Mixed-Effects Models

Introduction to Mixed-Effects Models

In statistics, an effect is anything that influences the value of a response variable at a particular setting of the predictor variables. Effects are translated into model parameters. In linear models, effects become coefficients, representing the proportional contributions of model terms. In nonlinear models, effects often have specific physical interpretations, and appear in more general nonlinear combinations.

Fixed effects represent population parameters, assumed to be the same each time data is collected. Estimating fixed effects is the traditional domain of regression modeling. Random effects, by comparison, are sample-dependent random variables. In modeling, random effects act like additional error terms, and their distributions and covariances must be specified.

For example, consider a model of the elimination of a drug from the bloodstream. The model uses time t as a predictor and the concentration of the drug C as the response. The nonlinear model term C₀e^–rt combines parameters C₀ and r, representing, respectively, an initial concentration and an elimination rate. If data is collected across multiple individuals, it is reasonable to assume that the elimination rate is a random variable r_i depending on individual i, varying around a population mean $\bar{r}$ . The term C₀e^–rt becomes

$C_{0} e^{- [\bar{r} + (r_{i} - \bar{r})] t} = C_{0} e^{- (β + b_{i}) t},$

where β = $\bar{r}$ is a fixed effect and b_i = $r_{i} - \bar{r}$ is a random effect.

Random effects are useful when data falls into natural groups. In the drug elimination model, the groups are simply the individuals under study. More sophisticated models might group data by an individual's age, weight, diet, etc. Although the groups are not the focus of the study, adding random effects to a model extends the reliability of inferences beyond the specific sample of individuals.

Mixed-effects models account for both fixed and random effects. As with all regression models, their purpose is to describe a response variable as a function of the predictor variables. Mixed-effects models, however, recognize correlations within sample subgroups. In this way, they provide a compromise between ignoring data groups entirely and fitting each group with a separate model.

Mixed-Effects Model Hierarchy

Suppose data for a nonlinear regression model falls into one of m distinct groups i = 1, ..., m. To account for the groups in a model, write response j in group i as:

$y_{i j} = f (φ, x_{i j}) + ε_{i j}$

y_i_j is the response, x_i_j is a vector of predictors, φ is a vector of model parameters, and ε_i_j is the measurement or process error. The index j ranges from 1 to n_i, where n_i is the number of observations in group i. The function f specifies the form of the model. Often, x_i_j is simply an observation time t_i_j. The errors are usually assumed to be independent and identically, normally distributed, with constant variance.

Estimates of the parameters in φ describe the population, assuming those estimates are the same for all groups. If, however, the estimates vary by group, the model becomes

$y_{i j} = f (φ_{i}, x_{i j}) + ε_{i j}$

In a mixed-effects model, φ_i may be a combination of a fixed and a random effect:

$φ_{i} = β + b_{i}$

The random effects b_i are usually described as multivariate normally distributed, with mean zero and covariance Ψ. Estimating the fixed effects β and the covariance of the random effects Ψ provides a description of the population that does not assume the parameters φ_i are the same across groups. Estimating the random effects b_i also gives a description of specific groups within the data.

Model parameters do not have to be identified with individual effects. In general, design matrices A and B are used to identify parameters with linear combinations of fixed and random effects:

$φ_{i} = A β + B b_{i}$

If the design matrices differ among groups, the model becomes

$φ_{i} = A_{i} β + B_{i} b_{i}$

If the design matrices also differ among observations, the model becomes

$\begin{array}{l} φ_{i j} = A_{i j} β + B_{i j} b_{i} \\ y_{i j} = f (φ_{i j}, x_{i j}) + ε_{i j} \end{array}$

Some of the group-specific predictors in x_i_j may not change with observation j. Calling those v_i, the model becomes

$y_{i j} = f (φ_{i j}, x_{i j}, v_{i}) + ε_{i j}$

Specifying Mixed-Effects Models

Suppose data for a nonlinear regression model falls into one of m distinct groups i = 1, ..., m. (Specifically, suppose that the groups are not nested.) To specify a general nonlinear mixed-effects model for this data:

Define group-specific model parameters φ_i as linear combinations of fixed effects β and random effects b_i.
Define response values y_i as a nonlinear function f of the parameters and group-specific predictor variables X_i.

