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MixTVEM.r
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MixTVEM.r
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TVEMMixNormal <- function( dep, # The dependent variable as a vector, one
# entry per assessment per subject
id, # The subject ID as a vector, one
# entry per assessment per subject
# (could be integers or strings)
numInteriorKnots, # The number of interior knots
# for the splines representing
# the time-varying coefficients.
# Assumed to be the same
# for each coefficient.
numClasses, # Number of classes in the mixture model
tcov, # Time-varying covariates as a matrix, one
# row per assessment per person. Usually,
# the first column should be a column of
# ones representing an intercept.
time, # Assessment time as a vector
# These arguments have default values, given after the equal sign for each:
convergenceCriterion=1e-6, # Convergence criterion for the
# maximum absolute deviation of parameter
# estimates between successive
# steps of the EM algorithm
deg=3, # Degree of the polynomial between
# successive knots for the time-
# varying coefficient functions.
doPlot=TRUE, # Whether to draw a plot of the time-
# varying coefficient functions.
getSEs=TRUE, # Whether to calculate standard errors.
# Setting this to FALSE would save
# computational time.
gridSize=1000, # The number of values of time for which
# to obtain estimates of the coefficient
# functions
maxIterations=1000, # Maximum number of EM iterations
# to attempt;
maxVarianceRatio=10, # Maximum ratio between estimated
# variances of different classes;
min.time=NA, # beginning of interval of interest; if left
# at NA, it will default to min(time) where
# time is the time variable given above
max.time=NA, # end of interval of interest; if left
# at NA, it will default to max(time) where
# time is the time variable given above
numStarts=50, # Number of random starting values to use.;
