# Normal-approximation-based test for one-way MANOVA problem proposed by Schott (2007) Schott, J. R. (2007)'s test for one-way MANOVA problem for high-dimensional data with assuming that underlying covariance matrices are the same. ```r S2007.ks.NABT(Y, n, p) ``` ## Arguments - `Y`: A list of $k$ data matrices. The $i$th element represents the data matrix ($n_i \times p$) from the $i$th population with each row representing a $p$-dimensional observation. - `n`: A vector of $k$ sample sizes. The $i$th element represents the sample size of group $i$, $n_i$. - `p`: The dimension of data. ## Returns A list of class `"NRtest"` containing the results of the hypothesis test. See the help file for `NRtest.object` for details. ## Details Suppose we have the following $k$ independent high-dimensional samples: $$ \boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma},i=1,\ldots,k. $$ It is of interest to test the following one-way MANOVA problem: $$ H_0: \boldsymbol{\mu}_1=\cdots=\boldsymbol{\mu}_k, \quad \text { vs. }\; H_1: H_0 \;\operatorname{is \; not\; ture}. $$ Schott (2007) proposed the following test statistic: $$ T_{S}=[\operatorname{tr}(\boldsymbol{H})/h-\operatorname{tr}(\boldsymbol{E})/e]/\sqrt{N-1}, $$ where $\boldsymbol{H}=\sum_{i=1}^kn_i(\bar{\boldsymbol{y}}_i-\bar{\boldsymbol{y}})(\bar{\boldsymbol{y}}_i-\bar{\boldsymbol{y}})^\top$, $\boldsymbol{E}=\sum_{i=1}^k\sum_{j=1}^{n_i}(\boldsymbol{y}_{ij}-\bar{\boldsymbol{y}}_{i})(\boldsymbol{y}_{ij}-\bar{\boldsymbol{y}}_{i})^\top$, $h=k-1$, and $e=N-k$, with $N=n_1+\cdots+n_k$. They showed that under the null hypothesis, $T_{S}$ is asymptotically normally distributed. ## Examples ```r library("HDNRA") data("corneal") dim(corneal) group1 <- as.matrix(corneal[1:43, ]) ## normal group group2 <- as.matrix(corneal[44:57, ]) ## unilateral suspect group group3 <- as.matrix(corneal[58:78, ]) ## suspect map group group4 <- as.matrix(corneal[79:150, ]) ## clinical keratoconus group p <- dim(corneal)[2] Y <- list() Y[[1]] <- group1 Y[[2]] <- group2 Y[[3]] <- group3 Y[[4]] <- group4 n <- c(nrow(Y[[1]]),nrow(Y[[2]]),nrow(Y[[3]]),nrow(Y[[4]])) S2007.ks.NABT(Y, n, p) ``` ## References \insertRef schott2007someHDNRA

S2007.ks.NABT function