The model is:

$\begin{array}{l} φ_{i} = A_{i} β + B_{i} b_{i} \\ y_{i} = f (φ_{i}, X_{i}) + ε_{i} \\ b_{i} \sim N (0, Ψ) \\ ε_{i} \sim N (0, σ^{2}) \end{array}$

This formulation of the nonlinear mixed-effects model uses the following notation:

φ_i	A vector of group-specific model parameters
β	A vector of fixed effects, modeling population parameters
b_i	A vector of multivariate normally distributed group-specific random effects
A_i	A group-specific design matrix for combining fixed effects
B_i	A group-specific design matrix for combining random effects
X_i	A data matrix of group-specific predictor values
y_i	A data vector of group-specific response values
f	A general, real-valued function of φ_i and X_i
ε_i	A vector of group-specific errors, assumed to be independent, identically, normally distributed, and independent of b_i
Ψ	A covariance matrix for the random effects
σ²	The error variance, assumed to be constant across observations

For example, consider a model of the elimination of a drug from the bloodstream. The model incorporates two overlapping phases:

An initial phase p during which drug concentrations reach equilibrium with surrounding tissues
A second phase q during which the drug is eliminated from the bloodstream

For data on multiple individuals i, the model is

$y_{i j} = C_{p i} e^{- r_{p i} t_{i j}} + C_{q i} e^{- r_{q i} t_{i j}} + ε_{i j},$

where y_ij is the observed concentration in individual i at time t_ij. The model allows for different sampling times and different numbers of observations for different individuals.

The elimination rates r_pi and r_qi must be positive to be physically meaningful. Enforce this by introducing the log rates R_pi = log(r_pi) and R_qi = log(r_qi) and reparameterizing the model:

$y_{i j} = C_{p i} e^{- \exp (R_{p i}) t_{i j}} + C_{q i} e^{- \exp (R_{q i}) t_{i j}} + ε_{i j}$

Choosing which parameters to model with random effects is an important consideration when building a mixed-effects model. One technique is to add random effects to all parameters, and use estimates of their variances to determine their significance in the model. An alternative is to fit the model separately to each group, without random effects, and look at the variation of the parameter estimates. If an estimate varies widely across groups, or if confidence intervals for each group have minimal overlap, the parameter is a good candidate for a random effect.

To introduce fixed effects β and random effects b_i for all model parameters, reexpress the model as follows:

$\begin{array}{l} y_{i j} & = & [{\bar{C}}_{p} + (C_{p i} - {\bar{C}}_{p})] e^{- \exp [{\bar{R}}_{p} + (R_{p i} - {\bar{R}}_{p})] t_{i j}} + \\ [{\bar{C}}_{q} + (C_{q i} - {\bar{C}}_{q})] e^{- \exp [{\bar{R}}_{q} + (R_{q i} - {\bar{R}}_{q})] t_{i j}} + ε_{i j} \\ = & (β_{1} + b_{1 i}) e^{- \exp (β_{2} + b_{2 i}) t_{i j}} + \\ (β_{3} + b_{3 i}) e^{- \exp (β_{4} + b_{4 i}) t_{i j}} + ε_{i j} \end{array}$

In the notation of the general model:

$β = (\begin{matrix} β_{1} \\ ⋮ \\ β_{4} \end{matrix}), b_{i} = (\begin{matrix} b_{i 1} \\ ⋮ \\ b_{i 4} \end{matrix}), y_{i} = (\begin{matrix} y_{i 1} \\ ⋮ \\ y_{i n_{i}} \end{matrix}), X_{i} = (\begin{matrix} t_{i 1} \\ ⋮ \\ t_{i n_{i}} \end{matrix}),$

where n_i is the number of observations of individual i. In this case, the design matrices A_i and B_i are, at least initially, 4-by-4 identity matrices. Design matrices may be altered, as necessary, to introduce weighting of individual effects, or time dependency.

Fitting the model and estimating the covariance matrix Ψ often leads to further refinements. A relatively small estimate for the variance of a random effect suggests that it can be removed from the model. Likewise, relatively small estimates for covariances among certain random effects suggests that a full covariance matrix is unnecessary. Since random effects are unobserved, Ψ must be estimated indirectly. Specifying a diagonal or block-diagonal covariance pattern for Ψ can improve convergence and efficiency of the fitting algorithm.

