gjrm: Generalised Joint Regression Models with...
In GJRM: Generalised Joint Regression Modelling
gjrm R Documentation
Generalised Joint Regression Models with Binary/Continuous/Discrete/Survival Margins
Description
gjrm fits flexible joint models with binary/continuous/discrete/survival margins, with several types of covariate
effects, copula and marginal distributions.
Usage
gjrm(formula, data = list(), weights = NULL, subset = NULL,
offset1 = NULL, offset2 = NULL,
copula = "N", copula2 = "N", margins, model, dof = 3, dof2 = 3,
cens1 = NULL, cens2 = NULL, cens3 = NULL, dep.cens = FALSE,
ub.t1 = NULL, ub.t2 = NULL, left.trunc1 = 0, left.trunc2 = 0,
uni.fit = FALSE, fp = FALSE, infl.fac = 1,
rinit = 1, rmax = 100, iterlimsp = 50, tolsp = 1e-07,
gc.l = FALSE, parscale, knots = NULL,
penCor = "unpen", sp.penCor = 3,
Chol = FALSE, gamma = 1, w.alasso = NULL,
drop.unused.levels = TRUE,
min.dn = 1e-40, min.pr = 1e-16, max.pr = 0.999999)
Arguments
formula
In the basic setup this will be a list of two (or three) formulas, one for equation 1, the other for equation 2 and another one
for equation 3 if a trivariate model is fitted to the data. Otherwise, more equations can be used depending on the
number of distributional parameters. s terms
are used to specify smooth functions of
predictors; see the documentation of mgcv for further
details on formula specifications. Note that if a selection model is employed (that is, model = "BSS"
or model = "TSS") then the first formula (and the second as well for trivariate models) MUST refer to the selection equation(s).
When one outcome is binary and the other continuous/discrete
then the first equation should refer to the binary outcome whereas
the second to the continuous/discrete one. When one outcome is discrete and the other continuous
then the first equation has to be the discrete one.
data
A data frame.
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.
offset1, offset2
They can be used to supply model offsets for use in fitting. These have been introduced for dealing with offsets in the case of discrete marginal distributions.
copula
Type of bivariate error distribution employed. Possible choices are "N", "C0", "C90", "C180", "C270", "GAL0", "GAL90", "GAL180", "GAL270", "J0", "J90", "J180", "J270",
"G0", "G90", "G180", "G270", "F", "AMH", "FGM", "T", "PL", "HO" which stand for bivariate normal, Clayton, rotated Clayton (90 degrees),
survival Clayton,
rotated Clayton (270 degrees), Galambos, rotated Galambos (90 degrees),
survival Galambos,
rotated Galambos (270 degrees), Joe, rotated Joe (90 degrees), survival Joe, rotated Joe (270 degrees),
Gumbel, rotated Gumbel (90 degrees), survival Gumbel, rotated Gumbel (270 degrees), Frank, Ali-Mikhail-Haq,
Farlie-Gumbel-Morgenstern, Student-t with dof, Plackett, Hougaard. Each of the Clayton, Galambos, Joe and Gumbel copulae is allowed to be mixed with a rotated version of the same
family. The options are: "C0C90", "C0C270", "C180C90", "C180C270", "GAL0GAL90", "GAL0GAL270", "GAL180GAL90", "GAL180GAL270", "G0G90", "G0G270", "G180G90",
"G180G270", "J0J90", "J0J270", "J180J90" and "J180J270". This allows the user to model negative and positive tail dependencies.
copula2
As above but used only for Roy models.
margins
It indicates the distributions used for margins. Possible distributions are normal ("N"), Tweedie ("TW"),
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 ("P"), truncated
Poisson ("tP"), negative binomial - type I ("NBI"), negative
binomial - type II ("NBII"), Poisson inverse Gaussian ("PIG"), truncated negative binomial - type I ("tNBI"), truncated negative
binomial - type II ("tNBII"), truncated Poisson inverse Gaussian ("tPIG"). If the responses are binary then
possible link functions are "probit", "logit", "cloglog". For survival models, the margins can be "-cloglog" (similar to generalised proportional
hazards), "-logit" (similar to generalised proportional odds), "-probit" (generalised probit). For ordinal marginals, the choices are "ord.probit" and
"ord.logit". For extreme value models, there are also options we are working on, which are already implemented in the univariate
gamlss() function. These are the generelised Pareto ("GP"),
generelised Pareto II ("GPII") where the shape parameter is forced to be > -0.5,
generelised Pareto (with orthogonal parametrisation) ("GPo") where the shape parameter is forced to be > -0.5,
discrete generelised Pareto ("DGP"),
discrete generelised Pareto II ("DGPII") where the shape parameter is forced to be positive, discrete generelised Pareto derived
under the scenario in which shape = 0 ("DGP0"). Regarding the Tweedie, this margin can currently only be used together with a binary margin; we
are working on the discrete/continuous margin extension.
model
Possible values are "B" (bivariate model), "T" (trivariate model),
"BSS" (bivariate model with non-random sample selection), "TSS" (trivariate model with double non-random sample selection),
"TESS" (trivariate model with endogeneity and non-random sample selection),
"BPO" (bivariate model with partial observability)
and "BPO0" (bivariate model with partial observability and zero correlation).
Options "T", "TESS" and "TSS" are currently for trivariate binary models only. "BPO" and "BPO0" are for bivariate binary models only.
"ROY" is for the Roy switching regression model.
dof
If copula = "T" then the degrees of freedom can be set to a value greater than 2 and smaller than 249. Only for continuous margins,
this will be taken as a starting value and the dof estimated from the data.
dof2
As above but used only for Roy models.
cens1
Censoring indicator for the first equation. For the case of right censored data only, this variable can be equal to 1 if the event occurred
and 0 otherwise. However, if there are several censoring mechanisms then cens will have to be specified as a factor
variable made up of four possible categories: I for interval, L for left, R for right, and U for uncensored.
cens2
Same as above but for the second equation.
cens3
Binary censoring indicator employed only when dep.cens = TRUE and administrative censoring is present.
dep.cens
If TRUE then the dependence censored model is employed.
ub.t1, ub.t2
Variable names of right/upper bounds when interval censoring is present.
left.trunc1, left.trunc2
Values of truncation at left. Currently done for count distributions only.
uni.fit
If uni.fit = TRUE then gamlss or gam univariate models are also fitted. This is useful for obtaining
starting values, for instance.
fp
If TRUE then a fully parametric model with unpenalised regression splines if fitted. See the Example 2 below.
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. See the documentation
of trust for further details.
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.
knots
Optional list containing user specified knot values to be used for basis construction.
penCor
This and the arguments below are only for trivariate binary models. Type of penalty for
correlation coefficients. Possible values are "unpen", "lasso", "ridge", "alasso".
sp.penCor
Starting value for smoothing parameter of penCor.
Chol
If TRUE then the Cholesky method instead of the eigenvalue method is employed for the correlation matrix.
gamma
Inflation factor used only for the alasso penalty.
w.alasso
When using the alasso penalty a weight vector made up of three values must be provided.
drop.unused.levels
By default unused levels are dropped from factors before fitting. For some smooths involving factor variables
this may have to be turned off (only use if you know what you are doing).
min.dn, min.pr, max.pr
These values are used to set, depending on the model used for modelling, the minimum and maximum allowed
for the densities and probabilities; recall that the margins of copula models have to be in the range (0,1). These
parameters are employed to avoid potential overflows/underflows in the calculations and the default
values seem to offer a good compromise. Function conv.check() provides some relevant
diagnostic information which can be used, for example, to check whether the lower bounds
of min.dn and min.pr have been reached. So based on this or if the user wishes to do some sensitivity
analysis then this can be easily carried out using these three arguments.
However, the user has to be cautious. For instance, it would not make much sense to choose for min.dn and min.pr
values bigger than the default ones. Bear in mind that the bounds can be reached for ill-defined models. For
certain distributions/models, if convergence failure occurs and the bounds have been reached then the user
can try a sensitivity analysis as mentioned above.
Details
The joint models considered by this function consist of two or three model equations which depend on flexible linear predictors and
whose dependence between the responses is modelled through one or more parameters of a chosen multivariate distribution. The additive predictors of the
equations are flexibly specified using
parametric components and smooth functions of covariates. The same can be done for the dependence parameter(s) if it makes sense.
Estimation is achieved within a penalized likelihood framework with integrated automatic multiple smoothing parameter selection. The use of
penalty matrices allows for the suppression of that part of smooth term complexity which has no support
from the data. The trade-off between smoothness
and fitness is controlled by smoothing parameters associated with the penalty matrices. Smoothing parameters are chosen to
minimise an approximate AIC.
For sample selection models, if there are factors in the model then before fitting the user has to ensure
that the numbers of factor variables' levels in the selected sample
are the same as those in the complete dataset. Even if a model could be fitted in such a situation,
the model may produce fits which are
not coherent with the nature of the correction sought. As an example consider the
situation in which the complete dataset contains a factor variable with five levels and that only three of them
appear in the selected sample. For the outcome equation (which is the one of interest) only three levels of such variable
exist in the population, but their effects will be corrected for non-random selection using a selection equation
in which five levels exist instead. Having differing numbers of factors' levels between complete and selected samples will also
make prediction not feasible (an aspect which may be particularly important for selection models);
clearly it is not possible to predict the response of interest for the missing entries using a
dataset that contains all levels of a factor variable but using an outcome model
estimated using a subset of these levels.
