**[ME] Glossary** -- Glossary of terms

__Description__

**ANOVA denominator degrees of freedom (DDF) method**. This method uses the
traditional ANOVA for computing DDF. According to this method, the
DDF for a test of a fixed effect of a given variable depends on
whether that variable is also included in any of the random-effects
equations. For traditional ANOVA models with balanced designs, this
method provides exact sampling distributions of the test statistics.
For more complex mixed-effects models or with unbalanced data, this
method typically leads to poor approximations of the actual sampling
distributions of the test statistics.

**approximation denominator degrees of freedom (DDF) methods**. The
Kenward-Roger and Satterthwaite DDF methods are referred to as
approximation methods because they approximate the sampling
distributions of test statistics using t and F distributions with the
DDF specific to the method for complicated mixed-effects models and
for simple mixed models with unbalanced data. Also see *exact*
*denominator degrees of freedom (DDF) methods*.

**between-within denominator degrees of freedom (DDF) method**. See *repeated*
*denominator degrees of freedom (DDF) method*.

**BLUPs**. BLUPs are best linear unbiased predictions of either random
effects or linear combinations of random effects. In linear models
containing random effects, these effects are not estimated directly
but instead are integrated out of the estimation. Once the fixed
effects and variance components have been estimated, you can use
these estimates to predict group-specific random effects. These
predictions are called BLUPs because they are unbiased and have
minimal mean squared errors among all linear functions of the
response.

**canonical link**. Corresponding to each family of distributions in a
generalized linear model (GLM) is a canonical link function for which
there is a sufficient statistic with the same dimension as the number
of parameters in the linear predictor. The use of canonical link
functions provides the GLM with desirable statistical properties,
especially when the sample size is small.

**conditional hazard function**. In the context of mixed-effects survival
models, the conditional hazard function is the hazard function
computed conditionally on the random effects. Even within the same
covariate pattern, the conditional hazard function varies among
individuals who belong to different random-effects clusters.

**conditional hazard ratio**. In the context of mixed-effects survival
models, the conditional hazard ratio is the ratio of two conditional
hazard functions evaluated at different values of the covariates.
Unless stated differently, the denominator corresponds to the
conditional hazard function at baseline, that is, with all the
covariates set to zero.

**conditional overdispersion**. In a negative binomial mixed-effects model,
conditional overdispersion is overdispersion conditional on random
effects. Also see *overdispersion*.

**containment denominator degrees of freedom (DDF) method**. See *ANOVA*
*denominator degrees of freedom (DDF) method*.

**continuous-time autoregressive structure**. A generalization of the
autoregressive structure that allows for unequally spaced and
noninteger time values.

**covariance structure**. In a mixed-effects model, covariance structure
refers to the variance-covariance structure of the random effects.

**crossed-effects model**. A crossed-effects model is a mixed-effects model
in which the levels of random effects are not nested. A simple
crossed-effects model for cross-sectional time-series data would
contain a random effect to control for panel-specific variation and a
second random effect to control for time-specific random variation.
Rather than being nested within panel, in this model a random effect
due to a given time is the same for all panels.

**crossed-random effects**. See *crossed-effects model*.

**EB**. See *empirical Bayes*.

**empirical Bayes**. In generalized linear mixed-effects models, empirical
Bayes refers to the method of prediction of the random effects after
the model parameters have been estimated. The empirical Bayes method
uses Bayesian principles to obtain the posterior distribution of the
random effects, but instead of assuming a prior distribution for the
model parameters, the parameters are treated as given.

**empirical Bayes mean**. See *posterior mean*.

**empirical Bayes mode**. See *posterior mode*.

**error covariance**, **error covariance structure**. Variance-covariance
structure of the errors within the lowest-level group. For example,
if you are modeling random effects for classes nested within schools,
then error covariance refers to the variance-covariance structure of
the observations within classes, the lowest-level groups. With a
slight abuse of the terminology, error covariance is sometimes also
referred to as residual covariance or residual error covariance in
the literature.

**exact denominator degrees of freedom (DDF) methods**. Residual, repeated,
and ANOVA DDF methods are referred to as exact methods because they
provide exact t and F sampling distributions of test statistics for
special classes of mixed-effects models -- linear regression,
repeated-measures designs, and traditional ANOVA models -- with
balanced data. Also see *approximation denominator degrees of freedom*
*(DDF) methods*.

**fixed effects**. In the context of multilevel mixed-effects models, fixed
effects represent effects that are constant for all groups at any
level of nesting. In the ANOVA literature, fixed effects represent
the levels of a factor for which the inference is restricted to only
the specific levels observed in the study. See also *fixed-effects*
*model* in **[XT] Glossary**.

**free parameter**. Free parameters are parameters that are not defined by a
linear form. Free parameters are displayed with a forward slash in
front of their names or their equation names.

**Gauss-Hermite quadrature**. In the context of generalized linear mixed
models, Gauss-Hermite quadrature is a method of approximating the
integral used in the calculation of the log likelihood. The
quadrature locations and weights for individual clusters are fixed
during the optimization process.

