Lehmacher's Test for Locating Asymmetry in Paired Multicategory Data

A nonparametric test for marginal homogeneity in square contingency tables for dependent samples, based on a normal approximation of the cell frequency differences.

Usage

lehmacherTest(x, y = NULL, p.adjust.method = "hochberg")

# S3 method for class 'MHTest'
print(x, digits = 1L, ...)

Arguments

x

either a two-dimensional square contingency table in matrix form, or a factor object.

y

a factor object; ignored if x is a matrix.

p.adjust.method

the method used to adjust the per-category p-values for multiple comparisons, passed to p.adjust(). Default is "hochberg", as recommended by Lehmacher (1980).

digits

a non-null value for digits specifies the minimum number of significant digits to be printed. See print.default().

...

further arguments to be passed to or from other methods, ignored in this function.

Value

A list with class c("MHTest", "htest") containing the following components:

statistic

a vector with the value of the test statistic for each category.

parameter

the degrees of freedom, which is always 1.

p.value

a vector with the p-values of the individual tests.

p.value.corr

a vector with the adjusted p-values of the individual tests (see p.adjust.method).

method

a character string indicating the test performed.

data.name

a character string giving the name of the data.

Details

Performs Lehmacher's chi-squared test for marginal homogeneity in a square two-dimensional contingency table.

Unlike Bowker's test of symmetry, which tests whether \(P(i,j) = P(j,i)\) for every off-diagonal cell pair, Lehmacher's test addresses marginal homogeneity: the null hypothesis is that, for every category \(i\), the row and column marginal probabilities agree, \(P(i \cdot) = P(\cdot i)\). One test statistic is computed per category and referred to a chi-squared distribution with 1 degree of freedom; the resulting p-values are adjusted for multiple comparisons.

If x is a matrix, it is taken as a two-dimensional contingency table, and hence its entries should be nonnegative integers. Otherwise, both x and y must be vectors or factors of the same length. Incomplete cases are removed, vectors are coerced into factors, and the contingency table is computed from these.

References

Lehmacher, W. (1980) Simultaneous sign tests for marginal homogeneity of square contingency tables. Biometrical Journal, 22 (8), 795-798.

See also

mcnemar.test() for the 2x2 case

Other test.categorical: barnardTest(), bhapkarTest(), breslowDayTest(), cochranQTest(), gTest(), mantelTrendTest(), stuartMaxwellTest(), woolfTest()

Examples

x <- matrix(c(400, 40, 20, 10,
              50, 300, 60, 20,
              10, 40, 120, 5,
              5, 90, 50, 80), nrow = 4, byrow = TRUE,
            dimnames = list(LETTERS[1:4], LETTERS[1:4]))

lehmacherTest(x)
#> 
#> 	Lehmacher test for marginal homogeneity
#> 
#> data:  
#> 
#>     Chi²   p-value     p-adj      
#> A    0.2     0.667     0.667      
#> B    5.3     0.021     0.042   *  
#> C   30.4   < 0.001   < 0.001   ***
#> D   67.2   < 0.001   < 0.001   ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>