dip {diptest}

Compute Hartigans' Dip Test Statistic for Unimodality

Description

Computes Hartigans' dip test statistic for testing unimodality, and additionally the modal interval.

Usage

dip(x, full.result = FALSE, min.is.0 = FALSE, debug = FALSE)

Arguments

x

numeric; the data.

full.result

logical or string; dip(., full.result=TRUE) returns the full result list; if "all" it additionally uses the mn and mj components to compute the initial GCM and LCM, see below.

min.is.0

logical indicating if the minimal value of the dip statistic D_n can be zero or not. Arguably should be set to TRUE for internal consistency reasons, but is false by default both for continuity and backwards compatibility reasons, see the examples below.

debug

logical; if true, some tracing information is printed (from the C routine).

Value

depending on full.result either a number, the dip statistic, or an object of class "dip" which is a list with components

x

the sorted unname()d data.

n

length(x).

dip

the dip statistic

lo.hi

indices into x for lower and higher end of modal interval

xl, xu

lower and upper end of modal interval

gcm, lcm

(last used) indices for greatest convex minorant and the least concave majorant.

mn, mj

index vectors of length n for the GC minorant and the LC majorant respectively.

For “full” results of class "dip", there are print and plot methods, the latter with its own manual page.

Note

For n \le 3 where n <- length(x), the dip statistic D_n is always the same minimum value, 1/(2n), i.e., there's no possible dip test. Note that up to May 2011, from Hartigan's original Fortran code, Dn was set to zero, when all x values were identical. However, this entailed discontinuous behavior, where for arbitrarily close data \tilde x, D_n(\tilde x) = \frac 1{2n}.

Yong Lu [e-mail] found in Oct 2003 that the code was not giving symmetric results for mirrored data (and was giving results of almost 1, and then found the reason, a misplaced ‘⁠")"⁠’ in the original Fortran code. This bug has been corrected for diptest version 0.25-0 (Feb 13, 2004).

Nick Cox (Durham Univ.) said (on March 20, 2008 on the Stata-list):
As it comes from a bimodal husband-wife collaboration, the name perhaps should be “Hartigan-Hartigan dip test”, but that does not seem to have caught on. Some of my less statistical colleagues would sniff out the hegemony of patriarchy there, although which Hartigan is being overlooked is not clear.

Martin Maechler, as a Swiss, and politician, would say:
Let's find a compromise, and call it “Hartigans' dip test”, so we only have to adapt orthography (:-).

Author(s)

Martin Maechler [e-mail], 1994, based on S (S-PLUS) and C code donated from Dario Ringach [e-mail] who had applied f2c on the original Fortran code available from Statlib.

In Aug.1993, recreated and improved Hartigans' "P-value" table, which later became qDiptab.

References

P. M. Hartigan (1985) Computation of the Dip Statistic to Test for Unimodality; Applied Statistics (JRSS C) 34, 320–325.
Corresponding (buggy!) Fortran code of ‘AS 217’ available from Statlib, http://lib.stat.cmu.edu/apstat/217

J. A. Hartigan and P. M. Hartigan (1985) The Dip Test of Unimodality; Annals of Statistics 13, 70–84.

See Also

dip.test to compute the dip and perform the unimodality test, based on P-values, interpolated from qDiptab; isoreg for isotonic regression.

Examples

data(statfaculty)
plot(density(statfaculty))
rug(statfaculty, col="midnight blue"); abline(h=0, col="gray")
dip(statfaculty)
(dS <- dip(statfaculty, full = TRUE, debug = TRUE))
plot(dS)
## even more output -- + plot showing "global" GCM/LCM:
(dS2 <- dip(statfaculty, full = "all", debug = 3))
plot(dS2)

data(faithful)
fE <- faithful$eruptions
plot(density(fE))
rug(fE, col="midnight blue"); abline(h=0, col="gray")
dip(fE, debug = 2) ## showing internal work
(dE <- dip(fE, full = TRUE)) ## note the print method
plot(dE, do.points=FALSE)

data(precip)
plot(density(precip))
rug(precip, col="midnight blue"); abline(h=0, col="gray")
str(dip(precip, full = TRUE, debug = TRUE))

##-----------------  The  'min.is.0' option :  ---------------------

##' dip(.) continuity and 'min.is.0' exploration:
dd <- function(x, debug=FALSE) {
   x_ <- x ; x_[1] <- 0.9999999999 * x[1]
   rbind(dip(x , debug=debug),
         dip(x_, debug=debug),
         dip(x , min.is.0=TRUE, debug=debug),
         dip(x_, min.is.0=TRUE, debug=debug), deparse.level=2)
}

dd( rep(1, 8) ) # the 3rd one differs ==> min.is.0=TRUE is *dis*continuous
dd( 1:7 )       # ditto

dd( 1:7, debug=TRUE)
## border-line case ..
dd( 1:2, debug=TRUE)

## Demonstrate that  'min.is.0 = TRUE'  does not change the typical result:
B.sim <- 1000 # or larger
D5  <- {set.seed(1); replicate(B.sim, dip(runif(5)))}
D5. <- {set.seed(1); replicate(B.sim, dip(runif(5), min.is.0=TRUE))}
stopifnot(identical(D5, D5.), all.equal(min(D5), 1/(2*5)))
hist(D5, 64); rug(D5)

D8  <- {set.seed(7); replicate(B.sim, dip(runif(8)))}
D8. <- {set.seed(7); replicate(B.sim, dip(runif(8), min.is.0=TRUE))}
stopifnot(identical(D8, D8.))