| Title: | Mixed Effect Model with the Box-Cox Transformation |
|---|---|
| Description: | Inference on the marginal model of the mixed effect model with the Box-Cox transformation and on the model median differences between treatment groups for longitudinal randomized clinical trials. These statistical methods are proposed by Maruo et al. (2017) <doi:10.1002/sim.7279>. |
| Authors: | Kazushi Maruo [aut, cre], Ryota Ishii [aut], Yusuke Yamaguchi [ctb], Masahiko Gosho [ctb] |
| Maintainer: | Kazushi Maruo <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 0.1.5 |
| Built: | 2025-03-27 03:03:07 UTC |
| Source: | https://github.com/kzkzmr/bcmixed |
The data are from a randomized, double-blind study of acquired immune deficiency syndrome (AIDS) patients with advanced immune suppression (cluster of differentiation 4 (CD4) cells count of less than or equal to 50 cells/mm3) (Henry et al., 1998; Fitzmaurice et al., 2011). Patients in the AIDS Clinical Trial Group Study 193A were randomized to dual or triple combinations of human immunodeficiency virus-1 reverse transcriptase inhibitors. Specifically, patients were randomized to one of four daily regimens.
A data frame with 4708 observations on the following 7 variables:
idpatient identifier; in total there are 1177 patients.
weekcnominal visit variable (weeks 8, 16, 24, 32).
treatmentallocated treatment regimens; 1 = zidovudine alternating monthly with 400mg didanosine, 2 = zidovudine plus 2.25mg of zalcitabine, 3 = zidovudine plus 400mg of didanosine, and 4 = zidovudine plus 400mg of didanosine plus 400mg of nevirapine.
agepatients' age (years).
sexpatients' sex (1 = male, 0 = female)
cd4.blbaseline value of CD4 cells count + 1.
cd4CD4 cells count + 1.
https://www.hsph.harvard.edu/fitzmaur/ala/
Henry, K., Erice, A., Tierney, C., Balfour, H.H. Jr., Fischl, M.A., Kmack, A., Liou, S.H., Kenton, A., Hirsch, M.S., Phair, J., Martinez, A., Kahn, J.O., for the AIDS Clinical Trial Group 193A Study Team. (1998). A randomized, controlled, double-blind study comparing the survival benefit of four different reverse transcriptase inhibitor therapies (three-drug, two-drug, and alternating drug) for the treatment of advanced AIDS. Journal of Acquired Immune Deficiency Syndromes and Human Retrovirology, 19, 339-349, doi:10.1097/00042560-199812010-00004.
Fitzmaurice, G.M., Laird, N.M., and Ware, J.H. (2011). Applied Longitudinal Analysis 2nd ed., Wiley, New York, doi:10.1002/9781119513469.
data(aidscd4)data(aidscd4)
bcmarg returns the inference results the parameters of
the marginal model of the linear mixed effect model with the Box-Cox
transformation proposed by Maruo et al. (2017). If time and id are not
specified, inference results reduce to the results for the context of
linear regression model provided by Maruo et al. (2015).
bcmarg( formula, data, time = NULL, id = NULL, structure = "UN", lmdint = c(-3, 3) )bcmarg( formula, data, time = NULL, id = NULL, structure = "UN", lmdint = c(-3, 3) )
formula |
a two-sided linear formula object describing the model, with the response on the left of a ~ operator and the terms, separated by + operators, on the right. |
data |
a data frame containing the variables used in the model. |
time |
time variable name for repeated measurements. The default is NULL. |
id |
subject id variable name for repeated measurements. The default is NULL. |
structure |
specify the covariance structure from c("UN", "CS", "AR(1)"). The default is "UN". |
lmdint |
a vector containing the end-points of the interval to be searched for a transformation parameter. The default is c(-3, 3). |
an object of class "bcmarg". Objects of this class have
methods for the generic functions coef, logLik,
print, and summary. The object includes following components
for the marginal model parameter inference:
lambdaa numeric value of the estimate of the transformation parameter.
betaa vector with the estimates of the regression parameters.
alphaa vector with the estimates of the covariance parameters.
Vvariance-covariance matrix for any subject with no missing values.
betainfa matrix containing the inference results for
beta under the assumption that lambda is known.
