Type:	Package
Title:	Density Regression via Dirichlet Process Mixtures of Normal Structured Additive Regression Models
Version:	1.0-1
Date:	2025-01-30
Imports:	stats, grDevices, graphics, splines, moments, Matrix, parallel, MASS
Depends:	R (≥ 3.5.0)
Description:	Implements a flexible, versatile, and computationally tractable model for density regression based on a single-weights dependent Dirichlet process mixture of normal distributions model for univariate continuous responses. The model assumes an additive structure for the mean of each mixture component and the effects of continuous covariates are captured through smooth nonlinear functions. The key components of our modelling approach are penalised B-splines and their bivariate tensor product extension. The proposed method can also easily deal with parametric effects of categorical covariates, linear effects of continuous covariates, interactions between categorical and/or continuous covariates, varying coefficient terms, and random effects. Please see Rodriguez-Alvarez, Inacio et al. (2025) for more details.
License:	GPL-2 | GPL-3 [expanded from: GPL]
NeedsCompilation:	no
Packaged:	2025-01-30 11:41:24 UTC; mrodriguez
Author:	Maria Xose Rodriguez-Alvarez [aut, cre], Vanda Inacio [aut]
Maintainer:	Maria Xose Rodriguez-Alvarez <mxrodriguez@uvigo.gal>
Repository:	CRAN
Date/Publication:	2025-01-31 11:10:09 UTC

Density Regression via Dirichlet Process Mixtures of Normal Structured Additive Regression Models

Description

Implements a flexible, versatile, and computationally tractable model for density regression based on a single-weights dependent Dirichlet process mixture of normal distributions model for univariate continuous responses. The model assumes an additive structure for the mean of each mixture component and the effects of continuous covariates are captured through smooth nonlinear functions. The key components of our modelling approach are penalised B-splines and their bivariate tensor product extension. The proposed method can also easily deal with parametric effects of categorical covariates, linear effects of continuous covariates, interactions between categorical and/or continuous covariates, varying coefficient terms, and random effects. Please see Rodriguez-Alvarez, Inacio et al. (2025) for more details.

Details

Index of help topics:

DDPstar                 Density Regression via Dirichlet Process
                        Mixtures (DDP) of Normal Structured Additive
                        Regression (STAR) Models
DDPstar-package         Density Regression via Dirichlet Process
                        Mixtures of Normal Structured Additive
                        Regression Models
dde                     Dichlorodiphenyldichloroethylene (DDE) and
                        preterm delivery data
f                       Defining smooth terms in DDPstar formulae
mcmccontrol             Markov chain Monte Carlo (MCMC) parameters
predict.DDPstar         Predictions from fitted DDPstar models
predictive.checks.DDPstar
                        Posterior predictive checks.
print.DDPstar           Print method for DDPstar objects
priorcontrol            Prior information for the DDPstar model
quantileResiduals       Quantile residuals.
rae                     Defining random effects in DDPstar formulae
summary.DDPstar         Summary method for DDPstar objects

Author(s)

Maria Xose Rodriguez-Alvarez [aut, cre] (<https://orcid.org/0000-0002-1329-9238>), Vanda Inacio [aut] (<https://orcid.org/0000-0001-8084-1616>)

Maintainer: Maria Xose Rodriguez-Alvarez <mxrodriguez@uvigo.gal>

References

Eilers, P.H.C. and Marx, B.D. (1996). Flexible smoothing with B-splines and penalties. Statistical Science, 11(2), 89-121.

Eilers, P.H.C. and Marx, B.D. (2003). Multidimensional calibration with temperature interaction using two- dimensional penalized signal regression. Chemometrics and Intelligence Laboratory Systems, 66, 159-174.

De Iorio, M., Muller, P., Rosner, G. L., and MacEachern, S. N. (2004). An anova model for dependent random measures. Journal of the American Statistical Association, 99(465), 205-215

De Iorio, M., Johnson, W. O., Muller, P., and Rosner, G. L. (2009). Bayesian nonparametric nonproportional hazards survival modeling. Biometrics, 65, 762–775.

Lang, S. and Brezger, A. (2004). Bayesian P-splines. Journal of Computational and Graphical Statistics, 13(1), 183-212.

