-3,-1)$, $(1,3)$ and $(5,7)$. Wring means are $-2,2,6$; $M{S}_W=2$ and $M{S}_B=32$, so the method-of-moments between-wring variance is $15\mathrm{n}{\mathrm{m}}^2$. These are arithmetic examples, not a recommended sample size or a NIST calibration result.

For the overall mean of the balanced study,With stipulated $\tau =6\mathrm{nm}$, $\sigma =2\mathrm{nm}$, five wrings and four readings per wring, its standard deviation is $\sqrt{7.4}\approx 2.72\mathrm{nm}$. Increasing to forty readings on the same five wrings lowers it only to about 2.69 nm. Fifty wrings with four readings each would instead give about 0.86 nm, if the same model and stable preparation remain valid.

A shared reference or persistent preparation contribution does not disappear under either averaging rule. An independent common uncertainty $u_0$ adds $u_0^2$ to the displayed variance. This connects the nested study with the wringing-film covariance model and the gauge-block reference uncertainty floor.

NIST's handbook documents contact-layer dependence on preparation and time, including evolution after thermal equilibration.^[ref] Record rewringing events, timestamps and surface preparation; use appropriate controls for drift and damage. A nested contact-study allocation should balance new wrings against repeat readings, rather than treating every reading as an independent contact experiment.

A normality assumption specifies a normal distribution for the relevant observations, errors, or latent effects. These are different assumptions: normal latent random effects do not imply that the observed response is normal. Estimated regression residuals or fitted random effects are not independent copies of the underlying errors or effects.

For observation $j$ of bird $i$ in year $t(i,j)$, a Gaussian linear mixed model with crossed random intercepts iswhere the $u_i$, $v_t$, and ${\epsilon }_{ij}$ are mutually independent random variables. Here $a_{ij}$ is age, ${\beta }_0$ is the population intercept at age zero, and ${\beta }_1$ is the change in conditional expected value per year of age. The $u_i$ capture persistent between-bird variation, the $v_t$ capture shared annual conditions, and the regression residuals capture remaining observation-level variation. Two observations from the same bird share covariance ${\tau }_b^2$; two from the same year share ${\tau }_y^2$, with both contributions present if both group labels agree. The model also assumes the random effects are independent of the age covariate.

Conditional predictions for observed birds and years add their fitted random intercepts to the fixed part of the Gaussian linear mixed model:These are predicted expected values, so they need not be integers. The first is likely more reliable: age three is within the displayed age range, whereas age seven requires extrapolation beyond ages one to six. A quantitative comparison would need the prediction intervals, including uncertainty in both fixed coefficients and random effects; the displayed coefficients alone do not supply them.

A frailty model represents unobserved individual heterogeneity by a random effect in the hazard function. In a proportional frailty model, an individual with frailty random variable $U=u$ has hazard $u{h}_0(t)$. The normalization $\mathbb{E}U=1$ identifies the scale of the baseline hazard: without it, multiplying all frailties by a constant and dividing $h_0$ by that constant would leave all individual hazards unchanged.

Let $H_0(t)={\int }_0^t{h}_0(v)dv$. Conditional survival follows by integrating the hazard function:Taking the expectation over the frailty law yieldsThis is the Laplace transform of the frailty density evaluated at the cumulative hazard. More generally, the same calculation integrates against a probability measure, allowing atoms. It averages conditional survival rather than substituting $\mathbb{E}U$ into its exponent. Indeed, the frailty distribution among survivors is proportional to $e^{-u{H}_0(t)}g(u)$, giving population hazardSurvival progressively selects smaller frailties, so the population hazard need not be the baseline or even retain the individual's proportional structure.

For the cure model, first take $0\le \pi <1$. The necessary mean-one frailty law and baseline hazard areThen $\mathbb{E}U=\pi \cdot 0+(1-\pi )/(1-\pi )=1$. A cured person's conditional hazard is zero, while a susceptible person's conditional hazard is $h_0/(1-\pi )=h_{\ast }$. Taking $U$ to be zero or one while leaving $h_0=h_{\ast }$ would instead have mean frailty $1-\pi$ and violate the required normalization.

The frailty law is the measurewhere the $\delta$ terms are Dirac measures. This is not an ordinary Lebesgue density; the cure construction requires permitting a discrete frailty law. Its Laplace transform isWriting $H_{\ast }(t)={\int }_0^t{h}_{\ast }(v)dv$, we have $H_0=(1-\pi )H_{\ast }$, and thereforeIf $H_{\ast }(t)\to \infty$, the susceptible class eventually experiences the event and $S(t)\to \pi$. If $H_{\ast }(\infty )<\infty$, the limiting survival is larger than $\pi$ because some susceptible subjects also never experience the event. At $\pi =1$, everyone is cured: set $h_0\equiv 0$ and, for example, $U\equiv 1$ to retain mean one, giving $S\equiv 1$ without using the singular formula $1/(1-\pi )$.

Model the count relative to exposure, for exampleHere $E_i$ is a measured exposure such as patient-days, $x_i$ contains hospital predictors, and $b_i$ is a random effect. This Poisson regression separates hospital size from rate variation and supplies partial pooling. Use proper, scale-appropriate priors on coefficients and variation; add regional or temporal effects if the data demand them. A gamma mixing law with an estimated shape is another way to allow more flexible overdispersion. Compare the fitted model's counts and dispersion through posterior predictive checks.

Model B replaces independent tree parameters by multivariate normal random effects. It permits correlated asymptote, timing and slope variation and supplies partial pooling, a useful improvement if the five trees can be regarded as exchangeable.

The Wishart distribution prior is on the precision matrix $\Omega$, not the covariance matrix. In the WinBUGS convention, dimension three and degrees of freedom three give a proper but very dispersed prior. The induced inverse-Wishart distribution has no finite mean at those degrees of freedom. Thus this is not a neutral choice of covariance prior, and estimating six covariance components from only five trees is weakly informed. The broad mean priors and possible negative growth slopes also remain. Check prior-predictive curves and sensitivity, and consider separate, interpretable scale and correlation priors.

The resulting posterior means of the second and third parameters are quite similar across trees, motivating a simpler shared-shape model; their uncertainty means this similarity is evidence to explore, rather than a proof of exact equality.

Keep a tree-specific log asymptote $a_i$ with a normal population distribution, but use one shared location $b$ and one shared positive growth coefficient $k$. For example, with fixed prior constants chosen on meaningful scales:

mean.a ~ dnorm(m.a, precision.a.prior)
sd.a ~ dnorm(0, precision.sd.a) T(0,)
precision.a <- 1 / pow(sd.a, 2)
b ~ dnorm(m.b, precision.b.prior)
k ~ dnorm(m.k, precision.k.prior) T(0,)
sigma ~ dnorm(0, precision.sigma.prior) T(0,)
precision.y <- 1 / pow(sigma, 2)
for (i in 1:5) {
    a[i] ~ dnorm(mean.a, precision.a)
    for (j in 1:7) {
        fitted[i,j] <- exp(a[i]) / (1 + exp(-b - k*t[j]))
        Y[i,j] ~ dnorm(fitted[i,j], precision.y)
    }
}

Here the transformed days t[j] are supplied as data. This is Model C: only the log asymptote is a random effect, while location and growth rate are shared. The prior constants are fixed inputs, and the truncations give positive standard deviations and an increasing growth curve. One can retain the original error-precision prior for a comparison matching the supplied fits; the code above illustrates a more interpretable positive-scale alternative.