Package 'skewlmm' reference manual

Title:	Scale Mixture of Skew-Normal Linear Mixed Models
Description:	It fits scale mixture of skew-normal linear mixed models using either an expectation–maximization (EM) type algorithm or its accelerated version (Damped Anderson Acceleration with Epsilon Monotonicity, DAAREM), including some possibilities for modeling the within-subject dependence. Details can be found in Schumacher, Lachos and Matos (2021) <doi:10.1002/sim.8870>.
Authors:	Fernanda L. Schumacher [aut, cre] , Larissa A. Matos [aut] , Victor H. Lachos [aut] , Katherine A. L. Valeriano [aut] , Nicholas Henderson [ctb], Ravi Varadhan [ctb]
Maintainer:	Fernanda L. Schumacher <[email protected]>
License:	MIT + file LICENSE
Version:	1.1.3
Built:	2025-02-19 02:58:36 UTC
Source:	https://github.com/fernandalschumacher/skewlmm

Autocorrelation function for smn.lmm or smsn.lmm residuals

Description

This function calculates the empirical autocorrelation function for the within-subject residuals from a smn.lmm or smsn.lmm fit. The autocorrelation values are calculated using pairs of residuals within-subjects. The autocorrelation function is useful for investigating serial correlation models for discrete-time data, preferably equally spaced.

Usage

acfresid(object, maxLag, resLevel = "marginal", resType = "normalized",
    calcCI = FALSE, levelCI, MCiter, seed)
acfresid(object, maxLag, resLevel = "marginal", resType = "normalized",
    calcCI = FALSE, levelCI, MCiter, seed)

Arguments

`object`	An object inheriting from class `SMN` or `SMSN`, representing a fitted scale mixture of (skew) normal linear mixed model.
`maxLag`	An optional integer giving the maximum lag for which the autocorrelation should be calculated. Defaults to maximum lag in the within-subject residuals.
`resLevel`	`"marginal"` (default) or `"conditional"`. An optional character string specifying which residual should be used. For details see `residuals.SMN`.
`resType`	`"response"`, `"normalized"` (default), or `"modified"`. An optional character string specifying which type of residual should be used. For details see `residuals.SMN`.
`calcCI`	`TRUE` or `FALSE` (default). A logical value indicating if Monte Carlo confidence intervals should be computed for the conditionally independent model, which can be used for testing if the autocorrelations are zero.
`levelCI`	An optional numeric value in $(0,1)$ indicating the confidence level that should be used in the Monte Carlo confidence intervals. Default is `0.95`.
`MCiter`	An optional discrete value indicating the number of Monte Carlo samples that should be used to compute the confidence intervals. Default is `300`.
`seed`	An optional value used to specify seeds inside the function. Default is to use a random seed.

Value

A data frame with columns lag, ACF, and n.used representing, respectively, the lag between residuals within a pair, the corresponding empirical autocorrelation, and the number of pairs used. If calcCI=TRUE, the data frame has two extra columns containing the confidence intervals for the conditionally independent model. The returned value inherits from class acfresid.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

References

Pinheiro, J. C. & Bates, D. M. (2000). Mixed-Effects Models in S and S-PLUS. Springer, New York, NY.

Schumacher, F. L., Lachos, V. H., and Matos, L. A. (2021). Scale mixture of skew-normal linear mixed models with within-subject serial dependence. Statistics in Medicine 40(7), 1790-1810.

Examples

fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
acf1 = acfresid(fm1)
acf1
plot(acf1)

## computing simulated bands
acfCI = acfresid(fm1, calcCI=TRUE)
plot(acfCI)
  
fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
acf1 = acfresid(fm1)
acf1
plot(acf1)

## computing simulated bands
acfCI = acfresid(fm1, calcCI=TRUE)
plot(acfCI)

Extract confidence intervals from `lmmBoot` object

Description

It extracts confidence intervals from parametric bootstrap results obtained using the boot_par() function.

Usage

boot_ci(object, conf = 0.95)
boot_ci(object, conf = 0.95)

Arguments

`object`	An object containing the results of the `boot_par()` function.
`conf`	Confidence level to be considered.

Value

A matrix containing the confidence intervals.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

Examples


fm1 = smn.lmm(nlme::Orthodont, formFixed=distance ~ age+Sex, groupVar="Subject")
b1 = boot_par(fm1, B=100)
boot_ci(b1)

fm1 = smn.lmm(nlme::Orthodont, formFixed=distance ~ age+Sex, groupVar="Subject")
b1 = boot_par(fm1, B=100)
boot_ci(b1)

Parametric bootstrap for SMSN/SMN objects

Description

It generates and estimates B Monte Carlo samples identical to the fitted model.

Usage

boot_par(object, B = 100, seed = 123)
boot_par(object, B = 100, seed = 123)

Arguments

`object`	A smsn.lmm or smn.lmm object containing the fitted model to be updated.
`B`	Number of samples to be used.
`seed`	Seed to be used.

Details

This function provides an alternative for the asymptotic standard errors and confidence intervals given in summary, which may be helpful for small samples. Nevertheless, the computational cost is higher and it may take several minutes to get the results.

Value

A tibble of class lmmBoot with B lines, where each line contains the estimated parameters from a simulated sample.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

Examples


fm1 = smn.lmm(nlme::Orthodont, formFixed=distance ~ age+Sex, groupVar="Subject")
b1 = boot_par(fm1, B=100)
boot_ci(b1)

fm1 = smn.lmm(nlme::Orthodont, formFixed=distance ~ age+Sex, groupVar="Subject")
b1 = boot_par(fm1, B=100)
boot_ci(b1)

Extract coefficients from smsn.lmm, smn.lmm and smn.clmm objects

Description

It extracts estimated coefficients from smsn.lmm, smn.lmm and smn.clmm objects. The estimated coefficients are obtained by adding together the fixed and random effects estimates.

Usage

## S3 method for class 'SMN'
coef(object, ...)
## S3 method for class 'SMSN'
coef(object, ...)
## S3 method for class 'SMNclmm'
coef(object, ...)
## S3 method for class 'SMN'
coef(object, ...)
## S3 method for class 'SMSN'
coef(object, ...)
## S3 method for class 'SMNclmm'
coef(object, ...)

Arguments

`object`	An object inheriting from class `SMN`, `SMSN`, or `SMNclmm`, representing a fitted scale mixture skew-normal linear mixed model.
`...`	Additional arguments

Value

Matrix of coefficients.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

References

Schumacher, F. L., Lachos, V. H., and Matos, L. A. (2021). Scale mixture of skew-normal linear mixed models with within-subject serial dependence. Statistics in Medicine 40(7), 1790-1810.

Examples

fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
coef(fm1)
fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
coef(fm1)

Computes confidence intervals from smn.lmm and smsn.lmm fitted models

Description

Computes either asymptotical (based on normality from maximum likelihood estimators) or parametric bootstraped confidence intervals from a model fit.

Usage

## S3 method for class 'SMN'
confint(object, parm, level = 0.95, method, ...)
## S3 method for class 'SMSN'
confint(object, parm, level = 0.95, method, ...)
## S3 method for class 'SMN'
confint(object, parm, level = 0.95, method, ...)
## S3 method for class 'SMSN'
confint(object, parm, level = 0.95, method, ...)

Arguments

`object`	An object inheriting from class `SMN` or `SMSN`
`parm`	A character indicating for which parameter the intervals should be returned. Available options: "beta" for fixed effects, or "all" for all parameters. Default is "all".
`level`	Confidence level to be used.
`method`	A character indicating with method should be used. "asymptotic" refers to traditional confidence intervals based on asymptotical normality from maximum likelihood estimators; "bootstrap" performs a parametric bootstrap method based on B samples (100 by default), and is only recommended to small to moderate sample sizes.
`...`	Additional arguments to be passed to `boot_par`.

Value

A table containing the estimate and the respective confidence interval.

Examples

fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
confint(fm1)
fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
confint(fm1)

Extracts criteria for model comparison of SMSN/SMN/SMNclmm objects

Description

It extracts criteria for model comparison of sereval SMSN-LMM, SMN-LMM and/or SMN-CLMM (for censored responses).

Usage

criteria(lobjects)
criteria(lobjects)

Arguments

lobjects

A list containg the smsn.lmm, smn.lmm or smn.clmm objects to be compared.

Value

A data.frame containing for each model the maximum log-likelihood value, the number of parameters, the AIC, and the BIC.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

References

Schumacher, F. L., Lachos, V. H., and Matos, L. A. (2021). Scale mixture of skew-normal linear mixed models with within-subject serial dependence. Statistics in Medicine 40(7), 1790-1810.

Examples

fm_norm = smn.lmm(nlme::Orthodont, formFixed=distance~age+Sex, groupVar="Subject")
fm_t = update(fm_norm, distr="t")
fm_sn = smsn.lmm(nlme::Orthodont, formFixed=distance~age+Sex, groupVar="Subject")
fm_cens = smn.clmm(nlme::Orthodont, formFixed=distance~age+Sex, groupVar="Subject")
criteria(list(fm_norm=fm_norm, fm_t=fm_t, fm_sn=fm_sn, fm_cens=fm_cens))
fm_norm = smn.lmm(nlme::Orthodont, formFixed=distance~age+Sex, groupVar="Subject")
fm_t = update(fm_norm, distr="t")
fm_sn = smsn.lmm(nlme::Orthodont, formFixed=distance~age+Sex, groupVar="Subject")
fm_cens = smn.clmm(nlme::Orthodont, formFixed=distance~age+Sex, groupVar="Subject")
criteria(list(fm_norm=fm_norm, fm_t=fm_t, fm_sn=fm_sn, fm_cens=fm_cens))

Error scale matrix associated with `times`

Description

It returns a scale matrix associated with the error term at time times. Can be applied to a smn.lmm or smsn.lmm object or to a specific dependence structure with chosen parameter values.

Usage

errorVar(times, object = NULL, sigma2 = NULL, depStruct = NULL,
    phi = NULL)
errorVar(times, object = NULL, sigma2 = NULL, depStruct = NULL,
    phi = NULL)

Arguments

`times`	A vector containing the times for which the matrix should be calculated.
`object`	A smn.lmm or smsn.lmm object for which the variance should be extracted.
`sigma2`	Common variance parameter, such that $\Sigma = sigma2*R$ . Only evaluated if object `is.null(object)`.
`depStruct`	Dependence structure. `"UNC"` for conditionally uncorrelated (`"CI"` is also accepted), `"ARp"` for AR(p) – p is `length(phi)`–, `"CS"` for compound symmetry, `"DEC"` for DEC, and `"CAR1"` for continuous-time AR(1). Only evaluated if object `is.null(object)`.
`phi`	Parameter vector indexing the dependence structure. Only evaluated if object `is.null(object)`.

Value

Matrix of dimension length(times).

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

References

Schumacher, F. L., Lachos, V. H., and Matos, L. A. (2021). Scale mixture of skew-normal linear mixed models with within-subject serial dependence. Statistics in Medicine 40(7), 1790-1810.

