The within-mouth covariance matrix with conditional autoregressive structure

Description

The covariance matrix of individual teeth measures for each subject follows a Conditional Autoregressive model (CAR) density

Usage

CAR_cov_teeth(m, rho, tau)

Arguments

Details

CAR_cov_teeth gives the covariance matrix among the teeth within each mouth based on the CAR structure (Besag et al., 1991), given the maximum number of teeth for each subject (m), the variance (\tau), and the association (\rho) parameters.

The CAR covariance matrix can be expressed as \Sigma_{28\times 28} = \tau^2 (W - \rho D)^{-1}, where \tau^2 > 0, and \rho \in [0, 1] are the parameters that control the magnitude of variation and the degree of spatial association, respectively. For matrix D, the element D_{tt'} is 1 if locations t and t' are adjacent and 0 otherwise. The matrix W is diagonal with diagonal elements W_{tt} = \sum_{t'} D_{tt'}. Note, the argument \tau in CAR_cov_teeth is the variance, and not the standard deviation.

Value

The covariance matrix among the teeth in each mouth (assuming full dentition, i.e., 28 teeth) based on a CAR model.

Author(s)

Jing Xu, Dipankar Bandyopadhyay, Douglas Azevedo, Bibhas Chakraborty

References

Besag, J., York, J. & Mollie, A. (1991), "Bayesian image restoration, with two applications in spatial statistics (With Discussion)", Annals of the Institute of Statistical Mathematics 43, 159.

Reich, B. & Bandyopadhyay, D. (2010), "A latent factor model for spatial data with informative missingness", The Annals of Applied Statistics 4, 439–459.

See Also

MC_var_yibar_mis, SampleSize_SMARTp

Examples

m <- 28
rho <- 0.975
tau <- 0.85
Sigma <- CAR_cov_teeth(m = m, rho = rho, tau = tau)

Estimated mean and variance of the average change in CAL for each subject

Description

The estimated Monte Carlo mean and variance of the average change in clinical attachment level (CAL) for each subject

Usage

MC_var_yibar_mis(mu, Sigma, sigma1, lambda, nu, sigma0, Num, a0, b0, cutoff)

Arguments

Details

MC_var_yibar_mis computes the Monte-Carlo estimates of expectation and variance of the sample mean among the teeth within each mouth, i.e \bar{Y}_i = \sum Y_{it}(1 - M_{it})/\sum(1 - M_{it}), where Y_{it} is the change in CAL (measured in mm) for patient i and tooth t, and M_{it} is the misingness indicator, i.e., M_{it} = 1 implies tooth t in subject i is mising. The joint regression models for Y_{it} and M_{it} are available in Reich & Bandyopadhyay (2010, Annals of Applied Statistics).

Value

The simulated dataset of CAL change "Y_{it}", missingness "M_{it}" and function inside the indicator of "M_{it} I_{it}" for each tooth of each patient, with the corresponding estimated mean "mY_i", variance "VarY_i" and missing proportion "PM" for each patient

Author(s)

Jing Xu, Dipankar Bandyopadhyay, Douglas Azevedo, Bibhas Chakraborty

References

Besag, J., York, J. & Mollie, A. (1991), "Bayesian image restoration, with two applications in spatial statistics (With Discussion)", Annals of the Institute of Statistical Mathematics 43, 159.

Reich, B. & Bandyopadhyay, D. (2010), "A latent factor model for spatial data with informative missingness", The Annals of Applied Statistics 4, 439–459.

See Also

CAR_cov_teeth, SampleSize_SMARTp

Examples

m <- 28
Num <- 1000
cutoff <- 0
sigma1 <- 0.95
sigma0 <- 1
lambda <- 0
nu <- Inf
b0 <- 0.5
a0 <- -1.0
rho <- 0.975
tau <- 0.85
del1 <- 0.5
del2 <- 2

Sigma <- CAR_cov_teeth(m, rho, tau)
Sigma_comp <- array(Sigma, c(m, m, 4))
Sigma_sim <- array(Sigma, c(m, m, 10))

mu_comp <- array(0, c(2, m, 2))
mu_comp[, , 1] <- rbind(rep(0, m), rep(del1, m))
mu_comp[, , 2] <- rbind(rep(0, m), rep(del2, m))