Lee, D.-J., Durban, M., and Eilers, P. (2013). Efficient two-dimensional smoothing with P- spline ANOVA mixed models and nested bases. Computational Statistics & Data Analysis, 61, 22-37.

Rodriguez-Alvarez, M. X, Inacio, V. and Klein N. (2025). Density regression via Dirichlet process mixtures of normal structured additive regression models. Accepted for publication in Statistics and Computing (DOI: 10.1007/s11222-025-10567-0).

Density Regression via Dirichlet Process Mixtures (DDP) of Normal Structured Additive Regression (STAR) Models

Description

This function estimates the DDPstar model proposed by Rodriguez-Alvarez, Inacio, et al. (2025).

Usage

DDPstar(formula, data, standardise = TRUE, compute.lpml = FALSE, compute.WAIC = FALSE, 
  compute.DIC = FALSE, prior = priorcontrol(), mcmc = mcmccontrol(), verbose = FALSE)

Arguments

Details

Our DDPstar model for the conditional density takes the following form

p(y|\boldsymbol{x}) = \sum_{l=1}^{L}\omega_{l}\phi(y\mid\mu_{l}(\boldsymbol{x}),\sigma_{l}^2),

where \phi(y|\mu, \sigma^2) denotes the density function of the normal distribution, evaluated at y, with mean \mu and variance \sigma^2. The regression function \mu_{l}(\boldsymbol{x}) can incorportate both linear and nonlinear (through P-splines) effects of continuous covariates, random effects, categorical covariates (factors) as well as interactions. Interactions between categorical and (nonlinear) continuous covariates are also allowed (factor-by curve interactions).

The mean (regression function) of each mixture component, \mu_l(\cdot), is compactly specified (see Rodriguez-Alvarez, Inacio, et al. (2025) for details) as

\mu_{l}(\boldsymbol{x}) = \underbrace{\mathbf{x}^{\top}\boldsymbol{\beta}_{l}}_{\mbox{Parametric}} + \underbrace{\sum_{r = 1}^{R}\mathbf{z}_{r}(\boldsymbol{v}_r)^{\top}\boldsymbol{\gamma}_{lr}}_{\mbox{Smooth/Random}}

where \mathbf{x}^{\top}\boldsymbol{\beta}_{l} contains all parametric effects, including, by default, the intercept, parametric effects of categorical covariates and linear or parametric effects of continuous covariates and their interactions, as well as, the unpenalised terms associated with the P-splines formulation, and \sum_{r = 1}^{R}\mathbf{z}_{r}(\boldsymbol{v}_r)^{\top}\boldsymbol{\gamma}_{lr} contains the P-splines' penalised terms, as well as, the random effects.

In our DDPstar model, L is a pre-specified upper bound on the number of mixture components. The \omega_{l}'s result from a truncated version of the stick-breaking construction (\omega_{1} = \eta_{1}; \omega_{l} = \eta_{l}\prod_{r<l}(1-\eta{r}), l=2,\ldots,L; \eta_{1},\ldots,\eta_{L-1}\sim Beta (1,\alpha); \eta_{L} = 1). Further, for \boldsymbol{\beta}_{l} we consider

p\left(\boldsymbol{\beta}_{l} \mid \mathbf{m}, \mathbf{S}\right) \propto \exp\left(-\frac{1}{2}\left(\boldsymbol{\beta}_{l} - \mathbf{m}\right)^{\top}\mathbf{S}^{-1}\left(\boldsymbol{\beta}_{l} - \mathbf{m}\right)\right),

with conjugate hyperpriors \mathbf{m} \sim N_{Q}(\mathbf{m}_{0},\mathbf{S}_{0}) and \mathbf{S}^{-1}\sim W_{Q}(\nu,(\nu\Psi)^{-1}), where W(\nu,(\nu\Psi)^{-1}) denotes a Wishart distribution with \nu degrees of freedom and expectation \Psi^{-1}, and Q denotes the length of vector \boldsymbol{\beta}_{l}. Regarding the prior distributions for the P-splines' penalised coefficients and random effect coefficients we have