referenceClass=1, # Reference class. If it is not
# specified here, then the first class
# will be selected as the reference class.
seed=NA, # The initial random seed. It will be used
# to fit the model (if numStarts=1) or to
# generate more seeds (if numStarts>1).
scov=NULL, # Subject-level class membership prediction
# covariates, as a matrix. A column of 1's
# should not be provided for scov because,
# unlike in the case of tcov, an intercept
# column is included automatically
# by the code. This should have the same
# number of rows as tcov, but should be
# identical on all observations within the
# same person.
xcov=NULL # Optional matrix of covariates assumed
# to have time-invariant effects.
) {
##################################################################
# MixTVEM macro Version 1.2
# By John DZIAK, Xianming TAN, and Runze LI
# Fits a mixture of nonparametric trajectories to longitudinal data.
#
# Copyright:
# (c) 2019 The Pennsylvania State University
#
# License:
# This program is free software; you can redistribute it and/or
# modify it under the terms of the GNU General Public License as
# published by the Free Software Foundation; either version 2 of
# the License, or (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
# General Public License for more details.
#
# Acknowledgments and references:
# We fit a mixture of nonparametric varying-coefficient models, using
# a penalized B-spline approach. See
# Eilers, P.H.C. and Marx, B.D. (1996). Flexible smoothing using
# B-splines and penalized likelihood. Statistical Science
# 11(2): 89-121.
# Hastie, T., & Tibshirani, R. (1993). Varying-coefficient models.
# Journal of the Royal Statistical Society, Series B, 55, 757-796.
# Shiyko, M. P., Lanza, S. T., Tan, X., Li, R., Shiffman, S. (2012).
# Using the Time-Varying Effect Model (TVEM) to Examine Dynamic
# Associations between Negative Affect and Self Confidence on Smoking :
# Urges Differences between Successful Quitters and Relapsers.
# Prevention Science, 13, 288-299.
# Ramsay, J., Hooker, G., & Graves, S. (2009). Functional Data Analysis
# with R and MATLAB. New York: Springer.
# Tan, X., Shiyko, M. P., Li, R., Li, Y., & Dierker, L. (2011, November 21).
# A time-varying effect model for intensive longitudinal data.
# Psychological Methods. Advance online publication.
# doi: 10.1037/a0025814.
# Estimation is done using the EM algorithm for finite mixtures.
# McLachlan, G. J., and Peel, D. (2000). Finite mixture models. New York:
# Wiley.
# Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum
# likelihood from incomplete data via the EM algorithm. Journal of
# the Royal Statistical Society, B, 39, 1-38.
# The standard error calculations for the mixture approach are based on
# those used by Turner (2000) and Turner's mixreg R package, which are
# based on the ideas of Louis (1982).
# Louis, T. A. (1982). Finding the Observed Information Matrix when Using
# the EM Algorithm. Journal of the Royal Statistical Society, B, 44,
# 226-233.
# Turner, T. R. (2000) Estimating the rate of spread of a viral infection
# of potato plants via mixtures of regressions. Applied Statistics,
# 49, pp. 371-384.
# Turner, R. (2009). mixreg: Functions to fit mixtures of regressions.
# R package version 0.0-3. http://CRAN.R-project.org/package=mixreg
# The use of a sandwich formula to adjust for within-subject correlation
# when calculating the standard errors is inspired by
# Liang, K.-Y., and Zeger, S. L. (1986). Longitudinal data analysis
# using generalized linear models. Biometrika, 73, 13-22.
# Clustering functional data modeled with splines is described in
# James, G., and Sugar, C. (2003) Clustering for sparsely sampled
# functional data. Journal of the American Statistical
# Association 98, 397-408.
# The model fit criteria used are adapted versions of the standard AIC, BIC
# and GCV of:
# Akaike, H. (1973). Information theory and an extension of the maximum
# likelihood principle. In B. N. Petrov & F. Csaki (Eds.), Second
# international symposium on information theory (p. 267-281).
# Budapest, Hungary: Akademai Kiado.
# Schwarz, G. (1978). Estimating the dimension of a model. Annals of
# Statistics, 6, 461-464.
# Craven, P., and Wahba, G. (1978). Smoothing noisy data with spline
# functions: Estimating the correct degree of smoothing by the
# method of generalized cross-validation. Numerische
# Mathematik, 31, 377403.
##################################################################
library(splines);
library(nlme);
## Define required helper functions:
MixTVEMFitInner <- function(convergenceCriterion,
intId,
maxIterations=1000,
maxVarianceRatio,
proportionNugget=.5,
numClasses,
penaltyMatrix=NULL,
referenceClass,
rho=0,
roughnessPenalty=0,
S=S,
time,
X,
Y ) {
## Prepare to begin loop;
stopifnot(length(time)==length(intId));
theta <- matrix(0,ncol(X),numClasses);
oldGamma <- Inf;
oldtheta <- Inf;
maxAbsDev <- Inf;
numSubjects <- max(intId);
numTotal <- length(intId);
nonreferenceClasses <- (1:numClasses)[-referenceClass];
iteration <- 1;
numObsBySub <- table(intId); # assumes that intId
# consists of consecutive integers starting at 1;
stopifnot(length(Y)==numTotal);
stopifnot(length(unique(intId))==numSubjects);
stopifnot(all.equal(unique(intId),1:numSubjects));
stopifnot(as.integer(rownames(numObsBySub))==1:numSubjects);
stopifnot(identical(intId,sort(intId)));
stopifnot(length(numObsBySub)==numSubjects);
stopifnot(length(intId)==numTotal);
stopifnot(identical(intId,sort(intId)));
## Initial E step (generate random posterior probabilities;
temp <- matrix(rexp(numSubjects*numClasses),
numSubjects,
numClasses);
postProbsBySub <- temp/apply(temp,1,sum);
