coin (version 1.4-4)

MarginalHomogeneityTests: Marginal Homogeneity Tests

Description

Testing the marginal homogeneity of a repeated measurements factor in a complete block design.

Usage

# S3 method for formula
mh_test(formula, data, subset = NULL, weights = NULL, ...)
# S3 method for table
mh_test(object, ...)
# S3 method for SymmetryProblem
mh_test(object, ...)

Value

An object inheriting from class "IndependenceTest".

Arguments

formula

a formula of the form y ~ x | block where y and x are factors and block is an optional factor (which is generated automatically if omitted).

data

an optional data frame containing the variables in the model formula.

subset

an optional vector specifying a subset of observations to be used. Defaults to NULL.

weights

an optional formula of the form ~ w defining integer valued case weights for each observation. Defaults to NULL, implying equal weight for all observations. (Not yet implemented!)

object

an object inheriting from classes "table" (with identical dimnames components) or "SymmetryProblem".

...

further arguments to be passed to symmetry_test().

Details

mh_test() provides the McNemar test, the Cochran \(Q\) test, the Stuart(-Maxwell) test and the Madansky test of interchangeability. A general description of these methods is given by coin::agresti2002.

The null hypothesis of marginal homogeneity is tested. The response variable and the measurement conditions are given by y and x, respectively, and block is a factor where each level corresponds to exactly one subject with repeated measurements.

This procedure is known as the McNemar test coin::mcnemar_1947 when both y and x are binary factors, as the Cochran \(Q\) test coin::cochran_1954 when y is a binary factor and x is a factor with an arbitrary number of levels, as the Stuart(-Maxwell) test coin::stuart_1955,maxwell_1970 when y is a factor with an arbitrary number of levels and x is a binary factor, and as the Madansky test of interchangeability coin::madansky_1963, which implies marginal homogeneity, when both y and x are factors with an arbitrary number of levels.

If y and/or x are ordered factors, the default scores, 1:nlevels(y) and 1:nlevels(x), respectively, can be altered using the scores argument (see symmetry_test()); this argument can also be used to coerce nominal factors to class "ordered". If both y and x are ordered factors, a linear-by-linear association test is computed and the direction of the alternative hypothesis can be specified using the alternative argument. This extension was given by coin::birch_1965 who also discussed the situation when either the response or the measurement condition is an ordered factor; see also coin::white_1982.

The conditional null distribution of the test statistic is used to obtain \(p\)-values and an asymptotic approximation of the exact distribution is used by default (distribution = "asymptotic"). Alternatively, the distribution can be approximated via Monte Carlo resampling or computed exactly for univariate two-sample problems by setting distribution to "approximate" or "exact", respectively. See asymptotic(), approximate() and exact() for details.

References

*

Examples

## Performance of prime minister
## Agresti (2002, p. 409)
performance <- matrix(
    c(794, 150,
       86, 570),
    nrow = 2, byrow = TRUE,
    dimnames = list(
         "First" = c("Approve", "Disprove"),
        "Second" = c("Approve", "Disprove")
    )
)
performance <- as.table(performance)
diag(performance) <- 0 # speed-up: only off-diagonal elements contribute

## Asymptotic McNemar Test
mh_test(performance)

## Exact McNemar Test
mh_test(performance, distribution = "exact")


## Effectiveness of different media for the growth of diphtheria
## Cochran (1950, Tab. 2)
cases <- c(4, 2, 3, 1, 59)
n <- sum(cases)
cochran <- data.frame(
    diphtheria = factor(
        unlist(rep(list(c(1, 1, 1, 1),
                        c(1, 1, 0, 1),
                        c(0, 1, 1, 1),
                        c(0, 1, 0, 1),
                        c(0, 0, 0, 0)),
                   cases))
    ),
    media = factor(rep(LETTERS[1:4], n)),
    case =  factor(rep(seq_len(n), each = 4))
)

## Asymptotic Cochran Q test (Cochran, 1950, p. 260)
mh_test(diphtheria ~ media | case, data = cochran) # Q = 8.05

## Approximative Cochran Q test
mt <- mh_test(diphtheria ~ media | case, data = cochran,
              distribution = approximate(nresample = 10000))
pvalue(mt)             # standard p-value
midpvalue(mt)          # mid-p-value
pvalue_interval(mt)    # p-value interval
size(mt, alpha = 0.05) # test size at alpha = 0.05 using the p-value


## Opinions on Pre- and Extramarital Sex
## Agresti (2002, p. 421)
opinions <- c("Always wrong", "Almost always wrong",
              "Wrong only sometimes", "Not wrong at all")
PreExSex <- matrix(
    c(144, 33, 84, 126,
        2,  4, 14,  29,
        0,  2,  6,  25,
        0,  0,  1,   5),
    nrow = 4,
    dimnames = list(
          "Premarital Sex" = opinions,
        "Extramarital Sex" = opinions
    )
)
PreExSex <- as.table(PreExSex)

## Asymptotic Stuart test
mh_test(PreExSex)

## Asymptotic Stuart-Birch test
## Note: response as ordinal
mh_test(PreExSex, scores = list(response = 1:length(opinions)))


## Vote intention
## Madansky (1963, pp. 107-108)
vote <- array(
    c(120, 1,  8, 2,   2,  1, 2, 1,  7,
        6, 2,  1, 1, 103,  5, 1, 4,  8,
       20, 3, 31, 1,   6, 30, 2, 1, 81),
    dim = c(3, 3, 3),
    dimnames = list(
          "July" = c("Republican", "Democratic", "Uncertain"),
        "August" = c("Republican", "Democratic", "Uncertain"),
          "June" = c("Republican", "Democratic", "Uncertain")
    )
)
vote <- as.table(vote)

## Asymptotic Madansky test (Q = 70.77)
mh_test(vote)


## Cross-over study
## http://www.nesug.org/proceedings/nesug00/st/st9005.pdf
dysmenorrhea <- array(
    c(6, 2, 1,  3, 1, 0,  1, 2, 1,
      4, 3, 0, 13, 3, 0,  8, 1, 1,
      5, 2, 2, 10, 1, 0, 14, 2, 0),
    dim = c(3, 3, 3),
    dimnames =  list(
          "Placebo" = c("None", "Moderate", "Complete"),
         "Low dose" = c("None", "Moderate", "Complete"),
        "High dose" = c("None", "Moderate", "Complete")
    )
)
dysmenorrhea <- as.table(dysmenorrhea)

## Asymptotic Madansky-Birch test (Q = 53.76)
## Note: response as ordinal
mh_test(dysmenorrhea, scores = list(response = 1:3))

## Asymptotic Madansky-Birch test (Q = 47.29)
## Note: response and measurement conditions as ordinal
mh_test(dysmenorrhea, scores = list(response = 1:3,
                                    conditions = 1:3))