# Conditional Inference Trees

##### Conditional Inference Trees

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

- Keywords
- tree

##### Usage

```
ctree(formula, data, subset = NULL, weights = NULL,
controls = ctree_control(), xtrafo = ptrafo, ytrafo = ptrafo,
scores = NULL)
```

##### Arguments

- formula
a symbolic description of the model to be fit. Note that symbols like

`:`

and`-`

will not work and the tree will make use of all variables listed on the rhs of`formula`

.- 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.

- controls
an object of class

`TreeControl`

, which can be obtained using`ctree_control`

.- xtrafo
a function to be applied to all input variables. By default, the

`ptrafo`

function is applied.- ytrafo
a function to be applied to all response variables. By default, the

`ptrafo`

function is applied.- scores
an optional named list of scores to be attached to ordered factors.

##### Details

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 resonse. 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"`

or `testtype == "MonteCarlo"`

in `ctree_control`

),
on the univariate p-values (`testtype == "Univariate"`

),
or on values of the test statistic
(`testtype == "Teststatistic"`

). 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 and no form of pruning or cross-validation
or whatsoever is needed. 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.

Multiplicity-adjusted Monte-Carlo p-values are computed following a "min-p" approach. The univariate p-values based on the limiting distribution (chi-square or normal) are computed for each of the random permutations of the data. This means that one should use a quadratic test statistic when factors are in play (because the evaluation of the corresponding multivariate normal distribution is time-consuming).

By default, the scores for each ordinal factor `x`

are
`1:length(x)`

, this may be changed using ```
scores = list(x =
c(1,5,6))
```

, for example.

Predictions can be computed using `predict`

or
`treeresponse`

. The first function accepts arguments
`type = c("response", "node", "prob")`

where `type = "response"`

returns predicted means, predicted classes or median predicted survival
times, `type = "node"`

returns terminal node IDs (identical to
`where`

) and `type = "prob"`

gives more information about
the conditional distribution of the response, i.e., class probabilities or
predicted Kaplan-Meier curves and is identical to
`treeresponse`

. For observations with zero weights,
predictions are computed from the fitted tree when `newdata = NULL`

.

For a general description of the methodology see Hothorn, Hornik and Zeileis (2006) and Hothorn, Hornik, van de Wiel and Zeileis (2006). Introductions for novices can be found in Strobl et al. (2009) and at http://github.com/christophM/overview-ctrees.git.

##### Value

An object of class `BinaryTree-class`

.

##### References

Helmut Strasser and Christian Weber (1999). On the asymptotic theory of permutation
statistics. *Mathematical Methods of Statistics*, **8**, 220--250.

Torsten Hothorn, Kurt Hornik, Mark A. van de Wiel and Achim Zeileis (2006).
A Lego System for Conditional Inference. *The American Statistician*,
**60**(3), 257--263.

Torsten Hothorn, Kurt Hornik and Achim Zeileis (2006). Unbiased Recursive
Partitioning: A Conditional Inference Framework. *Journal of
Computational and Graphical Statistics*, **15**(3), 651--674.
Preprint available
from http://statmath.wu-wien.ac.at/~zeileis/papers/Hothorn+Hornik+Zeileis-2006.pdf

Carolin Strobl, James Malley and Gerhard Tutz (2009).
An Introduction to Recursive Partitioning: Rationale, Application, and Characteristics of
Classification and Regression Trees, Bagging, and Random forests.
*Psychological Methods*, **14**(4), 323--348.

##### Examples

```
# NOT RUN {
set.seed(290875)
### regression
airq <- subset(airquality, !is.na(Ozone))
airct <- ctree(Ozone ~ ., data = airq,
controls = ctree_control(maxsurrogate = 3))
airct
plot(airct)
mean((airq$Ozone - predict(airct))^2)
### extract terminal node ID, two ways
all.equal(predict(airct, type = "node"), where(airct))
### classification
irisct <- ctree(Species ~ .,data = iris)
irisct
plot(irisct)
table(predict(irisct), iris$Species)
### estimated class probabilities, a list
tr <- treeresponse(irisct, newdata = iris[1:10,])
### ordinal regression
data("mammoexp", package = "TH.data")
mammoct <- ctree(ME ~ ., data = mammoexp)
plot(mammoct)
### estimated class probabilities
treeresponse(mammoct, newdata = mammoexp[1:10,])
### survival analysis
if (require("TH.data") && require("survival")) {
data("GBSG2", package = "TH.data")
GBSG2ct <- ctree(Surv(time, cens) ~ .,data = GBSG2)
plot(GBSG2ct)
treeresponse(GBSG2ct, newdata = GBSG2[1:2,])
}
### if you are interested in the internals:
### generate doxygen documentation
# }
# NOT RUN {
### download src package into temp dir
tmpdir <- tempdir()
tgz <- download.packages("party", destdir = tmpdir)[2]
### extract
untar(tgz, exdir = tmpdir)
wd <- setwd(file.path(tmpdir, "party"))
### run doxygen (assuming it is there)
system("doxygen inst/doxygen.cfg")
setwd(wd)
### have fun
browseURL(file.path(tmpdir, "party", "inst",
"documentation", "html", "index.html"))
# }
```

*Documentation reproduced from package party, version 1.3-3, License: GPL-2*