gamlss: Generalised Additive Models for Location, Scale and Shape
In SemiParBIVProbit: Semiparametric Copula Regression Models

Description Usage Arguments Details Value WARNINGS Author(s) References See Also Examples

gamlss fits flexible univariate regression models with several continuous and discrete distributions, and types of covariate effects. The purpose of this function was only to provide, in some cases, starting values for the simultaneous models in the package, but it has now been made available in the form of a proper function should the user wish to fit univariate models using the general estimation approach of this package. The distributions implemented here have been parametrised according to Rigby and Stasinopoulos (2005).

gamlss(formula, data = list(), weights = NULL, subset = NULL,  
       margin = "N", surv = FALSE, cens = NULL,
       robust = FALSE, rc = 3, lB = NULL, uB = NULL, infl.fac = 1, 
       rinit = 1, rmax = 100, iterlimsp = 50, tolsp = 1e-07,
       gc.l = FALSE, parscale, extra.regI = "t", gev.par = -0.25,
       chunk.size = 10000)

`formula`	List of equations. This should contain one or more equations.
`data`	An optional data frame, list or environment containing the variables in the model. If not found in `data`, the variables are taken from `environment(formula)`, typically the environment from which `gamlss` is called.
`weights`	Optional vector of prior weights to be used in fitting.
`subset`	Optional vector specifying a subset of observations to be used in the fitting process.
`margin`	Possible distributions are normal ("N"), normal where sigma2 corresponds to the standard deviation instead of the variance ("N2"), log-normal ("LN"), Gumbel ("GU"), reverse Gumbel ("rGU"), logistic ("LO"), Weibull ("WEI"), inverse Gaussian ("iG"), gamma ("GA"), Dagum ("DAGUM"), Singh-Maddala ("SM"), beta ("BE"), Fisk ("FISK", also known as log-logistic distribution), Poisson ("PO"), zero truncated Poisson ("ZTP"), negative binomial - type I ("NBI"), negative binomial - type II ("NBII"), Poisson inverse Gaussian ("PIG"), generalised extreme value link function ("GEVlink", this is used for binary responses and is more stable and faster than the `R` package `bgeva`).
`surv`	If `TRUE` then a survival model is fitted. Here margin can be "PH" (generalised proportional hazards), "PO" (generalised proportional odds), "probit" (generalised probit).
`cens`	Binary censoring indicator. This is required when `surv = TRUE`.
`robust`	If `TRUE` then the robust version of the model is fitted.
`rc`	Robust constant.
`lB, uB`	Lower and upper bounds for integrals, when robust estimation is employed. If not provided then standard bounds are used.
`infl.fac`	Inflation factor for the model degrees of freedom in the approximate AIC. Smoother models can be obtained setting this parameter to a value greater than 1.
`rinit`	Starting trust region radius. The trust region radius is adjusted as the algorithm proceeds.
`rmax`	Maximum allowed trust region radius. This may be set very large. If set small, the algorithm traces a steepest descent path.
`iterlimsp`	A positive integer specifying the maximum number of loops to be performed before the smoothing parameter estimation step is terminated.
`tolsp`	Tolerance to use in judging convergence of the algorithm when automatic smoothing parameter estimation is used.
`gc.l`	This is relevant when working with big datasets. If `TRUE` then the garbage collector is called more often than it is usually done. This keeps the memory footprint down but it will slow down the routine.
`parscale`	The algorithm will operate as if optimizing objfun(x / parscale, ...) where parscale is a scalar. If missing then no rescaling is done. See the documentation of `trust` for more details.
`extra.regI`	If "t" then regularization as from `trust` is applied to the information matrix if needed. If different from "t" then extra regularization is applied via the options "pC" (pivoted Choleski - this will only work when the information matrix is semi-positive or positive definite) and "sED" (symmetric eigen-decomposition).
`gev.par`	GEV link parameter.
`chunk.size`	This is used for discrete robust models.

The underlying algorithm is based on the procedure used for SemiParBIVProbit(). For more details see ?SemiParBIVProbit.

There are many continuous/discrete distributions to choose from and we plan to include more options. Get in touch if you are interested in a particular distribution.

The "GEVlink" option is used for binary response additive models and is more stable and faster than the R package bgeva. This model has been incorporated into this package to take advantage of the richer set of smoother choices, and of the estimation approach. Details on the model can be found in Calabrese, Marra and Osmetti (2016).

The function returns an object of class gamlss as described in gamlssObject.

