segmented: Segmented relationships in regression models
In segmented: Regression Models with Break-Points / Change-Points Estimation (with Possibly Random Effects)

segmented

R Documentation

Segmented relationships in regression models

Description

Fits regression models with segmented relationships between the response and one or more explanatory variables. Break-point estimates are provided.

Usage

segmented(obj, seg.Z, psi, npsi, fixed.psi=NULL, control = seg.control(), 
    model = TRUE, ...)

## Default S3 method:
segmented(obj, seg.Z, psi, npsi, fixed.psi=NULL, control = seg.control(),
    model = TRUE, keep.class=FALSE, ...)

## S3 method for class 'lm'
segmented(obj, seg.Z, psi, npsi, fixed.psi=NULL, control = seg.control(),
    model = TRUE, keep.class=FALSE, ...)

## S3 method for class 'glm'
segmented(obj, seg.Z, psi, npsi, fixed.psi=NULL, control = seg.control(),
    model = TRUE, keep.class=FALSE, ...)

## S3 method for class 'Arima'
segmented(obj, seg.Z, psi, npsi, fixed.psi=NULL, control = seg.control(),
    model = TRUE, keep.class=FALSE, ...)
    
## S3 method for class 'numeric'
segmented(obj, seg.Z, psi, npsi, fixed.psi=NULL, control = seg.control(),
    model = TRUE, keep.class=FALSE, adjX=FALSE, weights=NULL, ...)

Arguments

`obj`	standard ‘linear’ model of class "lm", "glm" or "Arima", or potentially any regression fit may be supplied since version 0.5-0 (see 'Details'). `obj` can include any covariate understood to have a linear (i.e. no break-points) effect on the response. If `obj` also includes the segmented covariate specified in `seg.Z`, then all the slopes of the fitted segmented relationship will be estimated. On the other hand, if `obj` misses the segmented variable, then the 1st (the leftmost) slope is assumed to be zero. Since version 1.5.0, `obj` can be a simple numeric or `ts` object but with only a single segmented variable (`segmented.numeric`) see examples below.
`seg.Z`	the segmented variable(s), i.e. the continuous covariate(s) understood to have a piecewise-linear relationship with response. It is a formula with no response variable, such as `seg.Z=~x` or `seg.Z=~x1+x2`. It can be missing when `obj` includes only one covariate which is taken as segmented variable. Currently, formulas involving functions, such as `seg.Z=~log(x1)`, or selection operators, such as `seg.Z=~d[,"x1"]` or `seg.Z=~d$x1`, are not allowed. Also, variable names formed by `⁠U⁠` or `⁠V⁠` only (with or without numbers) are not permitted.
`psi`	starting values for the breakpoints to be estimated. If there is a single segmented variable specified in `seg.Z`, `psi` is a numeric vector, and it can be missing when 1 breakpoint has to be estimated (and the median of the segmented variable is used as a starting value). If `seg.Z` includes several covariates, `psi` has be specified as a named list of vectors whose names have to match the variables in the `seg.Z` argument. Each vector of such list includes starting values for the break-point(s) for the corresponding variable in `seg.Z`. A `NA` value means that '`K`' quantiles (or equally spaced values) are used as starting values; `K` is fixed via the `seg.control` auxiliary function. See `npsi` as an alternative to specify just the number of breakpoints.
`npsi`	A named vector or list meaning the number (and not locations) of breakpoints to be estimated. The starting values will be internally computed via the quantiles or equally spaced values, as specified in argument `quant` in `seg.control`. `npsi` can be missing and `npsi=1` is assumed for all variables specified in `seg.Z`. If `psi` is provided, `npsi` is ignored.
`fixed.psi`	An optional named list meaning the breakpoints to be kept fixed during the estimation procedure. The names should be a subset of (or even the same) variables specified in `seg.Z`. If there is a single variable in `seg.Z`, a simple numeric vector can be specified. Note that, in addition to the values specified here, `segmented` will estimate additional breakpoints. To keep fixed all breakpoints (to be specified in `psi`) use `it.max=0` in `seg.control`
`control`	a list of parameters for controlling the fitting process. See the documentation for `seg.control` for details.
`model`	logical value indicating if the model.frame should be returned.
`keep.class`	logical value indicating if the final fit returned by `segmented.default` should keep the class '`segmented`' (along with the class of the original fit `obj`). Ignored by the segmented methods.
`...`	optional arguments (to be ignored safely). Notice specific arguments relevant to the original call (via `lm` or `glm` for instance), such as `weights` or `offet`, have to be included in the starting model `obj`
`adjX`	if `obj` is a `ts`, the segmented variable (if not specified in `seg.Z`) is computed by taking information from the time series (e.g., years starting from 2000, say). If `adjX=TRUE`, the segmented variable is shifted such that its min equals zero. Default is using the unshifted values, but if there are several breakpoints to be estimated , it is strongly suggested to set `adjX=TRUE`.
`weights`	the weights if `obj` is a vector or a ts object, otherwise the weights should be specified in the starting fit `obj`.