**generalized linear mixed-effects model**. A generalized linear
mixed-effects model is an extension of a generalized linear model
allowing for the inclusion of random deviations (effects).

**generalized linear model**. The generalized linear model is an estimation
framework in which the user specifies a distributional family for the
dependent variable and a link function that relates the dependent
variable to a linear combination of the regressors. The distribution
must be a member of the exponential family of distributions. The
generalized linear model encompasses many common models, including
linear, probit, and Poisson regression.

**GHQ**. See *Gauss-Hermite quadrature*.

**GLM**. See *generalized linear model*.

**GLME model**. See *generalized linear mixed-effects model*.

**GLMM**. Generalized linear mixed model. See *generalized linear*
*mixed-effects model*.

**hierarchical model**. A hierarchical model is one in which successively
more narrowly defined groups are nested within larger groups. For
example, in a hierarchical model, patients may be nested within
doctors who are in turn nested within the hospital at which they
practice.

**intraclass correlation**. In the context of mixed-effects models,
intraclass correlation refers to the correlation for pairs of
responses at each nested level of the model.

**Kenward-Roger denominator degrees of freedom (DDF) method**. This method
implements the Kenward and Roger (1997) method, which is designed to
approximate unknown sampling distributions of test statistics for
complex linear mixed-effects models. This method is supported only
with restricted maximum-likelihood estimation.

**Laplacian approximation**. Laplacian approximation is a technique used to
approximate definite integrals without resorting to quadrature
methods. In the context of mixed-effects models, Laplacian
approximation is as a rule faster than quadrature methods at the cost
of producing biased parameter estimates of variance components.

**Lindstrom-Bates algorithm**. An algorithm used by the linearization method.

**linear form**. A linear combination is what we call a "linear form" as long
as you do not refer to its coefficients or any subset of the linear
combination anywhere in the expression. Linear forms are beneficial
for some nonlinear commands such as **nl** because they make derivative
computation faster and more accurate. In contrast to free parameters,
parameters of a linear form are displayed without forward slashes in
the output. Rather, they are displayed as parameters within an
equation whose name is the linear combination name. Also see *Linear*
*forms versus linear combinations* in **[ME] menl**.

**linear mixed model**. See *linear mixed-effects model*.

**linear mixed-effects model**. A linear mixed-effects model is an extension
of a linear model allowing for the inclusion of random deviations
(effects).

**linearization log likelihood**. Objective function used by the
linearization method for optimization. This is the log likelihood of
the linear mixed-effects model used to approximate the specified
nonlinear mixed-effects model.

**linearization method**, **Lindstrom-Bates method**. Method developed by
Lindstrom and Bates (1990) to approximate for fitting nonlinear
mixed-effects models. The linearization method uses a first-order
Taylor-series expansion of the specified nonlinear mean function to
approximate it with a linear function of fixed and random effects.
Thus a nonlinear mixed-effects model is approximated by a linear
mixed-effects model, in which the fixed-effects and random-effects
design matrices involve derivatives of the nonlinear mean function
with respect to fixed effects (coefficients) and random effects,
respectively. Also see *Introduction* in **[ME] menl**.

**link function**. In a generalized linear model or a generalized linear
mixed-effects model, the link function relates a linear combination
of predictors to the expected value of the dependent variable. In a
linear regression model, the link function is simply the identity
function.

**LME model**. See *linear mixed-effects model*.

**lowest-level group**. The second level of a multilevel model with the
observations composing the first level. For example, if you are
modeling random effects for classes nested within schools, then
classes are the lowest-level groups.

**MCAGH**. See *mode-curvature adaptive Gauss-Hermite quadrature*.

**mean-variance adaptive Gauss-Hermite quadrature**. In the context of
generalized linear mixed models, mean-variance adaptive Gauss-Hermite
quadrature is a method of approximating the integral used in the
calculation of the log likelihood. The quadrature locations and
weights for individual clusters are updated during the optimization
process by using the posterior mean and the posterior standard
deviation.

**mixed model**. See *mixed-effects model*.

**mixed-effects model**. A mixed-effects model contains both fixed and random
effects. The fixed effects are estimated directly, whereas the random
effects are summarized according to their (co)variances.
Mixed-effects models are used primarily to perform estimation and
inference on the regression coefficients in the presence of
complicated within-subject correlation structures induced by multiple
levels of grouping.

**mode-curvature adaptive Gauss-Hermite quadrature**. In the context of
generalized linear mixed models, mode-curvature adaptive
Gauss-Hermite quadrature is a method of approximating the integral
used in the calculation of the log likelihood. The quadrature
locations and weights for individual clusters are updated during the
optimization process by using the posterior mode and the standard
deviation of the normal density that approximates the log posterior
at the mode.

**MVAGH**. See *mean-variance adaptive Gauss-Hermite quadrature*.

**named substitutable expression**. A named substitutable expression is a
substitutable expression defined within **menl**'s **define()** option; see
*Substitutable expressions* in **[ME] menl**.

**nested random effects**. In the context of mixed-effects models, nested
random effects refer to the nested grouping factors for the random
effects. For example, we may have data on students who are nested in
classes that are nested in schools.

