Perform a one- or two-sample Kolmogorov-Smirnov test.
ks.test(x, y, …,
alternative = c("two.sided", "less", "greater"),
exact = NULL)a numeric vector of data values.
either a numeric vector of data values, or a character string
naming a cumulative distribution function or an actual cumulative
distribution function such as pnorm. Only continuous CDFs
are valid.
parameters of the distribution specified (as a character
string) by y.
indicates the alternative hypothesis and must be
one of "two.sided" (default), "less", or
"greater". You can specify just the initial letter of the
value, but the argument name must be give in full.
See ‘Details’ for the meanings of the possible values.
NULL or a logical indicating whether an exact
p-value should be computed. See ‘Details’ for the meaning of
NULL. Not available in the two-sample case for a one-sided
test or if ties are present.
A list with class "htest" containing the following components:
the value of the test statistic.
the p-value of the test.
a character string describing the alternative hypothesis.
a character string indicating what type of test was performed.
a character string giving the name(s) of the data.
If y is numeric, a two-sample test of the null hypothesis
that x and y were drawn from the same continuous
distribution is performed.
Alternatively, y can be a character string naming a continuous
(cumulative) distribution function, or such a function. In this case,
a one-sample test is carried out of the null that the distribution
function which generated x is distribution y with
parameters specified by ….
The presence of ties always generates a warning, since continuous distributions do not generate them. If the ties arose from rounding the tests may be approximately valid, but even modest amounts of rounding can have a significant effect on the calculated statistic.
Missing values are silently omitted from x and (in the
two-sample case) y.
The possible values "two.sided", "less" and
"greater" of alternative specify the null hypothesis
that the true distribution function of x is equal to, not less
than or not greater than the hypothesized distribution function
(one-sample case) or the distribution function of y (two-sample
case), respectively. This is a comparison of cumulative distribution
functions, and the test statistic is the maximum difference in value,
with the statistic in the "greater" alternative being
\(D^+ = \max_u [ F_x(u) - F_y(u) ]\).
Thus in the two-sample case alternative = "greater" includes
distributions for which x is stochastically smaller than
y (the CDF of x lies above and hence to the left of that
for y), in contrast to t.test or
wilcox.test.
Exact p-values are not available for the two-sample case if one-sided
or in the presence of ties. If exact = NULL (the default), an
exact p-value is computed if the sample size is less than 100 in the
one-sample case and there are no ties, and if the product of
the sample sizes is less than 10000 in the two-sample case.
Otherwise, asymptotic distributions are used whose approximations may
be inaccurate in small samples. In the one-sample two-sided case,
exact p-values are obtained as described in Marsaglia, Tsang & Wang
(2003) (but not using the optional approximation in the right tail, so
this can be slow for small p-values). The formula of Birnbaum &
Tingey (1951) is used for the one-sample one-sided case.
If a single-sample test is used, the parameters specified in
… must be pre-specified and not estimated from the data.
There is some more refined distribution theory for the KS test with
estimated parameters (see Durbin, 1973), but that is not implemented
in ks.test.
Z. W. Birnbaum and Fred H. Tingey (1951). One-sided confidence contours for probability distribution functions. The Annals of Mathematical Statistics, 22/4, 592--596. 10.1214/aoms/1177729550.
William J. Conover (1971). Practical Nonparametric Statistics. New York: John Wiley & Sons. Pages 295--301 (one-sample Kolmogorov test), 309--314 (two-sample Smirnov test).
Durbin, J. (1973). Distribution theory for tests based on the sample distribution function. SIAM.
George Marsaglia, Wai Wan Tsang and Jingbo Wang (2003). Evaluating Kolmogorov's distribution. Journal of Statistical Software, 8/18. 10.18637/jss.v008.i18.
shapiro.test which performs the Shapiro-Wilk test for
normality.
# NOT RUN {
require(graphics)
x <- rnorm(50)
y <- runif(30)
# Do x and y come from the same distribution?
ks.test(x, y)
# Does x come from a shifted gamma distribution with shape 3 and rate 2?
ks.test(x+2, "pgamma", 3, 2) # two-sided, exact
ks.test(x+2, "pgamma", 3, 2, exact = FALSE)
ks.test(x+2, "pgamma", 3, 2, alternative = "gr")
# test if x is stochastically larger than x2
x2 <- rnorm(50, -1)
plot(ecdf(x), xlim = range(c(x, x2)))
plot(ecdf(x2), add = TRUE, lty = "dashed")
t.test(x, x2, alternative = "g")
wilcox.test(x, x2, alternative = "g")
ks.test(x, x2, alternative = "l")
# }
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