There are many continuous/discrete/survival distributions and copula functions to choose from and we plan to include more
options. Get in touch if you are interested in a particular distribution.
Value
The function returns an object of class gjrm as described in gjrmObject.
WARNINGS
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.
gjrm() 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 rescaling (see parscale). Using uni.fit = TRUE is typically more effective than the first two options as
this will provide better calibrated starting values as compared to those obtained from the default starting value procedure.
The default option is, however, uni.fit = FALSE only because it tends to be computationally cheaper and because the
default procedure has typically been found to do a satisfactory job in most cases.
(The results obtained when using
uni.fit = FALSE and uni.fit = TRUE could also be compared to check if starting values make any difference.)
The above suggestions may help, especially the latter option. However, the user should also consider
re-specifying/simplifying the model, and/or using a diferrent dependence structure and/or checking that the chosen marginal
distributions fit the responses 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. Examples
of misspecification include using a Clayton copula rotated by 90 degrees when a positive
association between the margins is present instead, using marginal distributions that do not fit
the responses, and
employing a copula which does not accommodate the type and/or strength of
the dependence between the margins (e.g., using AMH when the association between the margins is strong). When using
smooth functions, if the covariate's values are too sparse then convergence may be affected by this.
It is also worth bearing in mind that the use of three parameter marginal distributions requires the data
to be more informative than a situation in which two parameter distributions are used instead.
In the contexts of endogeneity and non-random sample selection, extra attention is required when specifying
the dependence parameter as a function of covariates. This is because in these situations the dependence parameter mainly models the
association between the unobserved confounders in the two equations. Therefore, this option would make sense when it
is believed that the
strength of the association between the unobservables in the two equations varies based on some grouping factor or across geographical
areas, for instance. In any case, a clear rationale is typically needed in such cases.
Author(s)
Maintainer: Giampiero Marra giampiero.marra@ucl.ac.uk
References
See help("GJRM-package").
See Also
adjCov, VuongClarke, GJRM-package, gjrmObject, conv.check, summary.gjrm
Examples
library(GJRM)
####################################
# JOINT MODELS WITH BINARY MARGINS #
####################################
###############
## EXAMPLE 1 ##
set.seed(0)
n <- 400
Sigma <- matrix(0.5, 2, 2); diag(Sigma) <- 1
u <- rMVN(n, rep(0,2), Sigma)
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)
y1 <- ifelse(-1.55 + 2*x1 + f1(x2) + u[,1] > 0, 1, 0)
y2 <- ifelse(-0.25 - 1.25*x1 + f2(x2) + u[,2] > 0, 1, 0)
dataSim <- data.frame(y1, y2, x1, x2, x3)
## CLASSIC BIVARIATE PROBIT
out <- gjrm(list(y1 ~ x1 + x2 + x3,
y2 ~ x1 + x2 + x3),
data = dataSim,
margins = c("probit", "probit"),
model = "B")
conv.check(out)
summary(out)
AIC(out)
BIC(out)
## Not run:
## BIVARIATE PROBIT with Splines
out <- gjrm(list(y1 ~ x1 + s(x2) + s(x3),
y2 ~ x1 + s(x2) + s(x3)),
data = dataSim,
margins = c("probit", "probit"),
model = "B")
conv.check(out)
summary(out)
AIC(out)
## estimated smooth function plots
plot(out, eq = 1, pages = 1, seWithMean = TRUE, scale = 0)
plot(out, eq = 2, pages = 1, seWithMean = TRUE, scale = 0)
## BIVARIATE PROBIT with Splines and
## varying dependence parameter
eq.mu.1 <- y1 ~ x1 + s(x2)
eq.mu.2 <- y2 ~ x1 + s(x2)
eq.theta <- ~ x1 + s(x2)
fl <- list(eq.mu.1, eq.mu.2, eq.theta)
outD <- gjrm(fl, data = dataSim,
margins = c("probit", "probit"),
model = "B")
conv.check(outD)
summary(outD)
summary(outD$theta)
plot(outD, eq = 1, seWithMean = TRUE)
plot(outD, eq = 2, seWithMean = TRUE)
plot(outD, eq = 3, seWithMean = TRUE)
graphics.off()
###############
## EXAMPLE 2 ##
## Generate data with one endogenous variable
## and exclusion restriction (or instrument)
set.seed(0)
n <- 400
Sigma <- matrix(0.5, 2, 2); diag(Sigma) <- 1
u <- rMVN(n, rep(0,2), Sigma)
cov <- rMVN(n, rep(0,2), Sigma)
cov <- pnorm(cov)
x1 <- round(cov[,1]); x2 <- cov[,2]
f1 <- function(x) cos(pi*2*x) + sin(pi*x)
f2 <- function(x) x+exp(-30*(x-0.5)^2)
y1 <- ifelse(-1.55 + 2*x1 + f1(x2) + u[,1] > 0, 1, 0)
y2 <- ifelse(-0.25 - 1.25*y1 + f2(x2) + u[,2] > 0, 1, 0)
dataSim <- data.frame(y1, y2, x1, x2)
#
## Testing the hypothesis of absence of endogeneity...
#
LM.bpm(list(y1 ~ x1 + s(x2), y2 ~ y1 + s(x2)), dataSim, model = "B")
## CLASSIC RECURSIVE BIVARIATE PROBIT
out <- gjrm(list(y1 ~ x1 + x2,
y2 ~ y1 + x2),
data = dataSim,
margins = c("probit", "probit"), model = "B")
conv.check(out)
summary(out)
AIC(out); BIC(out)
## FLEXIBLE RECURSIVE BIVARIATE PROBIT
out <- gjrm(list(y1 ~ x1 + s(x2),
y2 ~ y1 + s(x2)),
data = dataSim,
margins = c("probit", "probit"),
model = "B")
conv.check(out)
summary(out)
AIC(out); BIC(out)
#
## Testing the hypothesis of absence of endogeneity post estimation...
gt.bpm(out)
#
## Causal effects
## average treatment effect, risk ratio and odds ratio with CIs
mb(y1, y2, model = "B")
ATE(out, trt = "y1")
RR(out, trt = "y1")
OR(out, trt = "y1")
ATE(out, trt = "y1", joint = FALSE)
## try a Clayton copula model...
outC <- gjrm(list(y1 ~ x1 + s(x2),
y2 ~ y1 + s(x2)),
data = dataSim, copula = "C0",
margins = c("probit", "probit"),
model = "B")
conv.check(outC)
summary(outC)
ATE(outC, trt = "y1")
## try a Joe copula model...
outJ <- gjrm(list(y1 ~ x1 + s(x2),
y2 ~ y1 + s(x2)),
data = dataSim, copula = "J0",
margins = c("probit", "probit"),
model = "B")
conv.check(outJ)
summary(outJ)
ATE(outJ, "y1")
VuongClarke(out, outJ)
#
## recursive bivariate probit modelling with unpenalized splines
## can be achieved as follows
outFP <- gjrm(list(y1 ~ x1 + s(x2, bs = "cr", k = 5),
y2 ~ y1 + s(x2, bs = "cr", k = 6)),
fp = TRUE, data = dataSim,
margins = c("probit", "probit"),
model = "B")
conv.check(outFP)
summary(outFP)
# in the above examples a third equation could be introduced
# as illustrated in Example 1
###############
## EXAMPLE 3 ##
## Generate data with a non-random sample selection mechanism
## and exclusion restriction
set.seed(0)
n <- 2000
Sigma <- matrix(0.5, 2, 2); diag(Sigma) <- 1
u <- rMVN(n, rep(0,2), Sigma)
SigmaC <- matrix(0.5, 3, 3); diag(SigmaC) <- 1
cov <- rMVN(n, rep(0,3), SigmaC)
cov <- pnorm(cov)
bi <- round(cov[,1]); x1 <- cov[,2]; x2 <- cov[,3]
f11 <- function(x) -0.7*(4*x + 2.5*x^2 + 0.7*sin(5*x) + cos(7.5*x))
f12 <- function(x) -0.4*( -0.3 - 1.6*x + sin(5*x))
f21 <- function(x) 0.6*(exp(x) + sin(2.9*x))
ys <- 0.58 + 2.5*bi + f11(x1) + f12(x2) + u[, 1] > 0
y <- -0.68 - 1.5*bi + f21(x1) + + u[, 2] > 0
yo <- y*(ys > 0)
dataSim <- data.frame(y, ys, yo, bi, x1, x2)