VarYitd1R = MC_var_yibar_mis(mu = mu_comp[1, , 1], Sigma = Sigma,
                             sigma1 = sigma1,
                             lambda = lambda, nu = nu,
                             sigma0 = sigma0, Num = Num, a0 = a0, b0 = b0,
                             cutoff = cutoff)
PM <- VarYitd1R$PM
VarYid1R <- VarYitd1R$VarYi
mYid1R <- VarYitd1R$mYi
VarYitd1NR <- MC_var_yibar_mis(mu = mu_comp[2, , 1], Sigma = Sigma,
                               sigma1 = sigma1,
                               lambda = lambda, nu = nu,
sigma0 = sigma0, Num = Num, a0 = a0, b0 = b0, cutoff = cutoff)

PM <- VarYitd1NR$PM
VarYid1NR <- VarYitd1NR$VarYi
mYid1NR <- VarYitd1NR$mYi
VarYitd3R <- MC_var_yibar_mis(mu = mu_comp[1, , 2], Sigma = Sigma,
                              sigma1 = sigma1,
                              lambda = lambda, nu = nu,
                              sigma0 = sigma0, Num = Num, a0 = a0, b0 = b0,
                              cutoff = cutoff)

PM <- VarYitd3R$PM
VarYid3R <- VarYitd3R$VarYi
mYid3R <- VarYitd3R$mYi
VarYitd3NR <- MC_var_yibar_mis(mu = mu_comp[2,,2], Sigma = Sigma,
                               sigma1 = sigma1,
                               lambda = lambda, nu = nu,
sigma0 = sigma0, Num = Num, a0 = a0, b0 = b0, cutoff = cutoff)

PM <- VarYitd3NR$PM
VarYid3NR <- VarYitd3NR$VarYi
mYid3NR <- VarYitd3NR$mYi

An object of "SMARTp" class

Description

An object of "SMARTp" class

Slots

N

The estimated sample size

sig.dd

N*the variance or covariance matrix of the estimated regime means correspond to "regime"

sig.e.sq

N*the variance or covariance matrix of the difference between first and rest of estimated regime means correspond to regime, sig.e.sq = sig.dd if the element number of regime is one

Del

Effect size

Del_std

Standardized effect size

ybar

The estimated regime means corresponding to "regime"

initr

column matrix with dimension = the number of treatment paths, the elements are the corresponding row number of st1

ga

The response rates of initial treatments corresponding to each treatment path

res

A vector with binary indicators represent responders, or non-responders corresponding to a treatment path

p_st1

The randomization probability of stage-1 for each treatment path

p_st2

The randomization probability of stage-2 for each treatment path

Sigma

The CAR covariance matrix of the latent Q_{it}

Sample size calculation under a clustered SMART design for non-surgical treatment of chronic periodontitis

Description

Sample size calculations to detect desired DTR effects, which includes (i) a single regime, (ii) difference between two regimes, and (iii) a specific regime is the best, based on CAL changes under the proposed clustered, two-stage, SMART trial given type I and type II error rates

Usage

SampleSize_SMARTp(mu, st1, dtr, regime, pow, a, rho, tau, sigma1,
                  lambda, nu, sigma0, Num, p_i, c_i, a0, b0, cutoff)

Arguments

Details

SampleSize_SMARTp computes the sample size required to detect the dynamic treatment regime (DTR) (Murphy, 2005, Statistics in Medicine) effects in a study comparing non-surgical treatments of chronic periodontitis, via the sequential multiple assignment randomized trial (SMART) design, with two-stages.

Outcome measures (i.e. change in CAL) are continuous and clustered (i.e. tooth within a subject’s mouth, where each subject/mouth is a cluster) with non-random missingness captured via a shared parameter setting, specified in Reich and Bandyopadhyay (2010, Annals of Applied Statistics). Each cluster sub-unit has a binary missingness indicator, which is associated to its corresponding change of CAL through a joint model. The covariance structure within a cluster is captured by the conditionally autoregressive (CAR) structure (Besag et al, 1991).

The DTR effect can be detected based on either a single treatment regime, or the difference between two treatment regimes (with or without sharing initial treatments), or when one regime is considered the best among others. The mean and variance of the CAL change for each DTR can be estimated by the inverse probability weighting method via method of moments.

