mmcm.mvt {mmcm}

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

Description

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

Usage

mmcm.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 modified 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

mmcm.mvt performs the modified maximum contrast method that is detecting a true response pattern under the unequal sample size situation.

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 modified 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)

S_{\max} is the modified 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}},

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),

S_{k}=\frac{\bm{c}^t_k \bar{\bm{Y}}}{\sqrt{V \bm{c}^t_k \bm{c}_k}},

S_{\max}=\max(S_1, S_2, \ldots, S_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 S_{\max} under the overall null hypothesis is

P\mbox{-value}=\Pr(S_{\max}>s_{\max} \mid H_0)

s_{\max} is observed value of statistics. This function gives distribution of S_{\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 modified maximum contrast statistics (row number of the coefficient matrix).

error

estimated absolute error and,

msg

status messages.

References

Nagashima, K., Sato, Y., Hamada, C. (2011). A modified maximum contrast method for unequal sample sizes in pharmacogenomic studies Stat Appl Genet Mol Biol. 10(1): Article 41. http://dx.doi.org/10.2202/1544-6115.1560

Sato, Y., Laird, N.M., Nagashima, K., et al. (2009). A new statistical screening approach for finding pharmacokinetics-related genes in genome-wide studies. Pharmacogenomics J. 9(2): 137–146. http://www.ncbi.nlm.nih.gov/pubmed/19104505

See Also

pmvt, GenzBretz, mmcm.resamp

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 <- mmcm.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)]
y