**NLME model**. See *nonlinear mixed-effects model*.

**nonlinear mixed-effects model**. A model in which the conditional mean
function given random effects is a nonlinear function of fixed and
random effects. A linear mixed-effects model is a special case of a
nonlinear mixed-effects model.

**one-level model**. A one-level model has no multilevel structure and no
random effects. Linear regression is a one-level model.

**overdispersion**. In count-data models, overdispersion occurs when there
is more variation in the data than would be expected if the process
were Poisson.

**posterior mean**. In generalized linear mixed-effects models, posterior
mean refers to the predictions of random effects based on the mean of
the posterior distribution.

**posterior mode**. In generalized linear mixed-effects models, posterior
mode refers to the predictions of random effects based on the mode of
the posterior distribution.

**QR decomposition**. QR decomposition is an orthogonal-triangular
decomposition of an augmented data matrix that speeds up the
calculation of the log likelihood; see *Methods and formulas* in **[ME]**
**mixed** for more details.

**quadrature**. Quadrature is a set of numerical methods to evaluate a
definite integral.

**random coefficient**. In the context of mixed-effects models, a random
coefficient is a counterpart to a slope in the fixed-effects
equation. You can think of a random coefficient as a randomly
varying slope at a specific level of nesting.

**random effects**. In the context of mixed-effects models, random effects
represent effects that may vary from group to group at any level of
nesting. In the ANOVA literature, random effects represent the
levels of a factor for which the inference can be generalized to the
underlying population represented by the levels observed in the
study. See also *random-effects model* in **[XT] Glossary**.

**random intercept**. In the context of mixed-effects models, a random
intercept is a counterpart to the intercept in the fixed-effects
equation. You can think of a random intercept as a randomly varying
intercept at a specific level of nesting.

**random-effects substitutable expression**. A random-effects substitutable
expression is a substitutable expression containing random-effects
terms; see *Random-effects substitutable expressions* in **[ME] menl**.

**REML**. See *restricted maximum likelihood*.

**repeated denominator degrees of freedom (DDF) method**. This method uses
the repeated-measures ANOVA for computing DDF. It is used with
balanced repeated-measures designs with spherical correlation error
structures. It partitions the residual degrees of freedom into the
between-subject degrees of freedom and the within-subject degrees of
freedom. The repeated method is supported only with two-level
models. For more complex mixed-effects models or with unbalanced
data, this method typically leads to poor approximations of the
actual sampling distributions of the test statistics.

**residual covariance**, **residual error covariance**. See *error covariance*.

**residual denominator degrees of freedom (DDF) method**. This method uses
the residual degrees of freedom, n - rank(X), as the DDF for all
tests of fixed effects. For a linear model without random effects
with independent and identically distributed errors, the
distributions of the test statistics for fixed effects are t or F
distributions with the residual DDF. For other mixed-effects models,
this method typically leads to poor approximations of the actual
sampling distributions of the test statistics.

**restricted maximum likelihood**. Restricted maximum likelihood is a method
of fitting linear mixed-effects models that involves transforming out
the fixed effects to focus solely on variance-component estimation.

**Satterthwaite denominator degrees of freedom (DDF) method**. This method
implements a generalization of the Satterthwaite (1946) approximation
of the unknown sampling distributions of test statistics for complex
linear mixed-effects models. This method is supported only with
restricted maximum-likelihood estimation.

**substitutable expression**. Substitutable expressions are like any other
mathematical expressions involving scalars and variables, such as
those you would use with Stata's **generate** command, except that the
parameters to be estimated are bound in braces. See *Substitutable*
*expressions* in **[ME] menl**.

**three-level model**. A three-level mixed-effects model has one level of
observations and two levels of grouping. Suppose that you have a
dataset consisting of patients overseen by doctors at hospitals, and
each doctor practices at one hospital. Then a three-level model
would contain a set of random effects to control for
hospital-specific variation, a second set of random effects to
control for doctor-specific random variation within a hospital, and a
random-error term to control for patients' random variation.

**two-level model**. A two-level mixed-effects model has one level of
observations and one level of grouping. Suppose that you have a
panel dataset consisting of patients at hospitals; a two-level model
would contain a set of random effects at the hospital level (the
second level) to control for hospital-specific random variation and a
random-error term at the observation level (the first level) to
control for within-hospital variation.

**variance components**. In a mixed-effects model, the variance components
refer to the variances and covariances of the various random effects.

**within-group errors**. In a two-level model with observations nested within
groups, within-group errors refer to error terms at the observation
level. In a higher-level model, they refer to errors within the
lowest-level groups.

__References__

Kenward, M. G., and J. H. Roger. 1997. Small sample inference for fixed
effects from restricted maximum likelihood. *Biometrics* 53: 983-997.

Lindstrom, M. J., and D. M. Bates. 1990. Nonlinear mixed effects models
for repeated measures data. *Biometrics* 46: 673-687.

Satterthwaite, F. E. 1946. An approximate distribution of estimates of
variance components. *Biometrics Bulletin* 2: 110-114.