p(\boldsymbol{\gamma}_{lr}\mid \tau_{lr}^{2}) \propto \exp\left(-\frac{1}{2\tau_{lr}^{2}}\boldsymbol{\gamma}_{lr}^{\top}\mathbf{K}_{r}\boldsymbol{\gamma}_{lr}\right),

where the specific forms of \mathbf{K}_{r} depend on the components type (see Rodriguez-Alvarez, Inacio, et al. (2025) for details). Finally, \sigma_{l}^{-2}\sim\Gamma(a_{\sigma^2},b_{\sigma^2}), \tau_{lr}^{-2}\sim\Gamma(a_{\tau^2},b_{\tau^2}) and \alpha \sim \Gamma(a_{\alpha}, b_{\alpha}), where \Gamma(a,b) denotes a Gamma distribution with shape parameter a and rate parameter b. For a detailed description, we refer to Rodriguez-Alvarez, Inacio, et al. (2025).

It is worth referring that with respect to the computation of the DIC, when L=1, it is computed as in Spiegelhalter et al. (2002), and when L>1, DIC3 as described in Celeux et al. (2006) is computed. Also, for the computation of the conditional predictive ordinates (CPO) we follow the stable version proposed by Gelman et al. (2014).

Value

An object of class DDPstar. It is a list containing the following objects:

Note

The input argument formula is similar to that used for the glm function, except that flexible specifications can be added by means of function f. For instance, specification y ~ x1 + f(x2) would assume a linear effect of x1 (if x1 is continuous) and the effect of x2 (assumed continuous)) would be modeled using Bayesian P-splines. Categorical variables (factors) can be also incorporated, as well as random effects (see rae) and interaction terms. For example, to include the factor-by-curve interaction between x4 (continuous covariate) and x3 (categorical covariate) we need to specify y ~ x3 + f(x4, by = x3).

References

Celeux, G., Forbes, F., Robert C. P., and Titerrington, D. M. (2006). Deviance information criteria for missing data models. Bayesian Analysis, 1, 651-674.

Eilers, P.H.C. and Marx, B.D. (1996). Flexible smoothing with B-splines and penalties. Statistical Science, 11(2), 89-121.

Eilers, P.H.C. and Marx, B.D. (2003). Multidimensional calibration with temperature interaction using two- dimensional penalized signal regression. Chemometrics and Intelligence Laboratory Systems, 66, 159-174.

De Iorio, M., Muller, P., Rosner, G. L., and MacEachern, S. N. (2004). An anova model for dependent random measures. Journal of the American Statistical Association, 99(465), 205-215

De Iorio, M., Johnson, W. O., Muller, P., and Rosner, G. L. (2009). Bayesian nonparametric nonproportional hazards survival modeling. Biometrics, 65, 762–775.

Geisser, S. and Eddy, W.F. (1979) A Predictive Approach to Model Selection, Journal of the American Statistical Association, 74, 153-160.

Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., and Rubin, D.B. (2014). Bayesian Data Analysis, 3rd ed. CRC Press: Boca Raton, FL.

Gelman, A., Hwang, J., and Vehtari, A. (2014). Understanding predictive information criteria for Bayesian models. Statistics and Computing, 24, 997-1010.

Lang, S. and Brezger, A. (2004). Bayesian P-splines. Journal of Computational and Graphical Statistics, 13(1), 183-212.

Lee, D.-J., Durban, M., and Eilers, P. (2013). Efficient two-dimensional smoothing with P- spline ANOVA mixed models and nested bases. Computational Statistics & Data Analysis, 61, 22-37.

Rodriguez-Alvarez, M. X, Inacio, V. and Klein N. (2025). Density regression via Dirichlet process mixtures of normal structured additive regression models. Accepted for publication in Statistics and Computing (DOI: 10.1007/s11222-025-10567-0).

Speigelhalter, D. J., Best, N. G., Carlin, B. P., and van der Linde, A. (2002). Bayesian measures of model comparison and fit. Journal of the Royal Statistical Society, Ser. B, 64, 583-639.

Watanabe, S. (2010). Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory. Journal of Machine Learning Research, 11, 3571-3594.