## Calculate inverse covariance matrices;
covMats <- list();
invCovMats <- list();
dets <- list();
sigsq.total <- rep(1,numClasses); # starting value;
for (i in 1:numSubjects) {
these <- which(intId==i);
covMats[[i]] <- (1 - proportionNugget) *
rho^(abs(outer(time[these],time[these],"-"))) +
proportionNugget*diag(length(these));
invCovMats[[i]] <- solve(covMats[[i]]);
dets[[i]] <- det(covMats[[i]]);
}
## Begin loop;
while ((iteration<maxIterations)&
(maxAbsDev>convergenceCriterion)) {
## Initial work;
iteration <- iteration+1;
postProbsByAssessment <- matrix(0,numTotal,numClasses);
for (k in 1:numClasses) {
postProbsByAssessment[,k] <-
rep(postProbsBySub[,k],numObsBySub);
}
## M Step;
## Get new beta estimates;
modelList <- list();
for (k in 1:numClasses) {
denominator <- matrix(0,ncol(X),ncol(X));
numerator <- matrix(0,ncol(X),1);
for (i in 1:numSubjects) {
these <- which(intId==i);
denominator <- denominator +
t(X[these,,drop=FALSE])%*%(invCovMats[[i]]/sigsq.total[k])%*%
(postProbsBySub[i,k]*X[these,,drop=FALSE]);
numerator <- numerator + t(X[these,,drop=FALSE])%*%
(invCovMats[[i]]/sigsq.total[k])%*%
(postProbsBySub[i,k]*Y[these]);
}
theta[,k] <- solve(denominator + # negative Hessian as usual
roughnessPenalty*penaltyMatrix, #+ # second order difference penalty
#(1e-6)*roughnessPenalty*diag(as.integer(ncol(penaltyMatrix))), # small additional ridge penalty;
numerator);
}
fittedY <- X%*%theta; # one column for each class;
stopifnot(nrow(postProbsByAssessment)==nrow(fittedY));
stopifnot(length(Y)==nrow(fittedY));
residsY <- kronecker(t(rep(1,numClasses)),Y)-fittedY;
## Get new sigma estimates;
f <- function(i) {
return(apply(residsY[which(intId==i),,drop=FALSE]^2,
2,sum))};
rssBySubjectAndClass <- t(sapply(1:numSubjects,f));
if (nrow(rssBySubjectAndClass)==1)
{rssBySubjectAndClass <- t(rssBySubjectAndClass);}
# corrects for the unusual fact that
# the result from sapply seems to come out
# transposed when there is only 1 column,
# relative to when there is >1;
for (k in 1:numClasses) {
sigsq.total[k] <- sum(rssBySubjectAndClass[,k]*
postProbsBySub[,k]) /
sum(numObsBySub*postProbsBySub[,k]);
}
if (!is.na(maxVarianceRatio)) {
if (max(sigsq.total)/min(sigsq.total)>
maxVarianceRatio) {
sigsq.total[which(sigsq.total<(max(sigsq.total)
/maxVarianceRatio))] <-
max(sigsq.total)/maxVarianceRatio;
}
}
## Get new gamma estimates;
SExtended <- kronecker(S,rep(1,numClasses));
colnames(SExtended) <- colnames(S);
classForLR <- rep(1:numClasses,times=numSubjects);
weightForLR <- as.vector(t(postProbsBySub));
dataForLR <- cbind(rep(1:numSubjects,each=numClasses),
classForLR,
weightForLR,
SExtended);
numGamma <- ncol(SExtended);
gamma <- matrix(0,numGamma,numClasses);
eta <- matrix(0,numSubjects,numClasses);
for (k in nonreferenceClasses) {
outcomeForLogisticRegression <- 1*(classForLR==k);
outcomeForLogisticRegression[(classForLR!=k)&
(classForLR!=referenceClass)] <- NA;
warnOption <- getOption("warn");
options(warn=-1);
thisModel <- glm(outcomeForLogisticRegression~
SExtended+0,
family=binomial,
weights=weightForLR);
options(warn=warnOption);
gamma[,k] <- thisModel$coef;
eta[,k] <- S%*%gamma[,k];
}
fittedProb <- exp(eta)/apply(exp(eta),1,sum);
## E Step (get new likelihood contributions
## and posterior probabilities);
logProbability <- matrix(0,numSubjects,numClasses);
wrssBySubjectAndClass <- matrix(0,numSubjects,numClasses);
for (k in 1:numClasses) {
f <- function(i,k) {
these <- which(intId==i);
return( drop(t(residsY[these,k,drop=FALSE])%*%
(invCovMats[[i]]/sigsq.total[k])%*%residsY[these,k,drop=FALSE]) );
};
wrssBySubjectAndClass[,k] <- drop(sapply(1:numSubjects,f,k));
f <- function(i,k) {
these <- which(intId==i);
return( (sigsq.total[k]^length(these))*dets[[i]] );
};
determinantsBySubject.class.k <-
drop(t(sapply(1:numSubjects,f,k)));
if (sum(is.nan(log(determinantsBySubject.class.k+1e-30)))>0) {
warning("Problem with covariance structure estimation");
print("log determinant NaN");
print("sigsq.total:");print(sigsq.total);
print("rho:");print(rho);
}
logProbability[,k] <- -(numObsBySub/2)*log(2*3.1415926535)-
(1/2)*(log(determinantsBySubject.class.k+1e-30)) -
wrssBySubjectAndClass[,k]/2;
}
tempMatrix <- logProbability;
for (k in 1:numClasses) {
tempMatrix[,k] = tempMatrix[,k] + log(fittedProb[,k]+1e-30);
}
logLikelihoodBySubject <- log(apply(exp(tempMatrix),1,sum));
logLik <- sum(logLikelihoodBySubject);
tempMax <- apply(tempMatrix,1,max);
for (k in 1:numClasses) {
tempMatrix[,k] <- tempMatrix[,k] - tempMax;
}
expTempMatrix <- exp(tempMatrix);
for (k in 1:numClasses) {
postProbsBySub[,k] <- expTempMatrix[,k]/apply(expTempMatrix,1,sum);
}
maxAbsDev <- max(abs(c(as.vector(theta)-oldtheta,
as.vector(gamma)-oldGamma)));
oldGamma <- as.vector(gamma);
oldtheta <- as.vector(theta);
}
## Done with main loop
enpByClass <- rep(0,numClasses);
for (k in 1:numClasses) {
denominator <- matrix(0,ncol(X),ncol(X));
for (i in 1:numSubjects) {
these <- which(intId==i);
denominator <- denominator + postProbsBySub[i,k]*t(X[these,,drop=FALSE])%*%
(invCovMats[[i]]/sigsq.total[k])%*%X[these,,drop=FALSE];
}
for (i in 1:numSubjects) {
these <- which(intId==i);
enpByClass[k] <- enpByClass[k] + postProbsBySub[i,k]*sum(diag(as.matrix((X[these,,drop=FALSE]%*%
solve(denominator +
roughnessPenalty*penaltyMatrix,# +
# (1e-6)*roughnessPenalty*diag(as.integer(ncol(penaltyMatrix))),
(t(X[these,,drop=FALSE])%*%(invCovMats[[i]]/sigsq.total[k])))))));
}
}
enp <- sum(enpByClass) +
ncol(S)*(numClasses-1)+
numClasses +2;
np <- ncol(X)*numClasses + ncol(S)*(numClasses-1)+
numClasses +2;
converged <- maxAbsDev<=convergenceCriterion;
weightedRSS <- sum(postProbsBySub*rssBySubjectAndClass);
