# Methods for Stratified Linear Transformation Models Methods for objects inheriting from class tram ```r ## S3 method for class 'tram' as.mlt(object) ## S3 method for class 'tram' model.frame(formula, ...) ## S3 method for class 'tram' model.matrix(object, data = object$data, with_baseline = FALSE, what = c("shifting", "interacting"), ...) ## S3 method for class 'stram' model.matrix(object, data = object$data, with_baseline = FALSE, what = c("shifting", "scaling", "interacting"), ...) ## S3 method for class 'tram' coef(object, with_baseline = FALSE, ...) ## S3 method for class 'Lm' coef(object, as.lm = FALSE, ...) ## S3 method for class 'Survreg' coef(object, as.survreg = FALSE, ...) ## S3 method for class 'tram' vcov(object, with_baseline = FALSE, complete = FALSE, ...) ## S3 method for class 'tram' logLik(object, parm = coef(as.mlt(object), fixed = FALSE), ...) ## S3 method for class 'tram' estfun(x, parm = coef(as.mlt(x), fixed = FALSE), ...) ## S3 method for class 'tram' predict(object, newdata = model.frame(object), type = c("lp", "trafo", "distribution", "logdistribution", "survivor", "logsurvivor", "density", "logdensity", "hazard", "loghazard", "cumhazard", "logcumhazard", "odds", "logodds", "quantile"), ...) ## S3 method for class 'stram' predict(object, newdata = model.frame(object), type = c("lp", "trafo", "distribution", "logdistribution", "survivor", "logsurvivor", "density", "logdensity", "hazard", "loghazard", "cumhazard", "logcumhazard", "odds", "logodds", "quantile"), what = c("shifting", "scaling"), ...) ## S3 method for class 'tram' plot(x, newdata = model.frame(x), which = c("QQ-PIT", "baseline only", "distribution"), confidence = c("none", "interval", "band"), level = 0.95, K = 50, cheat = K, col = "black", fill = "lightgrey", lwd = 1, ...) ## S3 method for class 'tram' residuals(object, ...) ## S3 method for class 'tram' PI(object, newdata = model.frame(object), reference = 0, one2one = FALSE, ...) ## Default S3 method: PI(object, prob, link = "logistic", ...) ## S3 method for class 'tram' OVL(object, newdata = model.frame(object), reference = 0, one2one = FALSE, ...) ## Default S3 method: OVL(object, link = "logistic", ...) ## S3 method for class 'tram' TV(object, newdata = model.frame(object), reference = 0, one2one = FALSE, ...) ## Default S3 method: TV(object, link = "logistic", ...) ## S3 method for class 'tram' L1(object, newdata = model.frame(object), reference = 0, one2one = FALSE, ...) ## Default S3 method: L1(object, link = "logistic", ...) ## S3 method for class 'tram' ROC(object, newdata = model.frame(object), reference = 0, prob = 1:99 / 100, one2one = FALSE, ...) ## Default S3 method: ROC(object, prob = 1:99 / 100, link = "logistic", ...) ## S3 method for class 'ROCtram' plot(x, lty = 1:ncol(x), col = "black", fill = "lightgrey", lwd = 1, ...) ``` ## Arguments - `object, formula, x`: a fitted stratified linear transformation model inheriting from class `tram`. `PI` also takes a numeric vector in the default method. - `data`: an optional data frame. - `with_baseline`: logical, if `TRUE` all model parameters are returned, otherwise parameters describing the baseline transformation are ignored. - `as.lm`: logical, return parameters in the `lm` parameterisation if `TRUE`. - `as.survreg`: logical, return parameters in the `survreg` parameterisation in `TRUE`. - `parm`: model parameters, including baseline parameters. - `complete`: currently ignored - `newdata`: an optional data frame of new observations. - `reference`: an optional data frame of reference observations, or a numeric vector of reference values. - `type`: type of prediction, current options include linear predictors (`"lp"`, of `x` variables in the formula `y | s ~ x`), transformation