(Exact) Page Test for Detecting Ordered Treatment Effects in Blocked Designs

A nonparametric test for ordered alternatives in a two-way layout with one observation per cell, assessing whether a monotonic trend exists across ordered treatment conditions in a randomized complete block design.

Usage

pageTest(y, ...)

# Default S3 method
pageTest(y, groups, blocks, ...)

# S3 method for class 'formula'
pageTest(formula, data, subset, na.action = na.pass, ...)

Arguments

y: either a numeric vector of data values, or a data matrix.
...: further arguments to be passed to or from methods.
groups: a vector giving the group for the corresponding elements of y if this is a vector; ignored if y is a matrix. If not a factor object, it is coerced to one.
blocks: a vector giving the block for the corresponding elements of y if this is a vector; ignored if y is a matrix. If not a factor object, it is coerced to one.
formula: a formula of the form y ~ groups | blocks.
data: an optional data frame containing the variables in formula.
subset: an optional vector specifying a subset of observations to be used.
na.action: a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

Value

A list with class "htest" containing:

statistic: the L-statistic with names attribute “L”.
p.value: the p-value of the test.
parameter: named vector with k (groups) and n (blocks).
method: the character string "Page test for ordered alternatives" with "(exact)" or "(asymptotic)" appended.
data.name: a character string giving the names of the data.

Details

Performs a Page test for ordered alternatives using an exact algorithm by Stefan Wellek (1989) with unreplicated blocked data.

pageTest can be used for analyzing unreplicated complete block designs (i.e., there is exactly one observation in y for each combination of levels of groups and blocks) where the normality assumption may be violated.

The null hypothesis is that apart from an effect of blocks, the location parameter of y is the same in each of the groups.

The alternative hypothesis is that the location parameter increases monotonically across groups: \(H_A: \theta_1 \le \theta_2 \le \theta_3 \ldots\) (where at least one inequality is strict).

If the decreasing direction is of interest, reverse the order of the groups before calling pageTest.

The Page test for ordered alternatives is slightly more powerful than the Friedman analysis of variance by ranks.

If y is a matrix, groups and blocks are obtained from the column and row indices, respectively. NA's are not allowed in groups or blocks; if y contains NA's, the corresponding blocks are removed.

For \(k \le 15\) (number of groups), exact p-values are computed from the pre-tabulated null distribution of Wellek (1989). For \(k > 15\), a normal approximation is used.

Note

Exact p-values are based on pre-tabulated distributions by Stefan Wellek (1989), valid for \(k = 3, \ldots, 15\) groups and any number of blocks.

References

Page, E. (1963): Ordered hypotheses for multiple treatments: A significance test for linear ranks. Journal of the American Statistical Association, 58, 216–230.

Siegel, S. & Castellan, N. J. Jr. (1988): Nonparametric statistics for the behavioral sciences. Boston, MA: McGraw-Hill.

Wellek, S. (1989): Computing exact p-values in Page's nonparametric test against trend. Biometrie und Informatik in Medizin und Biologie 20, 163–170.

Examples


# Craig's data from Siegel & Castellan, p. 186
soa.mat <- matrix(c(.797,.873,.888,.923,.942,.956,
                    .794,.772,.908,.982,.946,.913,
                    .838,.801,.853,.951,.883,.837,
                    .815,.801,.747,.859,.887,.902),
                  nrow = 4, byrow = TRUE)
pageTest(soa.mat)
#> 
#> 	Page test for ordered alternatives (exact)
#> 
#> data:  soa.mat
#> L = 342, k = 6, n = 4, p-value = 0.0005661
#> 

# Duller, pg. 236
pers <- matrix(c(
  1, 72, 72, 71.5, 69, 70, 69.5, 68, 68, 67, 68,
  2, 83, 81, 81, 82, 82.5, 81, 79, 80.5, 80, 81,
  3, 95, 92, 91.5, 89, 89, 90.5, 89, 89, 88, 88,
  4, 71, 72, 71, 70.5, 70, 71, 71, 70, 69.5, 69,
  5, 79, 79, 78.5, 77, 77.5, 78, 77.5, 76, 76.5, 76,
  6, 80, 78.5, 78, 77, 77.5, 77, 76, 76, 75.5, 75.5
), nrow = 6, byrow = TRUE)
colnames(pers) <- c("person", paste("week", 1:10))

# Alternative: week10 < week9 < ... < week1
pageTest(pers[, 11:2])
#> 
#> 	Page test for ordered alternatives (exact)
#> 
#> data:  pers[, 11:2]
#> L = 2226, k = 10, n = 6, p-value = 9.037e-14
#> 

# long format and formula interface
plng <- data.frame(
  expand.grid(block = 1:9, group = c("B","C","D","A")),
  x = as.vector(
    matrix(c(3,2,1,4, 4,2,3,1, 4,1,2,3, 4,2,3,1,
             3,2,1,4, 4,1,2,3, 4,3,2,1, 3,1,2,4,
             3,1,4,2),
           nrow = 9, byrow = TRUE,
           dimnames = list(1:9, LETTERS[1:4]))[, c("B","C","D","A")]
  )
)
pageTest(x ~ group | block, data = plng)
#> 
#> 	Page test for ordered alternatives (exact)
#> 
#> data:  
#> L = 252, k = 4, n = 9, p-value = 0.0007053
#> 


# Sachs, pg. 464: L = 252
pers <- matrix(c(
  3,2,1,4,
  4,2,3,1,
  4,1,2,3,
  4,2,3,1,
  3,2,1,4,
  4,1,2,3,
  4,3,2,1,
  3,1,2,4,
  3,1,4,2), 
  nrow=9, byrow=TRUE, dimnames=list(1:9, LETTERS[1:4]))  

# Alternative: B < C < D < A
pageTest(pers[, c("B","C","D","A")])
#> 
#> 	Page test for ordered alternatives (exact)
#> 
#> data:  pers[, c("B", "C", "D", "A")]
#> L = 252, k = 4, n = 9, p-value = 0.0007053
#>