vcrpart (version 0.2-1)

tvcm: Tree-based varying coefficient regression models

Description

TVCM is a tree-based algorithm that aims to estimate varying coefficient regression models. TVCM approximates varying coefficients by piecewise constant functions, i.e., it estimates linear models with stratum specific coefficients. The tvcm function implements the partitioning algorithms described in Buergin and Ritschard (2014b) (default) and Buergin and Ritschard (2014a).

Usage

tvcm(formula, data, fit, family,
     weights, subset, offset, na.action,
     control = tvcm_control(), ...)
tvcolmm(formula, data, family = cumulative(), 
        weights, subset, offset, na.action, 
        control = tvcm_control(), ...)
tvcglm(formula, data, family, 
       weights, subset, offset, na.action, 
       control = tvcm_control(), ...)

Arguments

formula

a symbolic description of the model to fit, e.g.,

y ~ vc(z1, z2) + vc(z1, z2, by = x) where vc specifies the varying coefficients. See vcrpart-formula.

fit

a character string or a function that specifies the fitting function, e.g., olmm or glm.

family

the model family, e.g., an object of class family.olmm or family.

data

a data frame containing the variables in the model.

weights

an optional numeric vector of weights to be used in the fitting process.

subset

an optional logical or integer vector specifying a subset of 'data' to be used in the fitting process.

offset

this can be used to specify an a priori known component to be included in the linear predictor during fitting.

na.action

a function that indicates what should happen if data contain NAs. See na.action.

control

a list with control parameters as returned by tvcm_control.

...

additional arguments passed to the fitting function fit.

Value

• An object of class tvcm. The tvcm class itself is based on the party class of the partykit package. The most important slots are:
• node
  an object of class partynode.
• data
  a (potentially empty) data.frame.
• fitted
  an optional data.frame with nrow(data) rows and containing at least the fitted terminal node identifiers as element (fitted). In addition, weights may be contained as element (weights) and responses as (response).
• info
  additional information includingcontrol and model.

Details

The TVCM partitioning algorithm works as follows: Starting with $M_k = 1$ stratum (i.e. node) for all $K$ vc terms, the algorithm splits in each iteration one of the current $K \sum_{k=1}^K M_k$ nodes into two new nodes. For selecting the vc term, the node, the variable and the cutpoint in each iteration, there are two procedures available.

The first and default procedure (cf. Buergin and Ritschard, 2014b) is a two-stage procedure which builds overly fine partitions in the first stage and selects the best-sized partitions by pruning in the second stage. For the second stage, which is automatically processed, we refer to tvcm-assessment. The partitioning stage selects, in each iteration, the split that maximizes the penalized loss reduction statistic that compares the current with the one-step ahead model (see also argument lossfun in tvcm_control). By default, the penalized loss reduction statistic is the Akaike Information Criterion (Akaike, 1974), but alternatives could be specified by the control argument. The algorithm is continued to build until the criteria specified in control are reached. By default, only the minsize (minimum node size) criteron are specified (andthat the penalized loss reduction is larger than 0). For large data sets with many partitioning variables, this can be slow and, therefore, tvcm_control provides further criteria. In particular, you may increase the minimum permitted penalized loss reduction by dfsplit.

The second procedure (cf. Buergin and Ritschard, 2014a) is, for technical reasons, restricted to the cases where a single vc term is used and non of the moderators (partitioning variables) intersects with predictors. On the other hand, the second procedure avoids the variable selection bias that exhibits the exhaustive search above and is computationally much less burdensome. The procedure selects the split by first choosing the node and the variable with M-fluctuation tests (cf. Zeileis and Hornik, 2007) and second choosing the cutpoint by the deviance reduction statistic, as above. To use this option it is necessary to set sctest=TRUE in tvcm_control. The algorithm is stopped as soon as all nodewise Bonferroni corrected p-values of M-fluctuation tests reach a prespecified threshold (e.g., 0.05), see argument alpha in tvcm_control and no pruning is necessary (or optional). Note that, as explained in Buergin and Ritschard (2014a), coefficient constancy tests are adjusted for intra-subject correlation for 2-stage models, see estfun.olmm. The procedure is illustrated below in example 2.