Statistics and Machine Learning Toolbox™ functions nlmefit and nlmefitsa fit the general nonlinear mixed-effects model to data, estimating the fixed and random effects. The functions also estimate the covariance matrix Ψ for the random effects. Additional diagnostic outputs allow you to assess tradeoffs between the number of model parameters and the goodness of fit.

Specifying Covariate Models

If the model in Specifying Mixed-Effects Models assumes a group-dependent covariate such as weight (w) the model becomes:

$(\begin{matrix} φ_{1} \\ φ_{2} \\ φ_{3} \end{matrix}) = (\begin{matrix} 1 & 0 & 0 & w_{i} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix}) (\begin{matrix} β_{1} \\ β_{2} \\ \begin{array}{l} β_{3} \\ β_{4} \end{array} \end{matrix}) + (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) (\begin{matrix} b_{1} \\ b_{2} \\ b_{3} \end{matrix})$

Thus, the parameter φ_i for any individual in the ith group is:

$(\begin{matrix} φ_{1_{i}} \\ φ_{2_{i}} \\ φ_{3_{i}} \end{matrix}) = (\begin{matrix} β_{1} + β_{4} w_{i} \\ β_{2} \\ β_{3} \end{matrix}) + (\begin{matrix} b_{1_{i}} \\ b_{2_{i}} \\ b_{3_{i}} \end{matrix})$

To specify a covariate model, use the 'FEGroupDesign' name-value argument of nlmefit or nlmefitsa.

'FEGroupDesign' is a p-by-q-by-m array specifying a different p-by-q fixed-effects design matrix for each of the m groups. Using the previous example, the array resembles the following:

Illustration showing the FEGroupDesign array as a stack of m 3x4 matrices.

% Number of parameters in the model (Phi)
num_params = 3;
% Number of covariates
num_cov = 1;
% Assuming number of groups in the data set is 7
num_groups = 7;
% Array of covariate values
covariates = [75; 52; 66; 55; 70; 58; 62 ];
A = repmat(eye(num_params, num_params+num_cov),...
[1,1,num_groups]);
A(1,num_params+1,1:num_groups) = covariates(:,1)

Specify 'FEGroupDesign',A to use the design matrix A.

Choosing `nlmefit` or `nlmefitsa`

Statistics and Machine Learning Toolbox provides two functions, nlmefit and nlmefitsa for fitting nonlinear mixed-effects models. Each function provides different capabilities, which may help you decide which to use.

Approximation Methods

nlmefit provides the following four approximation methods for fitting nonlinear mixed-effects models:

'LME' — Use the likelihood for the linear mixed-effects model at the current conditional estimates of beta and B. This is the default.
'RELME' — Use the restricted likelihood for the linear mixed-effects model at the current conditional estimates of beta and B.
'FO' — First-order Laplacian approximation without random effects.
'FOCE' — First-order Laplacian approximation at the conditional estimates of B.

nlmefitsa provides an additional approximation method, Stochastic Approximation Expectation-Maximization (SAEM) [1] with three steps :

Simulation: Generate simulated values of the random effects b from the posterior density p(b|Σ) given the current parameter estimates.
Stochastic approximation: Update the expected value of the log likelihood function by taking its value from the previous step, and moving part way toward the average value of the log likelihood calculated from the simulated random effects.
Maximization step: Choose new parameter estimates to maximize the log likelihood function given the simulated values of the random effects.

Both nlmefit and nlmefitsa attempt to find parameter estimates to maximize a likelihood function, which is difficult to compute. nlmefit deals with the problem by approximating the likelihood function in various ways, and maximizing the approximate function. It uses traditional optimization techniques that depend on things like convergence criteria and iteration limits.

nlmefitsa, on the other hand, simulates random values of the parameters in such a way that in the long run they converge to the values that maximize the exact likelihood function. The results are random, and traditional convergence tests don't apply. Therefore nlmefitsa provides options to plot the results as the simulation progresses, and to restart the simulation multiple times. You can use these features to judge whether the results have converged to the accuracy you desire.

Parameters Specific to `nlmefitsa`

The following parameters are specific to nlmefitsa. Most control the stochastic algorithm.