Examples


fm1 = smsn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
errorVar(times=1:4,fm1)
#
errorVar(times=1:5, sigma2=1, depStruct="ARp", phi=.5)
errorVar(times=1:5, sigma2=1, depStruct="DEC", phi=c(.5,.8))
fm1 = smsn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
errorVar(times=1:4,fm1)
#
errorVar(times=1:5, sigma2=1, depStruct="ARp", phi=.5)
errorVar(times=1:5, sigma2=1, depStruct="DEC", phi=c(.5,.8))

Extract smn.lmm fitted values

Description

The fitted values are obtained by adding together the population fitted values (based only on the fixed effects estimates) and the estimated contributions of the random effects to the fitted values at grouping levels.

Usage

## S3 method for class 'SMN'
fitted(object, ...)
## S3 method for class 'SMN'
fitted(object, ...)

Arguments

`object`	An object inheriting from class `SMN`, representing a fitted scale mixture normal linear mixed model.
`...`	Additional arguments.

Value

Vector of fitted values with length equal to nrow(data).

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

Examples

fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
fitted(fm1)
fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
fitted(fm1)

Extract smn.clmm fitted values

Description

Usage

  ## S3 method for class 'SMNclmm'
fitted(object, ...)
## S3 method for class 'SMNclmm'
fitted(object, ...)

Arguments

`object`	An object inheriting from class `SMNclmm`, representing a fitted scale mixture normal linear mixed model with censored responses.
`...`	Additional arguments.

Value

Vector of fitted values with length equal to nrow(data).

Author(s)

Fernanda L. Schumacher, Larissa A. Matos, Victor H. Lachos and Katherine L. Valeriano

Examples

nj1 = 5; m = 30
time = rep(1:nj1, times=m)
groups = as.factor(rep(1:m, each=nj1))
dat1 = rsmsn.clmm(time, groups, cbind(1,time), rep(1,m*nj1), sigma2=0.7,
                  D=0.6*diag(1), beta=c(1,2), depStruct="UNC")

fm1 = smn.clmm(dat1, formFixed=y~x, groupVar="ind", depStruct="UNC",
               ci="ci", lcl="lcl", ucl="ucl", control=lmmControl(max.iter=30))
fitted(fm1)
nj1 = 5; m = 30
time = rep(1:nj1, times=m)
groups = as.factor(rep(1:m, each=nj1))
dat1 = rsmsn.clmm(time, groups, cbind(1,time), rep(1,m*nj1), sigma2=0.7,
                  D=0.6*diag(1), beta=c(1,2), depStruct="UNC")

fm1 = smn.clmm(dat1, formFixed=y~x, groupVar="ind", depStruct="UNC",
               ci="ci", lcl="lcl", ucl="ucl", control=lmmControl(max.iter=30))
fitted(fm1)

Extract smsn.lmm fitted values

Description

Usage

## S3 method for class 'SMSN'
fitted(object, ...)
## S3 method for class 'SMSN'
fitted(object, ...)

Arguments

`object`	An object inheriting from class `SMSN`, representing a fitted scale mixture skew-normal linear mixed model.
`...`	Additional arguments.

Value

Vector of fitted values with length equal to nrow(data).

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

Examples

fm1 = smsn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject",
               control=lmmControl(tol=.0001))
fitted(fm1)
fm1 = smsn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject",
               control=lmmControl(tol=.0001))
fitted(fm1)

Extract estimated fixed effects from smsn.lmm, smn.lmm and smn.clmm objects

Description

It extracts fixed effects from smsn.lmm and smn.lmm objects.

Usage

## S3 method for class 'SMN'
fixef(object, ...)
## S3 method for class 'SMSN'
fixef(object, ...)
## S3 method for class 'SMN'
fixef(object, ...)
## S3 method for class 'SMSN'
fixef(object, ...)

Arguments

`object`	An object inheriting from class `SMN`, `SMSN`, or `SMNCens`, representing a fitted scale mixture skew-normal linear mixed model.
`...`	Additional arguments

Value

Matrix of estimated random effects.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

References

Schumacher, F. L., Lachos, V. H., and Matos, L. A. (2021). Scale mixture of skew-normal linear mixed models with within-subject serial dependence. Statistics in Medicine 40(7), 1790-1810.

Examples

fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
fixef(fm1)
fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
fixef(fm1)

Formula from an smn.lmm and smsn.lmm models

Description

It returns the formula used for both fixed and random terms of the linear mixed model represented by object.

Usage

## S3 method for class 'SMN'
formula(x, ...)
## S3 method for class 'SMSN'
formula(x, ...)
## S3 method for class 'SMN'
formula(x, ...)
## S3 method for class 'SMSN'
formula(x, ...)

Arguments

`x`	An object inheriting from class `SMN` or `SMSN`
`...`	Additional arguments

Value

`formFixed`	Fixed effects formula
`formRandom`	Random effects formula
`groupVar`	Variable identified subjects or clusters

Examples

fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
formula(fm1)
fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
formula(fm1)

Healy-type plot from a smn.lmm or smsn.lmm object

Description

It creates a Healy-type plot from a smn.lmm or smsn.lmm object, for goodness-of-fit assessment.

Usage

healy.plot(object, dataPlus = NULL, dotsize = 0.4, calcCI = FALSE,
           levelCI, MCiter, seed, ...)
healy.plot(object, dataPlus = NULL, dotsize = 0.4, calcCI = FALSE,
           levelCI, MCiter, seed, ...)

Arguments

`object`	An object inheriting from class `SMN` or `SMSN`, representing a fitted scale mixture of (skew) normal linear mixed model.
`dataPlus`	Optional. Expanded dataset that should be used instead the one used for fitting. This is necessary for unbalanced datasets, since Haley's plot requires all subject to have the same number of observations.
`dotsize`	Optional. Dotsize used in ggplot.
`calcCI`	`TRUE` or `FALSE` (default). A logical value indicating if Monte Carlo confidence intervals should be computed for the conditionally independent model, which can be used for testing if the autocorrelations are zero.
`levelCI`	An optional numeric value in $(0,1)$ indicating the confidence level that should be used in the Monte Carlo confidence intervals. Default is `0.95`.
`MCiter`	An optional discrete value indicating the number of Monte Carlo samples that should be used to compute the confidence intervals. Default is `300`.
`seed`	An optional value used to specify seeds inside the function. Default is to use a random seed.
`...`	Additional arguments.

Details

It constructs a Healy-type plot (Healy, 1968) by plotting the nominal probability values $1/n,2/n,...,n/n$ against the theoretical cumulative probabilities of the ordered observed Mahalanobis distances. If the fitted model is appropriate, the plot should resemble a straight line through the origin with unit slope. If calcCI=TRUE, the plot presents two dashed blue lines containing approximated confidence intervals by considering that the fitted model is correct.

Value

A ggplot object.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

References

Healy, M. J. R. (1968). Multivariate normal plotting. Journal of the Royal Statistical Society: Series C (Applied Statistics), 17(2), 157-161.

Schumacher, F. L., Lachos, V. H., and Matos, L. A. (2021). Scale mixture of skew-normal linear mixed models with within-subject serial dependence. Statistics in Medicine 40(7), 1790-1810.

Examples

fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
healy.plot(fm1)

## computing simulated bands
healy.plot(fm1, calcCI=TRUE)

fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
healy.plot(fm1)

## computing simulated bands
healy.plot(fm1, calcCI=TRUE)

Control options for `smsn.lmm()`, `smn.lmm()` and `smn.clmm()`

Description

The values supplied in the function call replace the defaults and a list with all possible arguments is returned. The returned list has class "lmmControl" and is used as the control argument to the smsn.lmm(), smn.lmm() and smn.clmm() functions.

Usage

lmmControl(tol = 1e-06, max.iter = 300, calc.se = TRUE, lb = NULL,
           lu = NULL, luDEC = 10,
           initialValues = list(beta = NULL, sigma2 = NULL, D = NULL,
                                lambda = NULL, phi = NULL, nu = NULL),
           quiet = !interactive(), showCriterium = FALSE, algorithm = "DAAREM",
           parallelphi = NULL, parallelnu = NULL, ncores = NULL,
           control.daarem = list())
lmmControl(tol = 1e-06, max.iter = 300, calc.se = TRUE, lb = NULL,
           lu = NULL, luDEC = 10,
           initialValues = list(beta = NULL, sigma2 = NULL, D = NULL,
                                lambda = NULL, phi = NULL, nu = NULL),
           quiet = !interactive(), showCriterium = FALSE, algorithm = "DAAREM",
           parallelphi = NULL, parallelnu = NULL, ncores = NULL,
           control.daarem = list())

Arguments

`tol`	Tolerance for the convergence criterion. Default = `1e-6`.
`max.iter`	Maximum number of iterations for the EM algorithm. Default = `200`.
`calc.se`	A logical value indicating if standard errors should be calculated.
`lb`	Optional. Bottom limit for estimating `nu`.
`lu`	Optional. Upper limit for estimating `nu`.
`luDEC`	Optional. Upper limit for estimating the "damping" parameter for DEC covariance. If `luDEC<=1`, only attenuation of the exponential decay can be obtained.
`initialValues`	Optional. A named list containing initial parameter values, with at most the following elements: `beta`, `sigma2`, `D`, `lambda`, `phi`, `nu`.
`quiet`	A logical value indicating if the iteration message should be suppressed. Useful when calling the function in R Markdown.
`showCriterium`	A logical value indicating if the criterium should be shown at each iteration.
`algorithm`	Algorithm to be used for estimation, either `"DAAREM"` (default) or `"EM"`. DAAREM is an acceleration method and usually converges with fewer iterations, but it may result in numerical errors (in this case, please use the `"EM"` option).
`parallelphi`	A logical value indicating if parallel optimization should be used in the numerical update of the parameters related to the within-subject dependence structure. Default is `TRUE` if the data contains more than `30` subjects, and `FALSE` otherwise. Meaningless if `depStruct = "UNC"`.
`parallelnu`	A logical value indicating if parallel optimization should be used in the numerical update of nu. Meaningless if `distr="norm"` or `distr="sn"`.
`ncores`	Number of cores to be used for the parallel optimization. Meaningless if `parallelphi = FALSE` and `parallelnu = FALSE`.
`control.daarem`	List of control for the daarem algorithm. See `help(daarem, package = "daarem")` for details. Meaningless if `algorithm = "EM"`

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

References

Henderson, N.C. and Varadhan, R. (2019) Damped Anderson acceleration with restarts and monotonicity control for accelerating EM and EM-like algorithms, Journal of Computational and Graphical Statistics, Vol. 28(4), 834-846.

Schumacher, F. L., Lachos, V. H., and Matos, L. A. (2021). Scale mixture of skew-normal linear mixed models with within-subject serial dependence. Statistics in Medicine 40(7), 1790-1810.

Examples

lmmControl(algorithm = "EM")

fm1 = smn.lmm(nlme::Orthodont, formFixed=distance ~ age+Sex,
              groupVar="Subject", control=lmmControl(tol=1e-7))
lmmControl(algorithm = "EM")

fm1 = smn.lmm(nlme::Orthodont, formFixed=distance ~ age+Sex,
              groupVar="Subject", control=lmmControl(tol=1e-7))

Log-likelihood of an smn.lmm and smsn.lmm models

Description

It returns the log-likelihood value of the linear mixed model represented by object evaluated at the estimated coefficients.

