ad.test: Anderson-Darling k-Sample Test
In kSamples: K-Sample Rank Tests and their Combinations
ad.test R Documentation
Anderson-Darling k-Sample Test
Description
This function uses the Anderson-Darling criterion to test
the hypothesis that k independent samples with sample sizes
n_1,\ldots, n_k arose
from a common unspecified distribution function F(x) and testing is
done conditionally given the observed tie pattern. Thus this is a permutation test.
Both versions of the AD statistic are computed.
Usage
ad.test(..., data = NULL, method = c("asymptotic", "simulated", "exact"),
dist = FALSE, Nsim = 10000)
Arguments
...
Either several sample vectors, say
x_1, \ldots, x_k,
with x_i containing n_i sample values.
n_i > 4 is recommended for reasonable asymptotic
P-value calculation. The pooled sample size is denoted
by N=n_1+\ldots+n_k,
or a list of such sample vectors,
or a formula y ~ g, where y contains the pooled sample values
and g is a factor (of same length as y) with levels identifying
the samples to which the elements of y belong.
data
= an optional data frame providing the variables in formula y ~ g.
method
= c("asymptotic","simulated","exact"), where
"asymptotic" uses only an asymptotic P-value approximation, reasonable
for P in [.00001, .99999] when all n_i > 4.
Linear extrapolation via \log(P/(1-P))
is used outside [.00001, .99999]. This calculation is always done.
See ad.pval for details.
The adequacy of the asymptotic P-value calculation
may be checked using pp.kSamples.
"simulated" uses Nsim simulated AD statistics, based on random
splits of the pooled samples into samples of sizes
n_1, \ldots, n_k, to estimate the exact conditional P-value.
"exact" uses full enumeration of all sample splits with
resulting AD statistics to obtain the exact conditional P-values.
It is used only when Nsim is at least as large as the number
ncomb = \frac{N!}{n_1!\ldots n_k!}
of full enumerations. Otherwise, method
reverts to "simulated" using the given Nsim. It also reverts
to "simulated" when ncomb > 1e8 and dist = TRUE.
dist
= FALSE (default) or TRUE. If TRUE, the
simulated or fully enumerated distribution vectors null.dist1 and
null.dist2 are returned for the respective test statistic versions.
Otherwise, NULL is returned. When dist = TRUE then
Nsim <- min(Nsim, 1e8), to limit object size.
Nsim
= 10000 (default), number of simulation sample splits to use.
It is only used when method = "simulated",
or when method = "exact" reverts to method =
"simulated", as previously explained.
Details
If AD is the Anderson-Darling criterion for the k samples,
its standardized test statistic is T.AD = (AD - \mu)/\sigma, with
\mu = k-1 and
\sigma representing mean and standard deviation of AD. This statistic
is used to test the hypothesis that the samples all come
from the same but unspecified continuous distribution function F(x).
According to the reference article, two versions
of the AD test statistic are provided.
The above mean and standard deviation are strictly
valid only for version 1 in the
continuous distribution case.
NA values are removed and the user is alerted with the total NA count.
It is up to the user to judge whether the removal of NA's is appropriate.
The continuity assumption can be dispensed with, if we deal with
independent random samples, or if randomization was used in allocating
subjects to samples or treatments, and if we view
the simulated or exact P-values conditionally, given the tie pattern
in the pooled samples. Of course, under such randomization any conclusions
are valid only with respect to the group of subjects that were randomly allocated
to their respective samples.
The asymptotic P-value calculation assumes distribution continuity. No adjustment
for lack thereof is known at this point. For details on the asymptotic
P-value calculation see ad.pval.
Value
A list of class kSamples with components
test.name
"Anderson-Darling"
k
number of samples being compared
ns
vector of the k sample sizes (n_1,\ldots,n_k)
N
size of the pooled sample = n_1+\ldots+n_k
n.ties
number of ties in the pooled samples
sig
standard deviations \sigma of version 1 of AD under the continuity assumption
ad
2 x 3 (2 x 4) matrix containing AD, T.AD, asymptotic P-value,
(simulated or exact P-value), for each version of the standardized test statistic T,
version 1 in row 1, version 2 in row 2.
warning
logical indicator, warning = TRUE when at least one
n_i < 5
null.dist1
simulated or enumerated null distribution of version 1
of the test statistic, given as vector of all generated AD statistics.
null.dist2
simulated or enumerated null distribution of version 2
of the test statistic, given as vector of all generated AD statistics.
method
The method used.