Note that standard errors might be underestimated
although statistical tests would be asymptotically valid.
Vtheta.modmodel-based variance-covariance matrix for MLE of
the vector of all parameters: c(lambda, beta, alpha).
Vtheta.robrobust variance-covariance matrix for MLE of the vector of all parameters.
logLika numeric value of the maximized likelihood.
adj.prma vector with parameters used for the empirical
small sample adjustment in bcmmrm:
c(number of subjects, number of completed subjects, number of
outcome observations, number of missing observations).
glsObjectan object of "gls" (or "lm" when
time and id are not specified) containing results of
gls (or lm) function on the transformed scale.
Maruo, K., Isogawa, N., Gosho, M. (2015). Inference of median
difference based on the Box-Cox model in randomized clinical trials.
Statistics in Medicine, 34, 1634-1644,
doi:10.1002/sim.6408.
Maruo, K., Yamaguchi, Y., Noma, H., Gosho, M. (2017). Interpretable inference on the mixed effect model with the Box-Cox transformation. Statistics in Medicine, 36, 2420-2434, doi:10.1002/sim.7279.
data(aidscd4) bcmarg(cd4 ~ as.factor(treatment) * as.factor(weekc) + age, data = aidscd4, time = weekc, id = id, structure = "AR(1)")data(aidscd4) bcmarg(cd4 ~ as.factor(treatment) * as.factor(weekc) + age, data = aidscd4, time = weekc, id = id, structure = "AR(1)")
bcmmrm provides inference results for the model median differences
between treatment groups proposed by Maruo et al. (2017), which focuses on
continuous and positive longitudinally observed outcomes and a situation
where the efficacy of some treatments is compared based on a randomized,
parallel group clinical trial. If time and id are not
specified, inference results reduce to the results for the context of
linear regression model provided by Maruo et al. (2015).
bcmmrm( outcome, group, data, time = NULL, id = NULL, covv = NULL, cfactor = NULL, structure = "UN", conf.level = 0.95, lmdint = c(-3, 3), glabel = NULL, tlabel = NULL )bcmmrm( outcome, group, data, time = NULL, id = NULL, covv = NULL, cfactor = NULL, structure = "UN", conf.level = 0.95, lmdint = c(-3, 3), glabel = NULL, tlabel = NULL )
outcome |
a name of positive outcome (dependent) variable included in
|
group |
a name of treatment group variable included in |
data |
a data frame that may include |
time |
a name of time variable for repeated measurements included
in |
id |
a name of subject id variable for repeated measurements included
in |
covv |
a character vector with names of covariate variables included
in |
cfactor |
an integer vector including nominal variable indicators for
covariate variables. Nominal variable: |
structure |
specify the covariance structure from |
conf.level |
a numeric value of the confidence level for the confidence intervals. The default is 0.95. |
lmdint |
a vector containing the end-points of the interval to be
searched for a transformation parameter. The default is |
glabel |
a vector of length number of treatment groups containing
the labels of |
tlabel |
a vector of length number of repeated measures containing
the labels of |
an object of class "bcmmrm" representing the results of model
median inference based on the Box-Cox transformed MMRM model.
Generic functions such as print, plot, and summary
have methods to show the results of the fit. See bcmmrmObject
for the components of the fit.
If baseline observation for the outcome variable is available, Box-Cox
transformed baseline should be included as a covariate for accuracy of
estimation.
Although this function can be applied to non-randomized
trial data, performances of the above approach have not evaluated in
context of non-randomized trials.
Maruo, K., Isogawa, N., Gosho, M. (2015). Inference of median
difference based on the Box-Cox model in randomized clinical trials.
Statistics in Medicine, 34, 1634-1644,
doi:10.1002/sim.6408.