Usage

## S3 method for class 'SMN'
logLik(object, ...)
## S3 method for class 'SMSN'
logLik(object, ...)
## S3 method for class 'SMN'
logLik(object, ...)
## S3 method for class 'SMSN'
logLik(object, ...)

Arguments

`object`	An object inheriting from class `SMN` or `SMSN`
`...`	Additional arguments.

Value

The log-likelihood value of the model represented by object evaluated at the estimated coefficients.

Examples

fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
logLik(fm1)
fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
logLik(fm1)

Likelihood-ratio test for SMSN/SMN objects

Description

It performs a likelihood-ratio test for two nested SMSN-LMM or SMN-LMM.

Usage

lr.test(obj1, obj2, level = 0.05)
lr.test(obj1, obj2, level = 0.05)

Arguments

`obj1`, `obj2`	smsn.lmm or smn.lmm objects containing the fitted models to be tested.
`level`	The significance level that should be used. If `quiet = TRUE`, this is ignored.

Value

`statistic`	The test statistic value.
`p.value`	The p-value from the test.
`df`	The degrees of freedom used on the test.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

References

Schumacher, F. L., Lachos, V. H., and Matos, L. A. (2021). Scale mixture of skew-normal linear mixed models with within-subject serial dependence. Statistics in Medicine 40(7), 1790-1810.

Examples

fm1 = smn.lmm(nlme::Orthodont, formFixed=distance ~ age+Sex, groupVar="Subject")
fm2 = smsn.lmm(nlme::Orthodont, formFixed=distance ~ age+Sex, groupVar="Subject",
               control=lmmControl(tol=.0001))
lr.test(fm1, fm2)
fm1 = smn.lmm(nlme::Orthodont, formFixed=distance ~ age+Sex, groupVar="Subject")
fm2 = smsn.lmm(nlme::Orthodont, formFixed=distance ~ age+Sex, groupVar="Subject",
               control=lmmControl(tol=.0001))
lr.test(fm1, fm2)

Mahalanobis distance from a smn.lmm or smsn.lmm object

Description

Returns the squared Mahalanobis distance from a fitted SMN-LMM or SMSN-LMM.

Usage

mahalDist(object, decomposed = FALSE, dataPlus = NULL)
mahalDist(object, decomposed = FALSE, dataPlus = NULL)

Arguments

`object`	An object inheriting from class `SMN` or `SMSN`, representing a fitted scale mixture of (skew) normal linear mixed model.
`decomposed`	Logical. If `TRUE`, the Mahalanobis distance is decomposed in an error term and a random effect term. Default is `FALSE`.
`dataPlus`	Optional. Expanded dataset that should be used instead the one used for fitting, useful for using Healy's plot with missing data.

Value

A vector containing the Mahalanobis distance, if decomposed = FALSE, or a data frame containing the Mahalanobis distance and its decomposition in error term and random effect (b) term, if decomposed = TRUE.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

References

Schumacher, F. L., Lachos, V. H., and Matos, L. A. (2021). Scale mixture of skew-normal linear mixed models with within-subject serial dependence. Statistics in Medicine 40(7), 1790-1810.

Zeller, C. B., Labra, F. V., Lachos, V. H. & Balakrishnan, N. (2010). Influence analyses of skew-normal/independent linear mixed models. Computational Statistics & Data Analysis, 54(5).

Examples

fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
mahalDist(fm1)
plot(mahalDist(fm1), nlabels=2)
fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
mahalDist(fm1)
plot(mahalDist(fm1), nlabels=2)

Mahalanobis distance from a smn.clmm object

Description

Returns the squared Mahalanobis distance from a fitted SMN-CLMM. Censored values are imputed using their conditional expectation from the fitting algorithm.

Usage

  mahalDistCens(object)
mahalDistCens(object)

Arguments

object

An object inheriting from class SMNclmm, representing a fitted scale mixture of normal censored linear mixed model.

Value

An object of class mahalDistCens containing the Mahalanobis distance.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos, Victor H. Lachos and Katherine L. Valeriano

References

Schumacher, F. L., Lachos, V. H., and Matos, L. A. (2021). Scale mixture of skew-normal linear mixed models with within-subject serial dependence. Statistics in Medicine 40(7), 1790-1810.

Zeller, C. B., Labra, F. V., Lachos, V. H. & Balakrishnan, N. (2010). Influence analyses of skew-normal/independent linear mixed models. Computational Statistics & Data Analysis, 54(5).

Examples

nj1 = 5; m = 30
time = rep(1:nj1, times=m)
groups = as.factor(rep(1:m, each=nj1))
dat1 = rsmsn.clmm(time, groups, cbind(1,time), rep(1,m*nj1), sigma2=0.7,
                  D=0.5*diag(1), beta=c(1,2), depStruct="CS", phi=0.4)
# Estimation
fm1 = smn.clmm(dat1, formFixed=y~x, groupVar="ind", depStruct="CS", ci="ci",
               lcl="lcl", ucl="ucl", control=lmmControl(max.iter=30))
distance = mahalDistCens(fm1)
plot(distance, level=0.95, nlabels=2)
nj1 = 5; m = 30
time = rep(1:nj1, times=m)
groups = as.factor(rep(1:m, each=nj1))
dat1 = rsmsn.clmm(time, groups, cbind(1,time), rep(1,m*nj1), sigma2=0.7,
                  D=0.5*diag(1), beta=c(1,2), depStruct="CS", phi=0.4)
# Estimation
fm1 = smn.clmm(dat1, formFixed=y~x, groupVar="ind", depStruct="CS", ci="ci",
               lcl="lcl", ucl="ucl", control=lmmControl(max.iter=30))
distance = mahalDistCens(fm1)
plot(distance, level=0.95, nlabels=2)

Data set for clinical trial measuring mice weight

Description

Sintetic longitudinal data set based on a clinical trial designed to test two diet treatments in comparison to a control group. The weight of 52 mice (reported in grams) is measured weekly from baseline until week 10 of treatment.

Usage

data(miceweight)
data(miceweight)

Format

This data frame contains the following columns:

treat: Treatment received (treatment 1 (T1), treatment 2 (T2), or control (C))
mouseID: Mouse ID
week: Week since treatment started
weight: Mouse weight in grams

Details

Dietary supplementations were administered for groups T1 and T2 starting at week 5.

This dataset was created using minor perturbations from a real study for which the original data is not publicly available. Data features such as covariances and skewness were preserved.

Examples

library(ggplot2)

data(miceweight)
ggplot(miceweight) + geom_line(aes(x=week, y=weight, group=mouseID)) +
  facet_wrap(~treat) + theme_bw()
library(ggplot2)

data(miceweight)
ggplot(miceweight) + geom_line(aes(x=week, y=weight, group=mouseID)) +
  facet_wrap(~treat) + theme_bw()

Extract the number of observations from smn.lmm and smsn.lmm fitted models

Description

Extract the total number of observations from a model fit (considering all repeated measurements from all subjects or clusters).

Usage

## S3 method for class 'SMN'
nobs(object, ...)
## S3 method for class 'SMSN'
nobs(object, ...)
## S3 method for class 'SMN'
nobs(object, ...)
## S3 method for class 'SMSN'
nobs(object, ...)

Arguments

`object`	An object inheriting from class `SMN` or `SMSN`
`...`	Additional arguments.

Value

A single integer, expected to be equal to nrow(data).

Examples

fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
nobs(fm1)
fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
nobs(fm1)

Plot a smn.lmm or smsn.lmm object

Description

Fitted values versus residuals plot.

Usage

## S3 method for class 'SMN'
plot(x, type = "response", level = "conditional",
     useweight = TRUE, alpha = 0.3, ...)

## S3 method for class 'SMSN'
plot(x, type = "response", level = "conditional",
     useweight = TRUE, alpha = 0.3, ...)
## S3 method for class 'SMN'
plot(x, type = "response", level = "conditional",
     useweight = TRUE, alpha = 0.3, ...)

## S3 method for class 'SMSN'
plot(x, type = "response", level = "conditional",
     useweight = TRUE, alpha = 0.3, ...)

Arguments

`x`	An object inheriting from class `SMN` or `SMSN`, representing a fitted scale mixture of (skew) normal linear mixed model.
`type`	Type of residual that should be used. For details see `residuals.SMN`. Default is "response", indicating raw residuals.
`level`	Level of residual that should be used. For details see `residuals.SMN`. Default is "conditional".
`useweight`	A logical value indicating if the estimated weights should be used as color in the plot.
`alpha`	Transparency parameter to be used (`0<alpha<1`). Meaningless if `useweight = TRUE`.
`...`	Additional arguments.

Value

A ggplot object.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

Examples

fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont,
              groupVar="Subject", distr="t")
plot(fm1)
plot(fm1, useweight=FALSE)

library(ggplot2)
plot(fm1) + ggtitle("t-LMM for orthodont data")
fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont,
              groupVar="Subject", distr="t")
plot(fm1)
plot(fm1, useweight=FALSE)

library(ggplot2)
plot(fm1) + ggtitle("t-LMM for orthodont data")

Plot ACF for smn.lmm or smsn.lmm residuals

Description

Plot method for objects of class "acfresid".

Usage

## S3 method for class 'acfresid'
plot(x,...)
## S3 method for class 'acfresid'
plot(x,...)

Arguments

`x`	An object inheriting from class `acfresid`, representing the empirical autocorrelation function from the residuals of a scale mixture of (skew) normal linear mixed model.
`...`	Additional arguments.

Value

A ggplot object.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

Examples

fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
plot(acfresid(fm1))

acfCI = acfresid(fm1, calcCI=TRUE)
plot(acfCI)
  
fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
plot(acfresid(fm1))

acfCI = acfresid(fm1, calcCI=TRUE)
plot(acfCI)

Plot Mahalanobis distance for a fitted smn.lmm or smsn.lmm

Description

Plot method for objects of class "mahalDist". For the total Mahalanobis distance, it gives a quantile for outlier detection, based on the Mahalanobis distance theoretical distribution.

Usage

## S3 method for class 'mahalDist'
plot(x, fitobject, type, level = 0.99, nlabels = 3, ...)
## S3 method for class 'mahalDist'
plot(x, fitobject, type, level = 0.99, nlabels = 3, ...)

Arguments

`x`	An object inheriting from class `mahalDist`, representing the Mahalanobis distance from a fitted scale mixture of (skew) normal linear mixed model.
`fitobject`	Optional. An object inheriting from class `SMN` or `SMSN`, representing the fitted scale mixture of (skew) normal linear mixed model that was used for calculating the Mahalanobis distance.
`type`	Optional. Either `"total"` (default), for the standard Mahalanobis distance, `"error"`, for the error term of the decomposition, or `"b"` for the random effect term of the decomposition. For details see `mahalDist`.
`level`	An optional numeric value in $(0,1)$ indicating the level of the quantile. It only has utility if `type="total"`. Default is `0.99`.
`nlabels`	Number of observations that should be labeled. Default is 3.
`...`	Additional arguments.