An alternative tree-based algorithm to tvcm are the MOB (Zeileis et al., 2008) and the PartReg (Wang and Hastie, 2014) algorithms. The MOB algorithm is implemented by the mob function in the packages party and partykit. For alternative, smoothing splines and kernel regression approaches to varying coefficients, see the packages mgcv, svcm,mboost or np.

The tvcm function builds on the software infrastructure of the partykit package. The authors are grateful for these codes.

References

Zeileis, A., Hothorn, T., and Hornik, K. (2008). Model-Based Recursive Partitioning. Journal of Computational and Graphical Statistics, 17(2), 492--514.

Zeileis, A., Hornik, K. (2007), Generalized M-Fluctuation Tests for Parameter Instability, Statistica Neerlandica, 61, 488--508. doi:10.1111/j.1467-9574.2007.00371.x.

Torsten Hothorn, Achim Zeileis (2014). partykit: A Modular Toolkit for Recursive Partytioning in R. Working Paper 2014-10. Working Papers in Economics and Statistics, Research Platform Empirical and Experimental Economics, Universitaet Innsbruck. URL http://EconPapers.RePEc.org/RePEc:inn:wpaper:2014-10

Wang, J. C., Hastie, T. (2013), Boosted Varying-Coefficient Regression Models for Product Demand Prediction, Journal of Computational and Graphical Statistics.

Buergin R. and Ritschard G. (2014a), Tree-based varying coefficient regression for longitudinal ordinal responses. Submitted article.

Buergin R. and Ritschard G. (2014b), Coefficient-wise tree-based varying coefficient regression with vcrpart. Article in progress.

Akaike H. (1974), A new look at the statistical model identification, IEEE Transactions on Automatic Control, 19, 716--723.

See Also

tvcm_control, tvcm-methods, tvcm-plot, tvcm-plot, tvcm-assessment

Examples

## ------------------------------------------------------------------- #  
## Example 1: Moderated effect of education on poverty
##
## The algorithm is used to find out whether the effect of high
## education 'EduHigh' on poverty 'Poor' is moderated by the civil
## status 'CivStat'. We specify two 'vc' terms in the logistic
## regression model for 'Poor': a first that accounts for the direct
## effect of 'CivStat' and a second that accounts for the moderation of
## 'CivStat' on the relation between 'EduHigh' and 'Poor'. We use here
## the 2-stage procedure with a partitioning- and a pruning stage as
## described in Buergin and Ritschard (2014b). 
## ------------------------------------------------------------------- #

data(poverty)
poverty$EduHigh <- 1 * (poverty$Edu == "high")

## fit the model
model.Pov <-
  tvcglm(Poor ~ -1 +  vc(CivStat) + vc(CivStat, by = EduHigh) + NChild, 
         family = binomial(), data = poverty, subset = 1:200,
         control = tvcm_control(verbose = TRUE,
           folds = folds_control(K = 1, type = "subsampling", seed = 4)))

## diagnosis
plot(model.Pov, "cv")
plot(model.Pov, "coef")
summary(model.Pov)
splitpath(model.Pov, steps = 1:3)
prunepath(model.Pov, steps = 1)


## ------------------------------------------------------------------- # 
## Example 2: Moderated effect effect of unemployment
##
## Here we fit a varying coefficient ordinal linear mixed on the 
## synthetic ordinal longitudinal data 'unemp'. The interest is whether 
## the effect of unemployment 'UNEMP' on happiness 'GHQL' is moderated 
## by 'AGE', 'FISIT', 'GENDER' and 'UEREGION'. 'FISIT' is the only true  
## moderator. For the the partitioning we coefficient constancy tests,
## as described in Buergin and Ritschard (2014a)
## ------------------------------------------------------------------- #

data(unemp)

## fit the model
model.UE <-
  tvcolmm(GHQL ~ -1 + 
          vc(AGE, FISIT, GENDER, UEREGION, by = UNEMP, intercept = TRUE) +
          re(1|PID), data = unemp, control = tvcm_control(sctest = TRUE))

## diagnosis (no cross-validation was performed since 'sctest = TRUE')
plot(model.UE, "coef")
summary(model.UE)
splitpath(model.UE, steps = 1, details = TRUE)