return(list(aic=-2*logLik+2*enp,
# A statistic similar to AIC (Akaike, 1973)
# but using the effective (penalized) number of
# parameters instead of a count of parameters
# (see Eilers and Marx, 1996)
bic=-2*logLik+log(numSubjects)*enp,
# A statistic somewhat analogous to BIC
# (Schwarz, 1978). The effective (penalized)
# number of parameters is used instead of a
# count of parameters (see Eilers and Marx,
# 1996).
converged=converged,
# Indicates whether the EM algorithm converged
dep=Y,
# The observed response for each assessment,
# copied here for convenience when doing
# fit diagnostics, follow-up analyses, etc.
enp=enp,
# The effective number of regression
# parameters (see Eilers and Marx, 1996)
# per class plus the number of free
# gamma parameters plus the number of free
# sigma parameters, summed across classes.
fittedProb=fittedProb,
# The fitted values for class probabilities
# given the S covariates for each subject,
# as given by the logistic regression model
# for class membership.
fittedY=fittedY,
# The fitted values for the dependent
# variable at each assessment.
gamma=gamma,
# The estimated logistic regression
# parameters in the model for class
# membership.
intId=intId,
# The internally assigned subject identification
# number for each assessment,
# copied here for convenience when doing
# fit diagnostics, follow-up analyses, etc.
iteration=iteration,
# The number of iterations run by the EM
# algorithm.
lambda=roughnessPenalty,
# The current candidate value of the penalty
# weight for the roughness penalty
logLik=logLik,
# The fitted log-likelihood
np=np,
# The counted number of parameters,
# ignoring shrinkage imposed by the penalty
postProbsBySub=postProbsBySub,
# Each subject's estimated posterior
# probability of belonging to each class.
postProbsByAssessment=postProbsByAssessment,
# The same as postProbsBySub, except that
# they are repeated within subject so that
# there is one row per assessment rather than
# only one row per subject.
proportionNugget=proportionNugget,
# The value assumed for the within-subject
# proportion of error variance which is
# idiosyncratic to a single time point
residsY = residsY,
# Residuals for each observation under the
# model for each class;
rho = rho,
# The value assumed for the autocorrelation
# parameter;
sigsq.total=sigsq.total,
# Estimated sigma squared (variance)
# parameters for each class.
theta=theta,
# Estimated regression coefficients for each
# of the regression coefficients, including
# spline basis coefficients
weightedGCV=weightedRSS/(numTotal*((1-(enp/numTotal))**2)),
# A statistic similar to GCV (see Wahba, 1990;
# Eilers and Marx, 1996) but calculated
# with the weightedRSS instead of an ordinary
# residual sum of squares.
weightedRSS=weightedRSS
# The weighted residual sum of squares
# error measure. It is like the classic
# regression sum of squared residuals
# (sum(y-yhat)^2), but it is weighted
# by the posterior probabilities estimated
# for each individual and class.
));
}
estimateRho <- function (thisFit, time) {
## Get new negLogRho estimate;
sigsq.total <- thisFit$sigsq.total;
numClasses <- ncol(thisFit$postProbsBySub);
crossprods <- NULL;
lags <- NULL;
weights<- NULL;
for (k in 1:numClasses) {
for (i in unique(intId)) {
these <- which(intId==i);
temp1 <- outer(thisFit$residsY[these,k,drop=TRUE],thisFit$residsY[these,k,drop=TRUE],"*");
new.crossprods <- as.vector(temp1[upper.tri(temp1,diag=FALSE)] ) / sigsq.total[k];
crossprods <- c(crossprods,new.crossprods);
temp1 <- abs(outer(time[these],time[these],"-"));
new.lags <- as.vector(temp1[upper.tri(temp1,diag=FALSE)]);
lags <- c(lags,new.lags);
stopifnot(length(new.crossprods)==length(new.lags));