Note that the first three inputs "mu", "st1" and "dtr" define the SMART design in term of matrices. From Xu et al. (2019+, Under Review), stage-1 includes two treatments, e.g., treatments "3" and "8". Participants who respond to the stage-1 treatment will receive same treatment at stage-2, while non-responders will be randomly allocated to other treatments, i.e. non-responders who received treatment "3" at stage-1 will be randomly allocated to treatments "4"-"7" at stage-2, while non-responders receiving treatment "8" at stage-1 will be randomly allocated to treatments "4"-"7" at stage-2.

There are 8 treatment regimes for this design. They are 1 (treatment "3" at stage-1 and treatment "3" at stage- 2 if responder, otherwise treatment "4"), 2 (treatment "3" at stage-1 and treatment "3" at stage-2 if responder, otherwise treatment "5"), 3 (treatment "3" at stage-1 and treatment "3" at stage-2 if responder, otherwise treatment "6"), 4 (treatment "3" at stage-1 and treatment "3" at stage-2 if responder, otherwise treatment "7"), 5 (treatment "8" at stage-1 and treatment "8" at stage-2 if responder, otherwise treatment "4"), 6 (treatment "8" at stage-1 and treatment "8" at stage-2 if responder, otherwise treatment "5"), 7 (treatment "8" at stage-1 and treatment "8" at stage-2 if responder, otherwise treatment "6") and 8 (treatment "8" at stage-1 and treatment "8" at stage-2 if responder, otherwise treatment "7"). See Figure 2 in Xu et al. (2019+, Under Review)

Value

Author(s)

Jing Xu, Dipankar Bandyopadhyay, Douglas Azevedo, Bibhas Chakraborty

References

Besag, J., York, J. & Mollie, A. (1991) "Bayesian image restoration, with two applications in spatial statistics (with discussion)", Annals of the Institute of Statistical Mathematics 43, 159.

Murphy, S. A. (2005), "An experimental design for the development of adaptive treatment strategies", Statistics in Medicine 24, 1455–1481.

Reich, B. & Bandyopadhyay, D. (2010), A latent factor model for spatial data with informative missingness, The Annals of Applied Statistics 4, 439–459.

Xu, J., Bandyopadhyay, D., Mirzaei, S., Michalowicz, B and Bibhas Chakraborty. (2019+), "SMARTp: A SMART design for non-surgical treatments of chronic periodontitis with spatially-referenced and non-randomly missing skewed outcomes", Under Review

See Also

CAR_cov_teeth, MC_var_yibar_mis

Examples

m <- 28
pow <- 0.8
a <- 0.05
Num <- 1000
cutoff <- 0
sigma1 <- 0.95
sigma0 <- 1
lambda <- 0
nu <- Inf
b0 <- 0.5
a0 <- -1.0
rho <- 0.975
tau <- 0.85

Sigma <- CAR_cov_teeth(m = m, rho = rho, tau = tau)
p_i <- SMARTp:::pifun(cutoff = cutoff, a0 = a0, b0 = b0,
                     Sigma = Sigma, sigma0 = sigma0)
cit4 <- b0*diag(Sigma)/sqrt((diag(Sigma) +
            (sigma1^2 - 2/pi*sigma1^2*(0^2/(1+0^2))))*(b0^2*diag(Sigma) +
            sigma0^2))
c_i <- mean(cit4)

del1 <- 5
del2 <- 0
del3 <- 0

mu_sim <- matrix(0, 10, m)
mu_sim[2, ] <- rep(del1, m)
mu_sim[4, ] <- rep(del2, m)
mu_sim[7, ] <- rep(del3, m)

st1 <- cbind(c(1, 1), c(4, 4), c(0.25, 0.5), 1:2)

##-- Stage-1 information
dtr <- cbind(1:8, c(rep(1, 4), rep(6, 4)),
             c(2, 3, 4, 5, 7, 8, 9, 10), c(rep(1, 4), rep(2, 4)))

