This function combines several independent Anderson-Darling \(k\)-sample tests into one overall test of the hypothesis that the independent samples within each block come from a common unspecified distribution, while the common distributions may vary from block to block. Both versions of the Anderson-Darling test statistic are provided.
ad.test.combined(..., data = NULL,
method = c("asymptotic", "simulated", "exact"),
dist = FALSE, Nsim = 10000)A list of class kSamples with components
\(=\) "Anderson-Darling"
number of blocks of samples being compared
list of M vectors, each vector giving the sample sizes for
each block of samples being compared
\(= (N_1,\ldots,N_M)\)
vector giving the number of ties in each the M
comparison blocks
list of M matrices giving the ad results
for ad.test applied to the samples in each of
the M blocks
vector of means of the \(AD\) statistic for the M blocks
vector of standard deviations of the \(AD\) statistic for the M blocks
2 x 3 (2 x 4) matrix containing \(AD_{comb}, T_{comb}\), asymptotic \(P\)-value, (simulated or exact \(P\)-value), for each version of the combined test statistic, version 1 in row 1 and version 2 in row 2
mean of \(AD_{comb}\)
standard deviation of \(AD_{comb}\)
logical indicator, warning = TRUE when at least one
\(n_{ij} < 5\)
simulated or enumerated null distribution of version 1 of \(AD_{comb}\)
simulated or enumerated null distribution of version 2 of \(AD_{comb}\)
the method used.
the number of simulations used for each block of samples.
Either a sequence of several lists, say \(L_1, \ldots, L_M\) (\(M > 1\)) where list \(L_i\) contains \(k_i > 1\) sample vectors of respective sizes \(n_{i1}, \ldots, n_{ik_i}\), where \(n_{ij} > 4\) is recommended for reasonable asymptotic \(P\)-value calculation. \(N_i=n_{i1}+\ldots+n_{ik_i}\) is the pooled sample size for block \(i\),
or a list of such lists,
or a formula, like y ~ g | b, where y is a numeric response vector, g is a factor with levels indicating different treatments and b is a factor indicating different blocks; y, g, b are or equal length. y is split separately for each block level into separate samples according to the g levels. The same g level may occur in different blocks. The variable names may correspond to variables in an optionally supplied data frame via the data = argument,
= an optional data frame providing the variables in formula input
= c("asymptotic","simulated","exact"), where
"asymptotic" uses only an asymptotic \(P\)-value approximation, reasonable
for P in [0.00001, .99999], linearly extrapolated via
\(\log(P/(1-P))\) outside
that range. See ad.pval for details.
The adequacy of the asymptotic \(P\)-value calculation may be checked using
pp.kSamples.
"simulated" uses simulation to get Nsim simulated \(AD\) statistics
for each block of samples, adding them across blocks component wise to get Nsim
combined values. These are compared with the observed combined value to obtain the
estimated \(P\)-value.
"exact" uses full enumeration of the test statistic values
for all sample splits of the pooled samples within each block.
The test statistic vectors for the first 2 blocks are added
(each component against each component, as in the R outer(x,y, "+") command)
to get the convolution enumeration for the combined test statistic. The resulting
vector is convoluted against the next block vector in the same fashion, and so on.
It is possible only for small problems, and is attempted only when Nsim
is at least the (conservatively maximal) length
$$\frac{N_1!}{n_{11}!\ldots n_{1k_1}!}\times\ldots\times
\frac{N_M!}{n_{M1}!\ldots n_{Mk_M}!}$$
of the final distribution vector. Otherwise, it reverts to the
simulation method using the provided Nsim.
FALSE (default) or TRUE. If TRUE, the
simulated or fully enumerated convolution vectors
null.dist1 and null.dist2 are returned for the respective
test statistic versions. Otherwise, NULL is returned for each.
= 10000 (default), number of simulation splits to use within
each block of samples. It is only used when method = "simulated"
or when method = "exact" reverts to method = "simulated",
as previously explained. Simulations are independent across blocks,
using Nsim for each block. Nsim is limited by 1e7.
If \(AD_i\) is the Anderson-Darling criterion for the i-th block of \(k_i\) samples, its standardized test statistic is \(T_i = (AD_i - \mu_i)/\sigma_i\), with \(\mu_i\) and \(\sigma_i\) representing mean and standard deviation of \(AD_i\). This statistic is used to test the hypothesis that the samples in the i-th block all come from the same but unspecified continuous distribution function \(F_i(x)\).
The combined Anderson-Darling criterion is \(AD_{comb}=AD_1 + \ldots + AD_M\) and \(T_{comb} = \) \((AD_{comb} - \mu_c)/\sigma_c\) is the standardized form, where \(\mu_c=\mu_1+\ldots+\mu_M\) and \(\sigma_c = \sqrt{\sigma_1^2 +\ldots+\sigma_M^2}\) represent the mean and standard deviation of \(AD_{comb}\). The statistic \(T_{comb}\) is used to simultaneously test whether the samples in each block come from the same continuous distribution function \(F_i(x), i=1,\ldots,M\). The unspecified common distribution function \(F_i(x)\) may change from block to block. According to the reference article, two versions of the test statistic and its corresponding combinations are provided.
The \(k_i\) for each block of \(k_i\) independent samples may change from block to block.
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, independently from block to block, and if we view the simulated or exact \(P\)-values conditionally, given the tie patterns within each block. Of course, under such randomization any conclusions are valid only with respect to the blocks of subjects that were randomly allocated. The asymptotic \(P\)-value calculation assumes distribution continuity. No adjustment for lack thereof is known at this point. The same comment holds for the means and standard deviations of respective statistics.
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.
ad.test, ad.pval
## Create two lists of sample vectors.
x1 <- list( c(1, 3, 2, 5, 7), c(2, 8, 1, 6, 9, 4), c(12, 5, 7, 9, 11) )
x2 <- list( c(51, 43, 31, 53, 21, 75), c(23, 45, 61, 17, 60) )
# and a corresponding data frame datx1x2
x1x2 <- c(unlist(x1),unlist(x2))
gx1x2 <- as.factor(c(rep(1,5),rep(2,6),rep(3,5),rep(1,6),rep(2,5)))
bx1x2 <- as.factor(c(rep(1,16),rep(2,11)))
datx1x2 <- data.frame(A = x1x2, G = gx1x2, B = bx1x2)
## Run ad.test.combined.
set.seed(2627)
ad.test.combined(x1, x2, method = "simulated", Nsim = 1000)
# or with same seed
# ad.test.combined(list(x1, x2), method = "simulated", Nsim = 1000)
# ad.test.combined(A~G|B,data=datx1x2,method="simulated",Nsim=1000)
Run the code above in your browser using DataLab