Value

A ggplot object, plotting the index versus the Mahalanobis distance, if all subject have the same number of observations; or plotting the number of observations per subject versus the Mahalanobis, otherwise.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

Examples

fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
plot(mahalDist(fm1), nlabels=2)

#the estimated quantile is stored at the attribute "info" of the plot object
plotMD = plot(mahalDist(fm1))
attr(plotMD, "info")
fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
plot(mahalDist(fm1), nlabels=2)

#the estimated quantile is stored at the attribute "info" of the plot object
plotMD = plot(mahalDist(fm1))
attr(plotMD, "info")

Plot Mahalanobis distance for a fitted smn.clmm

Description

Plot method for objects of class "mahalDistCens". It also gives a quantile for outlier detection, based on the Mahalanobis distance theoretical distribution.

Usage

  ## S3 method for class 'mahalDistCens'
plot(x, fitobject, level = 0.99, nlabels = 3, ...)
## S3 method for class 'mahalDistCens'
plot(x, fitobject, level = 0.99, nlabels = 3, ...)

Arguments

`x`	An object inheriting from class `mahalDistCens`, representing the Mahalanobis distance from a fitted scale mixture of normal censored linear mixed model.
`fitobject`	Optional. An object inheriting from class `SMNclmm`, representing the fitted scale mixture of normal linear mixed model that was used for calculating the Mahalanobis distance.
`level`	An optional numeric value in $(0,1)$ indicating the level of the quantile. Default is `0.99`.
`nlabels`	Number of observations that should be labeled. Default is 3.
`...`	Additional arguments.

Value

Author(s)

Fernanda L. Schumacher, Larissa A. Matos, Victor H. Lachos and Katherine L. Valeriano

Examples

nj1 = 5; m = 30
time = rep(1:nj1, times=m)
groups = as.factor(rep(1:m, each=nj1))
dat1 = rsmsn.clmm(time, groups, cbind(1,time), rep(1,m*nj1), sigma2=0.7,
                  D=0.5*diag(1), beta=c(1,2), depStruct="CS", phi=0.4)
# Estimation
fm1 = smn.clmm(dat1, formFixed=y~x, groupVar="ind", depStruct="CS", ci="ci",
               lcl="lcl", ucl="ucl", control=lmmControl(max.iter=30))
distance = mahalDistCens(fm1)
plot(distance, level=0.95, nlabels=2)
nj1 = 5; m = 30
time = rep(1:nj1, times=m)
groups = as.factor(rep(1:m, each=nj1))
dat1 = rsmsn.clmm(time, groups, cbind(1,time), rep(1,m*nj1), sigma2=0.7,
                  D=0.5*diag(1), beta=c(1,2), depStruct="CS", phi=0.4)
# Estimation
fm1 = smn.clmm(dat1, formFixed=y~x, groupVar="ind", depStruct="CS", ci="ci",
               lcl="lcl", ucl="ucl", control=lmmControl(max.iter=30))
distance = mahalDistCens(fm1)
plot(distance, level=0.95, nlabels=2)

Plot a smn.clmm object

Description

Fitted values versus residuals plot. Censored values are imputed using their conditional expectation from the fitting algorithm.

Usage

  ## S3 method for class 'SMNclmm'
plot(x, level = "conditional", useweight = TRUE,
                         alpha = 0.3, ...)
## S3 method for class 'SMNclmm'
plot(x, level = "conditional", useweight = TRUE,
                         alpha = 0.3, ...)

Arguments

`x`	An object inheriting from class `SMNclmm`, representing a fitted scale mixture of normal censored linear mixed model.
`level`	Level of residual that should be used. For details see `residuals.SMNclmm`. Default is "conditional".
`useweight`	A logical value indicating if the estimated weights should be used as color in the plot.
`alpha`	Transparency parameter to be used (`0<alpha<1`). Meaningless if `useweight = TRUE`.
`...`	Additional arguments.

Value

A ggplot object.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos, Victor H. Lachos and Katherine L. Valeriano

Examples

nj1 = 5; m = 30
time = rep(1:nj1, times=m)
groups = as.factor(rep(1:m, each=nj1))
dat1 = rsmsn.clmm(time, groups, cbind(1,time), rep(1,m*nj1), sigma2=0.7,
                  D=0.5*diag(1), beta=c(1,2), depStruct="UNC", distr="t", nu=4)
# Estimation
fm1 = smn.clmm(dat1, formFixed=y~x, groupVar="ind", depStruct="UNC", ci="ci",
               lcl="lcl", ucl="ucl", distr="t", control=lmmControl(max.iter=30))
plot(fm1)
nj1 = 5; m = 30
time = rep(1:nj1, times=m)
groups = as.factor(rep(1:m, each=nj1))
dat1 = rsmsn.clmm(time, groups, cbind(1,time), rep(1,m*nj1), sigma2=0.7,
                  D=0.5*diag(1), beta=c(1,2), depStruct="UNC", distr="t", nu=4)
# Estimation
fm1 = smn.clmm(dat1, formFixed=y~x, groupVar="ind", depStruct="UNC", ci="ci",
               lcl="lcl", ucl="ucl", distr="t", control=lmmControl(max.iter=30))
plot(fm1)

Prediction of future observations from an smn.lmm object

Description

Predicted values are obtained through conditional expectation. For details, see Schumacher, Lachos, and Matos (2021).

Usage

## S3 method for class 'SMN'
predict(object, newData, ...)
## S3 method for class 'SMN'
predict(object, newData, ...)

Arguments

`object`	An object inheriting from class `SMN`, representing a fitted scale mixture normal linear mixed model.
`newData`	A data frame for which response variable should be predicted, including covariates, groupVar and possibly timeVar. If missing or `NULL`, fitted values are returned.
`...`	Additional arguments.

Value

A data frame with covariates, groupVar and ypred, where ypred contains the predicted values from the response variable.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

References

Schumacher, F. L., Lachos, V. H., and Matos, L. A. (2021). Scale mixture of skew-normal linear mixed models with within-subject serial dependence. Statistics in Medicine 40(7), 1790-1810.

Examples

dat1 = nlme::Orthodont
fm1 = smn.lmm(distance ~ age+Sex, data=subset(dat1, age<14), groupVar="Subject")
predict(fm1, subset(dat1, age==14))
dat1 = nlme::Orthodont
fm1 = smn.lmm(distance ~ age+Sex, data=subset(dat1, age<14), groupVar="Subject")
predict(fm1, subset(dat1, age==14))

Prediction of future observations from an smn.clmm object

Description

Predicted values are obtained through conditional expectation. For details, see Schumacher, Lachos, and Matos (2021).

Usage

  ## S3 method for class 'SMNclmm'
predict(object, newData, ...)
## S3 method for class 'SMNclmm'
predict(object, newData, ...)

Arguments

`object`	An object inheriting from class `SMNclmm`, representing a fitted scale mixture normal linear mixed model.
`newData`	A data frame for which response variable should be predicted, including covariates, groupVar and possibly timeVar. If missing or `NULL`, fitted values are returned.
`...`	Additional arguments.

Value

A data frame with covariates, groupVar and ypred, where ypred contains the predicted values from the response variable.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos, Victor H. Lachos and Katherine L. Valeriano

References

Schumacher, F. L., Lachos, V. H., and Matos, L. A. (2021). Scale mixture of skew-normal linear mixed models with within-subject serial dependence. Statistics in Medicine 40(7), 1790-1810.

Examples

# Simulated example: 20 individuals with 5 times for estimation and
# 1 time for prediction
set.seed(963)
nj1 = 6; m = 20
time = rep(1:nj1, times=m)
groups = as.factor(rep(1:m, each=nj1))
dat1 = rsmsn.clmm(time, groups, cbind(1,time), rep(1,m*nj1), sigma2=0.4,
                  D=0.6*diag(1), beta=c(1,2), depStruct="CS", phi=0.25)
# Estimation
fm1 = smn.clmm(subset(dat1,time<6), formFixed=y~x, groupVar="ind",
               depStruct="CS", ci="ci", lcl="lcl", ucl="ucl",
               control=lmmControl(max.iter=30, tol=1e-4))
# Prediction
pred = predict(fm1, subset(dat1,time==6))
# MSPE
mean((subset(dat1,time==6)$y - pred$ypred)^2)
# Simulated example: 20 individuals with 5 times for estimation and
# 1 time for prediction
set.seed(963)
nj1 = 6; m = 20
time = rep(1:nj1, times=m)
groups = as.factor(rep(1:m, each=nj1))
dat1 = rsmsn.clmm(time, groups, cbind(1,time), rep(1,m*nj1), sigma2=0.4,
                  D=0.6*diag(1), beta=c(1,2), depStruct="CS", phi=0.25)
# Estimation
fm1 = smn.clmm(subset(dat1,time<6), formFixed=y~x, groupVar="ind",
               depStruct="CS", ci="ci", lcl="lcl", ucl="ucl",
               control=lmmControl(max.iter=30, tol=1e-4))
# Prediction
pred = predict(fm1, subset(dat1,time==6))
# MSPE
mean((subset(dat1,time==6)$y - pred$ypred)^2)

Prediction of future observations from an smsn.lmm object

Description

Predicted values are obtained through conditional expectation. For details, see Schumacher, Lachos, and Matos (2021).

Usage

## S3 method for class 'SMSN'
predict(object, newData, ...)
## S3 method for class 'SMSN'
predict(object, newData, ...)

Arguments

`object`	An object inheriting from class `SMSN`, representing a fitted scale mixture skew-normal linear mixed model.
`newData`	A data frame for which response variable should be predicted, including covariates, groupVar and possibly timeVar. If missing or `NULL`, fitted values are returned.
`...`	Additional arguments.

Value

A data frame with covariates, groupVar and ypred, where ypred contains the predicted values from the response variable.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

References

Schumacher, F. L., Lachos, V. H., and Matos, L. A. (2021). Scale mixture of skew-normal linear mixed models with within-subject serial dependence. Statistics in Medicine 40(7), 1790-1810.

Examples


dat1 = nlme::Orthodont
fm1 = smsn.lmm(distance ~ age+Sex, data=subset(dat1, age<14), groupVar="Subject")
predict(fm1, subset(dat1, age==14))

dat1 = nlme::Orthodont
fm1 = smsn.lmm(distance ~ age+Sex, data=subset(dat1, age<14), groupVar="Subject")
predict(fm1, subset(dat1, age==14))

Print a smn.lmm object

Description

Print a smn.lmm object.

Usage

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

Arguments

`x`	An object inheriting from class `SMN`, representing a fitted scale mixture normal linear mixed model.
`...`	Additional print arguments.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

Examples

fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
fm1
fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
fm1

Print a smn.clmm object

Description

Print a smn.clmm object.

Usage

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

Arguments

`x`	An object inheriting from class `SMNclmm`, representing a fitted scale mixture normal linear mixed model with censored responses.
`...`	Additional print arguments.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos, Victor H. Lachos and Katherine L. Valeriano

Examples

nj1 = 5; m = 30
time = rep(1:nj1, times=m)
groups = as.factor(rep(1:m, each=nj1))
dat1 = rsmsn.clmm(time, groups, cbind(1,time), rep(1,m*nj1), sigma2=0.7,
                  D=0.6*diag(1), beta=c(1,2), depStruct="UNC")
fm1 = smn.clmm(dat1, formFixed=y~x, groupVar="ind", formRandom=~1,
               depStruct="UNC", ci="ci", lcl="lcl", ucl="ucl")
fm1
nj1 = 5; m = 30
time = rep(1:nj1, times=m)
groups = as.factor(rep(1:m, each=nj1))
dat1 = rsmsn.clmm(time, groups, cbind(1,time), rep(1,m*nj1), sigma2=0.7,
                  D=0.6*diag(1), beta=c(1,2), depStruct="UNC")
fm1 = smn.clmm(dat1, formFixed=y~x, groupVar="ind", formRandom=~1,
               depStruct="UNC", ci="ci", lcl="lcl", ucl="ucl")
fm1

Print a smsn.lmm object

Description

Print a smsn.lmm object.

