fitParaAR1: Performs the maximum likelihood estimation for the negative...
In lmeNB: Compute the Personalized Activity Index Based on a Negative Binomial Model
Description
Usage
Arguments
Details
Value
Author(s)
References
See Also
Examples
Description
This function fits a negative binomial mixed-effect AR(1) model in the formulation described Zhao et al. (2013).
The conditional distribution of response counts given random effect is modelled by Negative Binomial as described in description of lmeNB.
The conditional dependence among the response counts of a subject is modeled with AR(1) structure.
The random effects are modelled with either gamma or log-normal distributions.
See descriptions of lmeNB.
Usage
1
2
fitParaAR1(formula, data, ID, Vcode, p.ini = NULL, IPRT = FALSE,
RE = "G", i.tol = 1e-75, o.tol = 0.001)
Arguments
formula
See lmeNB.
data
See lmeNB.
ID
See lmeNB.
Vcode
See lmeNB.
p.ini
A vector of length 4 + # covariates, containing the initial values for the parameters
(log(α),
log(θ),
logit(δ),
β[0],
β[1]...).
NULL is accepted.
IPRT
See lmeNB.
RE
See fitParaIND.
i.tol
See lmeNB.
o.tol
See lmeNB.
Details
fitParaAR1 calls optim to minimize the negative log-likelihood of the negative binomial model with respect to the model parameters:
(log(α),
log(θ),
logit(δ),
β[0],
β[1]...).
The Nelder-Mead algorithm is employed.
The log-likelihood is obtained by marginalizing out the random effects.
The numerical integration is carried out using adaptive quadrature.
When missing visits are present, an approximation of the likelihood is used (see Zhao et al. (2013) for details.)
All the computations are done in R.
Value
opt
See lmeNB.
nlk
See lmeNB.
V
See lmeNB.
est
See lmeNB.
AR
TRUE
Author(s)
Zhao, Y. and Kondo, Y.
References
Detection of unusual increases in MRI lesion counts in individual multiple sclerosis patients. (2013) Zhao, Y., Li, D.K.B., Petkau, A.J., Riddehough, A., Traboulsee, A., Journal of the American Statistical Association.
See Also
The main function to fit the Negative Binomial mixed-effect model:
lmeNB,
The functions to fit the other models:
fitParaIND,
fitSemiIND,
fitSemiAR1,
The subroutines of index.batch to compute the conditional probability index:
jCP.ar1,
CP1.ar1,
MCCP.ar1,
CP.ar1.se,
CP.se,
jCP,
The functions to generate simulated datasets:
rNBME.R.
Examples
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## Not run:
## ==================================================================================
## generate a data based on the negative binomial mixed-effect AR(1) model.
## Under this model, the response counts follows the negative binomial:
## Y_ij | G_i = g_i ~ NB(r_ij,p_i) where r_ij = exp(X^T beta)/a , p_i =1/(a*g_i+1)
## with G_i ~ Gamma(scale=th,shape=1/th)
##
## The adjacent repeated measures of the same subject are correlated
## with correlation structure:
## cov(Y_ij,Y_ij'|G_i=g_i)=d^{j-j'} E(Y_ij')*(a*g_i^2+g_i)
loga <- -0.5
logtheta<- 1.3
logitd <- -0.2
b0 <- 0.5 ## no covariates;
## 80 subjects each with 5 scans
n <- 80
sn <- 5
set.seed(1)
DT2 <- rNBME.R(gdist = "G",
n = n, ## the total number of subjectss
sn = sn,
th=exp(logtheta),
u1 = rep(exp(b0),sn),
u2 = rep(exp(b0),sn),
