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 sampledependent 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 C_{0}e^{–rt} combines parameters C_{0} 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 $$\overline{r}$$. The term C_{0}e^{–rt} becomes
$${C}_{0}{e}^{[\overline{r}+({r}_{i}\overline{r})]t}={C}_{0}{e}^{(\beta +{b}_{i})t}\text{,}$$
where β = $$\overline{r}$$ is a fixed effect and b_{i} = $${r}_{i}\overline{r}$$ 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.
Mixedeffects 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. Mixedeffects 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.
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:
$${y}_{ij}=f(\phi ,{x}_{ij})+{\epsilon}_{ij}$$
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
$${y}_{ij}=f({\phi}_{i},{x}_{ij})+{\epsilon}_{ij}$$
In a mixedeffects model, φ_{i} may be a combination of a fixed and a random effect:
$${\phi}_{i}=\beta +{b}_{i}$$
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:
$${\phi}_{i}=A\beta +B{b}_{i}$$
If the design matrices differ among groups, the model becomes
$${\phi}_{i}={A}_{i}\beta +{B}_{i}{b}_{i}$$
If the design matrices also differ among observations, the model becomes
$$\begin{array}{l}{\phi}_{ij}={A}_{ij}\beta +{B}_{ij}{b}_{i}\\ {y}_{ij}=f({\phi}_{ij},{x}_{ij})+{\epsilon}_{ij}\end{array}$$
Some of the groupspecific predictors in x_{i}_{j} may not change with observation j. Calling those v_{i}, the model becomes
$${y}_{ij}=f({\phi}_{ij},{x}_{ij},{v}_{i})+{\epsilon}_{ij}$$
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 mixedeffects model for this data:
Define groupspecific 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 groupspecific predictor variables X_{i}.
The model is:
$$\begin{array}{l}{\phi}_{i}={A}_{i}\beta +{B}_{i}{b}_{i}\\ {y}_{i}=f({\phi}_{i},{X}_{i})+{\epsilon}_{i}\\ {b}_{i}\sim N(0,\Psi )\\ {\epsilon}_{i}\sim N(0,{\sigma}^{2})\end{array}$$
This formulation of the nonlinear mixedeffects model uses the following notation:
φ_{i}  A vector of groupspecific model parameters 
β  A vector of fixed effects, modeling population parameters 
b_{i}  A vector of multivariate normally distributed groupspecific random effects 
A_{i}  A groupspecific design matrix for combining fixed effects 
B_{i}  A groupspecific design matrix for combining random effects 
X_{i}  A data matrix of groupspecific predictor values 
y_{i}  A data vector of groupspecific response values 
f  A general, realvalued function of φ_{i} and X_{i} 
ε_{i}  A vector of groupspecific errors, assumed to be independent, identically, normally distributed, and independent of b_{i} 
Ψ  A covariance matrix for the random effects 
σ^{2}  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
$${y}_{ij}={C}_{pi}{e}^{{r}_{pi}{t}_{ij}}+{C}_{qi}{e}^{{r}_{qi}{t}_{ij}}+{\epsilon}_{ij},$$
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 reparametrizing the model:
$${y}_{ij}={C}_{pi}{e}^{\mathrm{exp}({R}_{pi}){t}_{ij}}+{C}_{qi}{e}^{\mathrm{exp}({R}_{qi}){t}_{ij}}+{\epsilon}_{ij}$$
Choosing which parameters to model with random effects is an important consideration when building a mixedeffects 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:
$$\begin{array}{lll}{y}_{ij}\hfill & =\hfill & [{\overline{C}}_{p}+({C}_{pi}{\overline{C}}_{p})]{e}^{\mathrm{exp}[{\overline{R}}_{p}+({R}_{pi}{\overline{R}}_{p})]{t}_{ij}}+\hfill \\ \hfill & \hfill & [{\overline{C}}_{q}+({C}_{qi}{\overline{C}}_{q})]{e}^{\mathrm{exp}[{\overline{R}}_{q}+({R}_{qi}{\overline{R}}_{q})]{t}_{ij}}+{\epsilon}_{ij}\hfill \\ \hfill & =\hfill & ({\beta}_{1}+{b}_{1i}){e}^{\mathrm{exp}({\beta}_{2}+{b}_{2i}){t}_{ij}}+\hfill \\ \hfill & \hfill & ({\beta}_{3}+{b}_{3i}){e}^{\mathrm{exp}({\beta}_{4}+{b}_{4i}){t}_{ij}}+{\epsilon}_{ij}\hfill \end{array}$$
In the notation of the general model:
$$\beta =\left(\begin{array}{c}{\beta}_{1}\\ \vdots \\ {\beta}_{4}\end{array}\right),{b}_{i}=\left(\begin{array}{c}{b}_{i1}\\ \vdots \\ {b}_{i4}\end{array}\right),{y}_{i}=\left(\begin{array}{c}{y}_{i1}\\ \vdots \\ {y}_{i{n}_{i}}\end{array}\right),{X}_{i}=\left(\begin{array}{c}{t}_{i1}\\ \vdots \\ {t}_{i{n}_{i}}\end{array}\right),$$
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, 4by4 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 blockdiagonal 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 mixedeffects
