# The maximum contrast method by using the randomized quasi-Monte Carlo method This function gives $P$-value for the maximum contrast statistics by using randomized quasi-Monte Carlo method from `pmvt` function of package `mvtnorm`. ```r mcm.mvt( x, g, contrast, alternative = c("two.sided", "less", "greater"), algorithm = GenzBretz() ) ``` ## Arguments - `x`: a numeric vector of data values - `g`: a integer vector giving the group for the corresponding elements of x - `contrast`: a numeric contrast coefficient matrix for the maximum contrast statistics - `alternative`: a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less". You can specify just the initial letter. - `algorithm`: an object of class `GenzBretz` defining the hyper parameters of this algorithm ## Returns - **statistic**: the value of the test statistic with a name describing it. - **p.value**: the p-value for the test. - **alternative**: a character string describing the alternative hypothesis. - **method**: the type of test applied. - **contrast**: a character string giving the names of the data. - **contrast.index**: a suffix of coefficient vector of the $k$th pattern that gives maximum contrast statistics (row number of the coefficient matrix). - **error**: estimated absolute error and, - **msg**: status messages. ## Details `mcm.mvt` performs the maximum contrast method that is detecting a true response pattern. c("$Y_ij (i = 1, 2, ...;\n$", "$j = 1, 2, ..., n_i)$") is an observed response for $j$-th individual in $i$-th group. $C$ is coefficient matrix for the maximum contrast statistics ($i x k$ matrix, $i$: No. of groups, $k$: No. of pattern). $$ \bm{C}=(\bm{c}_1, \bm{c}_2, \ldots, \bm{c}_k)^{\rm{T}}C = (c_1, c_2, ..., c_k)^T $$ $c_k$ is coefficient vector of $k$th pattern. $$ \bm{c}_k=(c_{k1}, c_{k2}, \ldots, c_{ki})^{\rm{T}} \qquad (\textstyle \sum_i c_{ki}=0)c_k = (c_k1, c_k2, ..., c_ki)^{\rm{T}} (sum from i of c_ki = 0) $$ $T_max$ is the maximum contrast statistic. $$ \bar{Y}_i=\frac{\sum_{j=1}^{n_i} Y_{ij}}{n_{i}},\bar{\bm{Y}}=(\bar{Y}_1, \bar{Y}_2, \ldots, \bar{Y}_i, \ldots, \bar{Y}_a)^{\rm{T}},Ybar_i = (sum from j of Y_ij) / n_i,Ybar = (Ybar_1 Ybar_2 ... Ybar_i ... Ybar_a)^T (a x 1 vector), $$ $$ \bm{D}=diag(n_1, n_2, \ldots, n_i, \ldots, n_a),V=\frac{1}{\gamma}\sum_{j=1}^{n_i}\sum_{i=1}^{a} (Y_{ij}-\bar{Y}_i)^2,D = diag(n_1, n_2, ..., n_i, ..., n_a) (a x a matrix),V = 1/gamma * sum_{j=1}^{n_i} sum_{i=1}^{a} (Y_ij-Ybar_i)^2, $$ $$ \gamma=\sum_{i=1}^{a} (n_i-1),T_{k}=\frac{\bm{c}^t_k \bar{\bm{Y}}}{\sqrt{V\bm{c}^t_k \bm{D} \bm{c}_k}},gamma = sum_{i=1}^{a} (n_i-1),T_k = c_k^t Ybar / (V c_k^t D c_k)^(1/2), $$ $$ T_{\max}=\max(T_1, T_2, \ldots, T_k).T_max = max(T_1, T_2, ..., T_k). $$ Consider testing the overall null hypothesis $H_0: \mu_1=\mu_2=\ldots=\mu_i$, versus alternative hypotheses $H_1$ for response petterns (c("$H_1: \\mu_1<\\mu_2<\\ldots<\\mu_i,~ \\mu_1=\\mu_2<\\ldots<\\mu_i,~\n$", "$\\mu_1<\\mu_2<\\ldots=\\mu_i$")). The $P$-value for the probability distribution of $T_max$ under the overall null hypothesis is $$ P\mbox{-value}=\Pr(T_{\max}>t_{\max} \mid H_0)P-value = Pr(T_max > t_max | H0) $$ $t_max$ is observed value of statistics. This function gives distribution of $T_max$ by using randomized quasi-Monte Carlo method from package `mvtnorm`. ## Examples ```r ## Example 1 ## # true response pattern: dominant model c=(1, 1, -2) set.seed(136885) x <- c( rnorm(130, mean = 1 / 6, sd = 1), rnorm( 90, mean = 1 / 6, sd = 1), rnorm( 10, mean = -2 / 6, sd = 1) ) g <- rep(1:3, c(130, 90, 10)) boxplot( x ~ g, width = c(length(g[g==1]), length(g[g==2]), length(g[g==3])), main = "Dominant model (sample data)", xlab = "Genotype", ylab = "PK parameter" ) # coefficient matrix # c_1: additive, c_2: recessive, c_3: dominant contrast <- rbind( c(-1, 0, 1), c(-2, 1, 1), c(-1, -1, 2) ) y <- mcm.mvt(x, g, contrast) y ## Example 2 ## # for dataframe # true response pattern: # pos = 1 dominant model c=( 1, 1, -2) # 2 additive model c=(-1, 0, 1) # 3 recessive model c=( 2, -1, -1) set.seed(3872435) x <- c( rnorm(130, mean = 1 / 6, sd = 1), rnorm( 90, mean = 1 / 6, sd = 1), rnorm( 10, mean = -2 / 6, sd = 1), rnorm(130, mean = -1 / 4, sd = 1), rnorm( 90, mean = 0 / 4, sd = 1), rnorm( 10, mean = 1 / 4, sd = 1), rnorm(130, mean = 2 / 6, sd = 1), rnorm( 90, mean = -1 / 6, sd = 1), rnorm( 10, mean = -1 / 6, sd = 1) ) g <- rep(rep(1:3, c(130, 90, 10)), 3) pos <- rep(c("rsXXXX", "rsYYYY", "rsZZZZ"), each=230) xx <- data.frame(pos = pos, x = x, g = g) # coefficient matrix # c_1: additive, c_2: recessive, c_3: dominant contrast <- rbind( c(-1, 0, 1), c(-2, 1, 1), c(-1, -1, 2) ) y <- by(xx, xx$pos, function(x) mmcm.mvt(x$x, x$g, contrast)) y <- do.call(rbind, y)[,c(3,7,9)] # miss-detection! y ``` ## References Yoshimura, I., Wakana, A., Hamada, C. (1997). A performance comparison of maximum contrast methods to detect dose dependency. **Drug Information J.** 31 : 423--432. ## See Also `pmvt`, `GenzBretz`, `mmcm.mvt`

mcm.mvt function