functions (`"trafo"`) or distribution functions on the scale of the cdf (`"distribution"`), survivor function, density function, log-density function, hazard function, log-hazard function, cumulative hazard function or quantile function. - `which`: type of plot, either a QQ plot of the probability-integral transformed observations (`"QQ-PIT"`), of the baseline transformation of the whole distribution. - `what`: type of model matrix / linear predictor: `shifting` returns model model matrix / linear predictor for shift term, `scaling` for the scale term. - `confidence`: type of uncertainty assessment. - `level`: confidence level. - `K`: number of grid points in the response, see `plot.ctm`. - `cheat`: reduced number of grid points for the computation of confidence bands, see `confband`. - `col`: line color. - `fill`: fill color. - `lwd`: line width. - `lty`: line type. - `prob`: a numeric vector of probabilities.. - `link`: a character identifying a link function. - `one2one`: logical, compute the ROC curve (and derived measures) comparing each row in `newdata` with each row in `reference` (`FALSE`, the default), or compare observations rowwise (`TRUE`). - ``...``: additional arguments to the underlying methods for class `mlt`, see `mlt-methods`. ## Details `coef` can be used to get (and set) model parameters, `logLik` evaluates the log-likelihood (also for parameters other than the maximum likelihood estimate); `vcov` returns the estimated variance-covariance matrix (possibly taking `cluster` into account) and and `estfun` gives the score contribution by each observation. `predict` and `plot` can be used to inspect the model on different scales. `PI` computes the probabilistic index (or concordance probability or AUC) for all observations in `newdata`, relative to `reference`, ie the probability $$ P(Y_1 \le Y_0 \mid x_0, x_1) $$ of observing a smaller value of a randomly sampled observation conditional on $x_1$ compared to a randomly sampled reference observation, which is conditional on $x_0$. This is equivalent to the area under the receiver operating curve (ROC). The probability only applies within strata, response-varying coefficients are not allowed. Under the same setup, `OVL` gives the overlap coefficient, which is one minus the total variation and one minus half the $L_1$ distance between the two conditional densities. The overlap coefficient is identical to the Youden index and the Smirnov statistic. `PI` and friends also accept an argument `conf.level` which triggers computation of simultaneous Wald confidence intervals for these measures. Arguments in `...` are forwarded to `glht`. ## See Also `mlt-methods`, `plot.ctm` ## References Torsten Hothorn, Lisa Moest, Peter Buehlmann (2018), Most Likely Transformations, **Scandinavian Journal of Statistics**, 45 (1), 110--134, tools:::Rd_expr_doi("10.1111/sjos.12291") . ## Examples ```r data("BostonHousing2", package = "mlbench") ### fit non-normal Box-Cox type linear model with two ### baseline functions (for houses near and off Charles River) BC_BH_2 <- BoxCox(cmedv | 0 + chas ~ crim + zn + indus + nox + rm + age + dis + rad + tax + ptratio + b + lstat, data = BostonHousing2) logLik(BC_BH_2) ### classical likelihood inference summary(BC_BH_2) ### coefficients of the linear predictor coef(BC_BH_2) ### plot linear predictor (mean of _transformed_ response) ### vs. observed values plot(predict(BC_BH_2, type = "lp"), BostonHousing2$cmedv) ### all coefficients coef(BC_BH_2, with_baseline = TRUE) ### compute predicted median along with 10% and 90% quantile for the first ### observations predict(BC_BH_2, newdata = BostonHousing2[1:3,], type = "quantile", prob = c(.1, .5, .9)) ### plot the predicted density for these