Nsim
The number of simulations.
warning
method = "exact" should only be used with caution.
Computation time is proportional to the number of enumerations. In most cases
dist = TRUE should not be used, i.e.,
when the returned distribution vectors null.dist1 and null.dist2
become too large for the R work space. These vectors are limited in length by 1e8.
Note
For small sample sizes and small k exact null distribution
calculations are possible (with or without ties), based on a recursively extended
version of Algorithm C (Chase's sequence) in Knuth (2011), Ch. 7.2.1.3, which allows the
enumeration of all possible splits of the pooled data into samples of
sizes of n_1, \ldots, n_k, as appropriate under treatment randomization. The
enumeration and simulation are both done in C.
Note
It has recently come to our attention that the Anderson-Darling test, originally
proposed by Pettitt (1976) in the 2-sample case and generalized to k samples by
Scholz and Stephens, has a close relative created by Baumgartner et al (1998) in
the 2 sample case and populatized by Neuhaeuser (2012) with at least 6 papers
among his cited references and generalized by Murakami (2006) to k samples.
References
Baumgartner, W., Weiss, P. and Schindler, H. (1998), A nonparametric test for the
general two-sample problem, Bionetrics, 54, 1129-1135.
Knuth, D.E. (2011), The Art of Computer Programming, Volume 4A
Combinatorial Algorithms Part 1, Addison-Wesley
Neuhaeuser, M. (2012), Nonparametric Statistical Tests, A Computational Approach,
CRC Press.
Murakami, H. (2006), A k-sample rank test based on modified Baumgartner statistic and
it power comparison,
Jpn. Soc. Comp. Statist., 19, 1-13.
Murakami, H. (2012), Modified Baumgartner statistic for the two-sample and multisample
problems: a numerical comparison.
J. of Statistical Comput. and Simul., 82:5, 711-728.
Pettitt, A.N. (1976), A two-sample Anderson_Darling rank statistic, Biometrika,
63, 161-168.
Scholz, F. W. and Stephens, M. A. (1987), K-sample Anderson-Darling Tests,
Journal of the American Statistical Association,
Vol 82, No. 399, 918–924.
See Also
ad.test.combined, ad.pval
Examples
u1 <- c(1.0066, -0.9587, 0.3462, -0.2653, -1.3872)
u2 <- c(0.1005, 0.2252, 0.4810, 0.6992, 1.9289)
u3 <- c(-0.7019, -0.4083, -0.9936, -0.5439, -0.3921)
y <- c(u1, u2, u3)
g <- as.factor(c(rep(1, 5), rep(2, 5), rep(3, 5)))
set.seed(2627)
ad.test(u1, u2, u3, method = "exact", dist = FALSE, Nsim = 1000)
# or with same seed
# ad.test(list(u1, u2, u3), method = "exact", dist = FALSE, Nsim = 1000)
# or with same seed
# ad.test(y ~ g, method = "exact", dist = FALSE, Nsim = 1000)
kSamples documentation built on Oct. 8, 2023, 1:07 a.m.
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| ad.test | R Documentation |
Anderson-Darling k-Sample Test
Description
This function uses the Anderson-Darling criterion to test
the hypothesis that k independent samples with sample sizes
n_1,\ldots, n_k arose
from a common unspecified distribution function F(x) and testing is
done conditionally given the observed tie pattern. Thus this is a permutation test.
Both versions of the AD statistic are computed.
Usage
ad.test(..., data = NULL, method = c("asymptotic", "simulated", "exact"),
dist = FALSE, Nsim = 10000)
Arguments
... |
Either several sample vectors, say
or a list of such sample vectors, or a formula y ~ g, where y contains the pooled sample values and g is a factor (of same length as y) with levels identifying the samples to which the elements of y belong. |
data |
= an optional data frame providing the variables in formula y ~ g. |
method |
=
of full enumerations. Otherwise, |
dist |
|
Nsim |
|
Details
If AD is the Anderson-Darling criterion for the k samples,
its standardized test statistic is T.AD = (AD - \mu)/\sigma, with
\mu = k-1 and
\sigma representing mean and standard deviation of AD. This statistic
is used to test the hypothesis that the samples all come
from the same but unspecified continuous distribution function F(x).