Usage

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

Arguments

`x`	An object inheriting from class `SMSN`, representing a fitted scale mixture skew-normal linear mixed model.
`...`	Additional print arguments.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

Examples


fm1 = smsn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
fm1
  
fm1 = smsn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
fm1

Extract random effects from smsn.lmm, smn.lmm and smn.clmm objects

Description

It extracts random effects from smsn.lmm, smn.lmm and smn.clmm objects.

Usage

## S3 method for class 'SMN'
ranef(object, ...)
## S3 method for class 'SMSN'
ranef(object, ...)
## S3 method for class 'SMNclmm'
ranef(object, ...)
## S3 method for class 'SMN'
ranef(object, ...)
## S3 method for class 'SMSN'
ranef(object, ...)
## S3 method for class 'SMNclmm'
ranef(object, ...)

Arguments

`object`	An object inheriting from class `SMN`, `SMSN`, or `SMNclmm`, representing a fitted scale mixture skew-normal linear mixed model.
`...`	Additional arguments

Value

Matrix of estimated random effects.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

References

Schumacher, F. L., Lachos, V. H., and Matos, L. A. (2021). Scale mixture of skew-normal linear mixed models with within-subject serial dependence. Statistics in Medicine 40(7), 1790-1810.

Examples

fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
ranef(fm1)
fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
ranef(fm1)

Extract model residuals from smn.lmm and smsn.lmm objects

Description

The conditional residuals are obtained by subtracting the fitted values from the response vector, while the marginal residuals are obtained by subtracting only the fixed effects from the response vector.

Usage

## S3 method for class 'SMN'
residuals(object, level = "conditional", type = "response", ...)

## S3 method for class 'SMSN'
residuals(object, level = "conditional", type = "response", ...)
## S3 method for class 'SMN'
residuals(object, level = "conditional", type = "response", ...)

## S3 method for class 'SMSN'
residuals(object, level = "conditional", type = "response", ...)

Arguments

`object`	An object inheriting from class `SMN` or `SMSN`, representing a fitted scale mixture of (skew) normal linear mixed model.
`level`	Either `"conditional"`, for obtaining conditional residuals, or `"marginal"`, for marginal residuals.
`type`	An optional character string specifying the type of residuals to be used. If `"response"`, as by default, the "raw" residuals (observed - fitted) are used; if `"normalized"`, the standardized residuals (residuals pre-multiplied by the inverse square-root of the estimated variance matrix) are used; else, if `"modified"`, modified residuals (residuals pre-multiplied by the inverse square-root of the estimated scale matrix) are used.
`...`	Additional arguments.

Details

Modified residuals are useful when the variance is not finite, such as when $\nu \le 2$ for t or ST distributions, or when $\nu \le 1$ for SL or SSL distributions.

Value

Vector with the residuals of length equal to nrow(data).

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

Examples

fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
residuals(fm1)
plot(fm1, useweight=FALSE)
fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
residuals(fm1)
plot(fm1, useweight=FALSE)

Extract model residuals from smn.clmm objects

Description

Censored values are imputed using their conditional expectation from the fitting algorithm.

Usage

  ## S3 method for class 'SMNclmm'
residuals(object, level = "conditional", ...)
## S3 method for class 'SMNclmm'
residuals(object, level = "conditional", ...)

Arguments

`object`	An object inheriting from class `SMNclmm`, representing a fitted scale mixture of normal censored linear mixed model.
`level`	Either `"conditional"`, for obtaining conditional residuals, or `"marginal"`, for marginal residuals.
`...`	Additional arguments.

Value

Vector with the residuals of length equal to nrow(data).

Note

The residuals are computed after imputing the censored observations.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos, Victor H. Lachos and Katherine L. Valeriano

Examples

nj1 = 5; m = 30
time = rep(1:nj1, times=m)
groups = as.factor(rep(1:m, each=nj1))
dat1 = rsmsn.clmm(time, groups, cbind(1,time), rep(1,m*nj1), sigma2=0.7,
                  D=0.6*diag(1), beta=c(1,2), depStruct="CS", phi=0.4)

fm1 = smn.clmm(dat1, formFixed=y~x, groupVar="ind", depStruct="CS", ci="ci",
               lcl="lcl", ucl="ucl", control=lmmControl(max.iter=30))
residuals(fm1)
nj1 = 5; m = 30
time = rep(1:nj1, times=m)
groups = as.factor(rep(1:m, each=nj1))
dat1 = rsmsn.clmm(time, groups, cbind(1,time), rep(1,m*nj1), sigma2=0.7,
                  D=0.6*diag(1), beta=c(1,2), depStruct="CS", phi=0.4)

fm1 = smn.clmm(dat1, formFixed=y~x, groupVar="ind", depStruct="CS", ci="ci",
               lcl="lcl", ucl="ucl", control=lmmControl(max.iter=30))
residuals(fm1)

Generate data from SMSN-CLMM with censored responses

Description

It creates a simulated data set from SMSN-CLMM with several possible dependence structures, with an established censoring rate or a fixed limit of detectation (LOD).

Usage

  rsmsn.clmm(time, ind, x, z, sigma2, D, beta, lambda=rep(0, nrow(D)),
             depStruct="UNC", phi=NULL, distr="norm", nu=NULL, type="left",
             pcens=0.10, LOD=NULL)
rsmsn.clmm(time, ind, x, z, sigma2, D, beta, lambda=rep(0, nrow(D)),
             depStruct="UNC", phi=NULL, distr="norm", nu=NULL, type="left",
             pcens=0.10, LOD=NULL)

Arguments

`time`	Vector of length $N$ containing times that should be used in data generation, where $N$ indicates the total number of observations.
`ind`	Vector of length $N$ containing the variable which represents the subjects or groups.
`x`	Design matrix for fixed effects of dimension $N\times p$ .
`z`	Design matrix for random effects of dimension $N \times q$ .
`sigma2`	Common variance parameter, such that $\Sigma=\sigma^2*R$ .
`D`	Variance matrix for random effects.
`beta`	Vector of fixed effects parameter.
`lambda`	Skewness parameter of random effects.
`depStruct`	Dependence structure. `"UNC"` for conditionally uncorrelated (`"CI"` is also accepted), `"ARp"` for AR(p) – p is `length(phi)`–, `"CS"` for compound symmetry, `"DEC"` for DEC, `"CAR1"` for continuous-time AR(1), and `"MA1"` for moving average of order 1.
`phi`	Parameter vector indexing the dependence structure.
`distr`	Distribution that should be used. `"norm"` for normal, `"t"` for Student-t, `"sn"` for skew-normal, and `"st"` for skew-t.
`nu`	Degrees of freedom for Student-t and skew-t distributions. It must be greater than 2.
`type`	`left` for left censoring and `right` for right censoring.
`pcens`	Desired censoring rate.
`LOD`	Desired limit of detectation. If `LOD` is provided, then `pcens` will be discarded.

Value

A data frame containing time, the variable indicating groups (ind), the generated response variable (y), the censoring indicator variable (ci), the lower censoring limit (lcl), the upper censoring limit (ucl), and possible covariates.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos, Victor H. Lachos and Katherine L. Valeriano

References

Matos, L. A., Prates, M. O., Chen, M. H., and Lachos, V. H. (2013). Likelihood-based inference for mixed-effects models with censored response using the multivariate-t distribution. Statistica Sinica 23(3), 1323-1345.

Lachos, V. H., A. Matos, L., Castro, L. M., and Chen, M. H. (2019). Flexible longitudinal linear mixed models for multiple censored responses data. Statistics in medicine, 38(6), 1074-1102.

Examples

library(ggplot2)

# Generating a sample for m=25 individuals with 5 times
nj1 = 5
m = 25
time = rep(1:nj1, times=m)
groups = as.factor(rep(1:m, each=nj1))

# Considering 10% of right censoring responses and normal distribution
dat1 = rsmsn.clmm(time, groups, cbind(1,time), rep(1,m*nj1), sigma2=.25,
                  D=0.5*diag(1), beta=c(1,2), depStruct="ARp", phi=0.5,
                  type="right")
head(dat1)
ggplot(dat1, aes(x=x, y=y, group=ind)) + geom_line() +
  stat_summary(aes(group=1), geom="line", fun=mean, col="blue", size=1.5) +
  geom_line(aes(x=x,y=lcl), color="red", linetype="dashed")

# Considering LOD=4, left censoring, and Student-t distribution
dat2 = rsmsn.clmm(time, groups, cbind(1,time), rep(1,m*nj1), sigma2=.25,
                  D=0.5*diag(1), beta=c(1,2), depStruct="CS", phi=0.2,
                  distr="t", nu=4, LOD=4)
mean(dat2$ci) #censoring rate
ggplot(dat2, aes(x=x, y=y, group=ind)) + geom_line() +
  stat_summary(aes(group=1), geom="line", fun=mean, col="blue", size=1.5) +
  geom_line(aes(x=x,y=ucl), color="red", linetype="dashed")
library(ggplot2)

# Generating a sample for m=25 individuals with 5 times
nj1 = 5
m = 25
time = rep(1:nj1, times=m)
groups = as.factor(rep(1:m, each=nj1))

# Considering 10% of right censoring responses and normal distribution
dat1 = rsmsn.clmm(time, groups, cbind(1,time), rep(1,m*nj1), sigma2=.25,
                  D=0.5*diag(1), beta=c(1,2), depStruct="ARp", phi=0.5,
                  type="right")
head(dat1)
ggplot(dat1, aes(x=x, y=y, group=ind)) + geom_line() +
  stat_summary(aes(group=1), geom="line", fun=mean, col="blue", size=1.5) +
  geom_line(aes(x=x,y=lcl), color="red", linetype="dashed")

# Considering LOD=4, left censoring, and Student-t distribution
dat2 = rsmsn.clmm(time, groups, cbind(1,time), rep(1,m*nj1), sigma2=.25,
                  D=0.5*diag(1), beta=c(1,2), depStruct="CS", phi=0.2,
                  distr="t", nu=4, LOD=4)
mean(dat2$ci) #censoring rate
ggplot(dat2, aes(x=x, y=y, group=ind)) + geom_line() +
  stat_summary(aes(group=1), geom="line", fun=mean, col="blue", size=1.5) +
  geom_line(aes(x=x,y=ucl), color="red", linetype="dashed")

Generate data from SMSN-LMM

Description

It creates a simulated data set from SMSN-LMM (or from SMN-LMM, if lambda = 0) with several possible dependence structures, for one subject.