observations plot(BC_BH_2, newdata = BostonHousing2[1:3, -1], which = "distribution", type = "density", K = 1000) ### evaluate the two baseline transformations, with confidence intervals nd <- model.frame(BC_BH_2)[1:2, -1] nd$chas <- factor(c("0", "1")) library("colorspace") col <- diverge_hcl(2, h = c(246, 40), c = 96, l = c(65, 90)) fill <- diverge_hcl(2, h = c(246, 40), c = 96, l = c(65, 90), alpha = .3) plot(BC_BH_2, which = "baseline only", newdata = nd, col = col, confidence = "interval", fill = fill, lwd = 2, xlab = "Median Value", ylab = expression(h[Y])) legend("bottomright", lty = 1, col = col, title = "Near Charles River", legend = c("no", "yes"), bty = "n") ### cars data; with quantile functions plot(dist ~ speed, data = cars) m <- Colr(dist ~ speed, data = cars) q <- predict(as.mlt(m), newdata = data.frame(speed = s <- 6:25), type = "quantile", prob = c(1, 5, 9) / 10) lines(s, q[1,]) lines(s, q[2,]) lines(s, q[3,]) nd <- data.frame(speed = s <- as.double(1:5 * 5)) # Prob(dist at speed s > dist at speed 0) # speed 0 is reference, not a good choice here PI(m, newdata = nd) # Prob(dist at speed s > dist at speed 15) lp15 <- c(predict(m, newdata = data.frame(speed = 15))) PI(m, newdata = nd, reference = lp15) PI(m, newdata = nd, reference = nd[3,,drop = FALSE]) # Prob(dist at speed s' > dist at speed s) PI(m, newdata = nd, reference = nd) # essentially: lp <- predict(m, newdata = nd) PI(object = dist(lp)) # same, with simultaneous confidence intervals PI(m, newdata = nd, reference = nd, conf.level = .95) # plot ROC curves + confidence bands # compare speed 20 and 25 to speed 15 plot(ROC(m, newdata = nd[4:5,,drop = FALSE], reference = nd[3,,drop = FALSE], conf.level = 0.95)) # Overlap of conditional densities at speed s' and s OVL(m, newdata = nd, reference = nd) ### ROC analysis (takes too long for CRAN Windows) if (require("mlbench") && .Platform$OS.type != "windows") { layout(matrix(1:4, nrow = 2)) data("PimaIndiansDiabetes2", package = "mlbench") dia <- sort(unique(PimaIndiansDiabetes2$diabetes)) nd <- data.frame(diabetes = dia, age = 29, mass = 32) ### median values ### unconditional ROC analysis: glucose tolerance test m0 <- Colr(glucose ~ diabetes, data = PimaIndiansDiabetes2) # ROC curve + confidence band plot(ROC(m0, newdata = nd[2,,drop = FALSE], conf.level = .95)) # Wald interval for AUC PI(m0, newdata = nd[2,,drop = FALSE], conf.level = .95) # score interval for AUC PI(-c(coef(m0), score_test(m0)$conf.int[2:1])) ### adjusted ROC analysis for age and mass m1 <- Colr(glucose ~ diabetes + age + mass, data = PimaIndiansDiabetes2) # ROC curve + confidence band (this is the same for all ages / # masses) plot(ROC(m1, newdata = nd[2,,drop = FALSE], reference = nd[1,,drop = FALSE], conf.level = .95)) # Wald interval for adjusted AUC PI(m1, newdata = nd[2,,drop = FALSE], reference = nd[1,,drop = FALSE], conf.level = .95) # Score interval for adjusted AUC PI(-c(coef(m1)[1], score_test(m1, names(coef(m1))[1])$conf.int[2:1])) ### conditional ROC analysis: AUC regression ~ age + mass m2 <- Colr(glucose ~ diabetes * (age + mass), data = PimaIndiansDiabetes2) # ROC curve for a person with age = 29 and mass = 32 plot(ROC(m2, newdata = nd[2,,drop = FALSE], reference = nd[1,,drop = FALSE], conf.level = .95)) # AUC for persons ages 21:81, all with mass = 32 nd1 <- data.frame(diabetes = nd[1,"diabetes"], age = 21:81, mass = 32) nd2 <- data.frame(diabetes = nd[2,"diabetes"], age = 21:81, mass = 32) auc <- PI(m2, newdata = nd2, reference = nd1, one2one = TRUE, conf.level = 0.95) plot(nd1$age, auc[, "Estimate"], xlab = "Age (in years)", ylab = "AUC", ylim = c(0, 1), type = "l") lines(nd1$age, auc[, "lwr"], lty = 3) lines(nd1$age, auc[, "upr"], lty = 3) } ```

tram-methods function