MAS-II Linear Mixed Models

What is a linear mixed model (LMM)?

parametric linear model for clustered, longitudinal, or repeated measures data that quantifies the relationships between a continuous dependent variable and various predictor variables.

Zs are fixed or random

random

What do random effects represent?

random deviations from the relationships described by fixed effects.

What is a conditional residual?

the difference between the observed value and the conditional predicted value of the dependent variable

What is the "best" model?

the model that is parsimonious in terms of parameters used and is best at predicting the dependent variable

What does level 2 data represent?

the next level of the hierarchy

What does level 3 data represent?

the next level of the hierarchy, and generally refers to clusters of units in clustered longitudinal datasets, or clusters of level 2 units (clusters of clusters) in 3 level clustered data sets.

When using likelihood ratio tests, the nested model should differ from the reference model in...

the number of fixed effect parameters or the number of covariance parameters for random effects, but not both.

What do fixed effects describe?

the relationships between the dependent variable and the predictor variables for an entire population of units of analysis, or for a relatively small number of subpopulations defined by levels of a fixed factor.

What do fixed effect parameters describe?

the relationships of the covariates to the dependent variable for an entire population

What is level 1 data in a repeated-measures or longitudinal data set?

the repeated measures made on the same unit of analysis.

What is level 2 data in a repeated-measures or longitudinal data set?

the units of analysis

What is level 1 data in a clustered data set?

the units of analysis (or subjects) in a study.

What do the elements along the main diagonal of the D matrix represent?

the variances of each random effect in Ui

What does aliasing mean?

there are an infinite number of combinations of parameters that maximize the likelihood, which may cause estimation problems. there is ambiguity in the model due to excessive parameters.

What is the important difference between an LMM and the implied marginal model?

there are more restrictions imposed on the covariance parameter space in the LMM

What is level 2 data in clustered datasets?

they represent clusters of units.

Why do we consider scaling the residuals?

to alleviate problems with the interpretation of conditional residuals that may have unequal variances.

How are random effects explicitly used in LMMs?

to explain the between subject or between cluster variation

When is the AR(1) commonly used?

to fit models to data sets with equally spaced longitudinal observations on the same unit of analysis

What is multivariate ANOVA used for?

to handle missing data for repeated measures or longitudinal studies.

what is the ei vector?

vector of residuals

When is the compound symmetry commonly used?

when an assumption of equal correlation of residuals is plausible (repeated trials)

When is an F test used?

when testing linear hypotheses about multiple fixed effects in an LMM

How do we define repeated measures data?

where the dependent variable is measured more than once for a unit of analysis across levels of a repeated - measures factor (or factors) - also called within - subject factors

Are residuals independent of random effects for a specific subject?

yes

Can residuals associated with repeated observations on the same subject in an LMM be correlated?

yes

Do ML and REML use the same formula to estimate the fixed effect parameters (beta)?

yes

Do the approximate methods that apply to both t and F tests take into account the presence of random effects and correlated residuals in an LMM?

yes

N-R maximizes...

-2*loglikelihood = deviance.

What advantages do LMMs have over ANOVA?

1. ANOVA omits all observations for a subject if data is missing at some time points. LMMs do not. 2. ANOVA requires observations for all subjects at each time point, whereas LMMs can handle different time points for different subjects,

What are the alternative approaches for fitting the model when problems arise with estimation of the covariance parameters?

1. Choose alternative starting values for covariance parameter estimates 2. Rescale the covariates 3. Remove unnecessary random effects 4. Fit the implied marginal model 5. Fit marginal model w/ unstructured covariance matrix

What are the 3 steps of developing a LMM?

1. Description 2. Analysis 3. Diagnostics

Why is the concept of the implied marginal model important?

1. Estimation of fixed effects and covariance parameters is carried out in the framework of the IMM. 2. If software produces a nonpositive-definite (invalid) estimate of the D matrix, we may be able to fit the IMM as it has fewer restrictions.

What is the Top-Down strategy?

1. Start with a well specified mean structure for the model. 2. Select a structure for the random effects in the model 3. Select a covariance structure for the residuals in the model 4. Reduce the model

What are two disadvantages of no pooling?

1. There is a limited amount of data in each class, so the error in the estimates will be higher than it would be with more data. 2. There is no way to predict results for a new class.

What is a fixed effect factor?

A categorical or classification variable, for which the investigator has included all levels that are of interest in the study.