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.
If the model in Specifying MixedEffects Models assumes a groupdependent covariate such as weight (w) the model becomes:
$$\left(\begin{array}{c}{\phi}_{1}\\ {\phi}_{2}\\ {\phi}_{3}\end{array}\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\begin{array}{c}wi\\ 0\\ 0\end{array}\right)\left(\begin{array}{c}{\beta}_{1}\\ {\beta}_{2}\\ \begin{array}{l}{\beta}_{3}\\ {\beta}_{4}\end{array}\end{array}\right)+\text{\hspace{0.17em}}\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}{b}_{1}\\ {b}_{2}\\ {b}_{3}\end{array}\right)$$
Thus, the parameter φ_{i} for
any individual in the i
th group is:
$$\left(\begin{array}{c}{\phi}_{{1}_{i}}\\ {\phi}_{{2}_{i}}\\ {\phi}_{{3}_{i}}\end{array}\right)=\left(\begin{array}{c}{\beta}_{1}+{\beta}_{4}*{w}_{i}\\ {\beta}_{2}\\ {\beta}_{3}\end{array}\right)+\text{\hspace{0.17em}}\left(\begin{array}{c}{b}_{{1}_{i}}\\ {b}_{{2}_{i}}\\ {b}_{{3}_{i}}\end{array}\right)$$
To specify a covariate model, use
the 'FEGroupDesign'
option.
'FEGroupDesign'
is a pbyqbym
array
specifying a different pbyq
fixedeffects design
matrix for each of the m
groups. Using the previous
example, the array resembles the following:
Create the array.
% 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)
Create a struct
with the specified
design matrix.
options.FEGroupDesign = A;
Specify the arguments for nlmefit
(or nlmefitsa
)
as shown in MixedEffects Models Using nlmefit and nlmefitsa.
Statistics and Machine Learning Toolbox provides two functions, nlmefit
and nlmefitsa
for
fitting nonlinear mixedeffects models. Each function provides different
capabilities, which may help you decide which to use.
nlmefit
provides the following
four approximation methods for fitting nonlinear mixedeffects models:
'LME'
— Use the likelihood
for the linear mixedeffects model at the current conditional estimates
of beta
and B
. This is the default.
'RELME'
— Use the restricted
likelihood for the linear mixedeffects model at the current conditional
estimates of beta
and B
.
'FO'
— Firstorder Laplacian
approximation without random effects.
'FOCE'
— Firstorder Laplacian
approximation at the conditional estimates of B
.
nlmefitsa
provides
an additional approximation method, Stochastic Approximation ExpectationMaximization
(SAEM) [25] with three steps :
Simulation: Generate simulated values of the random effects b from the posterior density p(bΣ) 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.
The following parameters are specific to nlmefitsa
.
Most control the stochastic algorithm.
Cov0
— Initial value for
the covariance matrix PSI
. Must be an rbyr 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
burnin 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.
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 observationdependent design is not supported.
The random effect design must be constant for the entire data set, so neither an observationdependent design nor a groupdependent 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
.
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.
The function definition line of the output function has the following form:
stop = outfun(beta,status,state)
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.
The following table lists the fields of the status
structure:
Field  Description 

procedure 

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:

fval  The current log likelihood 
Psi  The current randomeffects covariance matrix 
theta  The current parameterization of Psi 
mse  The current error variance 
The following table lists the possible values for state
:
state  Description 

 The algorithm is in the initial state before the first iteration. 
 The algorithm is at the end of an iteration. 
 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
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
nmlefitoutputfcn
is the sample Statistics and Machine Learning Toolbox output
function for nlmefit
and nlmefitsa
.
It initializes or updates a plot with the fixedeffects (BETA
)
and variance of the random effects (diag
(STATUS.Psi
)).
For nlmefit
, the plot also includes the loglikelihood
(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, …)
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.