mcm.mvt {mmcm}

The maximum contrast method by using the randomized quasi-Monte Carlo method

Description

This function gives P-value for the maximum contrast statistics by using randomized quasi-Monte Carlo method from pmvt function of package mvtnorm.

Usage

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

Details

mcm.mvt performs the maximum contrast method that is detecting a true response pattern.

Y_{ij} (i=1, 2, \ldots; j=1, 2, \ldots, n_i) is an observed response for j-th individual in i-th group.

\bm{C} is coefficient matrix for the maximum contrast statistics (i \times k matrix, i: No. of groups, k: No. of pattern).

\bm{C}=(\bm{c}_1, \bm{c}_2, \ldots, \bm{c}_k)^{\rm{T}}

\bm{c}_k is coefficient vector of kth pattern.

\bm{c}_k=(c_{k1}, c_{k2}, \ldots, c_{ki})^{\rm{T}} \qquad (\textstyle \sum_i 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}},

\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,

\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}},

T_{\max}=\max(T_1, T_2, \ldots, 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 (H_1: \mu_1<\mu_2<\ldots<\mu_i,~ \mu_1=\mu_2<\ldots<\mu_i,~ \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)

t_{\max} is observed value of statistics. This function gives distribution of T_{\max} by using randomized quasi-Monte Carlo method from package mvtnorm.

Value

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 kth pattern that gives maximum contrast statistics (row number of the coefficient matrix).

error

estimated absolute error and,

msg

status messages.

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

Examples

## 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