This function generates a posterior density sample for a semiparametric ordinal linear mixed model, using a Mixture of Multivariate Polya Trees prior for the distribution of the random effects.
PTolmm(fixed,random,prior,mcmc,state,status,data=sys.frame(sys.parent()),
na.action=na.fail)a two-sided linear formula object describing the
fixed-effects part of the model, with the response on the
left of a ~ operator and the terms, separated by +
operators, on the right.
a one-sided formula of the form ~z1+...+zn | g, with
z1+...+zn specifying the model for the random effects and
g the grouping variable. The random effects formula will be
repeated for all levels of grouping.
a list giving the prior information. The list include the following
parameter: a0 and b0 giving the hyperparameters for
prior distribution of the precision parameter of the Polya Tree (PT)
prior, alpha giving the value of the precision parameter (it
must be specified if a0 and b0 are missing, see details
below), nu0 and tinv giving the hyperparameters of the
inverted Wishart prior distribution for the scale matrix of the normal
baseline distribution, sigma giving the value of the covariance
matrix of the centering distribution (it must be specified if
nu0 and tinv are missing),
mub and Sb giving the hyperparameters
of the normal prior distribution for the mean of the normal
baseline distribution, mu giving the value of the mean of the
centering distribution (it must be specified if
mub and Sb are missing), beta0 and Sbeta0
giving the
hyperparameters of the normal prior distribution for the fixed effects
(must be specified only if fixed effects are considered in the model),
M giving the finite level of the PT prior to be considered,
frstlprob a logical variable
indicating whether the first level probabilities of the PT are fixed
or not (the default is FALSE), and
typepr indicating whether the type of decomposition of the centering
covariance is random (1) or not (0).
a list giving the MCMC parameters. The list must include
the following integers: nburn giving the number of burn-in
scans, nskip giving the thinning interval, nsave giving
the total number of scans to be saved, ndisplay giving
the number of saved scans to be displayed on screen (the function reports
on the screen when every ndisplay iterations have been carried
out), nbase giving the number scans to be performed before the
parameters of the centering distribution and the precision parameter are
updated (i.e., the update of this parameters is invoked only once in every
nbase scans) (the default value is 1), tune1, tune2, and tune3,
giving the Metropolis tuning parameter for the baseline mean,
variance, and precision parameter, respectively. If tune1, tune2,
or tune3 are not specified or negative, an adpative Metropolis algorithm is performed.
Finally, the integer samplef indicates whether
the functional parameters must be sample (1) or not (0).
a list giving the current value of the parameters. This list is used if the current analysis is the continuation of a previous analysis.
a logical variable indicating whether this run is new (TRUE) or the
continuation of a previous analysis (FALSE). In the latter case
the current value of the parameters must be specified in the
object state.
data frame.
a function that indicates what should happen when the data
contain NAs. The default action (na.fail) causes
PTolmm to print an error message and terminate if there are any
incomplete observations.
An object of class PTolmm representing the linear
mixed-effects model fit. Generic functions such as print, plot,
and summary have methods to show the results of the fit.
The results include betaR, betaF, mu, the elements of
Sigma, alpha, and ortho.
The function PTrandom can be used to extract the posterior mean of the
random effects.
The list state in the output object contains the current value of the parameters
necessary to restart the analysis. If you want to specify different starting values
to run multiple chains set status=TRUE and create the list state based on
this starting values. In this case the list state must include the following objects:
giving the value of the precision parameter
a matrix of dimension (nsubjects)*(nrandom effects) giving the value of the random effects for each subject.
a real vector defining the cutoff points. Note that the first cutoff must be fixed at 0 in this function.
giving the value of the fixed effects.
giving the mean of the normal baseline distributions.
giving the variance matrix of the normal baseline distributions.
giving the orthogonal matrix H, used in the decomposition of the covariance matrix.