Cov0 — Initial value for the covariance matrix PSI. Must be an r-by-r positive definite matrix. If empty, the default value depends on the values of BETA0.
ComputeStdErrors — true to compute standard errors for the coefficient estimates and store them in the output STATS structure, or false (default) to omit this computation.
LogLikMethod — Specifies the method for approximating the log likelihood.
NBurnIn — Number of initial burn-in iterations during which the parameter estimates are not recomputed. Default is 5.
NIterations — Controls how many iterations are performed for each of three phases of the algorithm.
NMCMCIterations — Number of Markov Chain Monte Carlo (MCMC) iterations.

Model and Data Requirements

There are some differences in the capabilities of nlmefit and nlmefitsa. Therefore some data and models are usable with either function, but some may require you to choose just one of them.

Error models — nlmefitsa supports a variety of error models. For example, the standard deviation of the response can be constant, proportional to the function value, or a combination of the two. nlmefit fits models under the assumption that the standard deviation of the response is constant. One of the error models, 'exponential', specifies that the log of the response has a constant standard deviation. You can fit such models using nlmefit by providing the log response as input, and by rewriting the model function to produce the log of the nonlinear function value.
Random effects — Both functions fit data to a nonlinear function with parameters, and the parameters may be simple scalar values or linear functions of covariates. nlmefit allows any coefficients of the linear functions to have both fixed and random effects. nlmefitsa supports random effects only for the constant (intercept) coefficient of the linear functions, but not for slope coefficients. So in the example in Specifying Covariate Models, nlmefitsa can treat only the first three beta values as random effects.
Model form — nlmefit supports a very general model specification, with few restrictions on the design matrices that relate the fixed coefficients and the random effects to the model parameters. nlmefitsa is more restrictive:
- The fixed effect design must be constant in every group (for every individual), so an observation-dependent design is not supported.
- The random effect design must be constant for the entire data set, so neither an observation-dependent design nor a group-dependent design is supported.
- As mentioned under Random Effects, the random effect design must not specify random effects for slope coefficients. This implies that the design must consist of zeros and ones.
- The random effect design must not use the same random effect for multiple coefficients, and cannot use more than one random effect for any single coefficient.
- The fixed effect design must not use the same coefficient for multiple parameters. This implies that it can have at most one nonzero value in each column.
If you want to use nlmefitsa for data in which the covariate effects are random, include the covariates directly in the nonlinear model expression. Don't include the covariates in the fixed or random effect design matrices.
Convergence — As described in the Model form, nlmefit and nlmefitsa have different approaches to measuring convergence. nlmefit uses traditional optimization measures, and nlmefitsa provides diagnostics to help you judge the convergence of a random simulation.

In practice, nlmefitsa tends to be more robust, and less likely to fail on difficult problems. However, nlmefit may converge faster on problems where it converges at all. Some problems may benefit from a combined strategy, for example by running nlmefitsa for a while to get reasonable parameter estimates, and using those as a starting point for additional iterations using nlmefit.

Using Output Functions with Mixed-Effects Models

The Outputfcn field of the options structure specifies one or more functions that the solver calls after each iteration. Typically, you might use an output function to plot points at each iteration or to display optimization quantities from the algorithm. To set up an output function:

Write the output function as a MATLAB^® file function or local function.
Use statset to set the value of Outputfcn to be a function handle, that is, the name of the function preceded by the @ sign. For example, if the output function is outfun.m, the command
```
 options = statset('OutputFcn', @outfun);
```
specifies OutputFcn to be the handle to outfun. To specify multiple output functions, use the syntax:
```
 options = statset('OutputFcn',{@outfun, @outfun2});
```
Call the optimization function with options as an input argument.

For an example of an output function, see Sample Output Function.

Structure of the Output Function

The function definition line of the output function has the following form:

stop = outfun(beta,status,state)

where

beta is the current fixed effects.
status is a structure containing data from the current iteration. Fields in status describes the structure in detail.
state is the current state of the algorithm. States of the Algorithm lists the possible values.
stop is a flag that is true or false depending on whether the optimization routine should quit or continue. See Stop Flag for more information.

The solver passes the values of the input arguments to outfun at each iteration.