See Also

f, rae, predict.DDPstar

Examples

library(DDPstar)
data(dde)
dde$GAD <- dde$GAD/7 # GAD in weeks

formula.dde <- GAD ~ f(DDE, bdeg = 3, nseg = 20, pord = 2, atau = 1, btau = 0.005)
mcmc <- mcmccontrol(nburn = 20000, nsave = 15000, nskip = 1)
prior <- priorcontrol(a = 2, b = 0.5, aalpha = 2, balpha = 2, L = 20)

set.seed(10) # For reproducibility
fit_dde <- DDPstar(formula = formula.dde, 
  data = dde, mcmc = mcmc, prior = prior, 
  standardise = TRUE, compute.lpml = TRUE, compute.WAIC = TRUE, compute.DIC = TRUE)
summary(fit_dde)

Dichlorodiphenyldichloroethylene (DDE) and preterm delivery data

Description

Data set associated with a study aimed to evaluate the association between maternal serum concentration of the DDT metabolite DDE and preterm delivery. For details, see Longnecker et al. (2001).

Usage

data("dde")

Format

A data frame with 2312 observations on the following 2 variables.

DDE

the level of the Dichlorodiphenyldichloroethylene (DDE)

GAD

the gestational age at delivery (GAD), in days.

Source

The dataset can be downloaded from https://github.com/tommasorigon/LSBP/blob/master/Tutorial/dde.RData.

References

Longnecker, M. P., Klebanoff, M. A., Zhou, H., and Brock, J. W. (2001). Association between maternal serum concentration of the DDT metabolite DDE and preterm and small-for-gestational-age babies at birth. The Lancet, 358(9276):110-114.

Examples

library(DDPstar)
data(dde)
summary(dde)

Defining smooth terms in DDPstar formulae

Description

Auxiliary function used to define smooth terms using Bayesian P-splines within DDPstar model formulae. The function does not evaluate the smooth - it exists purely to help set up a model using P-spline based smoothers.

Usage

f(..., by = NULL, nseg = 5, bdeg = 3, pord = 2, atau = 1, btau = 0.005,
  prior.2D = c("anisotropic", "isotropic"))

Arguments

Details

The functions f() is designed to represent either one dimensional smooth functions for main effects of continuous explanatory variables or two dimensional smooth functions representing two way interactions of continuous variables using Bayesian P-splines. By default, the values of the arguments nseg, pord and bdeg are repeated to the length of the explanatory covariates. The two dimensional smooth terms are constructed using the tensor-product of marginal (one-dimensional) B-spline bases (Eilers and Marx, 2003). In this case, the Bayesian extension of the PS-ANOVA model by Lee et al. (2013) is considered when prior.2D = "anisotropic". It is also possible to include factor-by-curve interactions between a continuous covariate (e.g., x1) and a categorical covariate (e.g., x2) by means of argument by: y ~ x2 + f(x1, by = x2).

Value

The function is interpreted in the formula of a DDPstar model and creates the right framework for fitting the smoother. List containing (among others) the following elements:

References

Eilers, P.H.C. and Marx, B.D. (1996). Flexible smoothing with B-splines and penalties. Statistical Science, 11(2), 89-121.

Eilers, P.H.C. and Marx, B.D. (2003). Multidimensional calibration with temperature interaction using two- dimensional penalized signal regression. Chemometrics and Intelligence Laboratory Systems, 66, 159-174.

Lang, S. and Brezger, A. (2004). Bayesian P-splines. Journal of Computational and Graphical Statistics, 13(1), 183-212.

Lee, D.-J., Durban, M., and Eilers, P. (2013). Efficient two-dimensional smoothing with P- spline ANOVA mixed models and nested bases. Computational Statistics & Data Analysis, 61, 22-37.

See Also

rae, DDPstar

Examples

library(DDPstar)
data(dde)
dde$GAD <- dde$GAD/7 # GAD in weeks

formula.dde <- GAD ~ f(DDE, bdeg = 3, nseg = 20, pord = 2, atau = 1, btau = 0.005)

set.seed(10) # For reproducibility
fit_dde <- DDPstar(formula = formula.dde, 
  data = dde, mcmc = list(nburn = 20000, nsave = 15000, nskip = 1), 
  prior = list(a = 2, b = 0.5, aalpha = 2, balpha = 2, L = 20), 
  standardise = TRUE, compute.lpml = TRUE, compute.WAIC = TRUE, compute.DIC = TRUE)
summary(fit_dde)

Markov chain Monte Carlo (MCMC) parameters

Description

This function is used to set various parameters controlling the Markov chain Monte Carlo (MCMC) parameters.

