# Perform Monte Carlo hypothesis tests. Performs Monte Carlo hypothesis test with either a fixed number of resamples or a sequential stopping boundary on the number of resamples. Outputs p-value and confidence interval for p-value. The program is very general and different bootstrap or permutation tests may be done by defining the statistic function and the resample function. ```r MCtest(x, statistic, resample, bound, extreme = "geq", seed = 1234325) MCtest.fixed(x, statistic, resample, Nmax, extreme = "geq", conf.level=.99, seed = 1234325) ``` ## Arguments - `x`: data object - `statistic`: function that inputs data object, x, and outputs a scalar numeric - `resample`: function that inputs data object,x, and outputs a "resampling" of x, an object of the same type as x - `Nmax`: the number of resamples for the fixed boundary - `bound`: a object of class MCbound, can be created by `MCbound` see that help - `extreme`: character value either "geq" or "leq". Defines which Monte Carlo outputs from statistic (T1,T2,...) are denoted extreme with respect to the original output (T0) ; extreme values are either all Ti >= T0 ("geq") or all Ti <= T0 ("leq"). - `conf.level`: confidence level for interval about p-value - `seed`: a numeric value used in set.seed ## Details Performs Monte Carlo hypothesis test. `MCtest` allows any types of Monte Carlo boundary created by `MCbound`, while `MCtest.fixed` only performs Monte Carlo tests using fixed boundaries. The only advantage of `MCtest.fixed` is that one can do the test without first creating the fixed stopping boundary through `MCbound`. The default boundary is described in `MCbound.precalc1`. Let T0=statistic(x) and let T1,T2,... be statistic(resample(x)). Then in the simplest type of boundary, the "fixed" type, then the resulting p-value is p=(S+1)/(N+1), where S=(\# Ti >= T0) (if extreme is "geq") or S=(\# Ti <= T0) (if extreme is "leq"). The confidence interval on the p-value is calculated by an exact method. There are several different types of MC designs that may be used for the MCtest. These are described in the `MCbound` help. ## Returns A LIST of class "MCtest",with elements: - **Ti**: an N vector of the outputs of the resampled statistic - **type**: type of boundary: "fixed", "tsprt", "Bvalue" or "BC" - **parms**: vector of parameters that define boundary specified by type - **T0**: the value of the test statistic applied to the original data - **p.value**: p.value - **pvalue.ci**: confidence interval about the p-value ## References Fay, M.P., Kim, H-J. and Hachey, M. (2007). Using truncated sequential probability ratio test boundaries for Monte Carlo implementation of hypothesis tests. Journal of Computational and Graphical Statistics. 16(4):946-967. ## Author(s) M.P. Fay ## See Also `MCbound`,`MCbound.precalc1` ## Examples ```r x<-data.frame(y=1:100,z=rnorm(100),group=c(rep(1,50),rep(2,50))) stat<-function(x){ cor(x[,1],x[,2]) } ### nonparametric bootstrap test on correlation between y and z ### low p-value means that such a large correlation unlikely due to chance resamp<-function(x){ n<-dim(x)[[1]] ; x[sample(1:n,replace=TRUE),] } out<-MCtest(x,stat,resamp,extreme="geq") out$p.value out$p.value.ci ### permutation test, permuting y only within group resamp<-function(x){ ug<-unique(x[,"group"]) y<- x[,"y"] for (i in 1:length(ug)){ pick.strata<- x[,"group"]==ug[i] y[pick.strata]<-sample(y[pick.strata],replace=FALSE) } x[,1]<-y x } out<-MCtest.fixed(x,stat,resamp,N=199) out$p.value out$p.value.ci ```

MCtest function