#
## Testing the hypothesis of absence of non-random sample selection...
LM.bpm(list(ys ~ bi + s(x1) + s(x2), yo ~ bi + s(x1)), dataSim, model = "BSS")
# p-value suggests presence of sample selection
#
## SEMIPARAMETRIC SAMPLE SELECTION BIVARIATE PROBIT
## the first equation MUST be the selection equation
out <- gjrm(list(ys ~ bi + s(x1) + s(x2),
yo ~ bi + s(x1)),
data = dataSim, model = "BSS",
margins = c("probit", "probit"))
conv.check(out)
gt.bpm(out)
## compare the two summary outputs below
## the second output produces a summary of the results obtained when
## selection bias is not accounted for
summary(out)
summary(out$gam2)
## corrected predicted probability that 'yo' is equal to 1
mb(ys, yo, model = "BSS")
prev(out)
prev(out, joint = FALSE)
## estimated smooth function plots
## the red line is the true curve
## the blue line is the univariate model curve not accounting for selection bias
x1.s <- sort(x1[dataSim$ys>0])
f21.x1 <- f21(x1.s)[order(x1.s)]-mean(f21(x1.s))
plot(out, eq = 2, ylim = c(-1.65,0.95)); lines(x1.s, f21.x1, col="red")
par(new = TRUE)
plot(out$gam2, se = FALSE, col = "blue", ylim = c(-1.65,0.95),
ylab = "", rug = FALSE)
#
#
## try a Clayton copula model...
outC <- gjrm(list(ys ~ bi + s(x1) + s(x2),
yo ~ bi + s(x1)),
data = dataSim, model = "BSS", copula = "C0",
margins = c("probit", "probit"))
conv.check(outC)
summary(outC)
prev(outC)
################
## See also ?hiv
################
###############
## EXAMPLE 4 ##
## Generate data with partial observability
set.seed(0)
n <- 1000
Sigma <- matrix(0.5, 2, 2); diag(Sigma) <- 1
u <- rMVN(n, rep(0,2), Sigma)
x1 <- round(runif(n)); x2 <- runif(n); x3 <- runif(n)
y1 <- ifelse(-1.55 + 2*x1 + x2 + u[,1] > 0, 1, 0)
y2 <- ifelse( 0.45 - x3 + u[,2] > 0, 1, 0)
y <- y1*y2
dataSim <- data.frame(y, x1, x2, x3)
## BIVARIATE PROBIT with Partial Observability
out <- gjrm(list(y ~ x1 + x2,
y ~ x3),
data = dataSim, model = "BPO",
margins = c("probit", "probit"))
conv.check(out)
summary(out)
# first ten estimated probabilities for the four events from object out
cbind(out$p11, out$p10, out$p00, out$p01)[1:10,]
# case with smooth function
f1 <- function(x) cos(pi*2*x) + sin(pi*x)
y1 <- ifelse(-1.55 + 2*x1 + f1(x2) + u[,1] > 0, 1, 0)
y2 <- ifelse( 0.45 - x3 + u[,2] > 0, 1, 0)
y <- y1*y2
dataSim <- data.frame(y, x1, x2, x3)
out <- gjrm(list(y ~ x1 + s(x2),
y ~ x3),
data = dataSim, model = "BPO",
margins = c("probit", "probit"))
conv.check(out)
summary(out)
plot(out, eq = 1, scale = 0)
################
## See also ?war
################
######################################################
# JOINT MODELS WITH BINARY AND/OR CONTINUOUS MARGINS #
######################################################
###############
## EXAMPLE 5 ##
## Generate data
## Correlation between the two equations 0.5 - Sample size 400
set.seed(0)
n <- 400
Sigma <- matrix(0.5, 2, 2); diag(Sigma) <- 1
u <- rMVN(n, rep(0,2), Sigma)
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)
y1 <- -1.55 + 2*x1 + f1(x2) + u[,1]
y2 <- -0.25 - 1.25*x1 + f2(x2) + u[,2]
dataSim <- data.frame(y1, y2, x1, x2, x3)
resp.check(y1, "N")
resp.check(y2, "N")
eq.mu.1 <- y1 ~ x1 + s(x2) + s(x3)
eq.mu.2 <- y2 ~ x1 + s(x2) + s(x3)
eq.sigma1 <- ~ 1
eq.sigma2 <- ~ 1
eq.theta <- ~ x1
fl <- list(eq.mu.1, eq.mu.2, eq.sigma1, eq.sigma2, eq.theta)
# the order above is the one to follow when
# using more than two equations
out <- gjrm(fl, data = dataSim, margins = c("N", "N"),
model = "B")
conv.check(out)
res.check(out)
summary(out)
AIC(out)
BIC(out)
nd <- data.frame(x1 = 1, x2 = 0.4, x3 = 0.6)
copula.prob(out, y1 = 1.4, y2 = 2.3, newdata = nd, intervals = TRUE)
###############
## EXAMPLE 6 ##
## Generate data with one endogenous binary variable
## and continuous outcome
set.seed(0)
n <- 1000
Sigma <- matrix(0.5, 2, 2); diag(Sigma) <- 1
u <- rMVN(n, rep(0,2), Sigma)
cov <- rMVN(n, rep(0,2), Sigma)
cov <- pnorm(cov)
x1 <- round(cov[,1]); x2 <- cov[,2]
f1 <- function(x) cos(pi*2*x) + sin(pi*x)
f2 <- function(x) x+exp(-30*(x-0.5)^2)
y1 <- ifelse(-1.55 + 2*x1 + f1(x2) + u[,1] > 0, 1, 0)
y2 <- -0.25 - 1.25*y1 + f2(x2) + u[,2]
dataSim <- data.frame(y1, y2, x1, x2)
## RECURSIVE Model
rc <- resp.check(y2, margin = "N", print.par = TRUE, loglik = TRUE)
AIC(rc); BIC(rc)
out <- gjrm(list(y1 ~ x1 + x2,
y2 ~ y1 + x2),
data = dataSim, margins = c("probit","N"),
model = "B")
conv.check(out)
summary(out)
res.check(out)
## SEMIPARAMETRIC RECURSIVE Model
eq.mu.1 <- y1 ~ x1 + s(x2)
eq.mu.2 <- y2 ~ y1 + s(x2)
eq.sigma <- ~ 1
eq.theta <- ~ 1
fl <- list(eq.mu.1, eq.mu.2, eq.sigma, eq.theta)
out <- gjrm(fl, data = dataSim,
margins = c("probit","N"), uni.fit = TRUE,
model = "B")
conv.check(out)
summary(out)
res.check(out)
ATE(out, trt = "y1")
ATE(out, trt = "y1", joint = FALSE)
#
#
###############
## EXAMPLE 7 ##
## Generate data with one endogenous continuous exposure
## and binary outcome
set.seed(0)
n <- 1000
Sigma <- matrix(0.5, 2, 2); diag(Sigma) <- 1
u <- rMVN(n, rep(0,2), Sigma)
cov <- rMVN(n, rep(0,2), Sigma)
cov <- pnorm(cov)
x1 <- round(cov[,1]); x2 <- cov[,2]
f1 <- function(x) cos(pi*2*x) + sin(pi*x)
f2 <- function(x) x+exp(-30*(x-0.5)^2)
y1 <- -0.25 - 2*x1 + f2(x2) + u[,2]
y2 <- ifelse(-0.25 - 0.25*y1 + f1(x2) + u[,1] > 0, 1, 0)
dataSim <- data.frame(y1, y2, x1, x2)
eq.mu.1 <- y2 ~ y1 + s(x2)
eq.mu.2 <- y1 ~ x1 + s(x2)
eq.sigma <- ~ 1
eq.theta <- ~ 1
fl <- list(eq.mu.1, eq.mu.2, eq.sigma, eq.theta)
out <- gjrm(fl, data = dataSim,
margins = c("probit","N"),
model = "B")
conv.check(out)
summary(out)
res.check(out)
ATE(out, trt = "y1")
ATE(out, trt = "y1", joint = FALSE)
RR(out, trt = "y1")
RR(out, trt = "y1", joint = FALSE)
OR(out, trt = "y1")
OR(out, trt = "y1", joint = FALSE)
#
#
#####################
## EXAMPLE 8 ##
## SURVIVAL MODELS ##
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
}
delta1 <- ifelse(t < c, 1, 0)
u1 <- apply(cbind(t, c), 1, min)
dataSim <- data.frame(u1, delta1, z1, z2)
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
}
delta2 <- ifelse(t < c,1, 0)
u2 <- apply(cbind(t, c), 1, min)
dataSim$delta2 <- delta2
dataSim$u2 <- u2
dataSim$z <- z
eq1 <- u1 ~ s(log(u1), bs = "mpi") + z1 + s(z2)
eq2 <- u2 ~ s(log(u2), bs = "mpi") + z
eq3 <- ~ s(z2)
out <- gjrm(list(eq1, eq2), data = dataSim,
margins = c("-cloglog", "-logit"),
cens1 = delta1, cens2 = delta2, model = "B")
# PH margin fit can also be compared with cox.ph from mgcv
conv.check(out)
res <- res.check(out)
## martingale residuals
mr1 <- out$cens1 - res$qr1
mr2 <- out$cens2 - res$qr2
# can be plotted against covariates
# obs index, survival time, rank order of
# surv times
# to determine func form, one may use
# res from null model against covariate
# to test for PH, use:
# library(survival)
# fit <- coxph(Surv(u1, delta1) ~ z1 + z2, data = dataSim)
# temp <- cox.zph(fit)
# print(temp)
# plot(temp, resid = FALSE)
summary(out)
AIC(out); BIC(out)
plot(out, eq = 1, scale = 0, pages = 1)
plot(out, eq = 2, scale = 0, pages = 1)
haz.surv(out, eq = 1, newdata = data.frame(z1 = 0, z2 = 0),
shade = TRUE, n.sim = 100, baseline = TRUE)
haz.surv(out, eq = 1, newdata = data.frame(z1 = 0, z2 = 0),
shade = TRUE, n.sim = 100, type = "haz", baseline = TRUE,
intervals = FALSE)
haz.surv(out, eq = 2, newdata = data.frame(z = 0),