Usage

mcmccontrol(nsave = 8000, nburn = 2000, nskip = 1)

Arguments

Details

The value returned by this function is used as a control argument of the DDPstar function.

Value

A list with components for each of the possible arguments.

See Also

DDPstar

Examples

library(DDPstar)
data(dde)
dde$GAD <- dde$GAD/7 # GAD in weeks

mcmc <- mcmccontrol(nburn = 20000, nsave = 15000, nskip = 1)

set.seed(10) # For reproducibility
fit_dde <- DDPstar(formula = GAD ~ f(DDE, bdeg = 3, nseg = 20, pord = 2, atau = 1, btau = 0.005), 
  data = dde, mcmc = mcmc, prior = list(a = 2, b = 0.5, aalpha = 2, balpha = 2, L = 20), 
  standardise = TRUE, compute.lpml = TRUE, compute.WAIC = TRUE, compute.DIC = TRUE)
summary(fit_dde)

Predictions from fitted DDPstar models

Description

Takes a fitted DDPstar object produced by DDPstar() and produces predictions for different functionals given a new set of values for the model covariates.

Usage

## S3 method for class 'DDPstar'
predict(object, what = NULL, newdata, reg.select = NULL, den.grid = NULL, 
  quant.probs = NULL, q.value = NULL, 
  parallel = c("no", "multicore", "snow"), ncpus = 1, cl = NULL, ...)

Arguments

Value

A list with the following components (npred is the number of columns in newdata and nsave is the number of Gibbs sampler iterations saved)

See Also

DDPstar

Examples

library(DDPstar)
data(dde)
dde$GAD <- dde$GAD/7 # GAD in weeks

set.seed(10) # For reproducibility
fit_dde <- DDPstar(formula = GAD ~ f(DDE, bdeg = 3, nseg = 20, pord = 2, atau = 1, btau = 0.005), 
 data = dde, mcmc = list(nburn = 20000, nsave = 15000, nskip = 1), 
  prior = list(a = 2, b = 0.5, aalpha = 2, balpha = 2, L = 20), 
  standardise = TRUE, compute.lpml = TRUE, compute.WAIC = TRUE)
 
# Data frame for predictions (regression, variance, quartiles and probabilities functions)
df_pred <- data.frame(DDE = seq(min(dde$DDE), max(dde$DDE), length = 50))

# Data frame for conditional densities
# Compute the densities for 4 different DDE values
df_pred_den <- data.frame(DDE = quantile(dde$DDE, c(0.1, 0.6, 0.9, 0.99)))
sequenceGAD <- seq(min(dde$GAD), max(dde$GAD), length = 100)
 
# Regression and variance function
  reg_var_DDE <- predict(fit_dde, newdata = df_pred, 
    what = c("regfun", "varfun"), 
    parallel = "multicore", 
    ncpus = 2)
    # Regression function
    reg_m  <- apply(reg_var_DDE$regfun, 1, median) 
    reg_ql <- apply(reg_var_DDE$regfun, 1, quantile, 0.025) 
    reg_qh <- apply(reg_var_DDE$regfun, 1, quantile, 0.975)

    plot(dde$DDE, dde$GAD, xlab = "DDE (mg/L)", 
      ylab = "Gestational age at delivery (in weeks)",
      main = "Regression function")
    lines(df_pred$DDE, reg_m, lwd = 2)
    lines(df_pred$DDE, reg_ql, lty = 2)
    lines(df_pred$DDE, reg_qh, lty = 2)