Usage

rsmsn.lmm(time1, x1, z1, sigma2, D1, beta, lambda, depStruct = "UNC",
          phi = NULL, distr = "sn", nu = NULL)
rsmsn.lmm(time1, x1, z1, sigma2, D1, beta, lambda, depStruct = "UNC",
          phi = NULL, distr = "sn", nu = NULL)

Arguments

`time1`	Vector containing times that should be used in data generation.
`x1`	Design matrix for fixed effects.
`z1`	Design matrix for random effects.
`sigma2`	Common variance parameter, such that $\Sigma=\sigma^2*R$ .
`D1`	Variance matrix for random effects.
`beta`	Vector of fixed effects parameter.
`lambda`	Skewness parameter of random effects.
`depStruct`	Dependence structure. `"UNC"` for conditionally uncorrelated (`"CI"` is also accepted), `"ARp"` for AR(p) – p is `length(phi)`–, `"CS"` for compound symmetry, `"DEC"` for DEC, and `"CAR1"` for continuous-time AR(1).
`phi`	Parameter vector indexing the dependence structure.
`distr`	Distribution that should be used. `"sn"` for skew-normal, `"st"` for skew-t, `"ss"` for skew-slash, and `"scn"` for skew-contaminated normal.
`nu`	Parameter vector indexing `distr`. Should be `NULL` for `"sn"`, be a vector of length 1 for `"st"` and `"ss"`, and of length 2 for `"scn"`.

Value

A data frame containing time, the generated response variable (y), and possible covariates.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

References

Lachos, V. H., P. Ghosh, and R. B. Arellano-Valle (2010). Likelihood based inference for skew-normal independent linear mixed models. Statistica Sinica 20, 303-322.

Schumacher, F. L., Lachos, V. H., and Matos, L. A. (2021). Scale mixture of skew-normal linear mixed models with within-subject serial dependence. Statistics in Medicine 40(7), 1790-1810.

Examples

# Generating a sample for 1 individual at 5 times
nj1 = 5
rsmsn.lmm(1:nj1, cbind(1, 1:nj1), rep(1, nj1), sigma2=.25, D1=diag(1),
          beta=c(1, 2), lambda=2, depStruct="ARp", phi=.5,
          distr="st", nu=5)

# Generating a sample for m=25 individuals with 5 times
library(dplyr)
library(purrr)
library(ggplot2)
nj1 = 5
m = 25
gendatList = map(rep(nj1, m),
                 function(nj) rsmsn.lmm(1:nj, cbind(1, 1:nj), rep(1, nj),
                                        sigma2=.25, D1=.5*diag(1), beta=c(1, 2),
                                        lambda=2, depStruct="ARp", phi=.5))
gendat = bind_rows(gendatList, .id="ind")
ggplot(gendat, aes(x=x, y=y, group=ind)) + geom_line() +
  stat_summary(aes(group=1), geom="line", fun=mean, col="blue", size=2)
#
fm1 = smsn.lmm(y ~ x, data=gendat, groupVar="ind", depStruct="ARp",
                pAR=1)
summary(fm1)

# Generating a sample for 1 individual at 5 times
nj1 = 5
rsmsn.lmm(1:nj1, cbind(1, 1:nj1), rep(1, nj1), sigma2=.25, D1=diag(1),
          beta=c(1, 2), lambda=2, depStruct="ARp", phi=.5,
          distr="st", nu=5)

# Generating a sample for m=25 individuals with 5 times
library(dplyr)
library(purrr)
library(ggplot2)
nj1 = 5
m = 25
gendatList = map(rep(nj1, m),
                 function(nj) rsmsn.lmm(1:nj, cbind(1, 1:nj), rep(1, nj),
                                        sigma2=.25, D1=.5*diag(1), beta=c(1, 2),
                                        lambda=2, depStruct="ARp", phi=.5))
gendat = bind_rows(gendatList, .id="ind")
ggplot(gendat, aes(x=x, y=y, group=ind)) + geom_line() +
  stat_summary(aes(group=1), geom="line", fun=mean, col="blue", size=2)
#
fm1 = smsn.lmm(y ~ x, data=gendat, groupVar="ind", depStruct="ARp",
                pAR=1)
summary(fm1)

Residual standard deviation from smn.lmm and smsn.lmm objects

Description

Extract the estimated residual standard deviation from smn.lmm and smsn.lmm objects.

Usage

## S3 method for class 'SMN'
sigma(object, ...)
## S3 method for class 'SMSN'
sigma(object, ...)
## S3 method for class 'SMN'
sigma(object, ...)
## S3 method for class 'SMSN'
sigma(object, ...)

Arguments

`object`	An object inheriting from class `SMN` or `SMSN`
`...`	Additional arguments.

Value

A positive number.

Examples

fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
sigma(fm1)
fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
sigma(fm1)

ML estimation of scale mixture of normal linear mixed models with censored responses

Description

It fits left, right, or interval censored scale mixture of normal linear mixed model with possible within-subject dependence structure, using the EM algorithm. It provides estimates and standard errors of parameters.

Usage

smn.clmm(data, formFixed, groupVar, formRandom = ~1, depStruct = "UNC",
         ci, lcl, ucl, timeVar = NULL, distr = "norm",
         nufix = FALSE, pAR = 1, control = lmmControl())
smn.clmm(data, formFixed, groupVar, formRandom = ~1, depStruct = "UNC",
         ci, lcl, ucl, timeVar = NULL, distr = "norm",
         nufix = FALSE, pAR = 1, control = lmmControl())

Arguments

`data`	A data frame containing the variables named in `formFixed`, `formRandom`, `groupVar`, `timeVar`, `ci`, `lcl`, and `ucl`.
`formFixed`	A two-sided linear formula object describing the fixed effects part of the model, with the response on the left of a `~` operator and the covariates, separated by `+` operators, on the right.
`groupVar`	A character containing the name of the variable which represents the subjects or groups in `data`.
`formRandom`	A one-sided linear formula object describing the random effects part of the model, with the covariates, separated by `+` operators, on the right of a `~` operator. By default, a model with random intercept is considered.
`depStruct`	A character indicating which dependence structure should be used. `"UNC"` for conditionally uncorrelated (`"CI"` is also accepted), `"ARp"` for AR(p) – p is `length(phi)`–, `"CS"` for compound symmetry, `"DEC"` for DEC, `"CAR1"` for continuous-time AR(1), and `"MA1"` for moving average of order 1.
`ci`	A character containing the name of the censoring indicator variable in `data`, which should be 1 if the respective observation is censored or missing, and 1 otherwise. If missing, it is assumed that none of the observations is censored.
`lcl`	A character containing the name of the lower censoring limit in `data`. If missing, it is assumed `lcl=-Inf`, i.e., no left limit.
`ucl`	A character containing the name of the upper censoring limit in `data`. If missing, it is assumed `ucl=Inf`, i.e., no right limit.
`timeVar`	A character containing the name of the variable which represents the time in `data`. Meaningless if `depStruct="UNC"` or `depStruct="CS"`. For other structures, if `is.null(timeVar)` the observations are considered equally spaced and ordered. If `depStruct="ARp"`, timeVar must be an index, preferably starting at 1.
`distr`	A character indicating which distribution should be used. `"norm"` for normal and `"t"` for Student-t.
`nufix`	`TRUE` or `FALSE` indicating if `nu` should be estimated for t distribution. If `nufix=TRUE`, nu must be especified through `lmmControl()`.
`pAR`	If `depStruct="ARp"`, `pAR` indicates the order of the autoregressive process that should be used (1 by default). Otherwise, it is meaningless.
`control`	An object resulting from the function `lmmControl()`, containing additional options for the estimation algorithm.

Details

It fits the model $Y_i = X_i \beta + Z_i b_i + \epsilon_i$ , for $i=1,\dots,n$ , where $Y_i$ is a vector with $n_i$ observed continuous responses, $b_i ~ SMN(0, D; H)$ and $\epsilon_i ~ SMN(0, \Sigma_i; H)$ , indexed by the same mixing distribution.

For details see Matos et al. (2013) and Lachos et al. (2019).

Value

An object of class "SMNclmm" representing the SMN-CLMM fit. Generic functions such as print and summary have methods to show the results of the fit. The functions fitted and ranef can be used to extract some of its components.

Specifically, the following components are returned:

`theta`	Named vector with parameter estimates.
`iter`	Number of iterations runned.
`estimates`	A named list containing parameter estimates.
`yest`	Imputed values in the response variable.
`uhat`	Estimated weights.
`loglik.track`	Vector containing the log-likelihood value from each iteration of the estimation procedure.
`random.effects`	Estimated random effects.
`std.error`	A vector with standard errors.
`loglik`	Value of the log-likelihood at last iteration.
`elapsedTime`	Time elapsed in processing, in seconds.
`error`	Convergence criterion at last iteration.
`criteria`	A list with AIC, BIC, and SIC criterion.
`call`	The `smn.clmm` call that produced the object.
`data`	The data frame used on `smn.clmm` call.
`formula`	A list containing the formulas used on `smn.clmm` call.
`depStruct`	A character indicating which dependence structure was used.
`covRandom`	A character indicating which structure was used for the random effects scale matrix.
`distr`	A character indicating which distribution was used.
`N`	The number of observations used.
`n`	The number of individuals/groups used.
`groupVar`	A character indicating the name of the grouping variable.
`timeVar`	A character indicating the name of the time variable, if any.
`fitted`	A vector of fitted values.

Author(s)

Larissa A. Matos, Victor H. Lachos, Katherine L. Valeriano and Fernanda L. Schumacher

References

Henderson, N. C. and R. Varadhan (2019). Damped anderson acceleration with restarts and monotonicity control for accelerating EM and EM-like algorithms. Journal of Computational and Graphical Statistics 28(4), 834-846.

Lachos, V. H., A. Matos, L., Castro, L. M., and Chen, M. H. (2019). Flexible longitudinal linear mixed models for multiple censored responses data. Statistics in medicine, 38(6), 1074-1102.

Examples

# Generating a sample for m=30 individuals with 5 times
# Considering 10% of left censoring responses
# AR(1) and normal distribution
nj1 = 5
m = 30
time = rep(1:nj1, times=m)
groups = as.factor(rep(1:m, each=nj1))
dat1 = rsmsn.clmm(time = time, ind = groups, x = cbind(1,time), z = rep(1,m*nj1),
                  sigma2=0.6, D=0.5*diag(1), beta=c(1,2), depStruct="ARp",
                  phi=0.4, pcens = .1, type = "left")

# Estimation using UNC
fm0 = smn.clmm(dat1, formFixed=y~x, groupVar="ind", ci="ci", ucl="ucl")
summary(fm0)

# Estimation using AR(1)
fm1 = update(fm0, depStruct="ARp")

# Estimation using AR(1) and t distribution
fm2 = update(fm1, distr="t")

# Comparing fitted models
criteria(list(fm0= fm0, fm1=fm1, fm2=fm2))

# Generating a sample for m=30 individuals with 5 times
# Considering 10% of left censoring responses
# AR(1) and normal distribution
nj1 = 5
m = 30
time = rep(1:nj1, times=m)
groups = as.factor(rep(1:m, each=nj1))
dat1 = rsmsn.clmm(time = time, ind = groups, x = cbind(1,time), z = rep(1,m*nj1),
                  sigma2=0.6, D=0.5*diag(1), beta=c(1,2), depStruct="ARp",
                  phi=0.4, pcens = .1, type = "left")

# Estimation using UNC
fm0 = smn.clmm(dat1, formFixed=y~x, groupVar="ind", ci="ci", ucl="ucl")
summary(fm0)

# Estimation using AR(1)
fm1 = update(fm0, depStruct="ARp")

# Estimation using AR(1) and t distribution
fm2 = update(fm1, distr="t")

# Comparing fitted models
criteria(list(fm0= fm0, fm1=fm1, fm2=fm2))

ML estimation of scale mixture of normal linear mixed models

Description

It fits a scale mixture of normal linear mixed model with possible within-subject dependence structure, using an EM-type algorithm (via the DAAREM method, by default). It provides estimates and standard errors of parameters.