shade = FALSE, n.sim = 100, baseline = TRUE)
haz.surv(out, eq = 2, newdata = data.frame(z = 0),
shade = TRUE, n.sim = 100, type = "haz", baseline = TRUE)
newd0 <- newd1 <- data.frame(z = 0, z1 = mean(dataSim$z1),
z2 = mean(dataSim$z2),
u1 = mean(dataSim$u1) + 1,
u2 = mean(dataSim$u2) + 1)
newd1$z <- 1
copula.prob(out, newdata = newd0, intervals = TRUE)
copula.prob(out, newdata = newd1, intervals = TRUE)
out1 <- gjrm(list(eq1, eq2, eq3), data = dataSim,
margins = c("-cloglog", "-logit"),
cens1 = delta1, cens2 = delta2, uni.fit = TRUE,
model = "B")
####################################################
## Joint continuous and survival outcomes
####################################################
# this is complete, just testing, get in touch if interested
#
# eq1 <- z2 ~ z1
# eq2 <- u2 ~ s(u2, bs = "mpi") + z
# eq3 <- ~ s(z2)
# eq4 <- ~ s(z2)
#
# f.l <- list(eq1, eq2, eq3, eq4)
#
# out3 <- gjrm(f.l, data = dataSim,
# margins = c("N", "-logit"),
# cens1 = NULL, cens2 = delta2,
# uni.fit = TRUE, model = "B")
#
# conv.check(out3)
# res.check(out3)
# summary(out3)
# AIC(out3); BIC(out3)
# plot(out3, eq = 2, scale = 0, pages = 1)
# plot(out3, eq = 3, scale = 0, pages = 1)
# plot(out3, eq = 4, scale = 0, pages = 1)
#
# newd <- newd1 <- data.frame(z = 0, z1 = mean(dataSim$z1),
# z2 = mean(dataSim$z2),
# u2 = mean(dataSim$u2) + 1)
#
# copula.prob(out3, y1 = 0.6, newdata = newd, intervals = TRUE)
##########################################
# JOINT MODELS WITH THREE BINARY MARGINS #
##########################################
###############
## EXAMPLE 9 ##
## Generate data
## Correlation between the two equations 0.5 - Sample size 400
set.seed(0)
n <- 400
Sigma <- matrix(0.5, 3, 3); diag(Sigma) <- 1
u <- rMVN(n, rep(0,3), Sigma)
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)
y1 <- ifelse(-1.55 + 2*x1 - f1(x2) + u[,1] > 0, 1, 0)
y2 <- ifelse(-0.25 - 1.25*x1 + f2(x2) + u[,2] > 0, 1, 0)
y3 <- ifelse(-0.75 + 0.25*x1 + u[,3] > 0, 1, 0)
dataSim <- data.frame(y1, y2, y3, x1, x2)
f.l <- list(y1 ~ x1 + s(x2),
y2 ~ x1 + s(x2),
y3 ~ x1)
margs <- c("probit", "probit", "probit")
out <- gjrm(f.l, data = dataSim, model = "T", margins = margs)
out1 <- gjrm(f.l, data = dataSim, Chol = TRUE, model = "T", margins = margs)
conv.check(out)
summary(out)
plot(out, eq = 1)
plot(out, eq = 2)
AIC(out)
BIC(out)
margs <- c("probit","logit","cloglog")
out <- gjrm(f.l, data = dataSim, model = "T", margins = margs)
out1 <- gjrm(f.l, data = dataSim, Chol = TRUE, model = "T", margins = margs)
conv.check(out)
summary(out)
plot(out, eq = 1)
plot(out, eq = 2)
AIC(out)
BIC(out)
margs <- c("probit", "probit", "probit")
f.l <- list(y1 ~ x1 + s(x2),
y2 ~ x1 + s(x2),
y3 ~ x1,
~ 1, ~ 1, ~ 1)
out1 <- gjrm(f.l, data = dataSim, Chol = TRUE, model = "T", margins = margs)
f.l <- list(y1 ~ x1 + s(x2),
y2 ~ x1 + s(x2),
y3 ~ x1,
~ 1, ~ s(x2), ~ 1)
out2 <- gjrm(f.l, data = dataSim, Chol = TRUE, model = "T", margins = margs)
f.l <- list(y1 ~ x1 + s(x2),
y2 ~ x1 + s(x2),
y3 ~ x1,
~ x1, ~ s(x2), ~ x1 + s(x2))
out2 <- gjrm(f.l, data = dataSim, Chol = TRUE, model = "T", margins = margs)
f.l <- list(y1 ~ x1 + s(x2),
y2 ~ x1 + s(x2),
y3 ~ x1,
~ x1, ~ x1, ~ s(x2))
out2 <- gjrm(f.l, data = dataSim, Chol = TRUE, model = "T", margins = margs)
f.l <- list(y1 ~ x1 + s(x2),
y2 ~ x1 + s(x2),
y3 ~ x1,
~ x1, ~ x1 + x2, ~ s(x2))
out2 <- gjrm(f.l, data = dataSim, Chol = TRUE, model = "T", margins = margs)
f.l <- list(y1 ~ x1 + s(x2),
y2 ~ x1 + s(x2),
y3 ~ x1,
~ x1 + x2, ~ x1 + x2, ~ x1 + x2)
out2 <- gjrm(f.l, data = dataSim, Chol = TRUE, model = "T", margins = margs)
nw <- data.frame( x1 = 0, x2 = 0.7 )
copula.prob(out2, 1, 1, 1, newdata = nw, cond = 0, intervals = TRUE, n.sim = 100)
# with endogenous variable
f.l <- list(y1 ~ x1 + s(x2),
y2 ~ y1 + x1 + s(x2),
y3 ~ x1)
margs <- c("probit", "probit", "probit")
out <- gjrm(f.l, data = dataSim, model = "T", margins = margs)
conv.check(out)
summary(out)
ATE(out, trt = "y1", eq = 2, joint = TRUE)
ATE(out, trt = "y1", eq = 2, joint = FALSE)
################
## EXAMPLE 10 ##
## Generate data
## with double sample selection
set.seed(0)
n <- 5000
Sigma <- matrix(c(1, 0.5, 0.4,
0.5, 1, 0.6,
0.4, 0.6, 1 ), 3, 3)
u <- rMVN(n, rep(0,3), Sigma)
f1 <- function(x) cos(pi*2*x) + sin(pi*x)
f2 <- function(x) x+exp(-30*(x-0.5)^2)
x1 <- runif(n)
x2 <- runif(n)
x3 <- runif(n)
x4 <- runif(n)
y1 <- 1 + 1.5*x1 - x2 + 0.8*x3 - f1(x4) + u[, 1] > 0
y2 <- 1 - 2.5*x1 + 1.2*x2 + x3 + u[, 2] > 0
y3 <- 1.58 + 1.5*x1 - f2(x2) + u[, 3] > 0
dataSim <- data.frame(y1, y2, y3, x1, x2, x3, x4)
f.l <- list(y1 ~ x1 + x2 + x3 + s(x4),
y2 ~ x1 + x2 + x3,
y3 ~ x1 + s(x2))
out <- gjrm(f.l, data = dataSim, model = "TSS",
margins = c("probit", "probit", "probit"))
conv.check(out)
summary(out)
plot(out, eq = 1)
plot(out, eq = 3)
prev(out)
prev(out, joint = FALSE)
#############
## EXAMPLE 11
set.seed(0)
n <- 2000
rh <- 0.5
sigmau <- matrix(c(1, rh, rh, 1), 2, 2)
u <- rMVN(n, rep(0,2), sigmau)
sigmac <- matrix(rh, 3, 3); diag(sigmac) <- 1
cov <- rMVN(n, rep(0,3), sigmac)
cov <- pnorm(cov)
bi <- round(cov[,1]); x1 <- cov[,2]; x2 <- cov[,3]
f11 <- function(x) -0.7*(4*x + 2.5*x^2 + 0.7*sin(5*x) + cos(7.5*x))
f12 <- function(x) -0.4*( -0.3 - 1.6*x + sin(5*x))
f21 <- function(x) 0.6*(exp(x) + sin(2.9*x))
ys <- 0.58 + 2.5*bi + f11(x1) + f12(x2) + u[, 1] > 0
y <- -0.68 - 1.5*bi + f21(x1) + u[, 2]
yo <- y*(ys > 0)
dataSim <- data.frame(ys, yo, bi, x1, x2)
## CLASSIC SAMPLE SELECTION MODEL
## the first equation MUST be the selection equation
resp.check(yo[ys > 0], "N")
out <- gjrm(list(ys ~ bi + x1 + x2,
yo ~ bi + x1),
data = dataSim, model = "BSS",
margins = c("probit", "N"))
conv.check(out)
res.check(out)
summary(out)
AIC(out)
BIC(out)
## SEMIPARAMETRIC SAMPLE SELECTION MODEL
out <- gjrm(list(ys ~ bi + s(x1) + s(x2),
yo ~ bi + s(x1)),
data = dataSim, model = "BSS",
margins = c("probit", "N"))
conv.check(out)
res.check(out)
AIC(out)
## compare the two summary outputs
## the second output produces a summary of the results obtained when only
## the outcome equation is fitted, i.e. selection bias is not accounted for
summary(out)
summary(out$gam2)
## estimated smooth function plots
## the red line is the true curve
## the blue line is the naive curve not accounting for selection bias
x1.s <- sort(x1[dataSim$ys>0])
f21.x1 <- f21(x1.s)[order(x1.s)] - mean(f21(x1.s))
plot(out, eq = 2, ylim = c(-1, 0.8)); lines(x1.s, f21.x1, col = "red")
par(new = TRUE)
plot(out$gam2, se = FALSE, lty = 3, lwd = 2, ylim = c(-1, 0.8),
ylab = "", rug = FALSE)
##
## SEMIPARAMETRIC SAMPLE SELECTION MODEL with association
## and dispersion parameters
## depending on covariates as well
eq.mu.1 <- ys ~ bi + s(x1) + s(x2)
eq.mu.2 <- yo ~ bi + s(x1)
eq.sigma <- ~ bi
eq.theta <- ~ bi + x1
fl <- list(eq.mu.1, eq.mu.2, eq.sigma, eq.theta)
out <- gjrm(fl, data = dataSim, model = "BSS",
margins = c("probit", "N"))
conv.check(out)
res.check(out)
summary(out)
summary(out$sigma)
summary(out$theta)
nd <- data.frame(bi = 0, x1 = 0.2, x2 = 0.8)
copula.prob(out, 0, 0.3, newdata = nd, intervals = TRUE)
outC0 <- gjrm(fl, data = dataSim, copula = "C0", model = "BSS",
margins = c("probit", "N"))
conv.check(outC0)
res.check(outC0)
AIC(out, outC0)
BIC(out, outC0)
## End(Not run)
GJRM documentation built on Oct. 25, 2024, 5:07 p.m.