Details

Given a linear regression model usually of class "lm" or "glm" (or even a simple numeric/ts vector), segmented tries to estimate a new regression model having broken-line relationships with the variables specified in seg.Z. A segmented (or broken-line) relationship is defined by the slope parameters and the break-points where the linear relation changes. The number of breakpoints of each segmented relationship is fixed via the psi (or npsi) argument, where initial values for the break-points (or simply their number via npsi) must be specified. The model is estimated simultaneously yielding point estimates and relevant approximate standard errors of all the model parameters, including the break-points.

Since version 0.2-9.0 segmented implements the bootstrap restarting algorithm described in Wood (2001). The bootstrap restarting is expected to escape the local optima of the objective function when the segmented relationship is flat and the log likelihood can have multiple local optima.

Since version 0.5-0.0 the default method segmented.default has been added to estimate segmented relationships in general (besides "lm" and "glm" fits) regression models, such as Cox regression or quantile regression (for a single percentile). The objective function to be minimized is the (minus) value extracted by the logLik function or it may be passed on via the fn.obj argument in seg.control. See example below. While the default method is expected to work with any regression fit (where the usual coef(), update(), and logLik() returns appropriate results), it is not recommended for "lm" or "glm" fits (as segmented.default is slower than the specific methods segmented.lm and segmented.glm), although final results are the same. However the object returned by segmented.default is not of class "segmented", as currently the segmented methods are not guaranteed to work for ‘generic’ (i.e., besides "lm" and "glm") regression fits. The user could try each "segmented" method on the returned object by calling it explicitly (e.g. via plot.segmented() or confint.segmented() wherein the regression coefficients and relevant covariance matrix have to be specified, see .coef and .vcov in plot.segmented(), confint.segmented(), slope()).

Value

segmented returns an object of class "segmented" which inherits from the class of obj, for instance "lm" or "glm".

An object of class "segmented" is a list containing the components of the original object obj with additionally the followings:

`psi`	estimated break-points (sorted) and relevant (approximate) standard errors
`it`	number of iterations employed
`epsilon`	difference in the objective function when the algorithm stops
`model`	the model frame
`psi.history`	a list or a vector including the breakpoint estimates at each step
`seed`	the integer vector containing the seed just before the bootstrap resampling. Returned only if bootstrap restart is employed
`..`	Other components are not of direct interest of the user

Warning

At convergence, if the estimated breakpoints are too close each other or at the boundaries, the parameter point estimate could be returned, but without finite standard errors. To avoid that, segmented revises the final breakpoint estimates to allow that at least min.nj are within each interval of the segmented covariate. A warning message is printed if such adjustment is made. See min.nj in seg.control.

Note

The algorithm will start if the it.max argument returned by seg.control is greater than zero. If it.max=0 segmented will estimate a new linear model with break-point(s) fixed at the values reported in psi.Alternatively, it is also possible to set h=0 in seg.control(). In this case, bootstrap restarting is unncessary, then to have breakpoints at mypsi type

segmented(.., psi=mypsi, control=seg.control(h=0, n.boot=0, it.max=1))
In the returned fit object, ‘U.’ is put before the name of the segmented variable to mean the difference-in-slopes coefficient.
Methods specific to the class "segmented" are
- print.segmented
- summary.segmented
- print.summary.segmented
- plot.segmented
- lines.segmented
- confint.segmented
- vcov.segmented
- predict.segmented
- points.segmented
- coef.segmented
Others are inherited from the class "lm" or "glm" depending on the class of obj.

Author(s)

Vito M. R. Muggeo, vito.muggeo@unipa.it

References

Muggeo, V.M.R. (2003) Estimating regression models with unknown break-points. Statistics in Medicine 22, 3055–3071.

Muggeo, V.M.R. (2008) Segmented: an R package to fit regression models with broken-line relationships. R News 8/1, 20–25.

Examples


set.seed(12)
xx<-1:100
zz<-runif(100)
yy<-2+1.5*pmax(xx-35,0)-1.5*pmax(xx-70,0)+15*pmax(zz-.5,0)+rnorm(100,0,2)
dati<-data.frame(x=xx,y=yy,z=zz)
out.lm<-lm(y~x,data=dati)

#the simplest example: the starting model includes just 1 covariate 
#.. and 1 breakpoint has to be estimated for that
o<-segmented(out.lm) #1 breakpoint for x

#the single segmented variable is not in the starting model, and thus..
#... you need to specify it via seg.Z, but no starting value for psi
o<-segmented(out.lm, seg.Z=~z)
#note the leftmost slope is constrained to be zero (since out.lm does not include z)

#2 segmented variables, 1 breakpoint each (again no need to specify npsi or psi)
o<-segmented(out.lm,seg.Z=~z+x)


#1 segmented variable, but 2 breakpoints: you have to specify starting values (vector) for psi:
o<-segmented(out.lm,seg.Z=~x,psi=c(30,60), control=seg.control(display=FALSE))