What is a random factor?

A classification variable with levels that can be thought of as being randomly sampled from a population of levels being studied.

What is the Intraclass Correlation Coefficient?

A measure describing the similarity of the responses on the dependent variable within a cluster or unit of analysis.

What is Maximum Likelihood Estimation?

A method of obtaining estimates of unknown parameters by optimizing a likelihood function

What does the Satterthwaite method do?

Adjust the number of degrees of freedom

What is intrinsic aliasing?

Aliasing attributable to the model formula specification

What is extrinsic aliasing?

Aliasing attributable to the particular characteristics of a given data set

Why do we have REML?

As an alternative to ML that is frequently used to eliminate the bias in the ML estimates of the covariance parameters

(Maximum Likelihood) If theta is known, what is the estimate of beta?

BLUE: Best Linear Unbiased Estimator

Is EBLUE for Betas or Ui's?

Betas

In any given analysis, what do we try to do for the Ri matrix?

Choose the one that seems most appropriate and parsimonious, given the observed data and knowledge about the observations on an individual subject.

What do likelihood ratio tests do?

Compare reference models with nested models 2(ln(Ref)-ln(Nested))

What is an advantage of the step-down approach?

Covariances can be thought of as measuring variances rather than as measuring variation due to omitted fixed effects.

How do we define longitudinal data?

Data sets in which the dependent variable is measured at several points in time for each unit of analysis.

How do we define clustered data?

Data sets in which the dependent variable is measured once for each subject and the units of analysis are grouped into or nested within clusters of units.

What is the notable exception to being at least a two level dataset in an LMM?

Datasets with crossed random factors which do not have an explicit hierarchy due to the fact that the levels of one random factor are not nested within levels of other random factors.

(Maximum Likelihood) If theta is unknown, what is the estimate of beta?

EBLUE: Empirical Best Linear Unbiased Estimator

What is the diagonal structure?

Each random effect in ui has its own variance and all covariances in D is defined to be 0.

What is EBLUP?

Empirical Best Linear Unbiased Predictor.

What algorithms are used to optimize linear mixed models / carry out ML & REML Estimation?

Expectation Maximization, N-R, Fisher Scoring.

What is the primary difference between N-R and Fisher Scoring?

Fisher Scoring uses the expected Hessian Matrix rather than the observed one.

Betas are fixed or random

Fixed

What do we refer to clustered, repeated-measures, and longitudinal data as?

Hierarchical data

What does Kenward-Roger method do?

It adjusts the degrees of freedom using the Satterthwaite method, but also modifies the estimated covariance matrix to reflect uncertainty in using V hat as a substitute for V

What is the main drawback of EM?

It converges slowly, and precision of estimators is overly optimistic

Why is REML preferred to ML?

It has been shown to reduce the bias inherent in ML estimates of covariance parameters

Why is EBLUP Best?

It has minimum variance for all unbiased estimators.

Why is EBLUP empirical?

It uses estimated values of D, Vi, and Beta.

What is the restriction for the D and Ri matrices in LMMs vs Implied Marginal Models?

LMMs: D and Ri Must be Positive-Definite IMM: Only Vi Matrix needs to be positive definite

What is EM often used for?

Maximize complicated likelihood functions or find good starting values of the parameters to be used in other algorithms.

What is external studentization?

Means the standard deviation of a residual is estimated excluding that residual.

What is internal studentization?

Means the standard deviation of a residual is estimated including that residual.

In terms of missing data, what assumption is used for LMMs?

Missing data in Clustered or Longitudinal datasets are Missing at Random (MAR).

Are REML estimates of covariance parameters biased?

No

Do ML and REML produce the same results since they use the same estimation formula?

No - because the Vi matrix is different in each case.

What other terms are often used to refer to intrinsic aliasing?

Non-identifiability and Overparameterization

What is level 1 data?

Observations at the most detailed level.

Marginal Models are referred to as:

Population Averaged

The optimization algorithms used to implement ML and REML estimation need to ensure that the estimates of the D and Ri matrices are....

Positive-Definite

Is ML or REML preferred? Why?

REML because it produces unbiased estimates of covariance parameters by taking into account the loss of degrees of freedom that results from estimating the fixed effects in beta.

What are random effects?

Random values associated with the levels of a random factor (or factors) in an LMM.

What are Pearson Residuals?

Residuals divided by the standard deviation of the dependent variable.

What are studentized residuals?

