The segmented Package

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1 The segmented Package February 17, 2004 Version Date Title Segmented relationships in regression models Author Vito M. R. Muggeo Maintainer Vito M. R. Muggeo Depends R (>= 1.8.0) Description Functions to estimate break-points of segmented relationships in regression models (GLMs) License GPL version 2 or newer R topics documented: down segmented slope.segmented Index 6 down Down syndrome in babies Description The down data frame has 30 rows and 3 columns. Variable cases means the number of babies with Down syndrome out of total number of births births for mothers with mean age age. Usage data(down) 1

2 2 segmented Format Source A data frame with 30 observations on the following 3 variables. age the mothers mean age births count of total births cases count of babies with Down syndrome Davison, A.C. and Hinkley, D. V. (1997) Bootstrap Methods and their Application. Cambridge University Press. Originally from Geyer, C. J. (1991) Constrained maximum likelihood exemplified by isotonic convex logistic regression. Journal of the American Statistical Association 86, Examples data(down) fit.glm<-glm(cases/births~age, weight=births, family=binomial, data=down) fit.seg<-segmented(fit.glm, Z=age, psi=25) fit.seg segmented Segmented relationships in regression models Description Usage Fits regression models with segmented relationships between the response and one or more explanatory variables. Break-point estimates are provided. segmented(obj, Z, psi, W, it.max=20, toll=0.0001, visual=false, last=true,...) Arguments obj Z psi W it.max toll visual last a standard linear model of class "lm" or "glm". a vector or a matrix meaning the (continuous) explanatory variable(s) having segmented relationships with the response of obj. starting values for the break-point(s). an optional categorical variable specifying levels in which a segmented relationship is assumed. the maximum number of iterations allowed. the tolerance value used to stop the algorithm. logical value indicating whether the process fitting should be printed at each iteration. logical value indicating whether only the last object should be returned.... further arguments passed to lm or glm when the algorithm stops.

3 segmented 3 Details Value Given a linear regression model (of class "lm" or "glm"), segmented estimates a new model having a broken-line relationship with a specified variable Z. Such a segmented or brokenline relationship is defined by the slope parameters and the break-points where the linear relation changes. Initial guesses for the break-points must be specified in the psi argument. The model is estimated simultaneously yielding point estimates and relevant (Wald-based) standard errors of all the model parameters, including the break-points. The returned object depends on the last argument. 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. If last=false the returned object is just a list in which the ith component is the model fitted at the ith approximation. Therefore the last component is the fitted model when the algorithm converges. The function summary (i.e., summary.segmented) can be used to obtain or print a summary of the results. If Z is a vector and W is missing, psi specifies the number of break-points to be estimated with respect to the single variable Z; if Z is a vector and W is provided, then a segmented relationship within each level of W is assumed: in this case length(psi) has to equal the number of levels of W and the ith component of psi is assumed as starting value of the breakpoint inside the ith group. If Z is a matrix, W has to be missing and the ith component of psi is the starting value of the break-point for the variable Z[,i]. An object of class "segmented" is a list containing the components of the original object obj with additionally the followings: psi it epsilon Estimated break-points and relevant standard errors The number of iterations employed difference in the objective function when the algorithm stops Warning Note 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. 1. The algorithm will start if the it.max argument is greater than zero. If it.max=0 segmented will estimate a new linear model with break-point(s) fixed at psi. 2. In the returned object, the name of the difference-in-slopes parameter is labelled with U.name of variable. 3. Currently the functions can not handle missing values, thus if obj has been created with argument na.action=na.omit, segmented will not work. 4. Methods specific to the class "segmented" are ˆ print.segmented ˆ summary.segmented ˆ print.summary.segmented Others are inherited from the class "lm" or "glm" depending on the class of obj.