    # Variance function
    var_m  <- apply(reg_var_DDE$varfun, 1, median) 
    var_ql <- apply(reg_var_DDE$varfun, 1, quantile, 0.025) 
    var_qh <- apply(reg_var_DDE$varfun, 1, quantile, 0.975)

    plot(df_pred$DDE, var_m, type = "l", lwd = 2,
      xlab = "DDE (mg/L)", 
      ylab = "",
      main = "Variance function")
    lines(df_pred$DDE, var_ql, lty = 2)
    lines(df_pred$DDE, var_qh, lty = 2)

# Quartiles
  p <- c(0.25, 0.5, 0.75)
  quart_DDE <- predict(fit_dde, newdata = df_pred, 
    what = "quantfun", quant.probs = p, 
    parallel = "multicore", ncpus = 2)
  
  quart_m <- apply(quart_DDE$quantfun , c(1,3), median)
  quart_ql <- apply(quart_DDE$quantfun , c(1,3), quantile, 0.025)
  quart_qh <- apply(quart_DDE$quantfun , c(1,3), quantile, 0.975)
  
  plot(dde$DDE, dde$GAD, xlab = "DDE (mg/L)", 
      ylab = "Gestational age at delivery (in weeks)",
      main = "Conditional quartiles")
  
  # Q1    
  lines(df_pred$DDE, quart_m[,1], lwd = 2, col = 2)
  lines(df_pred$DDE, quart_ql[,1], lty = 2, col = 2)
  lines(df_pred$DDE, quart_qh[,1], lty = 2, col = 2)
  
  # Median
  lines(df_pred$DDE, quart_m[,2], lwd = 2)
  lines(df_pred$DDE, quart_ql[,2], lty = 2)
  lines(df_pred$DDE, quart_qh[,2], lty = 2)
  
  # Q3
  lines(df_pred$DDE, quart_m[,3], lwd = 2, col = 3)
  lines(df_pred$DDE, quart_ql[,3], lty = 2, col = 3)
  lines(df_pred$DDE, quart_qh[,3], lty = 2, col = 3)

# Conditional densities
  den_DDE <- predict(fit_dde, newdata = df_pred_den,
    what = "denfun", den.grid = sequenceGAD,
    parallel = "multicore", ncpus = 2)
  
  den_m  <- apply(den_DDE$denfun, c(1,2), median)
  den_ql <- apply(den_DDE$denfun, c(1,2), quantile, 0.025)
  den_qh <- apply(den_DDE$denfun, c(1,2), quantile, 0.975)
  
  op <- par(mfrow = c(2,2))
  plot(sequenceGAD, den_m[,1], type = "l", lwd = 2,
    xlab = "DDE (mg/L)",
    ylab = "",
    main = "DDE = 12.57 (10th percentile)")
  lines(sequenceGAD, den_ql[,1], lty = 2)
  lines(sequenceGAD, den_qh[,1], lty = 2)

  plot(sequenceGAD, den_m[,2], type = "l", lwd = 2,
  xlab = "DDE (mg/L)",
    ylab = "",
    main = "DDE = 28.44 (60th percentile)")
  lines(sequenceGAD, den_ql[,2], lty = 2)
  lines(sequenceGAD, den_qh[,2], lty = 2)

  plot(sequenceGAD, den_m[,3], type = "l", lwd = 2,
    xlab = "DDE (mg/L)",
    ylab = "",
    main = "DDE = 53.72 (90th percentile)")
  lines(sequenceGAD, den_ql[,3], lty = 2)
  lines(sequenceGAD, den_qh[,3], lty = 2)
  plot(sequenceGAD, den_m[,4], type = "l", lwd = 2,
    xlab = "DDE (mg/L)",
    ylab = "",
    main = "DDE = 105.47 (99th percentile)")
  lines(sequenceGAD, den_ql[,4], lty = 2)
  lines(sequenceGAD, den_qh[,4], lty = 2) 
  par(op)

# Conditional probabilities
  probs_DDE <- predict(fit_dde, newdata = df_pred, 
    what = "probfun", q.value = c(37), 
    parallel = "multicore", ncpus = 2)
  
  prob_m  <- apply(probs_DDE$probfun, c(1,2), median)
  prob_ql <- apply(probs_DDE$probfun, c(1,2), quantile, 0.025)
  prob_qh <- apply(probs_DDE$probfun, c(1,2), quantile, 0.975)
 
  plot(df_pred$DDE, prob_m, type = "l", lwd = 2, ylim = c(0, 1),
      xlab = "DDE (mg/L)", 
      ylab = "P(GAD < 37 | DDE)",
      main = "Conditional Probabilities (premature delivery)")
  lines(df_pred$DDE, prob_ql, lty = 2)
  lines(df_pred$DDE, prob_qh, lty = 2)

Posterior predictive checks.

