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, control = seg.control(), 
    model = TRUE, ...)

## Default S3 method:
segmented(obj, seg.Z, psi, control = seg.control(),
    model = TRUE, ...)

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

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

## S3 method for class 'Arima'
segmented(obj, seg.Z, psi, control = seg.control(),
    model = TRUE, ...)

Arguments

Details

Given a linear regression model (usually of class "lm" or "glm"), segmented tries to estimate a new 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 argument, where initial values for the break-points 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()).

Value

The returned object depends on the last component returned by seg.control. If last=TRUE, the default, segmented returns an object of class "segmented" which inherits from the class "lm" or "glm" depending on the class of obj. Otherwise a list is returned, where the last component is the fitted model at the final iteration, see seg.control.

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

Warning

It is well-known that the log-likelihood function for the break-point may be not concave, especially for poor clear-cut kink-relationships. In these circumstances the initial guess for the break-point, i.e. the psi argument, must be provided with care. For instance visual inspection of a, possibly smoothed, scatter-plot is usually a good way to obtain some idea on breakpoint location. However bootstrap restarting, implemented since version 0.2-9.0, is relatively more robust to starting values specified in psi. Alternatively an automatic procedure may be implemented by specifying psi=NA and stop.if.error=FALSE in seg.control: experience suggests to increase the number of iterations via it.max in seg.control(). This automatic procedure, however, is expected to overestimate the number of breakpoints.

Note

1. 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.

2. In the returned fit object, ‘U.’ is put before the name of the segmented variable to mean the difference-in-slopes coefficient.

3. 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

   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.

See Also

lm, glm

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)

#simple example: 1 segmented variable, 1 breakpoint: you do not need to specify 
# the starting value for psi
o<-segmented(out.lm,seg.Z=~z)

#1 segmented variable, 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))
slope(o)

#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))


#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
# Notice: bootstrap restart is not allowed!
o<-segmented.lm(out.lm,seg.Z=~x+z,psi=list(x=NA,z=.3), 
    control=seg.control(stop.if.error=FALSE,n.boot=0, it.max=20))

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

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


#An example using the default method:
# 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

## End(Not run)