Compute Hartigans' Dip Test Statistic for Unimodality

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

distribution, htest
dip(x, full.result = FALSE, min.is.0 = FALSE, debug = FALSE)
numeric; the data.
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.
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 compatibilit
logical; if true, some tracing information is printed (from the C routine).

  • depending on full.result either a number, the dip statistic, or an object of class "dip" which is a list with components
  • xthe sorted unname()d data.
  • nlength(x).
  • dipthe dip statistic
  • lo.hiindices into x for lower and higher end of modal interval
  • xl, xulower and upper end of modal interval
  • gcm, lcm(last used) indices for greatest convex minorant and the least concave majorant.
  • mn, mjindex 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.


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 lyongu+@cs.cmu.edu 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.

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 (:-).


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.

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

fE <- faithful$eruptions
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)

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.))
Documentation reproduced from package diptest, version 0.75-5, License: GPL (>= 2)

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