According to the reference article, two versions
of the AD test statistic are provided.
The above mean and standard deviation are strictly
valid only for version 1 in the
continuous distribution case.
NA values are removed and the user is alerted with the total NA count. It is up to the user to judge whether the removal of NA's is appropriate.
The continuity assumption can be dispensed with, if we deal with
independent random samples, or if randomization was used in allocating
subjects to samples or treatments, and if we view
the simulated or exact P-values conditionally, given the tie pattern
in the pooled samples. Of course, under such randomization any conclusions
are valid only with respect to the group of subjects that were randomly allocated
to their respective samples.
The asymptotic P-value calculation assumes distribution continuity. No adjustment
for lack thereof is known at this point. For details on the asymptotic
P-value calculation see ad.pval.
Value
A list of class kSamples with components
test.name |
|
k |
number of samples being compared |
ns |
vector of the |
N |
size of the pooled sample |
n.ties |
number of ties in the pooled samples |
sig |
standard deviations |
ad |
2 x 3 (2 x 4) matrix containing |
warning |
logical indicator, warning = TRUE when at least one
|
null.dist1 |
simulated or enumerated null distribution of version 1
of the test statistic, given as vector of all generated |
null.dist2 |
simulated or enumerated null distribution of version 2
of the test statistic, given as vector of all generated |
method |
The |
Nsim |
The number of simulations. |
warning
method = "exact" should only be used with caution.
Computation time is proportional to the number of enumerations. In most cases
dist = TRUE should not be used, i.e.,
when the returned distribution vectors null.dist1 and null.dist2
become too large for the R work space. These vectors are limited in length by 1e8.
Note
For small sample sizes and small k exact null distribution
calculations are possible (with or without ties), based on a recursively extended
version of Algorithm C (Chase's sequence) in Knuth (2011), Ch. 7.2.1.3, which allows the
enumeration of all possible splits of the pooled data into samples of
sizes of n_1, \ldots, n_k, as appropriate under treatment randomization. The
enumeration and simulation are both done in C.
Note
It has recently come to our attention that the Anderson-Darling test, originally proposed by Pettitt (1976) in the 2-sample case and generalized to k samples by Scholz and Stephens, has a close relative created by Baumgartner et al (1998) in the 2 sample case and populatized by Neuhaeuser (2012) with at least 6 papers among his cited references and generalized by Murakami (2006) to k samples.
References
Baumgartner, W., Weiss, P. and Schindler, H. (1998), A nonparametric test for the general two-sample problem, Bionetrics, 54, 1129-1135.
Knuth, D.E. (2011), The Art of Computer Programming, Volume 4A Combinatorial Algorithms Part 1, Addison-Wesley
Neuhaeuser, M. (2012), Nonparametric Statistical Tests, A Computational Approach, CRC Press.
Murakami, H. (2006), A k-sample rank test based on modified Baumgartner statistic and it power comparison, Jpn. Soc. Comp. Statist., 19, 1-13.
Murakami, H. (2012), Modified Baumgartner statistic for the two-sample and multisample problems: a numerical comparison. J. of Statistical Comput. and Simul., 82:5, 711-728.
Pettitt, A.N. (1976), A two-sample Anderson_Darling rank statistic, Biometrika, 63, 161-168.
Scholz, F. W. and Stephens, M. A. (1987), K-sample Anderson-Darling Tests, Journal of the American Statistical Association, Vol 82, No. 399, 918–924.
See Also
ad.test.combined, ad.pval
Examples
u1 <- c(1.0066, -0.9587, 0.3462, -0.2653, -1.3872)
u2 <- c(0.1005, 0.2252, 0.4810, 0.6992, 1.9289)
u3 <- c(-0.7019, -0.4083, -0.9936, -0.5439, -0.3921)
y <- c(u1, u2, u3)
g <- as.factor(c(rep(1, 5), rep(2, 5), rep(3, 5)))
set.seed(2627)
ad.test(u1, u2, u3, method = "exact", dist = FALSE, Nsim = 1000)
# or with same seed
# ad.test(list(u1, u2, u3), method = "exact", dist = FALSE, Nsim = 1000)
# or with same seed
# ad.test(y ~ g, method = "exact", dist = FALSE, Nsim = 1000)
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