Convergence can be checked using conv.check which provides some information about the score and information matrix associated with the fitted model. The former should be close to 0 and the latter positive definite. gamlss() will produce some warnings if there is a convergence issue.

Convergence failure may sometimes occur. This is not necessarily a bad thing as it may indicate specific problems with a fitted model. In such a situation, the user may use some extra regularisation (see extra.regI) and/or rescaling (see parscale). However, the user should especially consider re-specifying/simplifying the model, and/or checking that the chosen distribution fits the response well. In our experience, we found that convergence failure typically occurs when the model has been misspecified and/or the sample size is low compared to the complexity of the model. It is also worth bearing in mind that the use of three parameter distributions requires the data to be more informative than a situation in which two parameter distributions are used instead.

Maintainer: Giampiero Marra giampiero.marra@ucl.ac.uk

Marra G. and Radice R. (2017), Bivariate Copula Additive Models for Location, Scale and Shape. Computational Statistics and Data Analysis, 112, 99-113.

Rigby R.A. and Stasinopoulos D.M. (2005). Generalized additive models for location, scale and shape (with discussion). Journal of the Royal Statistical Society, Series C, 54(3), 507-554.

Calabrese R., Marra G. and Osmetti SA (2016), Bankruptcy Prediction of Small and Medium Enterprises Using a Flexible Binary Generalized Extreme Value Model. Journal of the Operational Research Society, 67(4), 604-615.

plot.SemiParBIVProbit, SemiParBIVProbit-package, gamlssObject, conv.check, summary.gamlss, predict.SemiParBIVProbit

## Not run:  

library(SemiParBIVProbit)

set.seed(0)

n <- 400

x1 <- round(runif(n))
x2 <- runif(n)
x3 <- runif(n)
f1 <- function(x) cos(pi*2*x) + sin(pi*x)
y1 <- -1.55 + 2*x1 + f1(x2) + rnorm(n)

dataSim <- data.frame(y1, x1, x2, x3)
resp.check(y1, "N")

eq.mu <- y1 ~ x1 + s(x2) + s(x3)
eq.s2 <-    ~ s(x3)
fl    <- list(eq.mu, eq.s2)

out <- gamlss(fl, data = dataSim)

conv.check(out)
post.check(out)

plot(out, eq = 1, scale = 0, pages = 1, seWithMean = TRUE)
plot(out, eq = 2, seWithMean = TRUE)

summary(out)

AIC(out)
BIC(out)

################
# Robust example
################

eq.mu <- y1 ~ x1 + x2 + x3
fl    <- list(eq.mu)

out <- gamlss(fl, data = dataSim, margin = "N", robust = TRUE, 
                  rc = 3, lB = -Inf, uB = Inf)

conv.check(out)
summary(out)

##

eq.s2 <-    ~ x3
fl    <- list(eq.mu, eq.s2)

out <- gamlss(fl, data = dataSim, margin = "N", robust = TRUE)

conv.check(out)
summary(out)

##

eq.mu <- y1 ~ x1 + s(x2) + s(x3)
eq.s2 <-    ~ s(x3)
fl    <- list(eq.mu, eq.s2)

out1 <- gamlss(fl, data = dataSim, margin = "N", robust = TRUE)

conv.check(out1)
summary(out1)
AIC(out, out1)

plot(out1, eq = 1, all.terms = TRUE, pages = 1, seWithMean = TRUE)
plot(out1, eq = 2, seWithMean = TRUE)

##########################
## GEV link binary example
##########################
# this incorporates the bgeva
# model implemented in the bgeva package
# however this implementation is more general 
# stable and efficient

set.seed(0)

n <- 400

x1 <- round(runif(n)); x2 <- runif(n); x3 <- runif(n)

f1 <- function(x) cos(pi*2*x) + sin(pi*x)
f2 <- function(x) x+exp(-30*(x-0.5)^2)   

y  <- ifelse(-3.55 + 2*x1 + f1(x2) + rnorm(n) > 0, 1, 0)

dataSim <- data.frame(y, x1, x2, x3)

out1 <- gamlss(list(y ~ x1 + x2 + x3), margin = "GEVlink", data = dataSim)
out2 <- gamlss(list(y ~ x1 + s(x2) + s(x3)), margin = "GEVlink", data = dataSim)

conv.check(out1)
conv.check(out2)
summary(out1)
summary(out2)
AIC(out1, out2)
BIC(out1, out2)

plot(out2, eq = 1, all.terms = TRUE, pages = 1, seWithMean = TRUE)

