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

View source: R/SemiParBIVProbit.r

`SemiParBIVProbit`

fits flexible copula bivariate binary models with several types of covariate
effects, copula distributions and link functions. During the model fitting process, the
possible presence of associated error equations, endogeneity, non-random sample selection or partial observability is accounted for.

1 2 3 4 5 6 7 | ```
SemiParBIVProbit(formula, data = list(), weights = NULL, subset = NULL,
Model = "B", BivD = "N",
margins = c("probit","probit"), dof = 3, gamlssfit = FALSE,
fp = FALSE, hess = TRUE, infl.fac = 1, theta.fx = NULL,
rinit = 1, rmax = 100,
iterlimsp = 50, tolsp = 1e-07,
gc.l = FALSE, parscale, extra.regI = "t", intf = FALSE)
``` |

`formula` |
In the basic setup this will be a list of two formulas, one for equation 1 and the other for equation 2. |

`data` |
An optional data frame, list or environment containing the variables in the model. If not found in |

`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. |

`Model` |
It indicates the type of model to be used in the analysis. Possible values are "B" (bivariate model), "BSS" (bivariate model with non-random sample selection), "BPO" (bivariate model with partial observability) and "BPO0" (bivariate model with partial observability and zero correlation). |

`margins` |
It indicates the link functions used for the two margins. Possible choices are "probit", "logit", "cloglog". |

`dof` |
If |

`gamlssfit` |
This is for internal purposes only. |

`BivD` |
Type of bivariate error distribution employed. Possible choices are "N", "C0", "C90", "C180", "C270", "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), 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 fixed |

`fp` |
If |

`hess` |
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. |

`theta.fx` |
If |

`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 |

`extra.regI` |
If "t" then regularization as from |

`intf` |
This is for internal use. |

The bivariate models considered in this package consist of two model equations which depend on flexible linear predictors and
whose association between the responses is modelled through parameter *θ* of a standardised bivariate normal
distribution or that of a bivariate copula distribution. The linear predictors of the two equations are flexibly specified using
parametric components and smooth functions of covariates. The same can be done for the dependence parameter 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.

Details of the underlying fitting methods are given in Radice, Marra and Wojtys (2016) and Marra et al. (in press). Releases previous to 3.2-7 were based on the algorithms detailed in Marra and Radice (2011, 2013).

For sample selection models, if there are factors in the model, 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.

The function returns an object of class `SemiParBIVProbit`

as described in `SemiParBIVProbitObject`

.

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.
`SemiParBIVProbit()`

will produce some warnings when 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`

). These 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 using different link functions.
In our experience, we found that convergence failure typically occurs
when the model has been misspecified and/or the sample size (and/or number of selected observations in selection models) 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 are not adequate, 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.

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.

Maintainer: Giampiero Marra [email protected]

Marra G. and Radice R. (2011), Estimation of a Semiparametric Recursive Bivariate Probit in the Presence of Endogeneity. *Canadian
Journal of Statistics*, 39(2), 259-279.

Marra G. and Radice R. (2013), A Penalized Likelihood Estimation Approach to Semiparametric Sample Selection Binary Response Modeling. *Electronic Journal of Statistics*, 7, 1432-1455.

Marra G., Radice R., Barnighausen T., Wood S.N. and McGovern M.E. (in press), A Simultaneous Equation Approach to Estimating HIV Prevalence with Non-Ignorable Missing Responses. *Journal of the American Statistical Association*.

McGovern M.E., Barnighausen T., Marra G. and Radice R. (2015), On the Assumption of Joint Normality in Selection Models: A Copula Approach Applied to Estimating HIV Prevalence. *Epidemiology*, 26(2), 229-237.

Radice R., Marra G. and Wojtys M. (2016), Copula Regression Spline Models for Binary Outcomes. *Statistics and Computing*, 26(5), 981-995.

Poirier D.J. (1980), Partial Observability in Bivariate Probit Models. *Journal of Econometrics*, 12, 209-217.

`copulaReg`

, `copulaSampleSel`

, `SemiParTRIV`

, `AT`

, `OR`

, `RR`

, `adjCov`

, `prev`

, `gt.bpm`

, `LM.bpm`

, `VuongClarke`

, `plot.SemiParBIVProbit`

, `SemiParBIVProbit-package`

, `SemiParBIVProbitObject`

, `conv.check`

, `summary.SemiParBIVProbit`

, `predict.SemiParBIVProbit`

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library(SemiParBIVProbit)
############
## EXAMPLE 1
############
## 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 <- 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 <- SemiParBIVProbit(list(y1 ~ x1 + x2 + x3,
y2 ~ x1 + x2 + x3),