#.. or you can specify just the *number* of breakpoints
#o<-segmented(out.lm,seg.Z=~x, npsi=2, control=seg.control(display=FALSE)) 

slope(o) #the slopes of the segmented relationship


#2 segmented variables: starting values requested via a named list
out.lm<-lm(y~z,data=dati)
o1<-update(o,seg.Z=~x+z,psi=list(x=c(30,60),z=.3))
#..or by specifying just the *number* of breakpoints
#o1<-update(o,seg.Z=~x+z, npsi=c(x=2,z=1))



#the default method leads to the same results (but it is slower)
#o1<-segmented.default(out.lm,seg.Z=~x+z,psi=list(x=c(30,60),z=.3))
#o1<-segmented.default(out.lm,seg.Z=~x+z,psi=list(x=c(30,60),z=.3), 
#    control=seg.control(fn.obj="sum(x$residuals^2)"))


#automatic procedure to estimate breakpoints in the covariate x (starting from K quantiles)
# Hint: increases number of iterations. Notice: bootstrap restart is not allowed!
# However see ?selgmented for a better approach
#o<-segmented.lm(out.lm,seg.Z=~x+z,psi=list(x=NA,z=.3), 
#    control=seg.control(fix.npsi=FALSE, n.boot=0, tol=1e-7, it.max = 50, K=5, display=TRUE))

#assess the progress of the breakpoint estimates throughout the iterations
## Not run: 
par(mfrow=c(1,2))
draw.history(o, "x")
draw.history(o, "z")

## End(Not run)
#try to increase the number of iterations and re-assess the 
#convergence diagnostics 


# A simple segmented model with continuous responses and no linear covariates
# No need to fit the starting lm model:
segmented(yy, npsi=2) #NOTE: subsetting the vector works ( segmented(yy[-1],..) ) 
#only a single segmented covariate is allowed in seg.Z, and if seg.Z is unspecified, 
#   the segmented variable is taken as 1:n/n 


# An example using the Arima method:
## Not run: 
n<-50
idt <-1:n #the time index

mu<-50-idt +1.5*pmax(idt-30,0)
set.seed(6969)
y<-mu+arima.sim(list(ar=.5),n)*3.5

o<-arima(y, c(1,0,0), xreg=idt)
os1<-segmented(o, ~idt, control=seg.control(display=TRUE))

#note using the .coef argument is mandatory!
slope(os1, .coef=os1$coef)
plot(y)
plot(os1, add=TRUE, .coef=os1$coef, col=2)


## End(Not run)

################################################################
################################################################
######Four examples using the default method:
################################################################
################################################################


################################################################
#==> 1. Cox regression with a segmented relationship  
################################################################
## Not run: 
library(survival)
data(stanford2)

o<-coxph(Surv(time, status)~age, data=stanford2)
os<-segmented(o, ~age, psi=40) #estimate the breakpoint in the age effect
summary(os) #actually it means summary.coxph(os)
plot(os) #it does not work
plot.segmented(os) #call explicitly plot.segmented() to plot the fitted piecewise lines


################################################################
# ==> 2. Linear mixed model via the nlme package
################################################################

dati$g<-gl(10,10) #the cluster 'id' variable
library(nlme)
o<-lme(y~x+z, random=~1|g, data=dati)
os<-segmented.default(o, ~x+z, npsi=list(x=2, z=1))

#summarizing results (note the '.coef' argument)
slope(os, .coef=fixef(os))
plot.segmented(os, "x", .coef=fixef(os), conf.level=.95)
confint.segmented(os, "x", .coef=fixef(os))
dd<-data.frame(x=c(20,50),z=c(.2,.6), g=1:2)
predict.segmented(os, newdata=dd, .coef=fixef(os)) 


################################################################
# ==> 3. segmented quantile regression via the quantreg  package
################################################################

library(quantreg)
data(Mammals)
y<-with(Mammals, log(speed))
x<-with(Mammals, log(weight))
o<-rq(y~x, tau=.9)
os<-segmented.default(o, ~x) #it does NOT work. It cannot compute the vcov matrix..

#Let's define the vcov.rq function.. (I don't know if it is the best option..)
vcov.rq<-function(x,...) {
  V<-summary(x,cov=TRUE,se="nid",...)$cov
  rownames(V)<-colnames(V)<-names(x$coef)
V}

os<-segmented.default(o, ~x) #now it does work
 plot.segmented(os, res=TRUE, col=2, conf.level=.95)


################################################################
# ==> 4. segmented regression with the svyglm() (survey  package)   
################################################################

library(survey)
data(api)
dstrat<-svydesign(id=~1,strata=~stype, weights=~pw, data=apistrat, fpc=~fpc)

o<-svyglm(api00~ell, design=dstrat)

#specify as a string the objective function to be minimized. It can be obtained via svyvar() 

fn.x<- 'as.numeric(svyvar(resid(x, "pearson"), x$survey.design, na.rm = TRUE))'
os<-segmented.default(o, ~ell, control=seg.control(fn.obj=fn.x, display=TRUE))
slope(os)
plot.segmented(os, res=TRUE, conf.level=.9, shade=TRUE)

## End(Not run)

segmented documentation built on Oct. 25, 2024, 5:07 p.m.