NullDistribution

Specification of the Reference Distribution

Specification of the asymptotic, approximative (Monte Carlo) and exact reference distribution.

Usage

asymptotic(maxpts = 25000, abseps = 0.001, releps = 0)
approximate(nresample = 10000L, parallel = c("no", "multicore", "snow"),
            ncpus = 1L, cl = NULL, B)
exact(algorithm = c("auto", "shift", "split-up"), fact = NULL)

Details

asymptotic, approximate and exact can be supplied to the distribution argument of, e.g., independence_test to provide control of the specification of the asymptotic, approximative (Monte Carlo) and exact reference distribution respectively.

The asymptotic reference distribution is computed using a randomised quasi-Monte Carlo method (Genz and Bretz, 2009) and is applicable to arbitrary covariance structures with dimensions up to 1000. See GenzBretz in package mvtnorm for details on maxpts, abseps and releps.

The approximative (Monte Carlo) reference distribution is obtained by a conditional Monte Carlo procedure, i.e., by computing the test statistic for nresample random samples from all admissible permutations of the response \(\bf{Y}\) within each block (Hothorn et al., 2008). By default, the distribution is computed using serial operation (parallel = "no"). The use of parallel operation is specified by setting parallel to either "multicore" (not available for MS Windows) or "snow". In the latter case, if cl = NULL (default) a cluster with ncpus processes is created on the local machine unless a default cluster has been registered (see setDefaultCluster in package parallel) in which case that gets used instead. Alternatively, the use of an optional parallel or snow cluster can be specified by cl. See ‘Examples’ and package parallel for details on parallel operation.

The exact reference distribution, currently available for univariate two-sample problems only, is computed using either the shift algorithm (Streitberg and R<U+00F6>hmel, 1984, 1986, 1987) or the split-up algorithm (van de Wiel, 2001). The shift algorithm handles blocks pertaining to, e.g., pre- and post-stratification, but can only be used with positive integer-valued scores \(h(\bf{Y})\). The split-up algorithm can be used with non-integer scores, but does not handle blocks. By default, an automatic choice is made (algorithm = "auto") but the shift and split-up algorithms can be selected by setting algorithm to either "shift" or "split-up" respectively.

Note

Starting with coin version 1.1-0, the default for algorithm is "auto", having identical behaviour to "shift" in previous versions. In earlier versions of the package, algorithm = "shift" silently switched to the split-up algorithm if non-integer scores were detected, whereas the current version exits with a warning.

In versions of coin prior to 1.3-0, the number of Monte Carlo replicates in approximate() was specified using the now deprecated B argument. This will be made defunct and removed in a future release. It has been replaced by the nresample argument (for conformity with the libcoin, party and partykit packages).

References

Genz, A. and Bretz, F. (2009). Computation of Multivariate Normal and t Probabilities. Heidelberg: Springer-Verlag.

Hothorn, T., Hornik, K., van de Wiel, M. A. and Zeileis, A. (2008). Implementing a class of permutation tests: The coin package. Journal of Statistical Software 28(8), 1--23. 10.18637/jss.v028.i08

Streitberg, B. and R<U+00F6>hmel, J. (1984). Exact nonparametrics in APL. APL Quote Quad 14(4), 313--325. 10.1145/384283.801115

Streitberg, B. and R<U+00F6>hmel, J. (1986). Exact distributions for permutations and rank tests: an introduction to some recently published algorithms. Statistical Software Newsletter 12(1), 10--17.

Streitberg, B. and R<U+00F6>hmel, J. (1987). Exakte verteilungen f<U+00FC>r rang- und randomisierungstests im allgemeinen c-stichprobenfall. EDV in Medizin und Biologie 18(1), 12--19.

van de Wiel, M. A. (2001). The split-up algorithm: a fast symbolic method for computing p-values of distribution-free statistics. Computational Statistics 16(4), 519--538. 10.1007/s180-001-8328-6

Aliases

• asymptotic
• approximate
• exact

Examples

# NOT RUN {
## Approximative (Monte Carlo) Cochran-Mantel-Haenszel test

## Serial operation
set.seed(123)
cmh_test(disease ~ smoking | gender, data = alzheimer,
         distribution = approximate(nresample = 100000))

# }
# NOT RUN {
## Multicore with 8 processes (not for MS Windows)
set.seed(123, kind = "L'Ecuyer-CMRG")
cmh_test(disease ~ smoking | gender, data = alzheimer,
         distribution = approximate(nresample = 100000,
                                    parallel = "multicore", ncpus = 8))

## Automatic PSOCK cluster with 4 processes
set.seed(123, kind = "L'Ecuyer-CMRG")
cmh_test(disease ~ smoking | gender, data = alzheimer,
         distribution = approximate(nresample = 100000,
                                    parallel = "snow", ncpus = 4))

## Registered FORK cluster with 12 processes (not for MS Windows)
fork12 <- parallel::makeCluster(12, "FORK") # set-up cluster
parallel::setDefaultCluster(fork12) # register default cluster
set.seed(123, kind = "L'Ecuyer-CMRG")
cmh_test(disease ~ smoking | gender, data = alzheimer,
         distribution = approximate(nresample = 100000,
                                    parallel = "snow"))
parallel::stopCluster(fork12) # clean-up

## User-specified PSOCK cluster with 8 processes
psock8 <- parallel::makeCluster(8, "PSOCK") # set-up cluster
set.seed(123, kind = "L'Ecuyer-CMRG")
cmh_test(disease ~ smoking | gender, data = alzheimer,
         distribution = approximate(nresample = 100000,
                                    parallel = "snow", cl = psock8))
parallel::stopCluster(psock8) # clean-up
# }