weights<- c(weights,
rep(thisFit$postProbsBySub[i,k],length(new.crossprods)));
}
}
initial.proportionNugget <- 1 / 2;
average.cross.product <- weighted.mean(x=crossprods, w=weights);
average.lag <- weighted.mean(x=lags, w=weights);
initial.rho <-exp((log(average.cross.product) - log(1 - initial.proportionNugget))/average.lag);
y.for.nlsfit <- crossprods;
x.for.nlsfit <- lags;
nlsfit1 <- nls(y.for.nlsfit ~ pnn*exp(logrho*x.for.nlsfit),
weights=weights,
control=nls.control(maxiter = 500, printEval=FALSE, warnOnly = TRUE),
start=list(logrho=log(initial.rho), pnn=1-initial.proportionNugget));
rho <- exp(summary(nlsfit1)$coefficients["logrho","Estimate"]);
proportionNugget <- 1-summary(nlsfit1)$coefficients["pnn","Estimate"];
return(list(rho=rho,proportionNugget =proportionNugget));
}
MixTVEMCovMats <- function( fittedProb,
gamma,
intId,
mu,
numSubjects,
numTotal,
penalty,
postProb,
proportionNugget,
referenceClass,
rho,
S,
sigsq.total,
theta,
X,
Y) {
numthetas <- nrow(theta);
numClasses <- ncol(theta);
stopifnot(numthetas==ncol(X));
numGammas <- nrow(gamma);
stopifnot(numClasses==ncol(mu));
nonreferenceClasses <- (1:numClasses)[-referenceClass];
numParams <- (numClasses*numthetas) +
(numClasses-1)*numGammas;
thetaIndex <- matrix(0,numthetas,numClasses);
if (numClasses>1) {
gammaIndex <- matrix(0,numGammas,numClasses-1);
} else {gammaIndex <- NULL;}
for (k in 1:numClasses) {
thetaIndex[,k] <- (((k-1)*numthetas+(k-1)*numGammas) +
(1:numthetas));
if (k < numClasses) {
gammaIndex[,k] <- (k*numthetas+(k-1)*numGammas) +
(1:numGammas);
}
}
y.inv.corrmats <- list();
for (i in 1:numSubjects) {
y.inv.corrmats[[i]] <- solve((1 - proportionNugget)*rho^(abs(outer(time[which(intId==i)],
time[which(intId==i)],"-")))
+ proportionNugget*diag(length(which(intId==i))));
}
# These calculations are based on the mixreg package in Turner
# (2009) with some modifications as described in our paper;
I1 <- matrix(0,numParams,numParams);
I2 <- matrix(0,numParams,numParams);
I3 <- matrix(0,numParams,numParams);
# Calculate matrix I1
for (c in 1:numClasses) {
for (i in 1:numSubjects) {
these <- which(intId==i);
xi <- X[which(intId==i),,drop=FALSE];
ni <- sum(intId==i);
contrib <- postProb[i,c]*
t(as.matrix(xi))%*%(y.inv.corrmats[[i]]/sigsq.total[c])%*%as.matrix(xi);
I1[thetaIndex[,c],thetaIndex[,c]] <-
I1[thetaIndex[,c],thetaIndex[,c]] + contrib;
}
I1[thetaIndex[,c],thetaIndex[,c]] <- I1[thetaIndex[,c],thetaIndex[,c]] + penalty;
}
if (numClasses>1) {
for (c in 1:(numClasses-1)) {
for (k in 1:(numClasses-1)) {
for (i in 1:numSubjects) {
nonrefc <- nonreferenceClasses[c];
nonrefk <- nonreferenceClasses[k];
Si <- S[i,,drop=FALSE];
contrib <- fittedProb[i,nonrefc]*
(1*(nonrefc==nonrefk)-fittedProb[i,nonrefk])*
crossprod(Si);
I1[gammaIndex[,c],gammaIndex[,k]] <-
I1[gammaIndex[,c],gammaIndex[,k]] + contrib;
}
}
}
}
# Calculate matrix I2 (information lost due to not knowing true class labels)
for (c in 1:numClasses) {
for (k in 1:numClasses) {
for (i in 1:numSubjects) {
sigsqS <- sigsq.total[c]*(1 - proportionNugget);
sigsqE <- sigsq.total[c]*proportionNugget;
ni <- sum(intId==i);
xi <- X[which(intId==i),,drop=FALSE];
yi <- Y[which(intId==i),drop=FALSE];
Si <- S[i,,drop=FALSE];
muic <- mu[which(intId==i),c,drop=FALSE];
hic <- t(xi)%*%(y.inv.corrmats[[i]]/sigsq.total[c])%*%(yi-muic);
for (j in 1:numSubjects) {
# Information loss in linear regression coefficients:
nj <- sum(intId==j);
xj <- X[which(intId==j),,drop=FALSE];
yj <- Y[which(intId==j),drop=FALSE];
Sj <- S[j,,drop=FALSE];
mujk <- mu[which(intId==j),k,drop=FALSE];
hjk <- t(xj)%*%(y.inv.corrmats[[j]]/sigsq.total[k])%*%(yj-mujk);
Gckij <- postProb[i,c]*postProb[j,k]*(i!=j) +
postProb[i,c]*(i==j)*(c==k);
I2[thetaIndex[,c],thetaIndex[,k]] <-