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| gjrm | R Documentation |
Generalised Joint Regression Models with Binary/Continuous/Discrete/Survival Margins
Description
gjrm fits flexible joint models with binary/continuous/discrete/survival margins, with several types of covariate
effects, copula and marginal distributions.
Usage
gjrm(formula, data = list(), weights = NULL, subset = NULL,
offset1 = NULL, offset2 = NULL,
copula = "N", copula2 = "N", margins, model, dof = 3, dof2 = 3,
cens1 = NULL, cens2 = NULL, cens3 = NULL, dep.cens = FALSE,
ub.t1 = NULL, ub.t2 = NULL, left.trunc1 = 0, left.trunc2 = 0,
uni.fit = FALSE, fp = FALSE, infl.fac = 1,
rinit = 1, rmax = 100, iterlimsp = 50, tolsp = 1e-07,
gc.l = FALSE, parscale, knots = NULL,
penCor = "unpen", sp.penCor = 3,
Chol = FALSE, gamma = 1, w.alasso = NULL,
drop.unused.levels = TRUE,
min.dn = 1e-40, min.pr = 1e-16, max.pr = 0.999999)
Arguments
formula |
In the basic setup this will be a list of two (or three) formulas, one for equation 1, the other for equation 2 and another one
for equation 3 if a trivariate model is fitted to the data. Otherwise, more equations can be used depending on the
number of distributional parameters. |
data |
A data frame. |
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. |
offset1, offset2 |
They can be used to supply model offsets for use in fitting. These have been introduced for dealing with offsets in the case of discrete marginal distributions. |
copula |
Type of bivariate error distribution employed. Possible choices are "N", "C0", "C90", "C180", "C270", "GAL0", "GAL90", "GAL180", "GAL270", "J0", "J90", "J180", "J270",
"G0", "G90", "G180", "G270", "F", "AMH", "FGM", "T", "PL", "HO" which stand for bivariate normal, Clayton, rotated Clayton (90 degrees),
survival Clayton,
rotated Clayton (270 degrees), Galambos, rotated Galambos (90 degrees),
survival Galambos,
rotated Galambos (270 degrees), Joe, rotated Joe (90 degrees), survival Joe, rotated Joe (270 degrees),
Gumbel, rotated Gumbel (90 degrees), survival Gumbel, rotated Gumbel (270 degrees), Frank, Ali-Mikhail-Haq,
Farlie-Gumbel-Morgenstern, Student-t with |
copula2 |
As above but used only for Roy models. |
margins |
It indicates the distributions used for margins. Possible distributions are normal ("N"), Tweedie ("TW"), 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 ("P"), truncated Poisson ("tP"), negative binomial - type I ("NBI"), negative binomial - type II ("NBII"), Poisson inverse Gaussian ("PIG"), truncated negative binomial - type I ("tNBI"), truncated negative binomial - type II ("tNBII"), truncated Poisson inverse Gaussian ("tPIG"). If the responses are binary then possible link functions are "probit", "logit", "cloglog". For survival models, the margins can be "-cloglog" (similar to generalised proportional hazards), "-logit" (similar to generalised proportional odds), "-probit" (generalised probit). For ordinal marginals, the choices are "ord.probit" and "ord.logit". For extreme value models, there are also options we are working on, which are already implemented in the univariate gamlss() function. These are the generelised Pareto ("GP"), generelised Pareto II ("GPII") where the shape parameter is forced to be > -0.5, generelised Pareto (with orthogonal parametrisation) ("GPo") where the shape parameter is forced to be > -0.5, discrete generelised Pareto ("DGP"), discrete generelised Pareto II ("DGPII") where the shape parameter is forced to be positive, discrete generelised Pareto derived under the scenario in which shape = 0 ("DGP0"). Regarding the Tweedie, this margin can currently only be used together with a binary margin; we are working on the discrete/continuous margin extension. |
model |
Possible values are "B" (bivariate model), "T" (trivariate model), "BSS" (bivariate model with non-random sample selection), "TSS" (trivariate model with double non-random sample selection), "TESS" (trivariate model with endogeneity and non-random sample selection), "BPO" (bivariate model with partial observability) and "BPO0" (bivariate model with partial observability and zero correlation). Options "T", "TESS" and "TSS" are currently for trivariate binary models only. "BPO" and "BPO0" are for bivariate binary models only. "ROY" is for the Roy switching regression model. |
dof |
If |
dof2 |
As above but used only for Roy models. |
cens1 |
Censoring indicator for the first equation. For the case of right censored data only, this variable can be equal to 1 if the event occurred
and 0 otherwise. However, if there are several censoring mechanisms then |
cens2 |
Same as above but for the second equation. |
cens3 |
Binary censoring indicator employed only when |
dep.cens |
If TRUE then the dependence censored model is employed. |
ub.t1, ub.t2 |
Variable names of right/upper bounds when interval censoring is present. |
left.trunc1, left.trunc2 |
Values of truncation at left. Currently done for count distributions only. |
uni.fit |
If |
fp |
If |
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. See the documentation
of |
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 |
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 |
knots |
Optional list containing user specified knot values to be used for basis construction. |
penCor |
This and the arguments below are only for trivariate binary models. Type of penalty for correlation coefficients. Possible values are "unpen", "lasso", "ridge", "alasso". |
sp.penCor |
Starting value for smoothing parameter of |
Chol |
If |
gamma |
Inflation factor used only for the alasso penalty. |
w.alasso |
When using the alasso penalty a weight vector made up of three values must be provided. |
drop.unused.levels |
By default unused levels are dropped from factors before fitting. For some smooths involving factor variables this may have to be turned off (only use if you know what you are doing). |
min.dn, min.pr, max.pr |
These values are used to set, depending on the model used for modelling, the minimum and maximum allowed
for the densities and probabilities; recall that the margins of copula models have to be in the range (0,1). These
parameters are employed to avoid potential overflows/underflows in the calculations and the default
values seem to offer a good compromise. Function |
Details
The joint models considered by this function consist of two or three model equations which depend on flexible linear predictors and whose dependence between the responses is modelled through one or more parameters of a chosen multivariate distribution. The additive predictors of the equations are flexibly specified using parametric components and smooth functions of covariates. The same can be done for the dependence parameter(s) if it makes sense. Estimation is achieved within a penalized likelihood framework with integrated automatic multiple smoothing parameter selection. The use of penalty matrices allows for the suppression of that part of smooth term complexity which has no support from the data. The trade-off between smoothness and fitness is controlled by smoothing parameters associated with the penalty matrices. Smoothing parameters are chosen to minimise an approximate AIC.
For sample selection models, if there are factors in the model then before fitting the user has to ensure that the numbers of factor variables' levels in the selected sample are the same as those in the complete dataset. Even if a model could be fitted in such a situation, the model may produce fits which are not coherent with the nature of the correction sought. As an example consider the situation in which the complete dataset contains a factor variable with five levels and that only three of them appear in the selected sample. For the outcome equation (which is the one of interest) only three levels of such variable exist in the population, but their effects will be corrected for non-random selection using a selection equation in which five levels exist instead. Having differing numbers of factors' levels between complete and selected samples will also make prediction not feasible (an aspect which may be particularly important for selection models); clearly it is not possible to predict the response of interest for the missing entries using a dataset that contains all levels of a factor variable but using an outcome model estimated using a subset of these levels.
There are many continuous/discrete/survival distributions and copula functions to choose from and we plan to include more options. Get in touch if you are interested in a particular distribution.
Value
The function returns an object of class gjrm as described in gjrmObject.
WARNINGS
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.
gjrm() 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 rescaling (see parscale). Using uni.fit = TRUE is typically more effective than the first two options as
this will provide better calibrated starting values as compared to those obtained from the default starting value procedure.
The default option is, however, uni.fit = FALSE only because it tends to be computationally cheaper and because the
default procedure has typically been found to do a satisfactory job in most cases.
(The results obtained when using
uni.fit = FALSE and uni.fit = TRUE could also be compared to check if starting values make any difference.)
The above suggestions may help, especially the latter option. However, the user should also consider re-specifying/simplifying the model, and/or using a diferrent dependence structure and/or checking that the chosen marginal distributions fit the responses 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. Examples of misspecification include using a Clayton copula rotated by 90 degrees when a positive association between the margins is present instead, using marginal distributions that do not fit the responses, and employing a copula which does not accommodate the type and/or strength of the dependence between the margins (e.g., using AMH when the association between the margins is strong). When using smooth functions, if the covariate's values are too sparse then convergence may be affected by this. It is also worth bearing in mind that the use of three parameter marginal distributions requires the data to be more informative than a situation in which two parameter distributions are used instead.