Description

Implements posterior predictive checks for objects of class DDPstar as produced by function DDPstar.

Usage

predictive.checks.DDPstar(object, statistics = c("min", "max", "kurtosis", "skewness"), 
  devnew = TRUE, ndensity = 512, trim = 0.025)

Arguments

Details

Compares a selected test statistic computed based on the observed response variable against the same test statistics computed based on simulated data from the posterior predictive distribution of the response variable obtained using a DDPstar model. The following graphics are depicted: (1) histograms of the desired statistics computed from simulated datasets (nsave of them) from the posterior predictive distribution of the response variable. In these plots, the estimated statistics from the observed response variable are also depicted. (2) Kernel density estimates computed computed from simulated datasets (nsave of them) from the posterior predictive distribution of the response variable. In these plots, the kernel density estimate of the observed response variable is also depicted. For a detailed discussion about predictive checks, see Gabry et al. (2019).

Value

As a result, the function provides a list with the following components:

References

Gabry, J., Simpson, D., Vehtari, A., Betancourt, M., and Gelman, A. (2019). Visualization in Bayesian workflow. Journal of the Royal Statistical Society, Series A, 182, 1-14.

See Also

DDPstar

Examples

library(DDPstar)
data(dde)
dde$GAD <- dde$GAD/7 # GAD in weeks

set.seed(10) # For reproducibility
fit_dde <- DDPstar(formula = GAD ~ f(DDE, bdeg = 3, nseg = 20, pord = 2, atau = 1, btau = 0.005), 
  data = dde, mcmc = list(nburn = 20000, nsave = 15000, nskip = 1), 
  prior = list(a = 2, b = 0.5, aalpha = 2, balpha = 2, L = 20), 
  standardise = TRUE, compute.lpml = TRUE, compute.WAIC = TRUE)
op <- par(mfrow = c(2,3))
predictive.checks.DDPstar(fit_dde)
par(op)

Print method for DDPstar objects

Description

Default print method for objects fitted with function DDPstar().

Usage

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

Arguments

Details

A short summary is printed.

Value

No return value, called for side effects

See Also

DDPstar

Examples

library(DDPstar)
data(dde)
dde$GAD <- dde$GAD/7 # GAD in weeks

set.seed(10) # For reproducibility
fit_dde <- DDPstar(formula = GAD ~ f(DDE, bdeg = 3, nseg = 20, pord = 2, atau = 1, btau = 0.005), 
  data = dde, mcmc = list(nburn = 20000, nsave = 15000, nskip = 1), 
  prior = list(a = 2, b = 0.5, aalpha = 2, balpha = 2, L = 20), 
  standardise = TRUE, compute.lpml = TRUE, compute.WAIC = TRUE, compute.DIC = TRUE)
fit_dde

Prior information for the DDPstar model

Description

This function is used to set various parameters controlling the prior information to be used in the DDPstar function.

Usage

priorcontrol(m0 = NA, S0 = NA, nu = NA, Psi = NA, 
  atau = 1, btau = 0.005, a = 2, b = NA, alpha.fixed = FALSE, 
  alpha = 1, aalpha = 2, balpha = 2, L = 10)

Arguments

Value

A list with components for each of the possible arguments.

See Also

DDPstar

Examples

library(DDPstar)
data(dde)
dde$GAD <- dde$GAD/7 # GAD in weeks

prior <- priorcontrol(a = 2, b = 0.5, aalpha = 2, balpha = 2, L = 20)

set.seed(10) # For reproducibility
fit_dde <- DDPstar(formula = GAD ~ f(DDE, bdeg = 3, nseg = 20, pord = 2, atau = 1, btau = 0.005), 
  data = dde, mcmc = list(nburn = 20000, nsave = 15000, nskip = 1), prior = prior, 
  standardise = TRUE, compute.lpml = TRUE, compute.WAIC = TRUE, compute.DIC = TRUE)
summary(fit_dde)

Quantile residuals.