I2[thetaIndex[,c],thetaIndex[,k]] +
Gckij * hic%*%t(hjk) -
(1/(numSubjects^2))*postProb[i,c]*hic%*%t(theta[,k])%*%penalty -
(1/(numSubjects^2))*postProb[j,k]*hjk%*%t(theta[,c])%*%penalty +
(1/(numSubjects^2))*penalty%*%theta[,c]%*%t(theta[,k])%*%penalty;
# Information loss in logistic regression coefficients:
if ((c < numClasses)&(k < numClasses)) {
nonrefc <- nonreferenceClasses[c];
nonrefk <- nonreferenceClasses[k];
Gcnonrefkij <- postProb[i,c]*postProb[j,nonrefk]*(i!=j) +
postProb[i,c]*(i==j)*(c==nonrefk);
Gnonrefcnonrefkij <- postProb[i,nonrefc]*postProb[j,nonrefk]*(i!=j) +
postProb[i,nonrefc]*(i==j)*(nonrefc==nonrefk);
contrib <- (Gnonrefcnonrefkij-
postProb[i,nonrefc]*fittedProb[j,nonrefk]-
postProb[j,nonrefk]*fittedProb[i,nonrefc] +
fittedProb[i,nonrefc]*fittedProb[j,nonrefk])*
(t(Si)%*%Sj);
I2[gammaIndex[,c],gammaIndex[,k]] <-
I2[gammaIndex[,c],gammaIndex[,k]] +
contrib;
}
# Information loss in covariance between linear and
# logistic regression coefficients:
if (k < numClasses) {
for (kprime in 1:numClasses) {
nonrefk <- nonreferenceClasses[k];
Gcnonrefkij <- postProb[i,c]*postProb[j,nonrefk]+
postProb[i,c]*(i==j)*
((c==nonrefk)-postProb[j,nonrefk]);
contrib = Gcnonrefkij*((k==kprime)-fittedProb[j,k])*hic%*%Sj +
( (1/numSubjects)*postProb[j,kprime] *
((k==kprime)-fittedProb[j,k])*penalty%*%theta[,c]%*%Sj) ;
I2[thetaIndex[,c],gammaIndex[,k]] <-
I2[thetaIndex[,c],gammaIndex[,k]] +
contrib;
I2[gammaIndex[,k],thetaIndex[,c]] <-
t(I2[thetaIndex[,c],gammaIndex[,k]]);
}
}
}
}
}
}
# Calculate matrix I3
for (i in 1:numSubjects) {
scorei <- matrix(0,numParams,1);
for (k in 1:numClasses) {
ni <- sum(intId==i);
xi <- X[which(intId==i),,drop=FALSE];
yi <- Y[which(intId==i),drop=FALSE];
Si <- S[i,,drop=FALSE];
muik <- mu[which(intId==i),k,drop=FALSE];
hik <- t(xi)%*%(y.inv.corrmats[[i]]/sigsq.total[k])%*%(yi-muik);
scorei[thetaIndex[,k]] <- postProb[i,k]*hik -
(1/numSubjects)*penalty%*%theta[,k];
if (k < numClasses) {
nonrefk <- nonreferenceClasses[k];
scorei[gammaIndex[,k]] <- (postProb[i,nonrefk]-
fittedProb[i,nonrefk])*Si;
}
}
I3 <- I3 + crossprod(t(scorei));
}
# Calculate naiveCovarianceMatrix (in theta and gamma)
if (kappa(I1)<Inf) {
uncorrectedCovarianceThetaGamma <- solve(I1)%*%I3%*%solve(I1);
} else {
uncorrectedCovarianceThetaGamma <- matrix(NA,nrow(I1),ncol(I1));
}
if (kappa(I1-I2)<Inf) {
correctedCovarianceThetaGamma <- solve(I1-I2);
} else {
correctedCovarianceThetaGamma <- matrix(NA,nrow(I1),ncol(I1));
}
# Calculate covarianceMatrix (in theta and gamma)
if (det(I3)>0) {
sandwichCovarianceThetaGamma <- correctedCovarianceThetaGamma%*%
I3%*%correctedCovarianceThetaGamma;
}
# Calculate covarianceForPrevalences
if (numClasses>1) {
allGammas <- as.vector(gammaIndex);
jacobian <- matrix(0,numClasses,(numClasses-1)*numGammas);
for (m in 1:(numClasses-1)) {
for (k in 1:numClasses) {
for (q in 1:numGammas) {
w <- (m-1)*numGammas + q;
nonrefm <- nonreferenceClasses[m];
# Jacobian[m,i] = derivative of lambda[k] in gamma[m,q], ;
# i.e., gamma[w] when the free gammas are all listed in a ;
# vector, one class on top of another, excluding the
# reference class. ;
temp <- sum(((1*(k==nonrefm))-fittedProb[,k])*
fittedProb[,nonrefm]*S[,q]);
jacobian[k,w] <- temp/numSubjects;
}
}
}
uncorrectedCovariancePrevalence <- jacobian %*%
uncorrectedCovarianceThetaGamma[allGammas,allGammas] %*%
t(jacobian);
correctedCovariancePrevalence <- jacobian %*%
correctedCovarianceThetaGamma[allGammas,allGammas] %*%
t(jacobian);
sandwichCovariancePrevalence <- jacobian %*%
sandwichCovarianceThetaGamma[allGammas,allGammas] %*%
t(jacobian);
} else {
uncorrectedCovariancePrevalence <- NULL;
correctedCovariancePrevalence <- NULL;
sandwichCovariancePrevalence <- NULL;