Residuals divided by their estimated standard deviation

What are standardized residuals?

Residuals divided by their standard deviation

LMMs are referred to as:

Subject Specific

What is an advantage of using the N-R algorithm?

The Hessian Matrix from the last iteration can be used to obtain an asymptotic variance covariance matrix for the estimated covariance parameters in theta, allowing for the calculation of standard errors of theta hat.

What is the important feature of repeated measures and longitudinal data?

The dependent variable is measured more than once for each unit of analysis, with the repeated measures likely to be correlated.

What is an advantage of the step-up approach?

The effect of each covariate on reducing the variance can be viewed separately for each level.

What is the mean of the implied marginal model?

The same as the predicted overall mean of the underlying linear mixed model.

What does the residual variance measure?

The variance WITHIN each level 2 entity.

What are the THETAi's?

The variances or covariances appearing in D and Ri Matrices.

What does the D matrix measure?

The variation BETWEEN two level two entities.

Why are EBLUPs known as shrinkage estimators?

They tend to be closer to zero than the estimated effects would be if they were computed by treating a random factor as if it were fixed.

Is EBLUP for Betas or Ui's?

Ui's

What is the covariance matrix of the Implied Marginal Model?

Vi = ZDZ' + Ri

Why is unknown theta EBLUE?

We estimated Vi and replaced it with Vi hat

What is partial pooling?

When parameters are fit taking into account all the data. Classes with more data create predictions based on mostly their own results. Smaller classes rely more on overall results. Tends to fit model best and have intermediate number of effective parameters.

When is a t test often used?

When testing hypotheses about a single fixed-effect parameter in an LMM

When do we use ML?

When testing hypotheses about fixed effect parameters in an LMM

When is an LMM referred to as a Hierarchical Linear Model (HLM) or Multi Level Model (MLM)?

When the LMM is specified in terms of an explicitly defined hierarchy of simpler models, which correspond to the levels of a clustered or longitudinal data set.

When is Pearson-type scaling appropriate?

When we assume the variability of Beta hat can be ignored.

What is no pooling?

When we fit different parameters for each class, using only that classes data. Tends to overfit model.

What is complete pooling?

When we fit parameters using the entire dataset and differences in classes are not accounted for. Worst model fit.

Can the D and Ri matrices be specified to allow heterogeneous variances for different groups of subjects (males and females)?

Yes

Does an LMM allow for subject specific inference?

Yes

Are the variances of the estimated fixed effects biased, and if so, which direction?

Yes, Biased Downward for both ML and REML.

Are ML estimates of theta biased? If so, why?

Yes, because they do not take into account the loss of degrees of freedom that results from estimating the fixed-effect parameters in Beta.

What is the unstructured D matrix

a D matrix with no additional constraints on the values of its elements (aside from positive definiteness and symmetry)

What does it mean for a factor to be nested within levels of the second factor?

a factor (fixed or random) can only be measured within a single level of another factor and not across multiple levels

What is the compound symmetry structure (Ri matrix)

add sigma to everything in the diagonal structure

WHat does the Toeplitz structure do?

allow more flexibility in the correlations, but at the expense of using more covariance parameters in the THETAr vector.

What do all data sets appropriate for an analysis using LMMs have?

at least two levels of data

How is the continuous dependent variable always measured?

at level 1 of the data

what are random effects specific to?

clusters or subjects within a population

When is dropout of subjects a concern?

in the analysis of longitudinal data

Marginal residuals are based on models that do not...

include explicit random effects

Why is EBLUP Linear?

it is a linear function of y

A model is said to be a nested model if...

it is a special case of another model.

Why is EBLUP unbiased?

it is an unbiased estimator of the random effect in that it's expected value is the expected value of the random effect for the entity.

Why is EBLUP predictor?

it predicts the random effects.

Why is EBLUP called a shrinkage estimator?

it shrinks the observed value of the marginal residual towards 0.

The BIC applies a greater penalty than the AIC for...

models with more parameters.

Are inferences from random effects valid in a marginal model?

no

Are random effects used in the specification of marginal models?

no

Do methods involving marginal models allow for inferences about random effects and their variances?

no

Does the Marginal Model allow for subject specific inference?

no

Is the Wald test recommended?

no

What does it mean for the Xi matrix to be of full-rank?

none of the columns or rows are a linear combination of the remaining ones.

What is the AR(1) structure?

observations closer together are more correlated. each column or row is multiplied by another p