In the contexts of endogeneity and non-random sample selection, extra attention is required when specifying the dependence parameter as a function of covariates. This is because in these situations the dependence parameter mainly models the association between the unobserved confounders in the two equations. Therefore, this option would make sense when it is believed that the strength of the association between the unobservables in the two equations varies based on some grouping factor or across geographical areas, for instance. In any case, a clear rationale is typically needed in such cases.
Author(s)
Maintainer: Giampiero Marra giampiero.marra@ucl.ac.uk
References
See help("GJRM-package").
See Also
adjCov, VuongClarke, GJRM-package, gjrmObject, conv.check, summary.gjrm
Examples
library(GJRM)
####################################
# JOINT MODELS WITH BINARY MARGINS #
####################################
###############
## EXAMPLE 1 ##
set.seed(0)
n <- 400
Sigma <- matrix(0.5, 2, 2); diag(Sigma) <- 1
u <- rMVN(n, rep(0,2), Sigma)
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)
y1 <- ifelse(-1.55 + 2*x1 + f1(x2) + u[,1] > 0, 1, 0)
y2 <- ifelse(-0.25 - 1.25*x1 + f2(x2) + u[,2] > 0, 1, 0)
dataSim <- data.frame(y1, y2, x1, x2, x3)
## CLASSIC BIVARIATE PROBIT
out <- gjrm(list(y1 ~ x1 + x2 + x3,
y2 ~ x1 + x2 + x3),
data = dataSim,
margins = c("probit", "probit"),
model = "B")
conv.check(out)
summary(out)
AIC(out)
BIC(out)
## Not run:
## BIVARIATE PROBIT with Splines
out <- gjrm(list(y1 ~ x1 + s(x2) + s(x3),
y2 ~ x1 + s(x2) + s(x3)),
data = dataSim,
margins = c("probit", "probit"),
model = "B")
conv.check(out)
summary(out)
AIC(out)
## estimated smooth function plots
plot(out, eq = 1, pages = 1, seWithMean = TRUE, scale = 0)
plot(out, eq = 2, pages = 1, seWithMean = TRUE, scale = 0)
## BIVARIATE PROBIT with Splines and
## varying dependence parameter
eq.mu.1 <- y1 ~ x1 + s(x2)
eq.mu.2 <- y2 ~ x1 + s(x2)
eq.theta <- ~ x1 + s(x2)
fl <- list(eq.mu.1, eq.mu.2, eq.theta)
outD <- gjrm(fl, data = dataSim,
margins = c("probit", "probit"),
model = "B")
conv.check(outD)
summary(outD)
summary(outD$theta)
plot(outD, eq = 1, seWithMean = TRUE)
plot(outD, eq = 2, seWithMean = TRUE)
plot(outD, eq = 3, seWithMean = TRUE)
graphics.off()
###############
## EXAMPLE 2 ##
## Generate data with one endogenous variable
## and exclusion restriction (or instrument)
set.seed(0)
n <- 400
Sigma <- matrix(0.5, 2, 2); diag(Sigma) <- 1
u <- rMVN(n, rep(0,2), Sigma)
cov <- rMVN(n, rep(0,2), Sigma)
cov <- pnorm(cov)
x1 <- round(cov[,1]); x2 <- cov[,2]
f1 <- function(x) cos(pi*2*x) + sin(pi*x)
f2 <- function(x) x+exp(-30*(x-0.5)^2)
y1 <- ifelse(-1.55 + 2*x1 + f1(x2) + u[,1] > 0, 1, 0)
y2 <- ifelse(-0.25 - 1.25*y1 + f2(x2) + u[,2] > 0, 1, 0)
dataSim <- data.frame(y1, y2, x1, x2)
#
## Testing the hypothesis of absence of endogeneity...
#
LM.bpm(list(y1 ~ x1 + s(x2), y2 ~ y1 + s(x2)), dataSim, model = "B")
## CLASSIC RECURSIVE BIVARIATE PROBIT
out <- gjrm(list(y1 ~ x1 + x2,
y2 ~ y1 + x2),
data = dataSim,
margins = c("probit", "probit"), model = "B")
conv.check(out)
summary(out)
AIC(out); BIC(out)
## FLEXIBLE RECURSIVE BIVARIATE PROBIT
out <- gjrm(list(y1 ~ x1 + s(x2),
y2 ~ y1 + s(x2)),
data = dataSim,
margins = c("probit", "probit"),
model = "B")
conv.check(out)
summary(out)
AIC(out); BIC(out)
#
## Testing the hypothesis of absence of endogeneity post estimation...
gt.bpm(out)
#
## Causal effects
## average treatment effect, risk ratio and odds ratio with CIs
mb(y1, y2, model = "B")
ATE(out, trt = "y1")
RR(out, trt = "y1")
OR(out, trt = "y1")
ATE(out, trt = "y1", joint = FALSE)
## try a Clayton copula model...
outC <- gjrm(list(y1 ~ x1 + s(x2),
y2 ~ y1 + s(x2)),
data = dataSim, copula = "C0",
margins = c("probit", "probit"),
model = "B")
conv.check(outC)
summary(outC)
ATE(outC, trt = "y1")
## try a Joe copula model...
outJ <- gjrm(list(y1 ~ x1 + s(x2),
y2 ~ y1 + s(x2)),
data = dataSim, copula = "J0",
margins = c("probit", "probit"),
model = "B")
conv.check(outJ)
summary(outJ)
ATE(outJ, "y1")
VuongClarke(out, outJ)
#
## recursive bivariate probit modelling with unpenalized splines
## can be achieved as follows
outFP <- gjrm(list(y1 ~ x1 + s(x2, bs = "cr", k = 5),
y2 ~ y1 + s(x2, bs = "cr", k = 6)),
fp = TRUE, data = dataSim,
margins = c("probit", "probit"),
model = "B")
conv.check(outFP)
summary(outFP)
# in the above examples a third equation could be introduced
# as illustrated in Example 1
###############
## EXAMPLE 3 ##
## Generate data with a non-random sample selection mechanism
## and exclusion restriction
set.seed(0)
n <- 2000
Sigma <- matrix(0.5, 2, 2); diag(Sigma) <- 1
u <- rMVN(n, rep(0,2), Sigma)
SigmaC <- matrix(0.5, 3, 3); diag(SigmaC) <- 1
cov <- rMVN(n, rep(0,3), SigmaC)
cov <- pnorm(cov)
bi <- round(cov[,1]); x1 <- cov[,2]; x2 <- cov[,3]
f11 <- function(x) -0.7*(4*x + 2.5*x^2 + 0.7*sin(5*x) + cos(7.5*x))
f12 <- function(x) -0.4*( -0.3 - 1.6*x + sin(5*x))
f21 <- function(x) 0.6*(exp(x) + sin(2.9*x))
ys <- 0.58 + 2.5*bi + f11(x1) + f12(x2) + u[, 1] > 0
y <- -0.68 - 1.5*bi + f21(x1) + + u[, 2] > 0
yo <- y*(ys > 0)
dataSim <- data.frame(y, ys, yo, bi, x1, x2)
#
## Testing the hypothesis of absence of non-random sample selection...
LM.bpm(list(ys ~ bi + s(x1) + s(x2), yo ~ bi + s(x1)), dataSim, model = "BSS")
# p-value suggests presence of sample selection
#
## SEMIPARAMETRIC SAMPLE SELECTION BIVARIATE PROBIT
## the first equation MUST be the selection equation
out <- gjrm(list(ys ~ bi + s(x1) + s(x2),
yo ~ bi + s(x1)),
data = dataSim, model = "BSS",
margins = c("probit", "probit"))
conv.check(out)
gt.bpm(out)
## compare the two summary outputs below
## the second output produces a summary of the results obtained when
## selection bias is not accounted for
summary(out)
summary(out$gam2)
## corrected predicted probability that 'yo' is equal to 1
mb(ys, yo, model = "BSS")
prev(out)
prev(out, joint = FALSE)
## estimated smooth function plots
## the red line is the true curve
## the blue line is the univariate model curve not accounting for selection bias
x1.s <- sort(x1[dataSim$ys>0])
f21.x1 <- f21(x1.s)[order(x1.s)]-mean(f21(x1.s))
plot(out, eq = 2, ylim = c(-1.65,0.95)); lines(x1.s, f21.x1, col="red")
par(new = TRUE)
plot(out$gam2, se = FALSE, col = "blue", ylim = c(-1.65,0.95),
ylab = "", rug = FALSE)