}
return(list( uncorrectedCovarianceThetaGamma=uncorrectedCovarianceThetaGamma,
# Similar to the covariance matrix, but without
# correcting for missing class labels as in Turner
correctedCovarianceThetaGamma=correctedCovarianceThetaGamma,
# The estimated covariance matrix for all of
# the model parameters, adjusting for the unknown
# class labels
sandwichCovarianceThetaGamma=sandwichCovarianceThetaGamma,
# The sandwich covariance matrix for the model
# parameters, which may be more robust to
# misspecification of the within-subject
# correlation (see Liang and Zeger 1986)
uncorrectedCovariancePrevalence=uncorrectedCovariancePrevalence,
# Estimators for the covariance of class prevalences (overall
# proportions) based on the respective estimators for the
# covariance of the model parameters.
correctedCovariancePrevalence=correctedCovariancePrevalence,
sandwichCovariancePrevalence=sandwichCovariancePrevalence,
thetaIndex=thetaIndex,
# Tells which rows and columns of the covariance
# matrix refer to thetas (linear regression
# parameters)
gammaIndex=gammaIndex,
# Tells which rows and columns of the covariance
# matrix refer to gammas (logistic regression
# parameters)
I1=I1,
I2=I2,
I3=I3)); # I1, I2, and I3 are described in our paper
# (Dziak et al) and are based on combining
# the approach of Turner (2000) with a
# sandwich estimation approach;
}
#####################################################################
## Main body of MixTVEM function
## Process the subject ID's;
if (!is.integer(id)) {
temporary.id <- as.integer(as.factor(id));
} else {
temporary.id <- id;
}
if (is.na(min.time)) {min.time <- min(time);}
if (is.na(max.time)) {max.time <- max(time);}
numTotal <- length(id);
intId <- rep(0,numTotal);
stopifnot(is.integer(unique(temporary.id)));
numSubjects <- length(unique(temporary.id));
for (i in 1:numSubjects) {
intId[which(temporary.id==unique(temporary.id)[i])] <- i;
}
stopifnot(length(unique(intId))==numSubjects);
stopifnot(all.equal(unique(intId),1:numSubjects));
## Decide on a reference class;
if (is.na(referenceClass)) {referenceClass <- 1;}
stopifnot(referenceClass>0);
stopifnot(referenceClass<=numClasses);
## Process the class membership predictor variables S;
if (!is.null(scov)) {
scov <- as.matrix(scov);
stopifnot(nrow(scov)==numTotal);
S <- matrix(0,numSubjects,1+ncol(scov));
for (i in 1:numSubjects) {
theseObs <- which(intId==i);
S[i,] <- c(1, scov[min(theseObs),]);
}
} else {
S <- matrix(1,numSubjects,1);
}
if (is.null(colnames(scov))) {
colnames(S) <- paste("S",0:(ncol(S)-1),sep="");
};
## Process the time variable;
stopifnot(!is.null(time));
stopifnot((deg==1)|(deg==2)|(deg==3));
time <- as.matrix(time);
stopifnot(nrow(time)==numTotal);
stopifnot(ncol(time)==1);
stopifnot(is.numeric(time));
if (is.null(colnames(time))) {
colnames(time) <- "Time";
}
timeBasis <- list();
designMatrix <- NULL;
whichBeta <- NULL;
## Construct the first part of the design matrix including ... ;
## ... the intercept column if there are no time-varying covariates;
if (is.null(tcov)) { interceptColumn <- matrix(1,numTotal,1);
colnames(interceptColumn) <- "Intercept";
designMatrix <- cbind(designMatrix,interceptColumn);
whichBeta <- c(whichBeta,0);
}
## ... and the covariates without time-varying effects;
if (!is.null(xcov)) {
if (min(apply(as.matrix(xcov),2,var))<1e-10) {
stop("Please do not include a constant column in xcov.");
}
xcov <- as.matrix(xcov);
stopifnot(nrow(xcov)==numTotal);
numXCov <- ncol(xcov);
if (is.null(colnames(xcov))) {
colnames(xcov) <- paste("X",1:ncol(xcov),sep="");
};
designMatrix <- cbind(designMatrix,xcov);
} else {
numXCov <- 0;
}
whichBeta <- c(whichBeta,rep(0,numXCov));
## Get the tcov matrix ready;
if (is.null(tcov)) {
numTCov <- 0;
stop(paste("tcov is null.",
"No time-varying betas or effects are in the model."));
} else {
tcov <- as.matrix(tcov);
stopifnot(nrow(tcov)==numTotal);
numTCov <- ncol(tcov);
if (is.null(colnames(tcov))) {
colnames(tcov) <- paste("TV",1:ncol(tcov),sep="");
};
## Create the scale vector to be used with the penalty for the time-varying
## covariates;
if (!identical(as.vector(tcov[,1]),rep(1,numTotal))) {
warning("tcov does not seem to contain an intercept (trajectory) column.");
}
tcov.scale <- apply(tcov,2,sd);
if (length(tcov.scale)>1) {
if (min(tcov.scale[-1])<1e-10) {
stop(paste("Please include the intercept column as the first",
"column in tcov, and do not include any other columns",
"which are constant across all",
"subjects and assessments."));
}
}
if (identical(as.vector(tcov[,1]),rep(1,numTotal))) {
tcov.scale[1] <- 1; # do not scale intercept column;
}
## Create the basis functions;
if (length(numInteriorKnots)!=1) {stop();}
stopifnot(numInteriorKnots>0);
num.intervals <- numInteriorKnots+1;
dx <- (max.time-min.time)/num.intervals;
all.knot.locations <- seq(min.time-(deg)*dx,
max.time+(deg)*dx,
by=dx);
all.knot.locations[1] <- all.knot.locations[2]-1e-8;
all.knot.locations[length(all.knot.locations)] <-
all.knot.locations[length(all.knot.locations)-1]+1e-8;
interior.knot.locations <- seq(min.time+dx,
max.time-dx,
by=dx);