a = exp(loga),
d = exp(logitd)/(1+exp(logitd))
)
Vcode <- rep(-1:(sn-2),n) # scan number -1, 0, 1, 2, 3
ID <- DT2$id
new <- Vcode > 0
dt2 <- data.frame(CEL=DT2$y)
## ================================================================================
## 1) Fit the negative binomial mixed-effect AR(1) model
## where the random effects are from the gamma distribution
## This is the true model
re.gamma.ar1 <- fitParaAR1(formula=CEL~1,data=dt2,ID=ID,
Vcode=Vcode,
p.ini=c(loga,logtheta,logitd,b0),
## log(a), log(theta), logit(d), b0
RE="G",
IPRT=TRUE)
## compute the estimates of the conditional probabilities
## with sum of the new repeated measure as a summary statistics
## Note C=TRUE with i.tol=1E-3 options compute the index faster
## i.se=TRUE requires more time
Psum <- index.batch(olmeNB=re.gamma.ar1,data=dt2,ID=ID,Vcode=Vcode,
labelnp=new,qfun="sum", IPRT=TRUE,i.se=FALSE,C=TRUE,i.tol=1E-3)
## compute the estimates of the conditional probabilities
## with max of the new repeated measure as a summary statistics
Pmax <-index.batch(olmeNB=re.gamma.ar1,data=dt2,ID=ID,Vcode=Vcode,
labelnp=new,qfun="max", IPRT=TRUE,i.se=FALSE,C=TRUE,i.tol=1E-3)
## Which patient's estimated probabilities based on the sum and max
## statistics disagrees the most?
( IDBigDif <- which(rank(abs(Pmax$condProbSummary[,1]-Psum$condProbSummary[,1]))==80) )
## Show the patient's CEL counts
dt2$CEL[ID==IDBigDif]
## Show the estimated conditional probabilities based on the sum summary statistics
Psum$condProbSummary[IDBigDif,]
## Show the estimated conditional probabilities based on the max summary statistics
Pmax$condProbSummary[IDBigDif,]
## 2) Fit the negative binomial mixed-effect AR(1) model
## where random effects is from the log-normal distribution
re.logn.ar1 <- fitParaAR1(formula=CEL~1,data=dt2,ID=ID,
Vcode=Vcode,
p.ini=c(loga,logtheta,logitd,b0), ## log(a), log(theta), logit(d), b0
RE="N",IPRT=TRUE)
Psum <- index.batch(olmeNB=re.logn.ar1,data=dt2,ID=ID,Vcode=Vcode,
labelnp=new,qfun="sum", IPRT=TRUE,i.se=FALSE,C=TRUE,i.tol=1E-3)
re.logn.ar1$Psum <- Psum
## 3) Fit the negative binomial independent model
## where random effects are from the gamma distribution
re.gamma.ind <- fitParaIND(formula=CEL~1,data=dt2,ID=ID,
RE="G",
p.ini=c(loga,logtheta,b0),
IPRT=TRUE)
Psum <- index.batch(olmeNB=re.gamma.ind,data=dt2,ID=ID,
labelnp=new,qfun="sum", IPRT=TRUE,i.se=TRUE)
## 4) Fit the negative binomial independent model
## where random effects are from the lognormal distribution
re.logn.ind <- fitParaIND(formula=CEL~1,data=dt2,ID=ID,
RE="N",
p.ini=c(loga,logtheta,b0),
IPRT=TRUE)
Psum <- index.batch(olmeNB=re.logn.ind, data=dt2,ID=ID,
labelnp=new,qfun="sum", IPRT=TRUE,i.se=TRUE)
## 5) Fit the semi-parametric negative binomial AR(1) model
logvarG <- -0.5
re.semi.ar1 <- fitSemiAR1(formula=CEL~1,data=dt2,ID=ID,
p.ini=c(loga, logvarG, logitd,b0),Vcode=Vcode)
Psum <- index.batch(olmeNB=re.semi.ar1,data=dt2,ID=ID, Vcode=Vcode,
labelnp=new,qfun="sum", IPRT=TRUE,i.se=FALSE)
## 6) Fit the semi-parametric negative binomial independent model
re.semi.ind <- fitSemiIND(formula=CEL~1,data=dt2,ID=ID, p.ini=c(loga, logvarG, b0))
Psum <- index.batch(olmeNB=re.semi.ind,data=dt2,ID=ID,
