summary.SemiParBIVProbit: SemiParBIVProbit summary

Description

It takes a fitted SemiParBIVProbit object produced by SemiParBIVProbit() and produces some summaries from it.

Usage

## S3 method for class 'SemiParBIVProbit':
summary(object, n.sim = 100, s.meth = "svd", prob.lev = 0.05, 
                                   thrs1 = 0.5, thrs2 = 0.5, cplot = FALSE, ...)

Arguments

object

A fitted SemiParBIVProbit object as produced by SemiParBIVProbit().

n.sim

The number of simulated coefficient vectors from the posterior distribution of the estimated model parameters. This is used to calculate intervals for the association parameter and other measures (e.g., gamma measure). It should be increased if

s.meth

Matrix decomposition used to determine the matrix root of the covariance matrix. See the documentation of mvtnorm for further details.

prob.lev

Probability of the left and right tails of the posterior distribution used for interval calculations.

thrs1, thrs2

Threshold to calculate the correct prediction ratios for the two binary responses. Default value is 0.5. It only works when Method = "B".

cplot

If TRUE then a bivariate contour meta plot corresponding to the assumed bivariate model with estimated association parameter is produced. See documentation of CDVine for details.

...

Other graphics parameters to pass on to plotting commands, when cplot=TRUE.

Value

tableP1Table containing parametric estimates, their standard errors, z-values and p-values for equation 1.
tableP2As above but for equation 2.
tableNP1Table of nonparametric summaries for each smooth component including effective degrees of freedom, estimated rank, approximate Wald statistic for testing the null hypothesis that the smooth term is zero and corresponding p-value, for equation 1.
tableNP2As above but for equation 2.
nSample size.
rhoEstimated correlation parameter between the two equations.
thetaEstimated copula parameter linking the two equations.
formula1, formula2Formulas used for equations 1 and 2.
l.sc1, l.sc2Number of smooth components in equations 1 and 2.
t.edfTotal degrees of freedom of the estimated bivariate model.
CIrs`Confidence' interval for either $\rho$ or $\theta$, depending of the model fitted.
CIl1, CIl2`Confidence' intervals for the shape parameters when an asymmetric link approach is employed.
n.selNumber of selected observations in the sample selection case.
goodIndicator variable indicating the observations actually used in model fitting.
OR, CIorOdds ratio and related CI. The odds ratio is a measure of association between binary random variables and is defined as p00p11/p10p01. In the case of independence this ratio is equal to 1. It can take values in the range (-Inf, Inf) and it does not depend on the marginal probabilities (Tajar et al., 2001). Interval is calculated using posterior simulation.
GM, CIgmGamma measure and related CI. This measure of association was proposed by Goodman and Kruskal (1954). It is defined as (OR - 1)/(OR + 1), can take values in the range (-1, 1) and does not depend on the marginal probabilities. Interval is calculated using posterior simulation.
xi1, xi2Estimated power or shape parameters of the link functions of the two equations if an asymmetric link approach is used.
table.RJoint absolute frequency table of observed binary value combinations. This as well as all remaining quantities below are not provided when Method = "BSS" or Method = "BPO".
table.PJoint absolute frequency table of predicted binary value combinations.
table.FIt is given by table.P/table.R.
MRBased on the comparison between table.P and table.R, it reports the matching rate as percentage.
P1, P2Marginal probabilities for the two binary outcomes y_1 = 1 and y_2 = 1.
QPS1, QPS2Quadratic probability scores for the two binary responses as suggested by Diebold and Rudebusch (1989). Values are on the interval [0,2], with 0 indicating a perfect fit.
CR1, CR2Correct prediction ratios for the two binary responses as percentage.

Details

As in the package mgcv, based on the results of Marra and Wood (2012), `Bayesian p-values' are returned for the smooth terms. These have better frequentist performance than their frequentist counterpart. Let $\hat{\bf f}$ and ${\bf V}_f$ denote the vector of values of a smooth term evaluated at the original covariate values and the corresponding Bayesian covariance matrix, and let ${\bf V}_f^{r-}$ denote the rank $r$ pseudoinverse of ${\bf V}_f$. The statistic used is $T=\hat{\bf f}^\prime {\bf V}_f^{r-} \hat{\bf f}$. This is compared to a chi-squared distribution with degrees of freedom given by $r$, which is obtained by biased rounding of the estimated degrees of freedom. See Marra (2013) for further details. Covariate selection can also be achieved using a single penalty shrinkage approach as shown in Marra and Wood (2011). Posterior simulation is used to obtain intervals of nonlinear functions of parameters, such as the odds ratio and gamma measure discussed by Tajar et al. (2001). For more details on some of the model summaries see Radice, Marra and Wojtys (submitted).

References

Diebold F.X. and Rudebusch G.D. (1989), Scoring the Leading Indicators. Journal of Business, 62(3), 369-391. Goodman L.A. and Kruskal W.H. (1954), Measures of Association for Cross Classification. Journal of the American Statistical Association, 49, 732-764. Marra G. (2013), On P-values for Semiparametric Bivariate Probit Models. Statistical Methodology, 10(1), 23-28. Radice R., Marra G. and M. Wojtys (submitted), Copula Regression Spline Models for Binary Outcomes. Marra G. and Wood S.N. (2011), Practical Variable Selection for Generalized Additive Models. Computational Statistics and Data Analysis, 55(7), 2372-2387. Marra G. and Wood S.N. (2012), Coverage Properties of Confidence Intervals for Generalized Additive Model Components. Scandinavian Journal of Statistics, 39(1), 53-74. Tajar M., Denuit M. and Lambert P. (2001), Copula-Type Representation for Random Couples with Bernoulli Margins. Discussion Papaer 0118, Universite Catholique De Louvain.

Examples

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## see examples for SemiParBIVProbit

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