Description

Produces quantile residuals for objects of class DDPstar as produced by function DDPstar.

Usage

## S3 method for class 'DDPstar'
quantileResiduals(object, parallel = c("no", "multicore", "snow"), 
  ncpus = 1, cl = NULL, ...)

Arguments

Details

Quantile residuals (Dunn and Smyth, 1996) are based on the well-known fact that for a continuous random variable, say Y, with CDF given by F, one has that F(Y) \sim U(0,1). As a consequence, quantile residuals defined by \hat{r}_j = \Phi^{-1}(\hat{F}(y_j)), j = 1, \ldots, n, should follow, approximately, a standard normal distribution if a correct model has been specified. A quantile-quantile (QQ) plot can then be used to determine deviations of the quantile residuals from the standard normal distribution.

Value

As a result, the function provides a list with the following components:

References

Dunn, P. K. and Smyth, G. K. (1996). Randomized Quantile Residuals. Journal of Computational and Graphical Statistics, 5(3), 236-244.

See Also

DDPstar

Examples

library(DDPstar)
data(dde)
dde$GAD <- dde$GAD/7 # GAD in weeks

set.seed(10) # For reproducibility
fit_dde <- DDPstar(formula = GAD ~ f(DDE, bdeg = 3, nseg = 20, pord = 2, atau = 1, btau = 0.005), 
  data = dde, mcmc = list(nburn = 20000, nsave = 15000, nskip = 1), 
  prior = list(a = 2, b = 0.5, aalpha = 2, balpha = 2, L = 20), 
  standardise = TRUE, compute.lpml = TRUE, compute.WAIC = TRUE)

qres <- quantileResiduals.DDPstar(fit_dde)

plot(qres$x, qres$y, xlab = "Theoretical Quantiles", ylab = "Sample Quantiles", 
  cex.main = 2, cex.lab = 1.5, cex.axis = 1.5)
lines(qres$x, qres$ymin, lty = 2, lwd = 2)
lines(qres$x, qres$ymax, lty = 2, lwd = 2)
abline(a = 0, b = 1, col ="red", lwd = 2) 

Defining random effects in DDPstar formulae

Description

Auxiliary function used to define random effects terms in a DDPstar model formula.

Usage

rae(x, atau = 1, btau = 0.005)

Arguments

Details

The functions is designed to represent random effects in DDPstar formulae.

Value

The function is interpreted in the formula of a DDPstar model and creates the right framework for fitting the random effect. List containing the following elements:

References

Rodriguez-Alvarez, M. X, Inacio, V. and Klein N. (2025). Density regression via Dirichlet process mixtures of normal structured additive regression models. Accepted for publication in Statistics and Computing (DOI: 10.1007/s11222-025-10567-0).

See Also

f, DDPstar

Examples

# For an example including random effects, we refer to Rodriguez-Alvarez, Inacio et al. (2025) 
# and associated codes (found in https://bitbucket.org/mxrodriguez/ddpstar)

Summary method for DDPstar objects

Description

Default summary method for objects fitted with function DDPstar.

Usage

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

Arguments

Details

The information printed includes: (1) the call to the function and (2) the sample size. If required, the function aso provides the log pseudo marginal likelihood (LPML), the widely applicable information criterion (WAIC) and/or the deviance information criterion (DIC).

Value

No return value, called for side effects

See Also

DDPstar

Examples

library(DDPstar)
data(dde)
dde$GAD <- dde$GAD/7 # GAD in weeks

set.seed(10) # For reproducibility
fit_dde <- DDPstar(formula = GAD ~ f(DDE, bdeg = 3, nseg = 20, pord = 2, atau = 1, btau = 0.005), 
  data = dde, mcmc = list(nburn = 20000, nsave = 15000, nskip = 1), 
  prior = list(a = 2, b = 0.5, aalpha = 2, balpha = 2, L = 20), 
  standardise = TRUE, compute.lpml = TRUE, compute.WAIC = TRUE, compute.DIC = TRUE)
summary(fit_dde)