This generic function fits an ordinal linear mixed-effects model with a probit link and a Mixture of Multivariate Polya Trees prior (see, Lavine 1992; 1994, for details about univariate PT) for the distribution of the random effects as described in Jara, Hanson and Lessaffre (2009):
$$Y_{ij} = k, \mbox{\ if } \gamma_{k-1} \leq W_{ij} < \gamma_{k}, k=1,\ldots,K$$ $$W_{ij} \mid \beta_F, \beta_R , b_i \sim N(X_{ij} \beta_F + Z_{ij} \beta_R + Z_{ij} b_i, 1), i=1,\ldots,N, j=1,\ldots,n_i$$ $$\theta_i | G \sim G$$ $$G | \alpha,\mu,\Sigma,O \sim PT^M(\Pi^{\mu,\Sigma,O},\mathcal{A})$$
where, \(\theta_i = \beta_R + b_i\), \(\beta = \beta_F\), and \(O\) is an orthogonal matrix defining the decomposition of the centering covariance matrix. As in Hanson (2006), the PT prior is centered around a \(N_d(\mu,\Sigma)\) distribution. However, we consider the class of partitions \(\Pi^{\mu,\Sigma, O}\). The partitions starts with base sets that are Cartesian products of intervals obtained as quantiles from the standard normal distribution. A multivariate location-scale transformation, \(\theta=\mu+\Sigma^{1/2} z\), is applied to each base set yielding the final sets. Here \(\Sigma^{1/2}=T'O'\) where \(T\) is the unique upper triangular Cholesky matrix of \(\Sigma\). The family \(\mathcal{A}=\{\alpha_e: e \in E^{*}\}\), where \(E^{*}=\bigcup_{m=0}^{M} E_d^m\), with \(E_d\) and \(E_d^m\) the \(d\)-fold product of \(E=\{0,1\}\) and the the \(m\)-fold product of \(E_d\), respectively. The family \(\mathcal{A}\) was specified as \(\alpha_{e_1 \ldots e_m}=\alpha m^2\).
To complete the model specification, independent hyperpriors are assumed, $$\alpha | a_0, b_0 \sim Gamma(a_0,b_0)$$ $$\beta | \beta_0, S_{\beta_0} \sim N(\beta_0,S_{\beta_0})$$ $$\mu | \mu_b, S_b \sim N(\mu_b,S_b)$$ $$\Sigma | \nu_0, T \sim IW(\nu_0,T)$$ $$O \sim Haar(q)$$
A uniform prior is used for the cutoff points. Note that the inverted-Wishart prior is parametrized such that \(E(\Sigma)= T^{-1}/(\nu_0-q-1)\).
The precision or total mass parameter, \(\alpha\), of the DP prior
can be considered as random, having a gamma distribution, \(Gamma(a_0,b_0)\),
or fixed at some particular value.
The computational implementation of the model is based on the marginalization of
the PT as descried in Jara, Hanson and Lessaffre (2009).
Hanson, T. (2006) Inference for Mixtures of Finite Polya Trees. Journal of the American Statistical Association, 101: 1548-1565.
Jara, A., Hanson, T., Lesaffre, E. (2009) Robustifying Generalized Linear Mixed Models using a New Class of Mixtures of Multivariate Polya Trees. Journal of Computational and Graphical Statistics, 18(4): 838-860.
Lavine, M. (1992) Some aspects of Polya tree distributions for statistical modelling. The Annals of Statistics, 20: 1222-11235.
Lavine, M. (1994) More aspects of Polya tree distributions for statistical modelling. The Annals of Statistics, 22: 1161-1176.
PTrandom,
PTlmm , PTglmm,
DPMglmm, DPMlmm, DPMolmm,
DPlmm , DPglmm, DPolmm
# NOT RUN {
# Schizophrenia Data
data(psychiatric)
attach(psychiatric)
# Prior information
prior <- list(M=4,
frstlprob=FALSE,
alpha=1,
nu0=4.01,
tinv=diag(1,1),
mub=rep(0,1),
Sb=diag(100,1),
beta0=rep(0,3),
Sbeta0=diag(1000,3))
# MCMC parameters
mcmc <- list(nburn=10000,
nsave=10000,
nskip=20,
ndisplay=100,
samplef=1)
# Initial state
state <- NULL
# Fitting the model
fit1 <- PTolmm(fixed=imps79o~sweek+tx+sweek*tx,random=~1|id,prior=prior,
mcmc=mcmc,state=state,status=TRUE)
fit1
# Summary with HPD and Credibility intervals
summary(fit1)
summary(fit1,hpd=FALSE)
# Plot model parameters
plot(fit1)
# Plot an specific model parameter
plot(fit1,ask=FALSE,nfigr=1,nfigc=2,param="sigma-(Intercept)")
# Extract random effects
PTrandom(fit1)
# Extract predictive information of random effects
aa<-PTrandom(fit1,predictive=TRUE)
aa
# Predictive marginal and joint distributions
plot(aa)
# }
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