SkiMack: The Skillings-Mack test statistic

Description

The Skillings-Mack test statistic is a generalization of the statistic used in Friedman's ANOVA method and in Durbin's rank test. This nonparametric statistical test is useful for the data obtained from block designs with missing observations occurring randomly. The results from the SkiMack function are the Skillings-Mack statistic, p-values based on the chi-squared distribution and Monte Carlo method. The Monte Carlo method is recommended for approximating p-values when there are many ties and/or small designs with missing values are conducted.

Usage

SkiMack(y, groups=NULL, blocks=NULL, simulate.p.value = FALSE, B = 10000)

Arguments

Either a numeric vector of data values, or a data matrix. If a matrix is used, columns and rows are correspondent to blocks and treatments (groups), respectively.

groups

A vector giving the group (treatment) for the corresponding elements of y if this is a vector; ignored if y is a matrix. Either a numeric or character vector is accepted.

blocks

A vector giving the block for the corresponding elements of y if this is a vector; ignored if y is a matrix. Either a numeric or character vector is accepted.

simulate.p.value

If TRUE, an estimated p-value based on the Monte Carlo approach is calculated. The default is FALSE.

If simulate.p.value = TRUE, the default number of replications is 10,000.

Value

$Nblocksthe number of blocks in a design
$Ntreatmentsthe number of groups (treatments) in a design
$rawdataa matrix containing the data that was read from y
$rankdataa matrix containing ranks of the data
$varCovarMatrixthe variance-covariance matrix for adjusted treatment sums
$adjustedSumthe adjusted treatment sum,Aj, for the jth treatment

Details

This function implements the Skilling-Mack test statistic for data obtained from block designs with missing observations. The skillings.mack function can also applied to balanced and partially balanced incomplete block designs. It reads a dataset from a vector or a matrix and returns a p-value based on the chi-squared distribution. If simulate.p.value = TRUE, an estimated p-value based on Monte Carlo will be provided. If there are ties in a particular block, the average rank for those will be generated in simulating data under the null hypothesis.

References

Skillings, J.H., Mack, G.A.: On the Use of a Friedman-Type Statistic in Balanced and Unbalanced Block Designs. Technometrics 23(2), 171-177 (1981)

Examples

Run this code

## Skilling and Mack (1981), Table 1 page 173
## Comparison of four methods of assembling a product
B <- rep(c(1,2,3,4,5,6,7,8,9),rep(4,9))
G <- rep(c('A','B','C','D'),9)
y <- c(3.2,4.1,3.8,4.2,    3.1,3.9,3.4,4.0,	 4.3,3.5,4.6,4.8, 
       3.5,3.6,3.9,4.0,   3.6,4.2,3.7,3.9,   4.5,4.7,3.7, NA,
	     NA ,4.2,3.4,NA ,   4.3,4.6,4.4,4.9,   3.5, NA,3.7, 3.9)
SkiMack(y,groups = G,blocks = B)

## y is a matrix
maty <- matrix(
   c(3.2,4.1,3.8,4.2,		3.1,3.9,3.4,4.0,	 4.3,3.5,4.6,4.8, 
     3.5,3.6,3.9,4.0,   3.6,4.2,3.7,3.9,   4.5,4.7,3.7, NA,
	   NA ,4.2,3.4,NA ,   4.3,4.6,4.4,4.9,   3.5, NA,3.7, 3.9),
    ncol=9,byrow=FALSE)
SkiMack(maty, simulate.p.value = TRUE, B = 1000)

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