R: Tree-based varying coefficient regression based on ordinal...

tvcolmm {vcrpart}

R Documentation

Tree-based varying coefficient regression based on ordinal and nominal two-stage linear mixed models.

Description

The tvcolmm function implements the tree-based longitudinal varying coefficient regression algorithm proposed in Burgin and Ritschard (2015). The algorithm approximates varying fixed coefficients in the cumulative logit mixed model by a (multivariate) piecewise constant function using recursive partitioning, i.e., it estimates the fixed effect component of the model separately for strata of the value space of partitioning variables.

Usage

tvcolmm(formula, data, family = cumulative(), 
        weights, subset, offset, na.action = na.omit, 
        control = tvcolmm_control(), ...)

tvcolmm_control(sctest = TRUE, alpha = 0.05, bonferroni = TRUE,
                minsize = 50, maxnomsplit = 5, maxordsplit = 9,
                maxnumsplit = 9, fast = TRUE,
                trim = 0.1, estfun.args = list(), nimpute = 5,
                seed = NULL, maxstep = 1e3, verbose = FALSE, ...)

Arguments

`formula`	a symbolic description of the model to fit, e.g., `y ~ -1 + vc(z1, ..., zL, by = x1 + ... + xP, intercept = TRUE) + re(1\|id)` where `vc` term specifies the varying fixed coefficients. Only one such `vc` term is allowed with `tvcolmm` (in contrast to `tvcglm` where multiple `vc` terms can be specified). The above example formula removes the global intercepts and adds locally varying intercepts, by adding a `-1` term and specfiying `intercept = TRUE` in the `vc` term. If varying intercepts are desired, we recommend to always remove the global intercepts. For more details on the formula specification, see `olmm` and `vcrpart-formula`.
`family`	the model family. An object of class `family.olmm`.
`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 `NA`s. The default `na.action = na.omit` is listwise deletion, i.e., observations with missings on any variable are dropped. See `na.action`.
`control`	a list with control parameters as returned by `tvcolmm_control`, or by `tvcm_control` for advanced users.
`sctest`	logical scalar. Defines whether coefficient constancy tests should be used for the variable and node selection in each iteration.
`alpha`	numeric significance threshold between 0 and 1. A node is splitted when the smallest (possibly Bonferroni-corrected) `p` value for any coefficient constancy test in the current step falls below `alpha`.
`bonferroni`	logical. Indicates if and how `p`-values of coefficient constancy tests must be Bonferroni corrected. See details.
`minsize`	numeric scalar. The minimum sum of weights in terminal nodes.
`maxnomsplit`, `maxordsplit`, `maxnumsplit`	integer scalars for split candidate reduction. See `tvcm_control`.
`fast`	logical scalar. Whether the approximative model should be used to search for the next split. See `tvcm_control`.
`trim`	numeric between 0 and 1. Specifies the trimming parameter in coefficient constancy tests for continuous partitioning variables. See also the argument `from` of function `supLM` in package strucchange.
`estfun.args`	list of arguments to be passed to `olmm_gefp`. See details.
`nimpute`	a positive integer scalar. The number of times coefficient constancy tests should be repeated in each iteration. See details.
`seed`	an integer specifying which seed should be set at the beginning.
`maxstep`	integer. The maximum number of iterations i.e. number of splits to be processed.
`verbose`	logical. Should information about the fitting process be printed to the screen?
`...`	additional arguments passed to the fitting function `fit` or to `tvcm_control`.

Details

The tvcolmm function iterates the following steps:

Fit the current mixed model

y ~ Node:x1 + ... + Node:xP + re(1 + w1 + ... |id)

with olmm, where Node is a categorical variable with terminal node labels 1, ..., M.
Test the constancy of the fixed effects Node:x1, ..., separately for each moderator z1, ..., zL in each node 1, ..., M. This yields L times M (possibly Bonferroni corrected) p-values for rejecting coefficient constancy.
If the minimum p-value is smaller than alpha, then select the node and the variable corresponding to the minimum p-value. Search and incorporate the optimal among the candidate splits in the selected node and variable by exhaustive likelihood search.
Else if minimum p-value is larger than alpha, stop the algorithm and return the current model.

The implemented coefficient constancy tests used for node and variable selection (step 2) are based on the M-fluctuation tests of Zeileis and Hornik (2007), using the observation scores of the fitted mixed model. The observation scores can be extracted by olmm_estfun for models fitted with olmm. To deal with intra-individual correlations between such observation scores, the olmm_estfun function decorrelates the observation scores. In cases of unbalanced data, the pre-decorrelation method requires imputation. nimpute gives the number of times the coefficient constancy tests are repeated in each iteration. The final p-values are then the averages of the repetations.

The algorithm combines the splitting technique of Zeileis (2008) with the technique of Hajjem et. al (2011) and Sela and Simonoff (2012) to incorporate regression trees into mixed models.

For the exhaustive search, the algorithm implements a number of split point reduction methods to decrease the computational complexity. See the arguments maxnomsplit, maxordsplit and maxnumsplit. By default, the algorithm also uses the approximative search model approach proposed in Burgin and Ritschard (2017). To disable this option to use the original algorithm, set fast = FALSE in tvcolmm_control.

Special attention is given to varying intercepts, i.e. the terms that account for the direct effects of the moderators. A common specification is

y ~ -1 + vc(z1, ..., zL, by = x1 + ... + xP, intercept = TRUE) + re(1 + w1 + ... |id)

Doing so replaces the globale intercept by local intercepts. As mentioned, if a varying intercepts are desired, we recommend to always remove the global intercept.

Value

An object of class tvcm

Author(s)

Reto Burgin

References

Zeileis, A., T. Hothorn, and K. Hornik (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(4), 488–508.

Burgin R. and Ritschard G. (2015), Tree-Based Varying Coefficient Regression for Longitudinal Ordinal Responses. Computational Statistics & Data Analysis, 86, 65–80.

Burgin, R. and G. Ritschard (2017), Coefficient-Wise Tree-Based Varying Coefficient Regression with vcrpart. Journal of Statistical Software, 80(6), 1–33.

Sela R. and J. S. Simonoff (2012). RE-EM trees: A Data Mining Approach for Longitudinal and Clustered data, Machine Learning 86(2), 169–207.

A. Hajjem, F. Bellavance and D. Larocque (2011), Mixed Effects Regression Trees for Clustered Data, Statistics & Probability Letters 81(4), 451–459.

Examples

## ------------------------------------------------------------------- # 
## Example: 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 Burgin and Ritschard (2015)
## ------------------------------------------------------------------- #

data(unemp)

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

## diagnosis
plot(model.UE, "coef")
summary(model.UE)
splitpath(model.UE, steps = 1, details = TRUE)

[Package vcrpart version 1.0-5 Index]