labelnp=new, qfun="sum", IPRT=TRUE,i.se=FALSE)
## ======================================================================== ##
## == Which model performed the best in terms of the estimation of beta0 == ##
## ======================================================================== ##
getpoints <- function(y,estb0,sdb0=NULL,crit=qnorm(0.975))
{
points(estb0,y,col="blue",pch=16)
if (!is.null(sdb0))
{
points(c(estb0-crit*sdb0,estb0+crit*sdb0),rep(y,2),col="red",type="l")
}
}
ordermethod <- c("gamma.ar1","logn.ar1","gamma.ind","logn.ind","semi.ar1","semi.ind")
estb0s <- c(
re.gamma.ar1$est[4,1],
re.logn.ar1$est[4,1],
re.gamma.ind$est[3,1],
re.logn.ind$est[3,1],
re.semi.ar1$est[4],
re.semi.ind$est[3]
)
## The true beta0 is:
b0
c <- 1.1
plot(0,0,type="n",xlim=c(min(estb0s)-0.5,max(estb0s)*c),ylim=c(0,7),yaxt="n",
main="Simulated from the AR(1) model \n with random effect ~ gamma")
legend("topright",
legend=ordermethod)
abline(v=b0,lty=3)
## 1) gamma.ar1
sdb0 <- re.gamma.ar1$est[4,2]
getpoints(6,estb0s[1],sdb0)
## 2)logn.ar1
sdb0 <- re.logn.ar1$est[4,2]
getpoints(5,estb0s[2],sdb0)
## 3) gamma.ind
sdb0 <- re.gamma.ind$est[3,2]
getpoints(4,estb0s[3],sdb0)
## 4) logn.ind
sdb0 <- re.logn.ind$est[3,2]
getpoints(3,estb0s[4],sdb0)
## 5) semi.ar1
getpoints(2,estb0s[5])
## 6) semi.ind
getpoints(1,estb0s[6])
## End(Not run)
lmeNB documentation built on May 2, 2019, 3:34 p.m.
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Description Usage Arguments Details Value Author(s) References See Also Examples
Description
This function fits a negative binomial mixed-effect AR(1) model in the formulation described Zhao et al. (2013).
The conditional distribution of response counts given random effect is modelled by Negative Binomial as described in description of lmeNB.
The conditional dependence among the response counts of a subject is modeled with AR(1) structure.
The random effects are modelled with either gamma or log-normal distributions.
See descriptions of lmeNB.
Usage
1 2 | fitParaAR1(formula, data, ID, Vcode, p.ini = NULL, IPRT = FALSE,
RE = "G", i.tol = 1e-75, o.tol = 0.001)
|
Arguments
formula |
See |
data |
See |
ID |
See |
Vcode |
See |
p.ini |
A vector of length 4 + # covariates, containing the initial values for the parameters
(log(α),
log(θ),
logit(δ),
β[0],
β[1]...).
|
IPRT |
See |
RE |
See |
i.tol |
See |
o.tol |
See |
Details
fitParaAR1 calls optim to minimize the negative log-likelihood of the negative binomial model with respect to the model parameters:
(log(α),
log(θ),
logit(δ),
β[0],
β[1]...).
The Nelder-Mead algorithm is employed.
The log-likelihood is obtained by marginalizing out the random effects.
The numerical integration is carried out using adaptive quadrature.
When missing visits are present, an approximation of the likelihood is used (see Zhao et al. (2013) for details.)
All the computations are done in R.
Value
opt |
See |
nlk |
See |
V |
See |
est |
See |
AR |
|
Author(s)
Zhao, Y. and Kondo, Y.
References
Detection of unusual increases in MRI lesion counts in individual multiple sclerosis patients. (2013) Zhao, Y., Li, D.K.B., Petkau, A.J., Riddehough, A., Traboulsee, A., Journal of the American Statistical Association.