Fields in status

The following table lists the fields of the status structure:

Field	Description
`procedure`	`'ALT'` — alternating algorithm for the optimization of the linear mixed effects or restricted linear mixed effects approximations `'LAP'` — optimization of the Laplacian approximation for first order or first order conditional estimation
`iteration`	An integer starting from 0.
`inner`	A structure describing the status of the inner iterations within the `ALT` and `LAP` procedures, with the fields: `procedure` — When `procedure` is `'ALT'`: `'PNLS'` (penalized nonlinear least squares) `'LME'` (linear mixed-effects estimation) `'none'` When `procedure` is `'LAP'`, `'PNLS'` (penalized nonlinear least squares) `'PLM'` (profiled likelihood maximization) `'none'` `state` — one of the following: `'init'` `'iter'` `'done'` `'none'` `iteration` — an integer starting from 0, or `NaN`. For `nlmefitsa` with burn-in iterations, the output function is called after each of those iterations with a negative value for `STATUS.iteration`.
`fval`	The current log likelihood
`Psi`	The current random-effects covariance matrix
`theta`	The current parameterization of `Psi`
`mse`	The current error variance

States of the Algorithm

The following table lists the possible values for state:

state	Description
`'init'`	The algorithm is in the initial state before the first iteration.
`'iter'`	The algorithm is at the end of an iteration.
`'done'`	The algorithm is in the final state after the last iteration.

The following code illustrates how the output function might use the value of state to decide which tasks to perform at the current iteration:

switch state
    case 'iter'
          % Make updates to plot or guis as needed
    case 'init'
          % Setup for plots or guis
    case 'done'
          % Cleanup of plots, guis, or final plot
otherwise
end

Stop Flag

The output argument stop is a flag that is true or false. The flag tells the solver whether it should quit or continue. The following examples show typical ways to use the stop flag.

Stopping an Optimization Based on Intermediate Results. The output function can stop the estimation at any iteration based on the values of arguments passed into it. For example, the following code sets stop to true based on the value of the log likelihood stored in the 'fval'field of the status structure:

stop = outfun(beta,status,state)
stop = false;
% Check if loglikelihood is more than 132.
if status.fval > -132
    stop = true;
end

Stopping an Iteration Based on GUI Input. If you design a GUI to perform nlmefit iterations, you can make the output function stop when a user clicks a Stop button on the GUI. For example, the following code implements a dialog to cancel calculations:

function retval = stop_outfcn(beta,str,status)
persistent h stop;
if isequal(str.inner.state,'none')
    switch(status)
        case 'init'
            % Initialize dialog
            stop = false;
            h = msgbox('Press STOP to cancel calculations.',...
                'NLMEFIT: Iteration 0 ');
            button = findobj(h,'type','uicontrol');
            set(button,'String','STOP','Callback',@stopper)
            pos = get(h,'Position');
            pos(3) = 1.1 * pos(3);
            set(h,'Position',pos)
            drawnow
        case 'iter'
            % Display iteration number in the dialog title
            set(h,'Name',sprintf('NLMEFIT: Iteration %d',...
                str.iteration))
            drawnow;
        case 'done'
            % Delete dialog
            delete(h);
    end
end
if stop
    % Stop if the dialog button has been pressed
    delete(h)
end
retval = stop;
 
    function stopper(varargin)
        % Set flag to stop when button is pressed
        stop = true;
        disp('Calculation stopped.')
    end
end

Sample Output Function

nmlefitoutputfcn is the sample Statistics and Machine Learning Toolbox output function for nlmefit and nlmefitsa. It initializes or updates a plot with the fixed-effects (BETA) and variance of the random effects (diag(STATUS.Psi)). For nlmefit, the plot also includes the log-likelihood (STATUS.fval).

nlmefitoutputfcn is the default output function for nlmefitsa. To use it with nlmefit, specify a function handle for it in the options structure:

opt = statset('OutputFcn', @nlmefitoutputfcn, …)
beta = nlmefit(…, 'Options', opt, …)

To prevent nlmefitsa from using of this function, specify an empty value for the output function:

opt = statset('OutputFcn', [], …)
beta = nlmefitsa(…, 'Options', opt, …)

nlmefitoutputfcn stops nlmefit or nlmefitsa if you close the figure that it produces.

References

[1] Delyon, B., M. Lavielle, and E. Moulines, Convergence of a stochastic approximation version of the EM algorithm, Annals of Statistics, 27, 94-128, 1999.