Usage

smn.lmm(data, formFixed, groupVar, formRandom = ~1, depStruct = "UNC",
        timeVar = NULL, distr = "norm", covRandom = "pdSymm",
        pAR = 1, control = lmmControl())
smn.lmm(data, formFixed, groupVar, formRandom = ~1, depStruct = "UNC",
        timeVar = NULL, distr = "norm", covRandom = "pdSymm",
        pAR = 1, control = lmmControl())

Arguments

`data`	A data frame containing the variables named in `formFixed`, `formRandom`, `groupVar`, and `timeVar`.
`formFixed`	A two-sided linear formula object describing the fixed effects part of the model, with the response on the left of a `~` operator and the covariates, separated by `+` operators, on the right.
`groupVar`	A character containing the name of the variable which represents the subjects or groups in `data`.
`formRandom`	A one-sided linear formula object describing the random effects part of the model, with the covariates, separated by `+` operators, on the right of a `~` operator. By default, a model with random intercept is considered.
`depStruct`	A character indicating which dependence structure should be used. `"UNC"` for conditionally uncorrelated (`"CI"` is also accepted), `"ARp"` for AR(p) – p is `length(phi)`–, `"CS"` for compound symmetry, `"DEC"` for DEC, and `"CAR1"` for continuous-time AR(1).
`timeVar`	A character containing the name of the variable which represents the time in `data`. Meaningless if `depStruct="UNC"` or `depStruct="CS"`. For other structures, if `is.null(timeVar)` the observations are considered equally spaced and ordered. If `depStruct="ARp"`, timeVar must be an index, preferably starting at 1.
`distr`	A character indicating which distribution should be used. `"norm"` for normal, `"t"` for t, `"sl"` for slash, and `"cn"` for contaminated normal.
`covRandom`	A character indicating which structure should be used for the random effects scale matrix (either `"pdSymm"` (default), for a general positive-definite matrix, or `"pdDiag"`, for a diagonal matrix).
`pAR`	If `depStruct="ARp"`, pAR indicates the order of the autoregressive process that should be used. Otherwise, it is meaningless.
`control`	An object resulting from the function `lmmControl()`, containing additional options for the estimation algorithm.

Details

For efficiency, the DAAREM method is used for parameter estimation. In case of numerical errors, please try passing to lmmControl(algorithm = "DAAREM") to the control argument.

For details see Schumacher, Lachos & Matos (2021).

Value

An object of class "SMN" representing the SMN-LMM fit. Generic functions such as print and summary have methods to show the results of the fit. The functions fitted and ranef can be used to extract some of its components.

Specifically, the following components are returned:

`theta`	Named vector with parameter estimates.
`iter`	Number of iterations runned.
`estimates`	A named list containing parameter estimates.
`uhat`	Estimated weights.
`loglik.track`	Vector containing the log-likelihood value from each iteration of the estimation procedure.
`random.effects`	Estimated random effects.
`std.error`	A vector with standard errors.
`loglik`	Value of the log-likelihood at last iteration.
`elapsedTime`	Time elapsed in processing, in seconds.
`error`	Convergence criterion at last iteration.
`call`	The `smn.lmm` call that produced the object.
`criteria`	A list with AIC and BIC criterion.
`data`	The data frame used on `smn.lmm` call.
`formula`	A list containing the formulas used on `smn.lmm` call.
`depStruct`	A character indicating which dependence structure was used.
`covRandom`	A character indicating which structure was used for the random effects scale matrix.
`distr`	A character indicating which distribution was used.
`N`	The number of observations used.
`n`	The number of individuals/groups used.
`groupVar`	A character indicating the name of the grouping variable.
`control`	The control list used in the function's call.
`timeVar`	A character indicating the name of the time variable, if any.
`fitted`	A vector of fitted values, if `calc.bi=TRUE`.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

References

Lachos, V. H., P. Ghosh, and R. B. Arellano-Valle (2010). Likelihood based inference for skew-normal independent linear mixed models. Statistica Sinica 20, 303-322.

Schumacher, F. L., Lachos, V. H., and Matos, L. A. (2021). Scale mixture of skew-normal linear mixed models with within-subject serial dependence. Statistics in Medicine 40(7), 1790-1810.

Examples

#simple example
dat1 = as.data.frame(nlme::Orthodont)
fm1 = smn.lmm(dat1, formFixed=distance ~ age, groupVar="Subject",
                control=lmmControl(max.iter=30))
fm1

#fitting for several distributions / dependence structures
fm1 = smn.lmm(dat1, formFixed=distance ~ age+Sex, groupVar="Subject")
fm2 = smn.lmm(dat1, formFixed=distance ~ age+Sex, groupVar="Subject",
              distr="t")
fm3 = smn.lmm(dat1, formFixed=distance ~ age+Sex, groupVar="Subject",
              distr="sl")
fm4 = smn.lmm(dat1, formFixed=distance ~ age+Sex, groupVar="Subject",
              depStruct="ARp", pAR=1)
criteria(list(fm1=fm1, fm2=fm2, fm3=fm3, fm4=fm4))
summary(fm3)

#some diagnostic tools
plot(fm3)
acf3 = acfresid(fm3, calcCI=TRUE, MCiter=100)
plot(acf3)
plot(mahalDist(fm3), nlabels=2)
healy.plot(fm3)

#simple example
dat1 = as.data.frame(nlme::Orthodont)
fm1 = smn.lmm(dat1, formFixed=distance ~ age, groupVar="Subject",
                control=lmmControl(max.iter=30))
fm1

#fitting for several distributions / dependence structures
fm1 = smn.lmm(dat1, formFixed=distance ~ age+Sex, groupVar="Subject")
fm2 = smn.lmm(dat1, formFixed=distance ~ age+Sex, groupVar="Subject",
              distr="t")
fm3 = smn.lmm(dat1, formFixed=distance ~ age+Sex, groupVar="Subject",
              distr="sl")
fm4 = smn.lmm(dat1, formFixed=distance ~ age+Sex, groupVar="Subject",
              depStruct="ARp", pAR=1)
criteria(list(fm1=fm1, fm2=fm2, fm3=fm3, fm4=fm4))
summary(fm3)

#some diagnostic tools
plot(fm3)
acf3 = acfresid(fm3, calcCI=TRUE, MCiter=100)
plot(acf3)
plot(mahalDist(fm3), nlabels=2)
healy.plot(fm3)

ML estimation of scale mixture of skew-normal linear mixed models

Description

It fits a scale mixture of skew-normal linear mixed model with possible within-subject dependence structure, using an EM-type algorithm (via the DAAREM method, by default). It provides estimates and standard errors of parameters.

Usage

smsn.lmm(data, formFixed, groupVar, formRandom = ~1, depStruct = "UNC",
         timeVar = NULL, distr = "sn", covRandom = "pdSymm",
         skewind, pAR = 1, control = lmmControl())
smsn.lmm(data, formFixed, groupVar, formRandom = ~1, depStruct = "UNC",
         timeVar = NULL, distr = "sn", covRandom = "pdSymm",
         skewind, pAR = 1, control = lmmControl())

Arguments

`data`	A data frame containing the variables named in `formFixed`, `formRandom`, `groupVar`, and `timeVar`.
`formFixed`	A two-sided linear formula object describing the fixed effects part of the model, with the response on the left of a `~` operator and the covariates, separated by `+` operators, on the right.
`groupVar`	A character containing the name of the variable which represents the subjects or groups in `data`.
`formRandom`	A one-sided linear formula object describing the random effects part of the model, with the covariates, separated by `+` operators, on the right of a `~` operator. By default, a model with random intercept is considered.
`depStruct`	A character indicating which dependence structure should be used. `"UNC"` for conditionally uncorrelated (`"CI"` is also accepted), `"ARp"` for AR(p) – p is `length(phi)`–, `"CS"` for compound symmetry, `"DEC"` for DEC, and `"CAR1"` for continuous-time AR(1).
`timeVar`	A character containing the name of the variable which represents the time in `data`. Meaningless if `depStruct="UNC"` or `depStruct="CS"`. For other structures, if `is.null(timeVar)` the observations are considered equally spaced and ordered. If `depStruct="ARp"`, timeVar must be an index, preferably starting at 1.
`distr`	A character indicating which distribution should be used. `"sn"` for skew-normal, `"st"` for skew-t, `"ssl"` for skew-slash, and `"scn"` for skew-contaminated normal.
`covRandom`	A character indicating which structure should be used for the random effects scale matrix (either `"pdSymm"` (default), for a general positive-definite matrix, or `"pdDiag"`, for a diagonal matrix).
`skewind`	A vector of length equal to the number of random effects, containing 0's and 1's, indicating which elements of the skewness parameter vector should be estimated (optional, default is an all-ones vector).
`pAR`	If `depStruct="ARp"`, pAR indicates the order of the autoregressive process that should be used. Otherwise, it is meaningless.
`control`	An object resulting from the function `lmmControl()`, containing additional options for the estimation algorithm.

Details

It fits the model $Y_i = X_i \beta + Z_i b_i + \epsilon_i$ , for $i=1,\dots,n$ , where $Y_i$ is a vector with $n_i$ observed continuous responses, $b_i ~ SMSN(c\Delta, D,\lambda;H)$ and $\epsilon_i ~ SMN(0, \Sigma_i; H)$ , indexed by the same mixing distribution.

For efficiency, the DAAREM method is used for parameter estimation. In case of numerical errors, please try passing to lmmControl(algorithm = "DAAREM") to the control argument.

For details see Schumacher, Lachos & Matos (2021).

Value

An object of class "SMSN" representing the SMSN-LMM fit. Generic functions such as print and summary have methods to show the results of the fit. The functions fitted and ranef can be used to extract some of its components.

Specifically, the following components are returned:

`theta`	Named vector with parameter estimates.
`iter`	Number of iterations runned.
`estimates`	A named list containing parameter estimates.
`uhat`	Estimated weights.
`loglik.track`	Vector containing the log-likelihood value from each iteration of the estimation procedure.
`random.effects`	Estimated random effects.
`std.error`	A vector with standard errors.
`loglik`	Value of the log-likelihood at last iteration.
`elapsedTime`	Time elapsed in processing, in seconds.
`error`	Convergence criterion at last iteration.
`call`	The `smsn.lmm` call that produced the object.
`criteria`	A list with AIC and BIC criterion.
`data`	The data frame used on `smsn.lmm` call.
`formula`	A list containing the formulas used on `smsn.lmm` call.
`depStruct`	A character indicating which dependence structure was used.
`covRandom`	A character indicating which structure was used for the random effects scale matrix.
`distr`	A character indicating which distribution was used.
`N`	The number of observations used.
`n`	The number of individuals/groups used.
`groupVar`	A character indicating the name of the grouping variable.
`control`	The control list used in the function's call.
`timeVar`	A character indicating the name of the time variable, if any.
`fitted`	A vector of fitted values, if `calc.bi=TRUE`.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

References

Lachos, V. H., P. Ghosh, and R. B. Arellano-Valle (2010). Likelihood based inference for skew-normal independent linear mixed models. Statistica Sinica 20, 303-322.

