# MarginalHomogeneityTests

##### Marginal Homogeneity Tests

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

- Keywords
- htest

##### Usage

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

##### 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`

.- 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 Agresti (2002).

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 (McNemar, 1947) when both `y`

and `x`

are binary factors, as the Cochran \(Q\) test (Cochran, 1950)
when `y`

is a binary factor and `x`

is a factor with an arbitrary
number of levels, as the Stuart(-Maxwell) test (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 (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 Birch
(1965) who also discussed the situation when either the response or the
measurement condition is an ordered factor; see also White, Landis and Cooper
(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.

##### Value

##### Note

This function is currently computationally inefficient for data with a large number of pairs or sets.

##### References

Agresti, A. (2002). *Categorical Data Analysis*, Second Edition.
Hoboken, New Jersey: John Wiley & Sons.

Birch, M. W. (1965). The detection of partial association, II: The general
case. *Journal of the Royal Statistical Society* B **27**(1),
111--124.

Cochran, W. G. (1950). The comparison of percentages in matched
samples. *Biometrika* **37**(3/4), 256--266.
10.1093/biomet/37.3-4.256

Madansky, A. (1963). Tests of homogeneity for correlated samples.
*Journal of the American Statistical Association* **58**(301),
97--119. 10.1080/01621459.1963.10500835

Maxwell, A. E. (1970). Comparing the classification of subjects by two
independent judges. *British Journal of Psychiatry* **116**(535),
651--655. 10.1192/bjp.116.535.651

McNemar, Q. (1947). Note on the sampling error of the difference between
correlated proportions or percentages. *Psychometrika* **12**(2),
153--157. 10.1007/BF02295996

Stuart, A. (1955). A test for homogeneity of the marginal distributions in a
two-way classification. *Biometrika* **42**(3/4), 412--416.
10.1093/biomet/42.3-4.412

White, A. A., Landis, J. R. and Cooper, M. M. (1982). A note on the
equivalence of several marginal homogeneity test criteria for categorical
data. *International Statistical Review* **50**(1), 27--34.
10.2307/1402457

##### Examples

```
# NOT RUN {
## 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))
# }
```

*Documentation reproduced from package coin, version 1.3-1, License: GPL-2*