See Also
The main function to fit the Negative Binomial mixed-effect model:
lmeNB,
The functions to fit the other models:
fitParaIND,
fitSemiIND,
fitSemiAR1,
The subroutines of index.batch to compute the conditional probability index:
jCP.ar1,
CP1.ar1,
MCCP.ar1,
CP.ar1.se,
CP.se,
jCP,
The functions to generate simulated datasets:
rNBME.R.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 | ## Not run:
## ==================================================================================
## generate a data based on the negative binomial mixed-effect AR(1) model.
## Under this model, the response counts follows the negative binomial:
## Y_ij | G_i = g_i ~ NB(r_ij,p_i) where r_ij = exp(X^T beta)/a , p_i =1/(a*g_i+1)
## with G_i ~ Gamma(scale=th,shape=1/th)
##
## The adjacent repeated measures of the same subject are correlated
## with correlation structure:
## cov(Y_ij,Y_ij'|G_i=g_i)=d^{j-j'} E(Y_ij')*(a*g_i^2+g_i)
loga <- -0.5
logtheta<- 1.3
logitd <- -0.2
b0 <- 0.5 ## no covariates;
## 80 subjects each with 5 scans
n <- 80
sn <- 5
set.seed(1)
DT2 <- rNBME.R(gdist = "G",
n = n, ## the total number of subjectss
sn = sn,
th=exp(logtheta),
u1 = rep(exp(b0),sn),
u2 = rep(exp(b0),sn),
a = exp(loga),
d = exp(logitd)/(1+exp(logitd))
)
Vcode <- rep(-1:(sn-2),n) # scan number -1, 0, 1, 2, 3
ID <- DT2$id
new <- Vcode > 0
dt2 <- data.frame(CEL=DT2$y)
## ================================================================================
## 1) Fit the negative binomial mixed-effect AR(1) model
## where the random effects are from the gamma distribution
## This is the true model
re.gamma.ar1 <- fitParaAR1(formula=CEL~1,data=dt2,ID=ID,
Vcode=Vcode,
p.ini=c(loga,logtheta,logitd,b0),
## log(a), log(theta), logit(d), b0
RE="G",
IPRT=TRUE)
## compute the estimates of the conditional probabilities
## with sum of the new repeated measure as a summary statistics
## Note C=TRUE with i.tol=1E-3 options compute the index faster
## i.se=TRUE requires more time
Psum <- index.batch(olmeNB=re.gamma.ar1,data=dt2,ID=ID,Vcode=Vcode,
labelnp=new,qfun="sum", IPRT=TRUE,i.se=FALSE,C=TRUE,i.tol=1E-3)
## compute the estimates of the conditional probabilities
## with max of the new repeated measure as a summary statistics
Pmax <-index.batch(olmeNB=re.gamma.ar1,data=dt2,ID=ID,Vcode=Vcode,
labelnp=new,qfun="max", IPRT=TRUE,i.se=FALSE,C=TRUE,i.tol=1E-3)
