Nonlinear mixedeffects estimation
beta = nlmefit(X,y,group,V,fun,beta0)
[beta,PSI] = nlmefit(X,y,group,V,fun,beta0)
[beta,PSI,stats] = nlmefit(X,y,group,V,fun,beta0)
[beta,PSI,stats,B] = nlmefit(X,y,group,V,fun,beta0)
[beta,PSI,stats,B] = nlmefit(X,y,group,V,fun,beta0,'Name
',value
)
beta = nlmefit(X,y,group,V,fun,beta0)
fits
a nonlinear mixedeffects regression model and returns estimates of
the fixed effects in beta
. By default, nlmefit
fits
a model in which each parameter is the sum of a fixed and a random
effect, and the random effects are uncorrelated (their covariance
matrix is diagonal).
X
is an nbyh matrix
of n observations on h predictors.
y
is an nby1 vector
of responses.
group
is a grouping variable indicating m groups in the
observations. group
is a categorical variable, a numeric vector, a character
matrix with rows for group names, a string array, or a cell array of character vectors. For more
information on grouping variables, see Grouping Variables.
V
is an mbyg matrix
or cell array of g groupspecific predictors. These
are predictors that take the same value for all observations in a
group. The rows of V
are assigned to groups using grp2idx
, according to the order specified
by grp2idx(group)
. Use a cell array for V
if
group predictors vary in size across groups. Use []
for V
if
there are no groupspecific predictors.
fun
is a handle to a function that accepts
predictor values and model parameters and returns fitted values. fun
has
the form
yfit = modelfun(PHI,XFUN,VFUN)
The arguments are:
PHI
— A 1byp vector
of model parameters.
XFUN
— A kbyh array
of predictors, where:
k = 1 if XFUN
is
a single row of X
.
k = n_{i} if XFUN
contains
the rows of X
for a single group of size n_{i}.
k = n if XFUN
contains
all rows of X
.
VFUN
— Groupspecific predictors
given by one of:
A 1byg vector corresponding to
a single group and a single row of V
.
An nbyg array,
where the jth row is V(I
,:)
if the jth observation is in group I
.
If V
is empty, nlmefit
calls modelfun
with
only two inputs.
yfit
— A kby1
vector of fitted values
When either PHI
or VFUN
contains
a single row, it corresponds to all rows in the other two input arguments.
Note
If modelfun
can compute yfit
for
more than one vector of model parameters per call, use the 'Vectorization'
parameter
(described later) for improved performance.
beta0
is a qby1 vector
with initial estimates for q fixed effects. By
default, q is the number of model parameters p.
nlmefit
fits the model by maximizing an
approximation to the marginal likelihood with random effects integrated
out, assuming that:
Random effects are multivariate normally distributed and independent between groups.
Observation errors are independent, identically normally distributed, and independent of the random effects.
[beta,PSI] = nlmefit(X,y,group,V,fun,beta0)
also
returns PSI
, an rbyr estimated
covariance matrix for the random effects. By default, r is
equal to the number of model parameters p.
[beta,PSI,stats] = nlmefit(X,y,group,V,fun,beta0)
also
returns stats
, a structure with fields:
dfe
— The error degrees
of freedom for the model
logl
— The maximized loglikelihood
for the fitted model
rmse
— The square root of
the estimated error variance (computed on the log scale for the exponential
error
model)
errorparam
— The estimated
parameters of the error variance model
aic
— The Akaike information
criterion, calculated as aic
= 2 * logl
+ 2 * numParam
, where numParam
is
the number of fitting parameters, including the degree of freedom
for covariance matrix of the random effects, the number of fixed effects
and the number of parameters of the error model, and logl
is
a field in the stats
structure
bic
— The Bayesian information
criterion, calculated as bic
= –2*logl
+
log(M
) * numParam
M
is the number of groups.
numParam
and logl
are
defined as in aic
.
Note that some literature suggests that the computation
of bic
should be , bic
= –2*logl
+
log(N
) * numParam
, where N
is
the number of observations.
covb
— The estimated covariance
matrix of the parameter estimates
sebeta
— The standard errors
for beta
ires
— The population residuals (yy_population)
,
where y_population
is the individual predicted
values
pres
— The population residuals (yy_population)
,
where y_population
is the population predicted
values
iwres
— The individual weighted
residuals
pwres
— The population weighted
residuals
cwres
— The conditional
weighted residuals
[beta,PSI,stats,B] = nlmefit(X,y,group,V,fun,beta0)
also
returns B
, an rbym matrix
of estimated random effects for the m groups. By
default, r is equal to the number of model parameters p.
[beta,PSI,stats,B] = nlmefit(X,y,group,V,fun,beta0,'
specifies
one or more optional parameter name/value pairs. Specify Name
',value
)Name
inside
single quotes.
Use the following parameters to fit a model different from the
default. (The default model is obtained by setting both FEConstDesign
and REConstDesign
to eye(p)
,
or by setting both FEParamsSelect
and REParamsSelect
to 1:p
.)
Use at most one parameter with an 'FE'
prefix
and one parameter with an 'RE'
prefix. The nlmefit
function
requires you to specify at least one fixed effect and one random effect.
Parameter  Value 

FEParamsSelect  A vector specifying which elements of the parameter vector 
FEConstDesign  A pbyq design
matrix 
FEGroupDesign  A pbyqbym array specifying a different pbyq fixedeffects design matrix for each of the m groups. 
FEObsDesign  A pbyqbyn array specifying a different pbyq fixedeffects design matrix for each of the n observations. 
REParamsSelect  A vector specifying which elements of the parameter vector 
REConstDesign  A pbyr design
matrix 
REGroupDesign  A pbyrbym array specifying a different pbyr randomeffects design matrix for each of m groups. 
REObsDesign  A pbyrbyn array specifying a different pbyr randomeffects design matrix for each of n observations. 
Use the following parameters to control the iterative algorithm for maximizing the likelihood:
Parameter  Value 

RefineBeta0  Determines whether 
ErrorModel  A character vector or string scalar specifying the form of the error term. Default is
If this parameter is given, the output

ApproximationType  The method used to approximate the likelihood of the model. Choices are:

Vectorization  Indicates acceptable sizes for the

CovParameterization  Specifies the parameterization used internally for the
scaled covariance matrix. Choices are 
CovPattern  Specifies an rbyr logical
or numeric matrix Alternatively, 
ParamTransform  A vector of pvalues specifying
a transformation function f() for each of the

Options  A structure of the form returned by

OptimFun  Optimization function for the estimation process that maximizes a likelihood function,
specified as 
[1] Lindstrom, M. J., and D. M. Bates. “Nonlinear mixedeffects models for repeated measures data.” Biometrics. Vol. 46, 1990, pp. 673–687.
[2] Davidian, M., and D. M. Giltinan. Nonlinear Models for Repeated Measurements Data. New York: Chapman & Hall, 1995.
[3] Pinheiro, J. C., and D. M. Bates. “Approximations to the loglikelihood function in the nonlinear mixedeffects model.” Journal of Computational and Graphical Statistics. Vol. 4, 1995, pp. 12–35.
[4] Demidenko, E. Mixed Models: Theory and Applications. Hoboken, NJ: John Wiley & Sons, Inc., 2004.