Conditional Inference Trees

Description

Recursive partitioning for continuous, censored, ordered, nominal and multivariate response variables in a conditional inference framework.

Usage

1
2
ctree(formula, data, weights, subset, na.action = na.pass,
    control = ctree_control(...), ytrafo = NULL, scores = NULL, ...)

Arguments

formula

a symbolic description of the model to be fit.

data

a data frame containing the variables in the model.

subset

an optional vector specifying a subset of observations to be used in the fitting process.

weights

an optional vector of weights to be used in the fitting process. Only non-negative integer valued weights are allowed.

na.action

a function which indicates what should happen when the data contain missing value.

control

a list with control parameters, see ctree_control.

ytrafo

an optional named list of functions to be applied to the response variable(s) before testing their association with the explanatory variables. Note that this transformation is only performed once for the root node and does not take weights into account. Alternatively, ytrafo can be a function of data and weights. In this case, the transformation is computed for every node. This feature is experimental and the user interface likely to change.

scores

an optional named list of scores to be attached to ordered factors.

...

arguments passed to ctree_control.

Details

Function partykit::ctree is a reimplementation of (most of) party::ctree employing the new party infrastructure of the partykit infrastructure. Although the new code was already extensively tested, it is not yet as mature as the old code. If you notice differences in the structure/predictions of the resulting trees, please contact the package maintainers. See also vignette("ctree", package = "partykit") for some remarks about the internals of the different implementations.

Conditional inference trees estimate a regression relationship by binary recursive partitioning in a conditional inference framework. Roughly, the algorithm works as follows: 1) Test the global null hypothesis of independence between any of the input variables and the response (which may be multivariate as well). Stop if this hypothesis cannot be rejected. Otherwise select the input variable with strongest association to the response. This association is measured by a p-value corresponding to a test for the partial null hypothesis of a single input variable and the response. 2) Implement a binary split in the selected input variable. 3) Recursively repeate steps 1) and 2).

The implementation utilizes a unified framework for conditional inference, or permutation tests, developed by Strasser and Weber (1999). The stop criterion in step 1) is either based on multiplicity adjusted p-values (testtype = "Bonferroni" in ctree_control) or on the univariate p-values (testtype = "Univariate"). In both cases, the criterion is maximized, i.e., 1 - p-value is used. A split is implemented when the criterion exceeds the value given by mincriterion as specified in ctree_control. For example, when mincriterion = 0.95, the p-value must be smaller than $0.05$ in order to split this node. This statistical approach ensures that the right-sized tree is grown without additional (post-)pruning or cross-validation. The level of mincriterion can either be specified to be appropriate for the size of the data set (and 0.95 is typically appropriate for small to moderately-sized data sets) or could potentially be treated like a hyperparameter (see Section~3.4 in Hothorn, Hornik and Zeileis, 2006). The selection of the input variable to split in is based on the univariate p-values avoiding a variable selection bias towards input variables with many possible cutpoints. The test statistics in each of the nodes can be extracted with the sctest method. (Note that the generic is in the strucchange package so this either needs to be loaded or sctest.constparty has to be called directly.) In cases where splitting stops due to the sample size (e.g., minsplit or minbucket etc.), the test results may be empty.

Predictions can be computed using predict, which returns predicted means, predicted classes or median predicted survival times and more information about the conditional distribution of the response, i.e., class probabilities or predicted Kaplan-Meier curves. For observations with zero weights, predictions are computed from the fitted tree when newdata = NULL.

By default, the scores for each ordinal factor x are 1:length(x), this may be changed for variables in the formula using scores = list(x = c(1, 5, 6)), for example.

For a general description of the methodology see Hothorn, Hornik and Zeileis (2006) and Hothorn, Hornik, van de Wiel and Zeileis (2006).

Value

An object of class party.

References

Hothorn T, Hornik K, Van de Wiel MA, Zeileis A (2006). A Lego System for Conditional Inference. The American Statistician, 60(3), 257–263.

Hothorn T, Hornik K, Zeileis A (2006). Unbiased Recursive Partitioning: A Conditional Inference Framework. Journal of Computational and Graphical Statistics, 15(3), 651–674.

Hothorn T, Zeileis A (2015). partykit: A Modular Toolkit for Recursive Partytioning in R. Journal of Machine Learning Research, 16, 3905–3909.

Strasser H, Weber C (1999). On the Asymptotic Theory of Permutation Statistics. Mathematical Methods of Statistics, 8, 220–250.

Examples

 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
### regression
airq <- subset(airquality, !is.na(Ozone))
airct <- ctree(Ozone ~ ., data = airq)
airct
plot(airct)
mean((airq$Ozone - predict(airct))^2)

### classification
irisct <- ctree(Species ~ .,data = iris)
irisct
plot(irisct)
table(predict(irisct), iris$Species)

### estimated class probabilities, a list
tr <- predict(irisct, newdata = iris[1:10,], type = "prob")

### survival analysis
if (require("TH.data") && require("survival") && require("coin")) {

  data("GBSG2", package = "TH.data")
  GBSG2ct <- ctree(Surv(time, cens) ~ ., data = GBSG2)
  predict(GBSG2ct, newdata = GBSG2[1:2,], type = "response")	  
  plot(GBSG2ct)

  ### with weight-dependent log-rank scores
  ctree(Surv(time, cens) ~ ., data = GBSG2)

  ### log-rank trafo for observations in this node only (= weights > 0)
  h <- function(data, weights) {
    s <- data[, "Surv(time, cens)"]
    s <- logrank_trafo(s[weights > 0,])
    r <- rep(0, nrow(data))
    r[weights > 0] <- s
    matrix(as.double(r), ncol = 1)
  }

  ### very much the same tree
  ctree(Surv(time, cens) ~ ., data = GBSG2, ytrafo = h) 
}

### multivariate responses
airct2 <- ctree(Ozone + Temp ~ ., data = airq)
airct2
plot(airct2)