##################
# prediction of Pr
##################

# Calculate eta (that is, X*model.coef)
# For a new data set the argument newdata should be used

eta <- predict(out2, eq = 1, type = "link")

# extract gev tail parameter

gev.par <- out2$gev.par

# multiply gev tail parameter by eta

gevpeta <- gev.par*eta 
  
# establish for which values the model is defined   

gevpetaIND <- ifelse(gevpeta < -1, FALSE, TRUE) 
gevpeta <- gevpeta[gevpetaIND]
    
# estimate probabilities  

pr <- exp(-(1 + gevpeta)^(-1/gev.par))

##################################
## Flexible survival model example
##################################

library(SemiParBIVProbit)

########################################
## Simulate proportional hazards data ##
########################################

set.seed(0)
n  <- 2000
c  <- runif(n, 3, 8)
u  <- runif(n, 0, 1)
z1 <- rbinom(n, 1, 0.5)
z2 <- runif(n, 0, 1)
t <- rep(NA, n)

beta_0 <- -0.2357
beta_1 <- 1

f <- function(t, beta_0, beta_1, u, z1, z2){ 
  S_0 <- 0.7 * exp(-0.03*t^1.9) + 0.3*exp(-0.3*t^2.5)
  exp(-exp(log(-log(S_0))+beta_0*z1 + beta_1*z2))-u
}


for (i in 1:n){
   t[i] <- uniroot(f, c(0, 8), tol = .Machine$double.eps^0.5, 
                   beta_0 = beta_0, beta_1 = beta_1, u = u[i], 
                   z1 = z1[i], z2 = z2[i], extendInt = "yes" )$root
}

delta   <- ifelse(t < c, 1, 0)
u       <- apply(cbind(t, c), 1, min)
dataSim <- data.frame(u, delta, z1, z2)
1-mean(delta) # average censoring rate

out <- gamlss(list(u ~ z1 + s(z2) + s(u, bs = "mpi") ), data = dataSim, 
              surv = TRUE, margin = "PH", cens = delta)
post.check(out)
summary(out)
AIC(out)
BIC(out)
plot(out, eq = 1, scale = 0, pages = 1)
hazsurv.plot(out, newdata = data.frame(z1 = 0, z2 = 0), shade = TRUE, n.sim = 1000)
hazsurv.plot(out, type = "hazard", newdata = data.frame(z1 = 0, z2 = 0), 
             shade = TRUE, n.sim = 1000)

out1 <- gam(u ~ z1 + s(z2), family = cox.ph(), 
            data = dataSim, weights = delta)
summary(out1)
# estimates of z1 and s(z2) are
# nearly identical between out and out1 

# note that the Weibull is implemented as AFT
# as using the PH parametrisation makes
# computation unstable
out2 <- gamlss(list(u ~ z1 + s(z2) ), data = dataSim, surv = TRUE, 
               margin = "WEI", cens = delta)
 
#####################################
## Simulate proportional odds data ##
#####################################

set.seed(0)

n <- 2000
c <- runif(n, 4, 8)
u <- runif(n, 0, 1)
z <- rbinom(n, 1, 0.5)
beta_0 <- -1.05
t <- rep(NA, n)

f <- function(t, beta_0, u, z){ 
  S_0 <- 0.7 * exp(-0.03*t^1.9) + 0.3*exp(-0.3*t^2.5)
  1/(1 + exp(log((1-S_0)/S_0)+beta_0*z))-u
}



for (i in 1:n){
    t[i] <- uniroot(f, c(0, 8), tol = .Machine$double.eps^0.5, 
                    beta_0 = beta_0, u = u[i], z = z[i], 
                    extendInt="yes" )$root
}

delta   <- ifelse(t < c,1, 0)
u       <- apply(cbind(t, c), 1, min)
dataSim <- data.frame(u, delta, z)
1-mean(delta) # average censoring rate

out <- gamlss(list(u ~ z + s(u, bs = "mpi") ), data = dataSim, surv = TRUE, 
              margin = "PO", cens = delta)
post.check(out)
summary(out)
AIC(out)
BIC(out)
plot(out, eq = 1, scale = 0)
hazsurv.plot(out, newdata = data.frame(z =0), shade = TRUE, n.sim = 1000)
hazsurv.plot(out, type = "hazard", newdata = data.frame(z = 0), 
             shade = TRUE, n.sim = 1000)

# note that the Fisk is implemented as AFT
# as using the PH parametrisation makes
# computation unstable
out1 <- gamlss(list(u ~ z), data = dataSim, surv = TRUE, 
               margin = "FISK", cens = delta)


## End(Not run)