lpeer: Longitudinal Functional Models with Structured Penalties

In refund: Regression with Functional Data

Longitudinal Functional Models with Structured Penalties

Description

Implements longitudinal functional model with structured penalties (Kundu et al., 2012) with scalar outcome, single functional predictor, one or more scalar covariates and subject-specific random intercepts through mixed model equivalence.

Usage

lpeer(
  Y,
  subj,
  t,
  funcs,
  argvals = NULL,
  covariates = NULL,
  comm.pen = TRUE,
  pentype = "Ridge",
  L.user = NULL,
  f_t = NULL,
  Q = NULL,
  phia = 10^3,
  se = FALSE,
  ...
)

Arguments

Y	vector of all outcomes over all visits or timepoints
subj	vector containing the subject number for each observation
t	vector containing the time information when the observation are taken
funcs	matrix containing observed functional predictors as rows. Rows with NA and Inf values will be deleted.
argvals	matrix (or vector) of indices of evaluations of X_i(t); i.e. a matrix with ith row (t_{i1},.,t_{iJ})
covariates	matrix of scalar covariates.
comm.pen	logical value indicating whether common penalty for all the components of regression function. Default is TRUE.
pentype	type of penalty: either decomposition based penalty ('DECOMP') or ridge ('RIDGE') or second-order difference penalty ('D2') or any user defined penalty ('USER'). For decomposition based penalty user need to specify Q matrix in Q argument (see details). For user defined penalty user need to specify L matrix in L argument (see details). For Ridge and second-order difference penalty, specification for arguments L and Q will be ignored. Default is 'RIDGE'.
L.user	penalty matrix. Need to be specified with pentype='USER'. When comm.pen=TRUE, Number of columns need to be equal with number of columns of matrix specified to funcs. When comm.pen=FALSE, Number of columns need to be equal with the number of columns of matrix specified to funcs times the number of components of regression function. Each row represents a constraint on functional predictor. This argument will be ignored when value of pentype is other than 'USER'.
f_t	vector or matrix with number of rows equal to number of total observations and number of columns equal to d (see details). If matrix then each column pertains to single function of time and the value in the column represents the realization corresponding to time vector t. The column with intercept or multiple of intercept will be dropped. A NULL value refers to time-invariant regression function. Default value is NULL.
Q	Q matrix to derive decomposition based penalty. Need to be specified with pentype='DECOMP'. When comm.pen=TRUE, number of columns must equal number of columns of matrix specified to funcs. When comm.pen=FALSE, Number of columns need to be equal with the number of columns of matrix specified to funcs times the number of components of regression function. Each row represents a basis function where functional predictor is expected lie according to prior belief. This argument will be ignored when value of pentype is other than 'DECOMP'.
phia	scalar value of a in decomposition based penalty. Needs to be specified with pentype='DECOMP'.
se	logical; calculate standard error when TRUE.
...	additional arguments passed to lme.

Details

If there are any missing or infinite values in Y, subj, t, covariates, funcs and f_t, the corresponding row (or observation) will be dropped, and infinite values are not allowed for these arguments. Neither Q nor L may contain missing or infinite values. lpeer() fits the following model:

y_{i(t)}=X_{i(t)}^T \beta+\int {W_{i(t)}(s)\gamma(t,s) ds} +Z_{i(t)}u_i + \epsilon_{i(t)}

where \epsilon_{i(t)} ~ N(0,\sigma ^2) and u_i ~ N(0, \sigma_u^2). For all the observations, predictor function W_{i(t)}(s) is evaluated at K sampling points. Here, regression function \gamma (t,s) is represented in terms of (d+1) component functions \gamma_0(s),..., \gamma_d(s) as follows

\gamma (t,s)= \gamma_0(s)+f_1(t) \gamma_1(s) + f_d(t) \gamma_d(s)

Values of y_{i(t)} , X_{i(t)} and W_{i(t)}(s) are passed through argument Y, covariates and funcs, respectively. Number of elements or rows in Y, t, subj, covariates (if not NULL) and funcs need to be equal.

Values of f_1(t),...,f_d(t) are passed through f_t argument. The matrix passed through f_t argument should have d columns where each column represents one and only one of f_1(t),..., f_d(t).

The estimate of (d+1) component functions \gamma_0(s),..., \gamma_d(s) is obtained as penalized estimated. The following 3 types of penalties can be used for a component function:

i. Ridge: I_K

ii. Second-order difference: [d_{i,j}] with d_{i,i} = d_{i,i+2} = 1, d_{i,i+1} = -2, otherwise d_{i,j} =0

iii. Decomposition based penalty: bP_Q+a(I-P_Q) where P_Q= Q^T (QQ^T)^{-1}Q

For Decomposition based penalty the user must specify pentype= 'DECOMP' and the associated Q matrix must be passed through the Q argument. Alternatively, one can directly specify the penalty matrix by setting pentype= 'USER' and using the L argument to supply the associated L matrix.

If Q (or L) matrix is similar for all the component functions then argument comm.pen should have value TRUE and in that case specified matrix to argument Q (or L) should have K columns. When Q (or L) matrix is different for all the component functions then argument comm.pen should have value FALSE and in that case specified matrix to argument Q (or L) should have K(d+1) columns. Here first K columns pertains to first component function, second K columns pertains to second component functions, and so on.