#
#
## try a Clayton copula model...
outC <- gjrm(list(ys ~ bi + s(x1) + s(x2),
yo ~ bi + s(x1)),
data = dataSim, model = "BSS", copula = "C0",
margins = c("probit", "probit"))
conv.check(outC)
summary(outC)
prev(outC)
################
## See also ?hiv
################
###############
## EXAMPLE 4 ##
## Generate data with partial observability
set.seed(0)
n <- 1000
Sigma <- matrix(0.5, 2, 2); diag(Sigma) <- 1
u <- rMVN(n, rep(0,2), Sigma)
x1 <- round(runif(n)); x2 <- runif(n); x3 <- runif(n)
y1 <- ifelse(-1.55 + 2*x1 + x2 + u[,1] > 0, 1, 0)
y2 <- ifelse( 0.45 - x3 + u[,2] > 0, 1, 0)
y <- y1*y2
dataSim <- data.frame(y, x1, x2, x3)
## BIVARIATE PROBIT with Partial Observability
out <- gjrm(list(y ~ x1 + x2,
y ~ x3),
data = dataSim, model = "BPO",
margins = c("probit", "probit"))
conv.check(out)
summary(out)
# first ten estimated probabilities for the four events from object out
cbind(out$p11, out$p10, out$p00, out$p01)[1:10,]
# case with smooth function
f1 <- function(x) cos(pi*2*x) + sin(pi*x)
y1 <- ifelse(-1.55 + 2*x1 + f1(x2) + u[,1] > 0, 1, 0)
y2 <- ifelse( 0.45 - x3 + u[,2] > 0, 1, 0)
y <- y1*y2
dataSim <- data.frame(y, x1, x2, x3)
out <- gjrm(list(y ~ x1 + s(x2),
y ~ x3),
data = dataSim, model = "BPO",
margins = c("probit", "probit"))
conv.check(out)
summary(out)
plot(out, eq = 1, scale = 0)
################
## See also ?war
################
######################################################
# JOINT MODELS WITH BINARY AND/OR CONTINUOUS MARGINS #
######################################################
###############
## EXAMPLE 5 ##
## Generate data
## Correlation between the two equations 0.5 - Sample size 400
set.seed(0)
n <- 400
Sigma <- matrix(0.5, 2, 2); diag(Sigma) <- 1
u <- rMVN(n, rep(0,2), Sigma)
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)
y1 <- -1.55 + 2*x1 + f1(x2) + u[,1]
y2 <- -0.25 - 1.25*x1 + f2(x2) + u[,2]
dataSim <- data.frame(y1, y2, x1, x2, x3)
resp.check(y1, "N")
resp.check(y2, "N")
eq.mu.1 <- y1 ~ x1 + s(x2) + s(x3)
eq.mu.2 <- y2 ~ x1 + s(x2) + s(x3)
eq.sigma1 <- ~ 1
eq.sigma2 <- ~ 1
eq.theta <- ~ x1
fl <- list(eq.mu.1, eq.mu.2, eq.sigma1, eq.sigma2, eq.theta)
# the order above is the one to follow when
# using more than two equations
out <- gjrm(fl, data = dataSim, margins = c("N", "N"),
model = "B")
conv.check(out)
res.check(out)
summary(out)
AIC(out)
BIC(out)
nd <- data.frame(x1 = 1, x2 = 0.4, x3 = 0.6)
copula.prob(out, y1 = 1.4, y2 = 2.3, newdata = nd, intervals = TRUE)
###############
## EXAMPLE 6 ##
## Generate data with one endogenous binary variable
## and continuous outcome
set.seed(0)
n <- 1000
Sigma <- matrix(0.5, 2, 2); diag(Sigma) <- 1
u <- rMVN(n, rep(0,2), Sigma)
cov <- rMVN(n, rep(0,2), Sigma)
cov <- pnorm(cov)
x1 <- round(cov[,1]); x2 <- cov[,2]
f1 <- function(x) cos(pi*2*x) + sin(pi*x)
f2 <- function(x) x+exp(-30*(x-0.5)^2)
y1 <- ifelse(-1.55 + 2*x1 + f1(x2) + u[,1] > 0, 1, 0)
y2 <- -0.25 - 1.25*y1 + f2(x2) + u[,2]
dataSim <- data.frame(y1, y2, x1, x2)
## RECURSIVE Model
rc <- resp.check(y2, margin = "N", print.par = TRUE, loglik = TRUE)
AIC(rc); BIC(rc)
out <- gjrm(list(y1 ~ x1 + x2,
y2 ~ y1 + x2),
data = dataSim, margins = c("probit","N"),
model = "B")
conv.check(out)
summary(out)
res.check(out)
## SEMIPARAMETRIC RECURSIVE Model
eq.mu.1 <- y1 ~ x1 + s(x2)
eq.mu.2 <- y2 ~ y1 + s(x2)
eq.sigma <- ~ 1
eq.theta <- ~ 1
fl <- list(eq.mu.1, eq.mu.2, eq.sigma, eq.theta)
out <- gjrm(fl, data = dataSim,
margins = c("probit","N"), uni.fit = TRUE,
model = "B")
conv.check(out)
summary(out)
res.check(out)
ATE(out, trt = "y1")
ATE(out, trt = "y1", joint = FALSE)
#
#
###############
## EXAMPLE 7 ##
## Generate data with one endogenous continuous exposure
## and binary outcome
set.seed(0)
n <- 1000
Sigma <- matrix(0.5, 2, 2); diag(Sigma) <- 1
u <- rMVN(n, rep(0,2), Sigma)
cov <- rMVN(n, rep(0,2), Sigma)
cov <- pnorm(cov)
x1 <- round(cov[,1]); x2 <- cov[,2]
f1 <- function(x) cos(pi*2*x) + sin(pi*x)
f2 <- function(x) x+exp(-30*(x-0.5)^2)
y1 <- -0.25 - 2*x1 + f2(x2) + u[,2]
y2 <- ifelse(-0.25 - 0.25*y1 + f1(x2) + u[,1] > 0, 1, 0)
dataSim <- data.frame(y1, y2, x1, x2)
eq.mu.1 <- y2 ~ y1 + s(x2)
eq.mu.2 <- y1 ~ x1 + s(x2)
eq.sigma <- ~ 1
eq.theta <- ~ 1
fl <- list(eq.mu.1, eq.mu.2, eq.sigma, eq.theta)
out <- gjrm(fl, data = dataSim,
margins = c("probit","N"),
model = "B")
conv.check(out)
summary(out)
res.check(out)
ATE(out, trt = "y1")
ATE(out, trt = "y1", joint = FALSE)
RR(out, trt = "y1")
RR(out, trt = "y1", joint = FALSE)
OR(out, trt = "y1")
OR(out, trt = "y1", joint = FALSE)
#
#
#####################
## EXAMPLE 8 ##
## SURVIVAL MODELS ##
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
}
delta1 <- ifelse(t < c, 1, 0)
u1 <- apply(cbind(t, c), 1, min)
dataSim <- data.frame(u1, delta1, z1, z2)
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
}
delta2 <- ifelse(t < c,1, 0)
u2 <- apply(cbind(t, c), 1, min)
dataSim$delta2 <- delta2
dataSim$u2 <- u2
dataSim$z <- z
eq1 <- u1 ~ s(log(u1), bs = "mpi") + z1 + s(z2)
eq2 <- u2 ~ s(log(u2), bs = "mpi") + z
eq3 <- ~ s(z2)
out <- gjrm(list(eq1, eq2), data = dataSim,
margins = c("-cloglog", "-logit"),
cens1 = delta1, cens2 = delta2, model = "B")
# PH margin fit can also be compared with cox.ph from mgcv
conv.check(out)
res <- res.check(out)
## martingale residuals
mr1 <- out$cens1 - res$qr1
mr2 <- out$cens2 - res$qr2
# can be plotted against covariates
# obs index, survival time, rank order of
# surv times
# to determine func form, one may use
# res from null model against covariate
# to test for PH, use:
# library(survival)
# fit <- coxph(Surv(u1, delta1) ~ z1 + z2, data = dataSim)
# temp <- cox.zph(fit)
# print(temp)
# plot(temp, resid = FALSE)
summary(out)
AIC(out); BIC(out)
plot(out, eq = 1, scale = 0, pages = 1)
plot(out, eq = 2, scale = 0, pages = 1)
haz.surv(out, eq = 1, newdata = data.frame(z1 = 0, z2 = 0),
shade = TRUE, n.sim = 100, baseline = TRUE)
haz.surv(out, eq = 1, newdata = data.frame(z1 = 0, z2 = 0),
shade = TRUE, n.sim = 100, type = "haz", baseline = TRUE,
intervals = FALSE)
haz.surv(out, eq = 2, newdata = data.frame(z = 0),
shade = FALSE, n.sim = 100, baseline = TRUE)
haz.surv(out, eq = 2, newdata = data.frame(z = 0),
shade = TRUE, n.sim = 100, type = "haz", baseline = TRUE)
newd0 <- newd1 <- data.frame(z = 0, z1 = mean(dataSim$z1),
z2 = mean(dataSim$z2),
u1 = mean(dataSim$u1) + 1,
u2 = mean(dataSim$u2) + 1)
newd1$z <- 1
copula.prob(out, newdata = newd0, intervals = TRUE)
copula.prob(out, newdata = newd1, intervals = TRUE)
out1 <- gjrm(list(eq1, eq2, eq3), data = dataSim,
margins = c("-cloglog", "-logit"),
cens1 = delta1, cens2 = delta2, uni.fit = TRUE,
model = "B")
####################################################