data = dataSim)
conv.check(out)
summary(out)
AIC(out)
BIC(out)
## Not run:
## SEMIPARAMETRIC BIVARIATE PROBIT
## "cr" cubic regression spline basis - "cs" shrinkage version of "cr"
## "tp" thin plate regression spline basis - "ts" shrinkage version of "tp"
## for smooths of one variable, "cr/cs" and "tp/ts" achieve similar results
## k is the basis dimension - default is 10
## m is the order of the penalty for the specific term - default is 2
## For COPULA models use BivD argument
out <- SemiParBIVProbit(list(y1 ~ x1 + s(x2, bs = "tp", k = 10, m = 2) + s(x3),
y2 ~ x1 + s(x2) + s(x3)),
data = dataSim)
conv.check(out)
summary(out)
AIC(out)
## estimated smooth function plots - red lines are true curves
x2 <- sort(x2)
f1.x2 <- f1(x2)[order(x2)] - mean(f1(x2))
f2.x2 <- f2(x2)[order(x2)] - mean(f2(x2))
f3.x3 <- rep(0, length(x3))
par(mfrow=c(2,2),mar=c(4.5,4.5,2,2))
plot(out, eq = 1, select = 1, seWithMean = TRUE, scale = 0)
lines(x2, f1.x2, col = "red")
plot(out, eq = 1, select = 2, seWithMean = TRUE, scale = 0)
lines(x3, f3.x3, col = "red")
plot(out, eq = 2, select = 1, seWithMean = TRUE, scale = 0)
lines(x2, f2.x2, col = "red")
plot(out, eq = 2, select = 2, seWithMean = TRUE, scale = 0)
lines(x3, f3.x3, col = "red")
## p-values suggest to drop x3 from both equations, with a stronger
## evidence for eq. 2. This can be also achieved using shrinkage smoothers
outSS <- SemiParBIVProbit(list(y1 ~ x1 + s(x2, bs = "ts") + s(x3, bs = "cs"),
y2 ~ x1 + s(x2, bs = "cs") + s(x3, bs = "ts")),
data = dataSim)
conv.check(outSS)
plot(outSS, eq = 1, select = 1, scale = 0, shade = TRUE)
plot(outSS, eq = 1, select = 2, ylim = c(-0.1,0.1))
plot(outSS, eq = 2, select = 1, scale = 0, shade = TRUE)
plot(outSS, eq = 2, select = 2, ylim = c(-0.1,0.1))
## SEMIPARAMETRIC BIVARIATE PROBIT with association parameter
## depending on covariates as well
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 <- SemiParBIVProbit(fl, data = dataSim)
conv.check(outD)
summary(outD)
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
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")
# p-value suggests presence of endogeneity, hence fit a bivariate model
## CLASSIC RECURSIVE BIVARIATE PROBIT
out <- SemiParBIVProbit(list(y1 ~ x1 + x2,
y2 ~ y1 + x2),
data = dataSim)
conv.check(out)
summary(out)
AIC(out); BIC(out)
## SEMIPARAMETRIC RECURSIVE BIVARIATE PROBIT
out <- SemiParBIVProbit(list(y1 ~ x1 + s(x2),
y2 ~ y1 + s(x2)),
data = dataSim)
conv.check(out)
summary(out)
AIC(out); BIC(out)
#
## Testing the hypothesis of absence of endogeneity post estimation...
gt.bpm(out)
#
## reatment effect, risk ratio and odds ratio with CIs
mb(y1, y2, Model = "B")
AT(out, nm.end = "y1", hd.plot = TRUE)
RR(out, nm.end = "y1")
OR(out, nm.end = "y1")
AT(out, nm.end = "y1", type = "univariate")
## try a Clayton copula model...
outC <- SemiParBIVProbit(list(y1 ~ x1 + s(x2),
y2 ~ y1 + s(x2)),
data = dataSim, BivD = "C0")
conv.check(outC)
summary(outC)
AT(outC, nm.end = "y1")
## try a Joe copula model...
outJ <- SemiParBIVProbit(list(y1 ~ x1 + s(x2),
y2 ~ y1 + s(x2)),
data = dataSim, BivD = "J0")
conv.check(outJ)
summary(outJ)
AT(outJ, "y1")
VuongClarke(out, outJ)
#
## recursive bivariate probit modelling with unpenalized splines
## can be achieved as follows
outFP <- SemiParBIVProbit(list(y1 ~ x1 + s(x2, bs = "cr", k = 5),
y2 ~ y1 + s(x2, bs = "cr", k = 6)),
fp = TRUE, data = dataSim)
conv.check(outFP)
summary(outFP)
# in the above examples a third equation could be introduced
# as illustrated in Example 1
#
#################
## See also ?meps
#################
############
## 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, hence fit a bivariate model
#
## SEMIPARAMETRIC SAMPLE SELECTION BIVARIATE PROBIT
## the first equation MUST be the selection equation
out <- SemiParBIVProbit(list(ys ~ bi + s(x1) + s(x2),
yo ~ bi + s(x1)),
data = dataSim, Model = "BSS")
conv.check(out)
gt.bpm(out)
## compare the two summary outputs
## 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, hd.plot = TRUE)
prev(out, type = "univariate", hd.plot = TRUE)
## 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 <- SemiParBIVProbit(list(ys ~ bi + s(x1) + s(x2),
yo ~ bi + s(x1)),
data = dataSim, Model = "BSS", BivD = "C0")
conv.check(outC)
summary(outC)
prev(outC)
# in the above examples a third equation could be introduced
# as illustrated in Example 1
#
################
## See also ?hiv
################
############
## EXAMPLE 4
############
## Generate data with partial observability
set.seed(0)
n <- 10000
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 <- SemiParBIVProbit(list(y ~ x1 + x2,
y ~ x3),
data = dataSim, Model = "BPO")
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
# (more computationally intensive)
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 <- SemiParBIVProbit(list(y ~ x1 + s(x2),
y ~ x3),
data = dataSim, Model = "BPO")
conv.check(out)
summary(out)
# plot estimated and true functions
x2 <- sort(x2); f1.x2 <- f1(x2)[order(x2)] - mean(f1(x2))
plot(out, eq = 1, scale = 0); lines(x2, f1.x2, col = "red")
#
################
## See also ?war
################
## End(Not run)
``` |

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