# Multivariate Conditional Transformation Models Conditional transformation models for multivariate continuous, discrete, or a mix of continuous and discrete outcomes ```r Mmlt(..., formula = ~ 1, data, conditional = FALSE, theta = NULL, fixed = NULL, scale = FALSE, optim = mltoptim(hessian = TRUE)[c("auglag", "constrOptim")], args = list(seed = 1, type = c("MC", "ghalton"), M = 1000), fit = c("jointML", "pseudo", "ACS", "sequential", "none"), ACSiter = 2) ``` ## Arguments - ``...``: marginal transformation models, one for each response, for `Mmlt`. Additional arguments for the methods. - `formula`: a model formula describing a model for the dependency structure via the lambda parameters. The default is set to `~ 1` for constant lambdas. - `data`: a data.frame. - `conditional`: logical; parameters are defined conditionally (only possible when all models are probit models). This is the default as described by Klein et al. (2022). If `FALSE`, parameters can be directly interpreted marginally, this is explained in Section 2.6 by Klein et al. (2022). Using `conditional = FALSE` with probit-only models gives the same likelihood but different parameter estimates. - `theta`: an optional vector of starting values. - `fixed`: an optional named numeric vector of predefined parameter values or a logical (for `coef`) indicating to also return fixed parameters (only when `type = "all"`). - `scale`: a logical indicating if (internal) scaling shall be applied to the model coefficients. - `optim`: a list of optimisers as returned by `mltoptim` - `args`: a list of arguments for `lpmvnorm`. - `fit`: character vector describing how to fit the model. The default is joint likelihood estimation of all parameters, `pseudo` fixes the marginal parameters, `sequential` starts with a univariate model and sequentially adds models, keeping the parameters of previously added models fit. `ACS` implements Alternate Convex Search, starting with `pseudo` and, in a second step, fixing the marginal parameters. This is iterated for `ACSiter` iterations. - `ACSiter`: number of iterations for `fit = "ACS"`. ## Details The function implements multivariate conditional transformation models as described by Klein et al (2020). Below is a simple example for an unconditional bivariate distribution. See `demo("undernutrition", package = "tram")` for a conditional three-variate example. ## Returns An object of class `Mmlt` with `coef` and `predict` methods. ## References Nadja Klein, Torsten Hothorn, Luisa Barbanti, Thomas Kneib (2022), Multivariate Conditional Transformation Models. **Scandinavian Journal of Statistics**, 49 , 116--142, tools:::Rd_expr_doi("10.1111/sjos.12501") . Torsten Hothorn (2024), On Nonparanormal Likelihoods. tools:::Rd_expr_doi("10.48550/arXiv.2408.17346") . ## Examples ```r data("cars") ### fit unconditional bivariate distribution of speed and distance to stop ## fit unconditional marginal transformation models m_speed <- BoxCox(speed ~ 1, data = cars, support = ss <- c(4, 25), add = c(-5, 5)) m_dist <- BoxCox(dist ~ 1, data = cars, support = sd <- c(0, 120), add = c(-5, 5)) ## fit multivariate unconditional transformation model m_speed_dist <- Mmlt(m_speed, m_dist, formula = ~ 1, data = cars) ## log-likelihood logLik(m_speed_dist) sum(predict(m_speed_dist, newdata = cars, type = "density", log = TRUE)) ## Wald test of independence of speed and dist (the "dist.speed.(Intercept)" ## coefficient) summary(m_speed_dist) ## LR test comparing to independence model LR <- 2 * (logLik(m_speed_dist) - logLik(m_speed) - logLik(m_dist)) pchisq(LR, df = 1, lower.tail = FALSE) ## constrain lambda to zero and fit independence model ## => log-likelihood is the sum of the marginal log-likelihoods mI <- Mmlt(m_speed, m_dist, formula = ~1, data = cars, fixed = c("dist.speed.(Intercept)" = 0)) logLik(m_speed) + logLik(m_dist) logLik(mI) ## linear correlation, ie Pearson correlation of speed and dist after ## transformation to bivariate normality (r <- coef(m_speed_dist, type = "Corr")) ## Spearman's rho (rank correlation) of speed and dist on original scale (rs <- coef(m_speed_dist, type = "Spearman")) ## evaluate joint and marginal densities (needs to be more user-friendly) nd <- expand.grid(c(nd_s <- mkgrid(m_speed, 100), nd_d <- mkgrid(m_dist, 100))) nd$d <- predict(m_speed_dist, newdata = nd, type = "density") ## compute marginal densities nd_s <- as.data.frame(nd_s) nd_s$d <- predict(m_speed_dist, newdata = nd_s, margins = 1L, type = "density") nd_d <- as.data.frame(nd_d) nd_d$d <- predict(m_speed_dist, newdata = nd_d, margins = 2L, type = "density") ## plot bivariate and marginal distribution col1 <- rgb(.1, .1, .1, .9) col2 <- rgb(.1, .1, .1, .5) w <- c(.8, .2) layout(matrix(c(2, 1, 4, 3), nrow = 2), width = w, height = rev(w)) par(mai = c(1, 1, 0, 0) * par("mai")) sp <- unique(nd$speed) di <- unique(nd$dist) d <- matrix(nd$d, nrow = length(sp)) contour(sp, di, d, xlab = "Speed (in mph)", ylab = "Distance (in ft)", xlim = ss, ylim = sd, col = col1) points(cars$speed, cars$dist, pch = 19, col = col2) mai <- par("mai") par(mai = c(0, 1, 0, 1) * mai) plot(d ~ speed, data = nd_s, xlim = ss, type = "n", axes = FALSE, xlab = "", ylab = "") polygon(nd_s$speed, nd_s$d, col = col2, border = FALSE) par(mai = c(1, 0, 1, 0) * mai) plot(dist ~ d, data = nd_d, ylim = sd, type = "n", axes = FALSE, xlab = "", ylab = "") polygon(nd_d$d, nd_d$dist, col = col2, border = FALSE) ### NOTE: marginal densities are NOT normal, nor is the joint ### distribution. The non-normal shape comes from the data-driven ### transformation of both variables to joint normality in this model. ```

mmlt function