4 4 segmented 5. Currently multiple break-points, namely a relationships with multiple changes, are allowed only if a single variable has to be analyzed, that is if Z is a vector and W is missing. Author(s) Vito M. R. Muggeo, vito.muggeo@giustizia.it References Muggeo, V.M.R. (2003) Estimating regression models with unknown break-points. Statistics in Medicine 22, See Also lm, glm Examples ## A linear model with a segmented relationship in each level of ## a categorical variable ## ## Simulate data x<-1:100 g<-rep(0:1,c(100,100)) set.seed(15) y1< *pmax(x-60,0)+rnorm(100,0,5) y2< *pmax(x-60,0)+rnorm(100,0,5) dati<-data.frame(xx=rep(x,2),yy=c(y1,y2),g=factor(g)) rm(x,g,y1,y2) ## Have a look at the plot ## Don't run: plot(dati$xx,dati$yy) ## Don't run: points(dati$xx[dati$g==0],dati$yy[dati$g==0],col=2) ## Fit the linear model obj<-lm(yy~0+g+xx:g,data=dati) ## Fit segmented models ## Model I: ignore the stratification, assume an equal difference-in-slopes ## parameter between groups ogg0<-segmented(obj,z=xx,psi=60) ## Model II: now stratificate by g. Here note that psi[i] refers to the ## segmented relationship in the ith level of W ogg1<-segmented(obj,z=xx,w=g,psi=c(50,70),visual=true) ## Results... ogg0 summary(ogg1) ## Have a look at the fitted models ## model I ## Don't run: points(dati$xx[dati$g==0],ogg0$fitted[dati$g==0],pch=3,col=2) ## Don't run: points(dati$xx[dati$g==1],ogg0$fitted[dati$g==1],pch=3,col=1) ## model II ## Don't run: points(dati$xx[dati$g==0],ogg1$fitted[dati$g==0],pch=20,col=2) ## Don't run: points(dati$xx[dati$g==1],ogg1$fitted[dati$g==1],pch=20,col=1) ## Don't run: legend(5,60,legend=c("model I","model II"),pch=c(3,20))

5 slope.segmented 5 slope.segmented Summary for slopes of segmented relationships Description Computes summary of the slopes of each segmented relationship in the fitted model. Usage slope.segmented(ogg, level = 0.95) Arguments ogg level an object of class "segmented", returned by any segmented method. the confidence level required. Details To fit broken-line relationships, segmented uses a parameterization whose coefficients are not the slopes. Therefore given an object "segmented", slope.segmented computes point estimates, standard errors, t-values and confidence intervals of the slopes of each segmented relationship in the fitted model. Value slope.segmented returns a list for each variable having a piecewise-linear relation with the response in the fitted model. Each list is a matrix with number of rows equal to number of segments and five columns summarizing the results. Note The returned summary is based on limiting Gaussian distribution for the model parameters involved in the computations. Sometimes even with large sample sizes such approximations are questionable (e.g., with small difference-in-slopes parameters) and the results returned by slope.segmented might be unreliable. Therefore is responsability of the user to gauge the applicability of such asymptotic approximations. Anyway, the t values may be not assumed for testing purposes and they should be used just as guidelines to assess the estimates uncertainty. Author(s) Vito M. R. Muggeo, vito.muggeo@giustizia.it References Muggeo, V.M.R. (2003) Estimating regression models with unknown break-points. Statistics in Medicine 22, See Also segmented

6 6 slope.segmented Examples ## A segmented relationship with three breakpoints ## x<-1:100 y<-2+1.5*pmax(x-35,0)-1.5*pmax(x-70,0)+rnorm(100,0,5) dati<-data.frame(x=x,y=y) out.lm<-lm(y~x,data=dati) out.seg<-segmented(out.lm,z=x,psi=c(20,80)) ## the slopes of the three segments... slope.segmented(out.seg) rm(x,y,out.lm,out.seg)

7 Index Topic datasets down, 1 Topic regression segmented, 2 slope.segmented, 4 down, 1 glm, 4 lm, 4 print.segmented (segmented), 2 print.summary.segmented (segmented), 2 segmented, 2, 5 slope.segmented, 4 summary.segmented (segmented), 2 7