# copulaReg: Semiparametric Copula Bivariate Models with... In SemiParBIVProbit: Semiparametric Copula Regression Models

## Description

`copulaReg` fits flexible copula bivariate models with continuous/discrete margins with several types of covariate effects, copula distributions and marginal distributions.

## Usage

 ```1 2 3 4 5 6 7``` ```copulaReg(formula, data = list(), weights = NULL, subset = NULL, BivD = "N", margins = c("N","N"), dof = 3, surv = FALSE, cens1 = NULL, cens2 = NULL, gamlssfit = FALSE, fp = FALSE, infl.fac = 1, rinit = 1, rmax = 100, iterlimsp = 50, tolsp = 1e-07, gc.l = FALSE, parscale, extra.regI = "t") ```

## Arguments

 `formula` In the basic setup this will be a list of two formulas, one for equation 1 and the other for equation 2. `s` terms are used to specify smooth functions of predictors. For the case of more than two equations see the example below and the documentation of `SemiParBIVProbit()` for more details. When one outcome is binary and the other continuous/discrete then the first equation MUST 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 MUST refer to the discrete one. `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 `copulaReg` 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. `margins` It indicates the distributions used for the two margins. 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"). When the first equation is binary then possible link functions are "probit", "logit", "cloglog"". `dof` If `BivD = "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 estiamted from the data. `surv` If `TRUE` then a bivariate survival model is fitted. `cens1` Binary censoring indicator 1. This is required when `surv = TRUE`. `cens2` Binary censoring indicator 2. This is required when `surv = TRUE`. `gamlssfit` If `gamlssfit = TRUE` then gamlss univariate models are also fitted. This is useful for obtaining starting values, for instance. `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 `dof`, Plackett, Hougaard. Each of the Clayton, Joe and Gumbel copulae is allowed to be mixed with a rotated version of the same family. The options are: "C0C90", "C0C270", "C180C90", "C180C270", "G0G90", "G0G270", "G180G90", "G180G270", "J0J90", "J0J270", "J180J90" and "J180J270". This allows the user to model negative and positive tail dependencies. `fp` If `TRUE` then a fully parametric model with unpenalised regression splines if fitted. See the example 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. `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).

## Details

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

There are many continuous/discrete 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 `copulaReg` as described in `copulaRegObject`.

## 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. `copulaReg()` 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`). Using `gamlssfit = 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, `gamlssfit = FALSE` only because it tends to be computationally cheaper and because the default starting value procedure has typically been found to do a satisfactory job in most cases. (The results obtained when using `gamlssfit = FALSE` and `gamlssfit = 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.

## Author(s)

Maintainer: Giampiero Marra [email protected]

## References

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

`adjCov`, `VuongClarke`, `plot.SemiParBIVProbit`, `SemiParBIVProbit-package`, `copulaRegObject`, `conv.check`, `summary.copulaReg`, `predict.SemiParBIVProbit`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256``` ```library(SemiParBIVProbit) ## Not run: ############ ## 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 <- -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.sigma2.1 <- ~ 1 eq.sigma2.2 <- ~ 1 eq.theta <- ~ x1 fl <- list(eq.mu.1, eq.mu.2, eq.sigma2.1, eq.sigma2.2, eq.theta) # the order above is the one to follow when # using more than two equations out <- copulaReg(fl, data = dataSim) conv.check(out) post.check(out) summary(out) AIC(out) BIC(out) jc.probs(out, 1.4, 2.3, intervals = TRUE)[1:4,] ############ ## EXAMPLE 2 ############ ## 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 <- copulaReg(list(y1 ~ x1 + x2, y2 ~ y1 + x2), data = dataSim, margins = c("probit","N")) conv.check(out) summary(out) post.check(out) ## SEMIPARAMETRIC RECURSIVE Model eq.mu.1 <- y1 ~ x1 + s(x2) eq.mu.2 <- y2 ~ y1 + s(x2) eq.sigma2 <- ~ 1 eq.theta <- ~ 1 fl <- list(eq.mu.1, eq.mu.2, eq.sigma2, eq.theta) out <- copulaReg(fl, data = dataSim, margins = c("probit","N"), gamlssfit = TRUE) conv.check(out) summary(out) post.check(out) jc.probs(out, 1, 1.5, intervals = TRUE)[1:4,] AT(out, nm.end = "y1") AT(out, nm.end = "y1", type = "univariate") # # ############ ## EXAMPLE 3 ############ ## 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.sigma2 <- ~ 1 eq.theta <- ~ 1 fl <- list(eq.mu.1, eq.mu.2, eq.sigma2, eq.theta) out <- copulaReg(fl, data = dataSim, margins = c("probit","N")) conv.check(out) summary(out) post.check(out) AT(out, nm.end = "y1") AT(out, nm.end = "y1", type = "univariate") RR(out, nm.end = "y1", rr.plot = TRUE) RR(out, nm.end = "y1", type = "univariate") OR(out, nm.end = "y1", or.plot = TRUE) OR(out, nm.end = "y1", type = "univariate") # # ############ ## EXAMPLE 4 ############ ## Survival model 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 eq1 <- u1 ~ z1 + s(z2) + s(u1, bs = "mpi") eq2 <- u2 ~ z + s(u2, bs = "mpi") eq3 <- ~ s(z2) out <- copulaReg(list(eq1, eq2), data = dataSim, surv = TRUE, margins = c("PH", "PO"), cens1 = delta1, cens2 = delta2) # PH margin fit can also be compared with cox.ph from mgcv conv.check(out) post.check(out) summary(out) AIC(out); BIC(out) plot(out, eq = 1, scale = 0, pages = 1) plot(out, eq = 2, scale = 0, pages = 1) hazsurv.plot(out, eq = 1, newdata = data.frame(z1 = 0, z2 = 0), shade = TRUE, n.sim = 1000) hazsurv.plot(out, eq = 1, newdata = data.frame(z1 = 0, z2 = 0), shade = TRUE, n.sim = 1000, type = "hazard") hazsurv.plot(out, eq = 2, newdata = data.frame(z = 0), shade = TRUE, n.sim = 1000) hazsurv.plot(out, eq = 2, newdata = data.frame(z = 0), shade = TRUE, n.sim = 1000, type = "hazard") out1 <- copulaReg(list(eq1, eq2, eq3), data = dataSim, surv = TRUE, margins = c("PH", "PO"), cens1 = delta1, cens2 = delta2, gamlssfit = TRUE) eq1 <- u1 ~ z1 + s(z2) eq2 <- u2 ~ z eq3 <- ~ s(z2) # note that Weibull is implemented as AFT model out2 <- copulaReg(list(eq1, eq2, ~ 1, ~ 1, eq3), data = dataSim, surv = TRUE, margins = c("WEI", "WEI"), cens1 = delta1, cens2 = delta2) ## End(Not run) ```