Maruo, K., Yamaguchi, Y., Noma, H., Gosho, M. (2017). Interpretable inference on the mixed effect model with the Box-Cox transformation. Statistics in Medicine, 36, 2420-2434, doi:10.1002/sim.7279.
data(aidscd4) # covariate: Box-Cox transformed baseline (continuous) and sex (nominal), # covariance structure: AR(1) structure # *Note: The UN structure is preferred although the AR(1) # structure is used in this example to reduce calculation time # Box-Cox transformation for the baseline aidscd4$cd4.bl.tr <- bct.v(aidscd4$cd4.bl)$transformed # Median inference for each group and week bcmmrm(outcome = cd4, group = treatment, data = aidscd4, time = weekc, id = id, covv = c("cd4.bl.tr", "sex"), cfactor = c(0, 1), structure = "AR(1)", glabel = c("Zid/Did", "Zid+Zal", "Zid+Did", "Zid+Did+Nev"))data(aidscd4) # covariate: Box-Cox transformed baseline (continuous) and sex (nominal), # covariance structure: AR(1) structure # *Note: The UN structure is preferred although the AR(1) # structure is used in this example to reduce calculation time # Box-Cox transformation for the baseline aidscd4$cd4.bl.tr <- bct.v(aidscd4$cd4.bl)$transformed # Median inference for each group and week bcmmrm(outcome = cd4, group = treatment, data = aidscd4, time = weekc, id = id, covv = c("cd4.bl.tr", "sex"), cfactor = c(0, 1), structure = "AR(1)", glabel = c("Zid/Did", "Zid+Zal", "Zid+Did", "Zid+Did+Nev"))
An object returned by the bcmmrm function, inheriting from
class "bcmmrm" and representing the Box-Cox transformed MMRM analysis.
Objects of this class have methods for the generic functions boxplot,
coef, logLik, plot, print, and summary.
The following components must be included in a legitimate "bcmmrm" object.
calla list containing an image of the bcmmrm
call that produced the object.
median.mod, median.rob, median.mod.adj,
median.rob.adj
lists including inference results for the
model medians for all groups. Levels of the list are time points,
where correspondence table is given as time.tbl$code.
mod: model-based inference, rob: robust inference,
adj: with empirical small sample adjustment.
meddif.mod, meddif.rob, meddif.mod.adj,
meddif.rob.adj
lists including inference results for the
for the model median differences between all pairs of groups
(group1 - group0). Levels of the list are time points,
where correspondence table is given as time.tbl$code.
mod: model-based inference, rob: robust inference,
adj: with empirical small sample adjustment.
lambdaa numeric value of estimates of the transformation parameter.
Ra correlation matrix for transformed outcomes.
betainfinference results for beta under the assumption that lambda is known. Note that standard errors might be underestimated although statistical tests would be asymptotically valid.
time.tbla data frame of a correspondence table for the timepoints.
group.tbla data frame of a correspondence table for treatment groups.
inf.marga list with results of bcmarg
function.
outdataa data frame where the transformed outcome
(ytr), the fitted values on the transformed scale
(ytr.fitted), and the residuals on the transformed scale
(res.tr) are added to input data.
conf.levela numeric value of the specified confidence level.
bcreg returns the maximum likelihood estimates for parameters of
the linear regression models with the Box-Cox transformation (Box and Cox,
1964).
bcreg(formula, data, lmdint = c(-3, 3))bcreg(formula, data, lmdint = c(-3, 3))
formula |
a two-sided linear formula object describing the model, with the response on the left of a ~ operator and the terms, separated by + operators, on the right. |
data |
a data frame in which to interpret the variables named in the formula. |
lmdint |
a vector containing the end-points of the interval to be searched for a transformation parameter. Default is c(-3, 3). |
bcreg returns a list including following components:
lambdaa numeric value with the estimate of the transformation parameter.
betaa vector with the estimates of the regression parameters.
sigmaa numeric value with the estimate of the scale parameter.
betainfa data frame with inference results for beta
under the assumption that lambda is known.
lika numeric value with the maximized likelihood.
lmObjectan object of "lm" containing the results of
lm function on the transformed scale
Box, G.E.P. and Cox, D.R. (1964). An analysis of transformations (with discussion). Journals of the Royal Statistical Society, Series B, 26, 211-246, doi:10.1111/j.2517-6161.1964.tb00553.x.
data(aidscd4) #Transformation of baseline observation for aid.cd4 data bcreg(cd4.bl ~ 1, aidscd4[aidscd4$weekc == 8, ])data(aidscd4) #Transformation of baseline observation for aid.cd4 data bcreg(cd4.bl ~ 1, aidscd4[aidscd4$weekc == 8, ])
bct returns the Box-Cox transformed numeric vector (Box and Cox,
1964).
bct(y, lambda)bct(y, lambda)
y |
a positive real number vector. |
lambda |
a scalar transformation parameter. |
bct returns the Box-Cox transformed numeric vector,
z = log(y) for lambda = 0,
z = (y ^ lambda - 1) / lambda for lambda ne 1.