##-- Detecting a single regime, e.g., Regime 1
regime <- 1
SampleSize <- SampleSize_SMARTp(mu = mu_sim, st1 = st1, dtr = dtr,
                                regime = regime,
                                pow = pow, a = a, rho = rho,
                                tau = tau, sigma1 = sigma1, lambda = 0,
                                nu = Inf, sigma0 = sigma0, Num = Num,
                                p_i = p_i, c_i = c_i,
                                cutoff = cutoff)

N <- ceiling(SampleSize$N)
sig.e.sq <- SampleSize$sig.e.sq

sqrt(diag(sig.e.sq)/N)
SampleSize$Del_std
SampleSize$Del
SampleSize$sig.dd
sqrt(diag(SampleSize$sig.dd)/N)
SampleSize$ybar

##-- Now using a0 and b0
SampleSize_SMARTp(mu = mu_sim, st1 = st1, dtr = dtr, regime = regime,
                  pow = pow, a = a, rho = rho,
                  tau = tau, sigma1 = sigma1, lambda = 0, nu = Inf,
                  sigma0 = sigma0, Num = Num, a0 = a0, b0 = b0,
                  cutoff = cutoff)
SampleSize_SMARTp(mu = mu_sim, st1 = st1, dtr = dtr, regime = regime,
                  p_i = p_i, c_i = c_i)

##-- Detecting the difference between two regimes that shares initial treatment,
##-- e.g., Regimes 1 vs 3
regime <- c(1, 3)
SampleSize = SampleSize_SMARTp(mu = mu_sim, st1 = st1, dtr = dtr, regime = regime,
                               pow = pow, a = a, rho = rho,
                               tau = tau, sigma1 = sigma1, lambda = 0, nu = Inf,
                               sigma0 = sigma0, Num = Num, a0 = a0, b0 = b0,
                               cutoff = cutoff)

N <- ceiling(SampleSize$N)
sig.e.sq <- SampleSize$sig.e.sq

sqrt(diag(sig.e.sq)/N)
SampleSize$Del_std
SampleSize$Del
SampleSize$sig.dd

##-- Detecting the difference between two regimes that do not share initial treatment,
##-- e.g., Regimes 1 vs 5
regime <- c(1, 5)

SampleSize <- SampleSize_SMARTp(mu = mu_sim, st1 = st1, dtr = dtr, regime = regime,
                                pow = pow, a = a, rho = rho,
                                tau = tau, sigma1 = sigma1, lambda = 0, nu = Inf,
                                sigma0 = sigma0, Num = Num, a0 = a0, b0 = b0,
                                cutoff = cutoff)

N <- ceiling(SampleSize$N)
sig.e.sq <- SampleSize$sig.e.sq

sqrt(diag(sig.e.sq)/N)
SampleSize$Del_std
SampleSize$Del
SampleSize$sig.dd

##-- Detecting when Regime 1 is the best, e.g., comparing Regimes 1 vs 2, 3, 4, 5, 6, 7 and 8, i.e.
##-- the alternative hypothesis is \mu_{d1}>\mu_{d2} & \mu_{d1}>\mu_{d3} ... & \mu_{d1}>\mu_{d8}
##-- Note that this is a one-side test with Type-1 error rate of 0.025.
regime <- c(1, 2, 3, 4, 5, 6, 7, 8)
##-- To detect Regime 2 is the best, just use regime = c(2, 1, 3, 4, 5, 6, 7, 8), and so on
SampleSize <- SampleSize_SMARTp(mu = mu_sim, st1 = st1, dtr = dtr, regime = regime,
                                pow = pow, a = a, rho = rho,
                                tau = tau, sigma1 = sigma1, lambda = 0, nu = Inf,
                                sigma0 = sigma0, Num = Num, a0 = a0, b0 = b0,
                                cutoff = cutoff)

N <- ceiling(SampleSize$N)
sig.e.sq <- SampleSize$sig.e.sq

sqrt(diag(sig.e.sq)/N)
SampleSize$Del_std
SampleSize$Del
SampleSize$sig.dd

Print for SMARTp class

Description

Print for SMARTp class

Usage

## S3 method for class 'SMARTp'
print(x, ...)

Arguments

Summary for SMARTp class

Description

Summary for SMARTp class

Usage

## S3 method for class 'SMARTp'
summary(object, ...)

Arguments