## Joint continuous and survival outcomes
####################################################
# this is complete, just testing, get in touch if interested
#
# eq1 <- z2 ~ z1
# eq2 <- u2 ~ s(u2, bs = "mpi") + z
# eq3 <- ~ s(z2)
# eq4 <- ~ s(z2)
#
# f.l <- list(eq1, eq2, eq3, eq4)
#
# out3 <- gjrm(f.l, data = dataSim,
# margins = c("N", "-logit"),
# cens1 = NULL, cens2 = delta2,
# uni.fit = TRUE, model = "B")
#
# conv.check(out3)
# res.check(out3)
# summary(out3)
# AIC(out3); BIC(out3)
# plot(out3, eq = 2, scale = 0, pages = 1)
# plot(out3, eq = 3, scale = 0, pages = 1)
# plot(out3, eq = 4, scale = 0, pages = 1)
#
# newd <- newd1 <- data.frame(z = 0, z1 = mean(dataSim$z1),
# z2 = mean(dataSim$z2),
# u2 = mean(dataSim$u2) + 1)
#
# copula.prob(out3, y1 = 0.6, newdata = newd, intervals = TRUE)
##########################################
# JOINT MODELS WITH THREE BINARY MARGINS #
##########################################
###############
## EXAMPLE 9 ##
## Generate data
## Correlation between the two equations 0.5 - Sample size 400
set.seed(0)
n <- 400
Sigma <- matrix(0.5, 3, 3); diag(Sigma) <- 1
u <- rMVN(n, rep(0,3), Sigma)
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)
y1 <- ifelse(-1.55 + 2*x1 - f1(x2) + u[,1] > 0, 1, 0)
y2 <- ifelse(-0.25 - 1.25*x1 + f2(x2) + u[,2] > 0, 1, 0)
y3 <- ifelse(-0.75 + 0.25*x1 + u[,3] > 0, 1, 0)
dataSim <- data.frame(y1, y2, y3, x1, x2)
f.l <- list(y1 ~ x1 + s(x2),
y2 ~ x1 + s(x2),
y3 ~ x1)
margs <- c("probit", "probit", "probit")
out <- gjrm(f.l, data = dataSim, model = "T", margins = margs)
out1 <- gjrm(f.l, data = dataSim, Chol = TRUE, model = "T", margins = margs)
conv.check(out)
summary(out)
plot(out, eq = 1)
plot(out, eq = 2)
AIC(out)
BIC(out)
margs <- c("probit","logit","cloglog")
out <- gjrm(f.l, data = dataSim, model = "T", margins = margs)
out1 <- gjrm(f.l, data = dataSim, Chol = TRUE, model = "T", margins = margs)
conv.check(out)
summary(out)
plot(out, eq = 1)
plot(out, eq = 2)
AIC(out)
BIC(out)
margs <- c("probit", "probit", "probit")
f.l <- list(y1 ~ x1 + s(x2),
y2 ~ x1 + s(x2),
y3 ~ x1,
~ 1, ~ 1, ~ 1)
out1 <- gjrm(f.l, data = dataSim, Chol = TRUE, model = "T", margins = margs)
f.l <- list(y1 ~ x1 + s(x2),
y2 ~ x1 + s(x2),
y3 ~ x1,
~ 1, ~ s(x2), ~ 1)
out2 <- gjrm(f.l, data = dataSim, Chol = TRUE, model = "T", margins = margs)
f.l <- list(y1 ~ x1 + s(x2),
y2 ~ x1 + s(x2),
y3 ~ x1,
~ x1, ~ s(x2), ~ x1 + s(x2))
out2 <- gjrm(f.l, data = dataSim, Chol = TRUE, model = "T", margins = margs)
f.l <- list(y1 ~ x1 + s(x2),
y2 ~ x1 + s(x2),
y3 ~ x1,
~ x1, ~ x1, ~ s(x2))
out2 <- gjrm(f.l, data = dataSim, Chol = TRUE, model = "T", margins = margs)
f.l <- list(y1 ~ x1 + s(x2),
y2 ~ x1 + s(x2),
y3 ~ x1,
~ x1, ~ x1 + x2, ~ s(x2))
out2 <- gjrm(f.l, data = dataSim, Chol = TRUE, model = "T", margins = margs)
f.l <- list(y1 ~ x1 + s(x2),
y2 ~ x1 + s(x2),
y3 ~ x1,
~ x1 + x2, ~ x1 + x2, ~ x1 + x2)
out2 <- gjrm(f.l, data = dataSim, Chol = TRUE, model = "T", margins = margs)
nw <- data.frame( x1 = 0, x2 = 0.7 )
copula.prob(out2, 1, 1, 1, newdata = nw, cond = 0, intervals = TRUE, n.sim = 100)
# with endogenous variable
f.l <- list(y1 ~ x1 + s(x2),
y2 ~ y1 + x1 + s(x2),
y3 ~ x1)
margs <- c("probit", "probit", "probit")
out <- gjrm(f.l, data = dataSim, model = "T", margins = margs)
conv.check(out)
summary(out)
ATE(out, trt = "y1", eq = 2, joint = TRUE)
ATE(out, trt = "y1", eq = 2, joint = FALSE)
################
## EXAMPLE 10 ##
## Generate data
## with double sample selection
set.seed(0)
n <- 5000
Sigma <- matrix(c(1, 0.5, 0.4,
0.5, 1, 0.6,
0.4, 0.6, 1 ), 3, 3)
u <- rMVN(n, rep(0,3), Sigma)
f1 <- function(x) cos(pi*2*x) + sin(pi*x)
f2 <- function(x) x+exp(-30*(x-0.5)^2)
x1 <- runif(n)
x2 <- runif(n)
x3 <- runif(n)
x4 <- runif(n)
y1 <- 1 + 1.5*x1 - x2 + 0.8*x3 - f1(x4) + u[, 1] > 0
y2 <- 1 - 2.5*x1 + 1.2*x2 + x3 + u[, 2] > 0
y3 <- 1.58 + 1.5*x1 - f2(x2) + u[, 3] > 0
dataSim <- data.frame(y1, y2, y3, x1, x2, x3, x4)
f.l <- list(y1 ~ x1 + x2 + x3 + s(x4),
y2 ~ x1 + x2 + x3,
y3 ~ x1 + s(x2))
out <- gjrm(f.l, data = dataSim, model = "TSS",
margins = c("probit", "probit", "probit"))
conv.check(out)
summary(out)
plot(out, eq = 1)
plot(out, eq = 3)
prev(out)
prev(out, joint = FALSE)
#############
## EXAMPLE 11
set.seed(0)
n <- 2000
rh <- 0.5
sigmau <- matrix(c(1, rh, rh, 1), 2, 2)
u <- rMVN(n, rep(0,2), sigmau)
sigmac <- matrix(rh, 3, 3); diag(sigmac) <- 1
cov <- rMVN(n, rep(0,3), sigmac)
cov <- pnorm(cov)
bi <- round(cov[,1]); x1 <- cov[,2]; x2 <- cov[,3]
f11 <- function(x) -0.7*(4*x + 2.5*x^2 + 0.7*sin(5*x) + cos(7.5*x))
f12 <- function(x) -0.4*( -0.3 - 1.6*x + sin(5*x))
f21 <- function(x) 0.6*(exp(x) + sin(2.9*x))
ys <- 0.58 + 2.5*bi + f11(x1) + f12(x2) + u[, 1] > 0
y <- -0.68 - 1.5*bi + f21(x1) + u[, 2]
yo <- y*(ys > 0)
dataSim <- data.frame(ys, yo, bi, x1, x2)
## CLASSIC SAMPLE SELECTION MODEL
## the first equation MUST be the selection equation
resp.check(yo[ys > 0], "N")
out <- gjrm(list(ys ~ bi + x1 + x2,
yo ~ bi + x1),
data = dataSim, model = "BSS",
margins = c("probit", "N"))
conv.check(out)
res.check(out)
summary(out)
AIC(out)
BIC(out)
## SEMIPARAMETRIC SAMPLE SELECTION MODEL
out <- gjrm(list(ys ~ bi + s(x1) + s(x2),
yo ~ bi + s(x1)),
data = dataSim, model = "BSS",
margins = c("probit", "N"))
conv.check(out)
res.check(out)
AIC(out)
## compare the two summary outputs
## the second output produces a summary of the results obtained when only
## the outcome equation is fitted, i.e. selection bias is not accounted for
summary(out)
summary(out$gam2)
## estimated smooth function plots
## the red line is the true curve
## the blue line is the naive curve not accounting for selection bias
x1.s <- sort(x1[dataSim$ys>0])
f21.x1 <- f21(x1.s)[order(x1.s)] - mean(f21(x1.s))
plot(out, eq = 2, ylim = c(-1, 0.8)); lines(x1.s, f21.x1, col = "red")
par(new = TRUE)
plot(out$gam2, se = FALSE, lty = 3, lwd = 2, ylim = c(-1, 0.8),
ylab = "", rug = FALSE)
##
## SEMIPARAMETRIC SAMPLE SELECTION MODEL with association
## and dispersion parameters
## depending on covariates as well
eq.mu.1 <- ys ~ bi + s(x1) + s(x2)
eq.mu.2 <- yo ~ bi + s(x1)
eq.sigma <- ~ bi
eq.theta <- ~ bi + x1
fl <- list(eq.mu.1, eq.mu.2, eq.sigma, eq.theta)
out <- gjrm(fl, data = dataSim, model = "BSS",
margins = c("probit", "N"))
conv.check(out)
res.check(out)
summary(out)
summary(out$sigma)
summary(out$theta)
nd <- data.frame(bi = 0, x1 = 0.2, x2 = 0.8)
copula.prob(out, 0, 0.3, newdata = nd, intervals = TRUE)
outC0 <- gjrm(fl, data = dataSim, copula = "C0", model = "BSS",
margins = c("probit", "N"))
conv.check(outC0)
res.check(outC0)
AIC(out, outC0)
BIC(out, outC0)
## End(Not run)
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