Stuart-Maxwell Test for Testing Marginal Homogeneity in Paired Multicategory Data

A nonparametric test for marginal homogeneity in square contingency tables for dependent samples, generalizing the McNemar test to more than two categories.

Usage

stuartMaxwellTest(x, y = NULL)

Arguments

x

either a 2-way \(k \times k\) contingency table in matrix form, or a factor.

y

a factor or vector of the same length as x; ignored if x is a matrix. Its levels are combined with those of x.

Value

A list with class "htest" containing the following components:

statistic

the value of the test statistic.

parameter

the degrees of freedom.

p.value

the p-value of the test.

method

a character string indicating what type of test was performed.

data.name

a character string giving the name of the data.

Details

This function computes the marginal homogeneity test for a \(k \times k\) matrix of assignments of objects to k categories or two vectors x, y of category scores for n data objects by two raters. The statistic is distributed as \(\chi^2\) with k-1 degrees of freedom.
It can be viewed as an extension of the McNemar test to \(k \times k\) table.

The null is that the probabilities of being classified into cells [i, j] and [j, i] are the same.

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. Since rows and columns must represent the same categories, both are given the union of the categories observed in x and y (those of x first): a category observed in only one of the two classifications becomes a row or column of zeros instead of making the table non-square. Levels observed in neither are dropped.

If there is perfect agreement for any category k, that category must be omitted in order to invert matrix S.

If for any category k, all frequencies in row k and column k are 0, except possibly for the main diagonal element (e.g., for perfect agreement for category k, in such cases also the corresponding row and column marginal frequencies would be equal), then the category is not included in the test and should be ignored, say the Stuart-Maxwell test is performed with respect to the remaining categories only. The degree of freedom df in this case can still be considered k - 1, where k is the number of original categories; this treats omitted categories as if they were included but contributed 0 to the value of \(\chi^2\) - a reasonable view since such categories have equal row and column marginals. (See: Stuart (1955).

Note

Based on code from Jim Lemon, adapted to conform to package standards.

References

Stuart, A (1955) A test for homogeneity of the marginal distributions in a two-way classification. Biometrika, 42, 412-416.

Maxwell, A.E. (1970) Comparing the classification of subjects by two independent judges. British Journal of Psychiatry, 116, 651-655.

Agresti, A. (2002) Categorical Data Analysis. John Wiley & Sons, pp 86 ff.

UCLA OARC (Stata textbook examples), An Introduction to Categorical Analysis, Chapter 9, Section 9.3.2 (Coffee Market Share Example): https://stats.oarc.ucla.edu/stata/examples/icda/an-introduction-to-categorical-analysis-by-alan-agrestichapter-9-models-for-matched-pairs/.

See also

mcnemar.test(), chisq.test()

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

Examples

# Hypothetical 3 x 3 contingency table
hyp <- as.table(matrix(c(20,3,0,10,30,5,5,15,40), nrow=3))
stuartMaxwellTest(hyp)
#> 
#> 	Stuart-Maxwell test for marginal homogeneity
#> 
#> data:  hyp
#> chi-squared = 13.765, df = 2, p-value = 0.001026
#> 

# same as defined with two vectors
d.hyp <- expand.grid(c("A","B","C"), c("A","B","C"))[
                     rep(1:9, times = hyp), ]
row.names(d.hyp) <- NULL

stuartMaxwellTest(x=d.hyp[,1], y=d.hyp[,2])
#> 
#> 	Stuart-Maxwell test for marginal homogeneity
#> 
#> data:  d.hyp[, 1] and d.hyp[, 2]
#> chi-squared = 13.765, df = 2, p-value = 0.001026
#> 


mc <- as.table(matrix(c(
         732, 1524, 1575, 1577, 1602, 837, 1554, 1437, 
         1672, 1600, 841, 1363, 1385, 1484, 1524, 791), nrow=4))

stuartMaxwellTest(mc)
#> 
#> 	Stuart-Maxwell test for marginal homogeneity
#> 
#> data:  mc
#> chi-squared = 0.089722, df = 3, p-value = 0.993
#> 


# UCLA OARC: Coffee Market Share Example (Agresti, Chapter 9)
# Stuart-Maxwell chi-squared = 12.29, df = 4, p = 0.0153 (rounded)
coffee <- matrix(c(
  93, 17, 44,  7, 10,
   9, 46, 11,  0,  9,
  17, 11,155,  9, 12,
   6,  4,  9, 15,  2,
  10,  4, 12,  2, 27
), nrow = 5, byrow = TRUE)
stuartMaxwellTest(coffee)
#> 
#> 	Stuart-Maxwell test for marginal homogeneity
#> 
#> data:  coffee
#> chi-squared = 12.291, df = 4, p-value = 0.01531
#>