Box, G.E.P. and Cox, D.R. (1964). An analysis of transformations (with discussion). Journals of the Royal Statistical Society, Series B, 26, 211-246, doi:10.1111/j.2517-6161.1964.tb00553.x.
y <- exp(rnorm(10)) z <- bct(y, 0) #log transformationy <- exp(rnorm(10)) z <- bct(y, 0) #log transformation
bct.v returns the Box-Cox transformed numeric vector (Box and Cox,
1964).
bct.v(y, lmdint = c(-3, 3))bct.v(y, lmdint = c(-3, 3))
y |
a positive real number vector. |
lmdint |
a vector containing the end-points of the interval to be searched for a transformation parameter. Default is c(-3, 3). |
bct.v returns the Box-Cox transformed numeric vector and
estimated transformation parameter.
transformedThe Box-Cox transformed numeric vector.
lambdaa numeric value of the estimate of the transformation parameter.
Box, G.E.P. and Cox, D.R. (1964). An analysis of transformations (with discussion). Journals of the Royal Statistical Society, Series B, 26, 211-246, doi:10.1111/j.2517-6161.1964.tb00553.x.
y <- exp(rnorm(50)) bct.v(y)y <- exp(rnorm(50)) bct.v(y)
Box-whisker plot for the transformed residuals of each treatment groups
at a specified time point with error bar plot (mean +- SD) using
bcmmrmObject.
## S3 method for class 'bcmmrm' boxplot( x, timepoint = NULL, xlab = NULL, ylab = NULL, main = TRUE, sub = NULL, verbose = FALSE, ... )## S3 method for class 'bcmmrm' boxplot( x, timepoint = NULL, xlab = NULL, ylab = NULL, main = TRUE, sub = NULL, verbose = FALSE, ... )
x |
an object inheriting from class " |
timepoint |
an numeric value of a specified level of |
xlab |
a title for the x axis. The default is |
ylab |
a title for the y axis. The default is |
main |
a main title for the plot. The default is |
sub |
a sub title for the plot. The default is |
verbose |
a logical optional value specifying to print the detailed
plot information in the console. The default is |
... |
some methods for this generic require additional arguments. |
a box-whisker plot for transformed residual.
data(aidscd4) lmd.bl <- bcreg(cd4.bl ~ 1, data = aidscd4[aidscd4$weekc == 8, ])$lambda aidscd4$cd4.bl.tr <- (aidscd4$cd4.bl ^ lmd.bl - 1) / lmd.bl resar <- bcmmrm(outcome = cd4, group = treatment, data = aidscd4, time = weekc, id = id, covv = c("cd4.bl.tr", "sex"), cfactor = c(0, 1), structure = "AR(1)", glabel = c("Zid/Did", "Zid+Zal", "Zid+Did", "Zid+Did+Nev")) boxplot(resar, xlab = "Treatment", col = 1:4)data(aidscd4) lmd.bl <- bcreg(cd4.bl ~ 1, data = aidscd4[aidscd4$weekc == 8, ])$lambda aidscd4$cd4.bl.tr <- (aidscd4$cd4.bl ^ lmd.bl - 1) / lmd.bl resar <- bcmmrm(outcome = cd4, group = treatment, data = aidscd4, time = weekc, id = id, covv = c("cd4.bl.tr", "sex"), cfactor = c(0, 1), structure = "AR(1)", glabel = c("Zid/Did", "Zid+Zal", "Zid+Did", "Zid+Did+Nev")) boxplot(resar, xlab = "Treatment", col = 1:4)
Plot for the model medians of each treatment groups with the 95 percent
confidence intervals stored in bcmmrmObject.