timeGrid <- seq(min.time,max.time,length=gridSize);
timeBasis <- spline.des(all.knot.locations,
time,
deg+1,
rep(0,length(time)),
outer.ok=TRUE)$design;
# Adapted from Eilers and Marx, 1996;
colnames(timeBasis) <- paste(colnames(time),".Spline.",
1:ncol(timeBasis),
sep="");
timeBasisByGrid <- spline.des(all.knot.locations,
timeGrid,
deg+1,
rep(0,length(timeGrid)),
outer.ok=TRUE)$design;
# Adapted from Eilers and Marx, 1996;
colnames(timeBasisByGrid) <- paste(colnames(time),".Spline.",
1:ncol(timeBasisByGrid),
sep="");
# Now generate regression matrix;
for (j in 1:numTCov) {
covariateTimesTimeBasis <- tcov[,j,drop=TRUE]*timeBasis;
# Elementwise product;
colnames(covariateTimesTimeBasis) <- paste(colnames(tcov)[j],
"times",
colnames(timeBasis),
sep=".");
designMatrix <- cbind(designMatrix, covariateTimesTimeBasis);
whichBeta <- c(whichBeta,rep(j,ncol(covariateTimesTimeBasis)));
}
stopifnot(sum(is.nan(designMatrix))==0);
stopifnot(sum(is.null(designMatrix))==0);
if(sum(is.na(designMatrix))>0) {
stop("Missing data is not yet supported in this software.");
}
}
## Create the penalty weight matrix (see Eilers and Marx, 1996)
diffMatrix <- crossprod(diff(diff(diag(rep(1,ncol(timeBasis))))));
penaltyMatrix <- matrix(0,ncol(designMatrix),ncol(designMatrix));
for (j in 1:numTCov) {
indices <- numXCov+(j-1)*ncol(timeBasis)+(1:ncol(timeBasis));
stopifnot(length(indices)==nrow(diffMatrix));
stopifnot(length(indices)==ncol(diffMatrix));
penaltyMatrix[indices,indices] <- diffMatrix/(tcov.scale[j]^2);
}
#######################################################################
## Run a loop to find the best seed;
logLikBySeed <- rep(NA,numStarts);
rhoBySeed <- rep(NA,numStarts);
proportionNuggetBySeed <- rep(NA,numStarts);
weightedRSSBySeed <- rep(NA,numStarts);
if (numStarts > 1) {
if (!is.na(seed)) {
};
seeds <- round(runif(numStarts)*1e8);
} else {
if (!is.na(seed)) {
seeds <- seed;
} else {
seeds <- round(runif(1)*1e8);
}
}
postProbsList <- list();
for (i in 1:numStarts) {
set.seed(seeds[i]);
print(paste("Trying seed=",seeds[i]));
thisFit <- MixTVEMFitInner(convergenceCriterion=convergenceCriterion,
intId=intId,
maxIterations=maxIterations/2,
maxVarianceRatio=maxVarianceRatio,
numClasses=numClasses,
penaltyMatrix=penaltyMatrix,
referenceClass=referenceClass,
roughnessPenalty=(1e9)*sd(dep),
rho=0,
S=S,
time=time,
X=designMatrix,
Y=dep);
estimates <- estimateRho(thisFit, time);
estimatedProportionNugget <- estimates$proportionNugget;
estimatedRho <- estimates$rho;
thisFit <- MixTVEMFitInner(convergenceCriterion=convergenceCriterion,
intId=intId,
maxIterations=maxIterations/2,
maxVarianceRatio=maxVarianceRatio,
numClasses=numClasses,
penaltyMatrix=penaltyMatrix,
proportionNugget=estimatedProportionNugget,
referenceClass=referenceClass,
roughnessPenalty=(1e9)*sd(dep),
rho=estimatedRho,
S=S,
time=time,
X=designMatrix,
Y=dep);
logLikBySeed[i] <- thisFit$logLik;
rhoBySeed[i] <- estimatedRho;
weightedRSSBySeed[i] <- thisFit$weightedRSS;
postProbsList[[i]] <- thisFit$postProbsBySub;
proportionNuggetBySeed[[i]] <- estimatedProportionNugget;
}
bestSeed <- seeds[which.max(logLikBySeed)];
bestRho <- rhoBySeed[which.max(logLikBySeed)];
bestProportionNugget <- proportionNuggetBySeed[which.max(logLikBySeed)];
#######################################################################
## Find the best tuning parameter;
f <- function(lambda) {
set.seed(bestSeed);
print(paste("Trying lambda=",round(lambda,5)));
thisFit <- MixTVEMFitInner(convergenceCriterion=convergenceCriterion,
intId=intId,
maxIterations=maxIterations,
maxVarianceRatio=maxVarianceRatio,
numClasses=numClasses,
penaltyMatrix=penaltyMatrix,
proportionNugget=bestProportionNugget,
referenceClass=referenceClass,
roughnessPenalty=lambda,
rho=bestRho,
S=S,
time=time,
X=designMatrix,
Y=dep);
return(thisFit$bic);
}
# Find the right order of magnitude;
lambdasSearch1 <- 10^(seq(-3,floor(log10(numTotal))))*sd(dep);
resultsSearch1 <- sapply(lambdasSearch1, f);
bestLambda1 <- lambdasSearch1[which.min(resultsSearch1)];
#print(cbind(lambdasSearch1,resultsSearch1));
# Find the right approximate value;
lambdasSearch2 <- (10^seq(-0.6,0.6,length=5))*bestLambda1;
resultsSearch2 <- sapply(lambdasSearch2, f);
bestLambda2 <- lambdasSearch2[which.min(resultsSearch2)];
#print(cbind(lambdasSearch1,resultsSearch1));
# Zoom in closer;
lambdasSearch3 <- (10^seq(-0.2,0.2,length=5))*bestLambda2;
resultsSearch3 <- sapply(lambdasSearch3, f);
bestLambda3 <- lambdasSearch3[which.min(resultsSearch3)];
#print(cbind(lambdasSearch1,resultsSearch1));
# Good enough;
lambda <- bestLambda3;
## Finally do the analysis;
set.seed(bestSeed);
bestFit <- MixTVEMFitInner(convergenceCriterion=convergenceCriterion,
intId=intId,
maxIterations=maxIterations,
maxVarianceRatio=maxVarianceRatio,
numClasses=numClasses,
penaltyMatrix=penaltyMatrix,
proportionNugget=bestProportionNugget,
referenceClass=referenceClass,
rho=bestRho,
roughnessPenalty=lambda,
S=S,
time=time,
X=designMatrix,
Y=dep);
## Now record the fitted values on the original assessment times;
fitted <- designMatrix%*%bestFit$theta;
fittedCoefficients <- list();
for (j in 1:numTCov) {