1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
|
\name{pscore.test}
\alias{pscore.test}
\title{ Testing for existence of one breakpoint}
\description{
Given a (generalized) linear model, the (pseudo) Score statistic tests for the existence of one breakpoint.
}
\usage{
pscore.test(obj, seg.Z, k = 10, alternative = c("two.sided", "less", "greater"),
values=NULL, dispersion=NULL, df.t=NULL, more.break=FALSE, n.break=1,
only.term=FALSE, break.type=c("break","jump"))
}
%- maybe also 'usage' for other objects documented here.
\arguments{
\item{obj}{ a fitted model typically returned by \code{glm} or \code{lm}. Even an object returned by
\code{segmented} can be set. Offset and weights are allowed.}
\item{seg.Z}{ a formula with no response variable, such as \code{seg.Z=~x1}, indicating the (continuous) segmented variable being tested. Only a single variable may be tested and an error is printed when \code{seg.Z} includes two or more terms. \code{seg.Z} can be omitted if i)\code{obj} is a segmented fit with a single segmented covariate (and that variable is taken), or ii)if it is a "lm" or "glm" fit with a single covariate (and that variable is taken).}
\item{k}{ optional. Number of points (equi-spaced from the min to max) used to compute the pseudo Score statistic. See Details. }
\item{alternative}{ a character string specifying the alternative hypothesis (relevant to the slope difference parameter). }
\item{values}{ optional. The evaluation points where the Score test is computed. See Details for default values.}
\item{dispersion}{ optional. the dispersion parameter for the family to be used to compute the test statistic.
When \code{NULL} (the default), it is inferred from \code{obj}. Namely it is taken as \code{1} for the
Binomial and Poisson families, and otherwise estimated by the residual Chi-squared statistic in the model \code{obj} (calculated from cases with non-zero weights divided by the residual degrees of freedom).}
\item{df.t}{ optional. The degress-of-freedom used to compute the p-value. When \code{NULL}, the df extracted from \code{obj} are used.}
\item{more.break}{ optional, logical. If \code{obj} is a 'segmented' fit, \code{more.break=FALSE} tests for the actual breakpoint for the variable 'seg.Z',
while \code{more.break=TRUE} tests for an \emph{additional} breakpoint(s) for the variable 'seg.Z'. Ignored when \code{obj} is not a segmented fit.}
\item{n.break}{optional. Number of breakpoints postuled under the alternative hypothesis.}
\item{only.term}{logical. If \code{TRUE}, only the pseudo covariate(s) relevant to the testing for the breakpoint is returned, and no test is computed.}
\item{break.type}{The kind of breakpoint being tested. \code{"break"} is for piecewise-linear relationships, \code{"jump"} means piecewise-constant, i.e. a step-function, relationships.}
}
\details{
\code{pscore.test} tests for a non-zero difference-in-slope parameter of a segmented
relationship. Namely, the null hypothesis is \eqn{H_0:\beta=0}{H_0:beta=0}, where \eqn{\beta}{beta} is the difference-in-slopes,
i.e. the coefficient of the segmented function \eqn{\beta(x-\psi)_+}{beta*(x-psi)_+}. The hypothesis of interest
\eqn{\beta=0}{beta=0} means no breakpoint. Simulation studies have shown that such Score test is more powerful than the Davies test (see reference) when the alternative hypothesis is `one changepoint'. If there are two or more breakpoints (for instance, a sinusoidal-like relationships), \code{pscore.test} can have lower power, and \code{\link{davies.test}} can perform better.
The \code{dispersion} value, if unspecified, is taken from \code{obj}. If \code{obj} represents the fit under the null hypothesis (no changepoint), the dispersion parameter estimate will be usually larger, leading to a (potentially severe) loss of power.
The \code{k} evaluation points are \code{k} equally spaced values in the range of the segmented covariate. \code{k} should not be small.
Specific values can be set via \code{values}, although I have found no important difference due to number and location of the evaluation points, thus default is \code{k=10} equally-spaced points. However, when the possible breakpoint is believed to lie into a specified narrower range, the user can specify \code{k} values in that range leading to higher power in detecting it, i.e. typically lower p-value.
If \code{obj} is a (segmented) \emph{lm} object, the returned p-value comes from the t-distribution with appropriate degrees of freedom. Otherwise, namely if \code{obj} is a (segmented) \emph{glm} object, the p-value is computed wrt the Normal distribution.
}
\value{
A list with class '\code{htest}' containing the following components:
\item{method}{title (character)}
\item{data.name}{the regression model and the segmented variable being tested}
\item{statistic }{the empirical value of the statistic}
\item{parameter }{number of evaluation points}
\item{p.value }{the p-value}
\item{process}{the alternative hypothesis set}
}
\references{
Muggeo, V.M.R. (2016) Testing with a nuisance parameter present only under the alternative:
a score-based approach with application to segmented modelling.
\emph{J of Statistical Computation and Simulation}, \bold{86}, 3059--3067.
}
\author{ Vito M.R. Muggeo }
\seealso{See also \code{\link{davies.test}}. }
\examples{
\dontrun{
set.seed(20)
z<-runif(100)
x<-rnorm(100,2)
y<-2+10*pmax(z-.5,0)+rnorm(100,0,3)
o<-lm(y~z+x)
#testing for one changepoint
#use the simple null fit
pscore.test(o,~z) #compare with davies.test(o,~z)..
#use the segmented fit
os<-segmented(o, ~z)
pscore.test(os,~z) #smaller p-value, as it uses the dispersion under the alternative (from 'os')
#test for the 2nd breakpoint in the variable z
pscore.test(os,~z, more.break=TRUE)
}
}
\keyword{ htest }
|