Schumacher, F. L., Lachos, V. H., and Matos, L. A. (2021). Scale mixture of skew-normal linear mixed models with within-subject serial dependence. Statistics in Medicine 40(7), 1790-1810.

Examples

#simple example
dat1 = as.data.frame(nlme::Orthodont)
fm1 = smsn.lmm(dat1, formFixed=distance ~ age, groupVar="Subject",
                control=lmmControl(max.iter=30))
fm1

#fitting for several distributions / dependence structures
fm1 = smsn.lmm(dat1, formFixed=distance ~ age+Sex, groupVar="Subject")
fm2 = smsn.lmm(dat1, formFixed=distance ~ age+Sex, groupVar="Subject",
               distr="st")
fm3 = smsn.lmm(dat1, formFixed=distance ~ age+Sex, groupVar="Subject",
               distr="ssl")
fm4 = smsn.lmm(dat1, formFixed=distance ~ age+Sex, groupVar="Subject",
               depStruct="ARp", pAR=1)
criteria(list(fm1=fm1, fm2=fm2, fm3=fm3, fm4=fm4))
summary(fm3)

#some diagnostic tools
plot(fm3)
acf3 = acfresid(fm3, calcCI=TRUE, MCiter=100)
plot(acf3)
plot(mahalDist(fm3), nlabels=2)
healy.plot(fm3)

#simple example
dat1 = as.data.frame(nlme::Orthodont)
fm1 = smsn.lmm(dat1, formFixed=distance ~ age, groupVar="Subject",
                control=lmmControl(max.iter=30))
fm1

#fitting for several distributions / dependence structures
fm1 = smsn.lmm(dat1, formFixed=distance ~ age+Sex, groupVar="Subject")
fm2 = smsn.lmm(dat1, formFixed=distance ~ age+Sex, groupVar="Subject",
               distr="st")
fm3 = smsn.lmm(dat1, formFixed=distance ~ age+Sex, groupVar="Subject",
               distr="ssl")
fm4 = smsn.lmm(dat1, formFixed=distance ~ age+Sex, groupVar="Subject",
               depStruct="ARp", pAR=1)
criteria(list(fm1=fm1, fm2=fm2, fm3=fm3, fm4=fm4))
summary(fm3)

#some diagnostic tools
plot(fm3)
acf3 = acfresid(fm3, calcCI=TRUE, MCiter=100)
plot(acf3)
plot(mahalDist(fm3), nlabels=2)
healy.plot(fm3)

Summary of a smn.lmm object

Description

summary method for class "SMN".

Usage

## S3 method for class 'SMN'
summary(object, confint.level = 0.95, ...)
## S3 method for class 'SMN'
summary(object, confint.level = 0.95, ...)

Arguments

`object`	An object inheriting from class `SMN`, representing a fitted scale mixture normal linear mixed model.
`confint.level`	Level of the approximate confidence intervals presented.
`...`	Additional arguments.

Value

`varRandom`	Estimated variance matrix from random effects ( $D$ ).
`varFixed`	Parameter estimates of variance from random errors ( $\Sigma$ ). For recovering the error variance matrix use `errorVar` function.
`tableFixed`	Estimated fixed effects, their standard errors and approximated confidence intervals.
`criteria`	Maximum log-likelihood value, AIC and BIC criteria.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

Examples


fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
summary(fm1)
fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject")
summary(fm1)

Summary of a smn.clmm object

Description

summary method for class "SMNclmm".

Usage

  ## S3 method for class 'SMNclmm'
summary(object, confint.level = 0.95, ...)
## S3 method for class 'SMNclmm'
summary(object, confint.level = 0.95, ...)

Arguments

`object`	An object inheriting from class `SMNclmm`, representing a fitted scale mixture normal linear mixed model with censored responses.
`confint.level`	Level of the approximate confidence intervals presented.
`...`	Additional arguments.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos, Victor H. Lachos and Katherine L. Valeriano

Examples

nj1 = 5; m = 30
time = rep(1:nj1, times=m)
groups = as.factor(rep(1:m, each=nj1))
dat1 = rsmsn.clmm(time, groups, cbind(1,time), rep(1,m*nj1), sigma2=0.7,
                  D=0.5*diag(1), beta=c(1,2), depStruct="UNC")
fm1 = smn.clmm(dat1, formFixed=y~x, groupVar="ind", formRandom=~1,
               depStruct="UNC", ci="ci", lcl="lcl", ucl="ucl")
summary(fm1)
nj1 = 5; m = 30
time = rep(1:nj1, times=m)
groups = as.factor(rep(1:m, each=nj1))
dat1 = rsmsn.clmm(time, groups, cbind(1,time), rep(1,m*nj1), sigma2=0.7,
                  D=0.5*diag(1), beta=c(1,2), depStruct="UNC")
fm1 = smn.clmm(dat1, formFixed=y~x, groupVar="ind", formRandom=~1,
               depStruct="UNC", ci="ci", lcl="lcl", ucl="ucl")
summary(fm1)

Summary of a smsn.lmm object

Description

summary method for class "SMSN".

Usage

## S3 method for class 'SMSN'
summary(object, confint.level = 0.95, ...)
## S3 method for class 'SMSN'
summary(object, confint.level = 0.95, ...)

Arguments

`object`	An object inheriting from class `SMSN`, representing a fitted scale mixture skew-normal linear mixed model.
`confint.level`	Level of the approximate confidence intervals presented.
`...`	Additional arguments.

Value

`varRandom`	Estimated variance matrix from random effects ( $D$ ).
`varFixed`	Parameter estimates of variance from random errors ( $\Sigma$ ). For recovering the error variance matrix use `errorVar` function.
`tableFixed`	Estimated fixed effects, their standard errors and approximated confidence intervals.
`criteria`	Maximum log-likelihood value, AIC and BIC criteria.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

Examples

fm1 = smsn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject",
               control=lmmControl(tol=.0001))
summary(fm1)
fm1 = smsn.lmm(distance ~ age+Sex, data=nlme::Orthodont, groupVar="Subject",
               control=lmmControl(tol=.0001))
summary(fm1)

Update for SMSN/SMN/SMNclmm objects

Description

It fits a SM(S)N-(C)LMM by updating a fitted object.

Usage

## S3 method for class 'SMN'
update(object, ..., evaluate = TRUE)

## S3 method for class 'SMSN'
update(object, ..., evaluate = TRUE)

## S3 method for class 'SMNclmm'
update(object, ..., evaluate = TRUE)
## S3 method for class 'SMN'
update(object, ..., evaluate = TRUE)

## S3 method for class 'SMSN'
update(object, ..., evaluate = TRUE)

## S3 method for class 'SMNclmm'
update(object, ..., evaluate = TRUE)

Arguments

`object`	A smsn.lmm, smn.lmm or smn.clmm object containing the fitted model to be updated.
`...`	Arguments to be changed.
`evaluate`	A logical value indicating if the new class should be evaluated. If `FALSE`, the call is returned.

Value

An object resulting from the smsm.lmm(), smn.lmm() or smn.clmm() function.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos, Victor H. Lachos and Katherine L. Valeriano

Examples

fm1 = smn.lmm(nlme::Orthodont, formFixed=distance ~ age+Sex, groupVar="Subject")
fm2 = update(fm1, distr="t")
fm2
fm1 = smn.lmm(nlme::Orthodont, formFixed=distance ~ age+Sex, groupVar="Subject")
fm2 = update(fm1, distr="t")
fm2

Data set for Unstructured Treatment Interruption Study

Description

Data set from a study of Unstructured Treatment Interruption in HIV-infected adolescents in four institutions in the USA. The main outcome is the HIV-1 RNA viral load, which is subject to censoring below the lower limit of detection of the assay (50 copies/mL). The censored observations are indicated by the variable RNAcens.

Usage

data(UTIdata)
data(UTIdata)

Format

This data frame contains the following columns:

Patid: patient ID
Days.after.TI: days after treatment interruption
Fup: follow-up months
RNA: viral load RNA
RNAcens: censoring indicator for viral load

Details

This dataset was copied from the lmec package, which was discontinued from CRAN in May 2022.

Source

Saitoh, A., Foca, M, et al. (2008), Clinical outcome in perinatally acquired HIV-infected children and adolescents after unstructured treatment interruption, Pediatrics,121, e513-e521.

Examples

library(ggplot2)

data(UTIdata)
ggplot(UTIdata) + geom_line(aes(x=Fup, y=log10(RNA), group=Patid)) +
  geom_hline(yintercept=log10(50), color="red", linetype="dashed") +
  geom_hline(yintercept=log10(400), color="red", linetype="dashed") +
  labs(x="Time", y=bquote(log["10"]("RNA"))) + theme_bw()

# Proportion of censoring
prop.table(table(UTIdata$RNAcens))
library(ggplot2)

data(UTIdata)
ggplot(UTIdata) + geom_line(aes(x=Fup, y=log10(RNA), group=Patid)) +
  geom_hline(yintercept=log10(50), color="red", linetype="dashed") +
  geom_hline(yintercept=log10(400), color="red", linetype="dashed") +
  labs(x="Time", y=bquote(log["10"]("RNA"))) + theme_bw()

# Proportion of censoring
prop.table(table(UTIdata$RNAcens))

Weight plot for smn.lmm or smsn.lmm object

Description

Estimated weights versus Mahalanobis distance plot

Usage

weight_plot(object)
weight_plot(object)

Arguments

object

An object inheriting from class SMN or SMSN, representing a fitted scale mixture of (skew) normal linear mixed model.

Value

A ggplot object.

Author(s)

Fernanda L. Schumacher, Larissa A. Matos and Victor H. Lachos

Examples

fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont,
              groupVar="Subject", distr="t")
weight_plot(fm1)
fm1 = smn.lmm(distance ~ age+Sex, data=nlme::Orthodont,
              groupVar="Subject", distr="t")
weight_plot(fm1)

Package 'skewlmm'

Help Index

Autocorrelation function for smn.lmm or smsn.lmm residuals

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Extract confidence intervals from lmmBoot object

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Parametric bootstrap for SMSN/SMN objects

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Examples

Extract coefficients from smsn.lmm, smn.lmm and smn.clmm objects

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Computes confidence intervals from smn.lmm and smsn.lmm fitted models

Description

Usage

Arguments

Value

See Also

Examples

Extracts criteria for model comparison of SMSN/SMN/SMNclmm objects

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Error scale matrix associated with times

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Extract smn.lmm fitted values

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Extract smn.clmm fitted values

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Extract smsn.lmm fitted values

Description

Extract confidence intervals from `lmmBoot` object

Error scale matrix associated with `times`

Control options for `smsn.lmm()`, `smn.lmm()` and `smn.clmm()`