## Which patient's estimated probabilities based on the sum and max
## statistics disagrees the most?
( IDBigDif <- which(rank(abs(Pmax$condProbSummary[,1]-Psum$condProbSummary[,1]))==80) )
## Show the patient's CEL counts
dt2$CEL[ID==IDBigDif]
## Show the estimated conditional probabilities based on the sum summary statistics
Psum$condProbSummary[IDBigDif,]
## Show the estimated conditional probabilities based on the max summary statistics
Pmax$condProbSummary[IDBigDif,]
## 2) Fit the negative binomial mixed-effect AR(1) model
## where random effects is from the log-normal distribution
re.logn.ar1 <- fitParaAR1(formula=CEL~1,data=dt2,ID=ID,
Vcode=Vcode,
p.ini=c(loga,logtheta,logitd,b0), ## log(a), log(theta), logit(d), b0
RE="N",IPRT=TRUE)
Psum <- index.batch(olmeNB=re.logn.ar1,data=dt2,ID=ID,Vcode=Vcode,
labelnp=new,qfun="sum", IPRT=TRUE,i.se=FALSE,C=TRUE,i.tol=1E-3)
re.logn.ar1$Psum <- Psum
## 3) Fit the negative binomial independent model
## where random effects are from the gamma distribution
re.gamma.ind <- fitParaIND(formula=CEL~1,data=dt2,ID=ID,
RE="G",
p.ini=c(loga,logtheta,b0),
IPRT=TRUE)
Psum <- index.batch(olmeNB=re.gamma.ind,data=dt2,ID=ID,
labelnp=new,qfun="sum", IPRT=TRUE,i.se=TRUE)
## 4) Fit the negative binomial independent model
## where random effects are from the lognormal distribution
re.logn.ind <- fitParaIND(formula=CEL~1,data=dt2,ID=ID,
RE="N",
p.ini=c(loga,logtheta,b0),
IPRT=TRUE)
Psum <- index.batch(olmeNB=re.logn.ind, data=dt2,ID=ID,
labelnp=new,qfun="sum", IPRT=TRUE,i.se=TRUE)
## 5) Fit the semi-parametric negative binomial AR(1) model
logvarG <- -0.5
re.semi.ar1 <- fitSemiAR1(formula=CEL~1,data=dt2,ID=ID,
p.ini=c(loga, logvarG, logitd,b0),Vcode=Vcode)
Psum <- index.batch(olmeNB=re.semi.ar1,data=dt2,ID=ID, Vcode=Vcode,
labelnp=new,qfun="sum", IPRT=TRUE,i.se=FALSE)
## 6) Fit the semi-parametric negative binomial independent model
re.semi.ind <- fitSemiIND(formula=CEL~1,data=dt2,ID=ID, p.ini=c(loga, logvarG, b0))
Psum <- index.batch(olmeNB=re.semi.ind,data=dt2,ID=ID,
labelnp=new, qfun="sum", IPRT=TRUE,i.se=FALSE)
## ======================================================================== ##
## == Which model performed the best in terms of the estimation of beta0 == ##
## ======================================================================== ##
getpoints <- function(y,estb0,sdb0=NULL,crit=qnorm(0.975))
{
points(estb0,y,col="blue",pch=16)
if (!is.null(sdb0))
{
points(c(estb0-crit*sdb0,estb0+crit*sdb0),rep(y,2),col="red",type="l")
}
}
ordermethod <- c("gamma.ar1","logn.ar1","gamma.ind","logn.ind","semi.ar1","semi.ind")
estb0s <- c(
re.gamma.ar1$est[4,1],
re.logn.ar1$est[4,1],
re.gamma.ind$est[3,1],
re.logn.ind$est[3,1],
re.semi.ar1$est[4],
re.semi.ind$est[3]
)
## The true beta0 is:
b0
c <- 1.1
plot(0,0,type="n",xlim=c(min(estb0s)-0.5,max(estb0s)*c),ylim=c(0,7),yaxt="n",
main="Simulated from the AR(1) model \n with random effect ~ gamma")
legend("topright",
legend=ordermethod)
abline(v=b0,lty=3)
## 1) gamma.ar1
sdb0 <- re.gamma.ar1$est[4,2]
getpoints(6,estb0s[1],sdb0)
## 2)logn.ar1
sdb0 <- re.logn.ar1$est[4,2]
getpoints(5,estb0s[2],sdb0)
## 3) gamma.ind
sdb0 <- re.gamma.ind$est[3,2]
getpoints(4,estb0s[3],sdb0)
## 4) logn.ind
sdb0 <- re.logn.ind$est[3,2]
getpoints(3,estb0s[4],sdb0)
## 5) semi.ar1
getpoints(2,estb0s[5])
## 6) semi.ind
getpoints(1,estb0s[6])
## End(Not run)
|
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