Default penalty is Ridge penalty for all the component functions and user needs to specify 'RIDGE'. For second-order difference penalty, user needs to specify 'D2'. When pentype is 'RIDGE' or 'D2' the value of comm.pen is always TRUE and comm.pen=FALSE will be ignored.

Value

A list containing:

fit	result of the call to lme
fitted.vals	predicted outcomes
BetaHat	parameter estimates for scalar covariates including intercept
se.Beta	standard error of parameter estimates for scalar covariates including intercept
Beta	parameter estimates with standard error for scalar covariates including intercept
GammaHat	estimates of components of regression functions. Each column represents one component function.
Se.Gamma	standard error associated with GammaHat
AIC	AIC value of fit (smaller is better)
BIC	BIC value of fit (smaller is better)
logLik	(restricted) log-likelihood at convergence
lambda	list of estimated smoothing parameters associated with each component function
V	conditional variance of Y treating only random intercept as random one.
V1	unconditional variance of Y
N	number of subjects
K	number of Sampling points in functional predictor
TotalObs	total number of observations over all subjects
Sigma.u	estimated sd of random intercept.
sigma	estimated within-group error standard deviation.

Author(s)

Madan Gopal Kundu [e-mail]

References

Kundu, M. G., Harezlak, J., and Randolph, T. W. (2012). Longitudinal functional models with structured penalties (arXiv:1211.4763 [stat.AP]).

Randolph, T. W., Harezlak, J, and Feng, Z. (2012). Structured penalties for functional linear models - partially empirical eigenvectors for regression. Electronic Journal of Statistics, 6, 323–353.

See Also

peer, plot.lpeer

Examples

## Not run: 
#------------------------------------------------------------------------
# Example 1: Estimation with Ridge penalty
#------------------------------------------------------------------------

##Load Data
data(DTI)

## Extract values for arguments for lpeer() from given data
cca = DTI$cca[which(DTI$case == 1),]
DTI = DTI[which(DTI$case == 1),]

##1.1 Fit the model with single component function
##    gamma(t,s)=gamm0(s)
t<- DTI$visit
fit.cca.lpeer1 = lpeer(Y=DTI$pasat, t=t, subj=DTI$ID, funcs = cca)
plot(fit.cca.lpeer1)

##1.2 Fit the model with two component function
##    gamma(t,s)=gamm0(s) + t*gamma1(s)
fit.cca.lpeer2 = lpeer(Y=DTI$pasat, t=t, subj=DTI$ID, funcs = cca,
                      f_t=t, se=TRUE)
plot(fit.cca.lpeer2)

#------------------------------------------------------------------------
# Example 2: Estimation with structured penalty (need structural
#            information about regression function or predictor function)
#------------------------------------------------------------------------

##Load Data
data(PEER.Sim)

## Extract values for arguments for lpeer() from given data
K<- 100
W<- PEER.Sim[,c(3:(K+2))]
Y<- PEER.Sim[,K+3]
t<- PEER.Sim[,2]
id<- PEER.Sim[,1]

##Load Q matrix containing structural information
data(Q)

##2.1 Fit the model with two component function
##    gamma(t,s)=gamm0(s) + t*gamma1(s)
Fit1<- lpeer(Y=Y, subj=id, t=t, covariates=cbind(t), funcs=W,
	    pentype='DECOMP', f_t=cbind(1,t), Q=Q, se=TRUE)

Fit1$Beta
plot(Fit1)

##2.2 Fit the model with three component function
##    gamma(t,s)=gamm0(s) + t*gamma1(s) + t^2*gamma1(s)
Fit2<- lpeer(Y=Y, subj=id, t=t, covariates=cbind(t), funcs=W,
		     pentype='DECOMP', f_t=cbind(1,t, t^2), Q=Q, se=TRUE)

Fit2$Beta
plot(Fit2)

##2.3 Fit the model with two component function with different penalties
##    gamma(t,s)=gamm0(s) + t*gamma1(s)
Q1<- cbind(Q, Q)
Fit3<- lpeer(Y=Y, subj=id, t=t, covariates=cbind(t), comm.pen=FALSE, funcs=W,
		     pentype='DECOMP', f_t=cbind(1,t), Q=Q1, se=TRUE)

##2.4 Fit the model with two component function with user defined penalties
##    gamma(t,s)=gamm0(s) + t*gamma1(s)
phia<- 10^3
P_Q <- t(Q)%*%solve(Q%*%t(Q))%*% Q
L<- phia*(diag(K)- P_Q) + 1*P_Q
Fit4<- lpeer(Y=Y, subj=id, t=t, covariates=cbind(t), funcs=W,
		     pentype='USER', f_t=cbind(1,t), L=L, se=TRUE)

L1<- adiag(L, L)
Fit5<- lpeer(Y=Y, subj=id, t=t, covariates=cbind(t), comm.pen=FALSE, funcs=W,
		     pentype='USER', f_t=cbind(1,t), L=L1, se=TRUE)

## End(Not run)

refund documentation built on Sept. 21, 2024, 1:07 a.m.