## S3 method for class 'bcmmrm' plot( x, robust = TRUE, ssadjust = TRUE, dt = 1, timepoint = NULL, tnom = TRUE, xlab = NULL, ylab = NULL, xlim = NULL, ylim = NULL, lwd = 2, col = NULL, lty = NULL, main = TRUE, sub = NULL, legend = TRUE, loc = "topright", verbose = FALSE, ... )## S3 method for class 'bcmmrm' plot( x, robust = TRUE, ssadjust = TRUE, dt = 1, timepoint = NULL, tnom = TRUE, xlab = NULL, ylab = NULL, xlim = NULL, ylim = NULL, lwd = 2, col = NULL, lty = NULL, main = TRUE, sub = NULL, legend = TRUE, loc = "topright", verbose = FALSE, ... )
x |
an object inheriting from class " |
robust |
an optional logical value used to specify whether to apply
the robust inference. The default is |
ssadjust |
an optional logical value used to specify whether to apply
the empirical small sample adjustment. The default is |
dt |
an numeric value indicating shift length between groups
in the longitudinal median plot. A multiplying factor for the default
settings specified (e.g. if |
timepoint |
an numeric value of a specified level of |
tnom |
a optional logical value indicating the scale of x axis
of the longitudinal median plot. When |
xlab |
a title for the x axis. The default is |
ylab |
a title for the y axis. The default is |
xlim |
a numeric vector with length of 2 indicating limits of x
axis. The default is |
ylim |
a numeric vector with length of 2 indicating limits of y
axis. The default is |
lwd |
an optional positive numeric value indicating line width.
The default is |
col |
an integer or a character vector with length of the number of
groups indicating colors of lines for each treatment group. The default is
|
lty |
an optional integer or a character vector with length of the
number of groups indicating line types of lines for each treatment group.
The default is |
main |
a main title for the plot. The default is |
sub |
a sub title for the plot. The default is |
legend |
a logical optional value specifying to add legends to
plots. When |
loc |
a character value indicating the location of the legends.
The location can be specified by setting |
verbose |
a logical optional value specifying to print the detailed
analysis information in the console. The default is |
... |
some methods for this generic require additional arguments. |
a median plot.
data(aidscd4) resar <- bcmmrm(outcome = cd4, group = treatment, data = aidscd4, time = weekc, id = id, structure = "AR(1)", glabel = c("Zid/Did", "Zid+Zal", "Zid+Did", "Zid+Did+Nev")) plot(resar, xlab = "Week", ylab = "CD4+1", col = 1:4, main = NULL) plot(resar, timepoint = 32, xlab = "Treatment", ylab = "CD4+1")data(aidscd4) resar <- bcmmrm(outcome = cd4, group = treatment, data = aidscd4, time = weekc, id = id, structure = "AR(1)", glabel = c("Zid/Did", "Zid+Zal", "Zid+Did", "Zid+Did+Nev")) plot(resar, xlab = "Week", ylab = "CD4+1", col = 1:4, main = NULL) plot(resar, timepoint = 32, xlab = "Treatment", ylab = "CD4+1")
Additional information about the Box-Cox transformed MMRM analysis
represented by object is extracted and included as components
of object.
## S3 method for class 'bcmmrm' summary(object, robust = TRUE, ssadjust = TRUE, ...)## S3 method for class 'bcmmrm' summary(object, robust = TRUE, ssadjust = TRUE, ...)
object |
an object inheriting from class " |
robust |
an optional logical value used to specify whether to apply
the robust inference. The default is |
ssadjust |
an optional logical value used to specify whether to apply
the empirical small sample adjustment. The default is |
... |
some methods for this generic require additional arguments. None are used in this method. |
an object inheriting from class summary.bcmmrm with all
components included in object (see glsObject for
a full description of the components) plus the following components:
mediana list including inference results of the model median
for specified values of robust and ssadjust.
meddifa list including inference results of the model median
difference for specified values of robust and
ssadjust.
robusta specified value of robust.
ssadjusta specified value of ssadjust.
data(aidscd4) resar <- bcmmrm(outcome = cd4, group = treatment, data = aidscd4, time = weekc, id = id, structure = "AR(1)", glabel = c("Zid/Did", "Zid+Zal", "Zid+Did", "Zid+Did+Nev")) summary(resar)data(aidscd4) resar <- bcmmrm(outcome = cd4, group = treatment, data = aidscd4, time = weekc, id = id, structure = "AR(1)", glabel = c("Zid/Did", "Zid+Zal", "Zid+Did", "Zid+Did+Nev")) summary(resar)