Testing the symmetry of set of repeated measurements variables measured on arbitrary scales in a complete block design.
# S3 method for formula
symmetry_test(formula, data, subset = NULL, weights = NULL, …)
# S3 method for table
symmetry_test(object, …)
# S3 method for SymmetryProblem
symmetry_test(object, teststat = c("maximum", "quadratic", "scalar"),
distribution = c("asymptotic", "approximate",
"exact", "none"),
alternative = c("two.sided", "less", "greater"),
xtrafo = trafo, ytrafo = trafo, scores = NULL,
check = NULL, paired = FALSE, …)a formula of the form y1 + ... + yq ~ x | block where y1,
…, yq are measured on arbitrary scales (nominal, ordinal or
continuous with or without censoring), x is a factor and block
is an optional factor (which is generated automatically if omitted).
an optional data frame containing the variables in the model formula.
an optional vector specifying a subset of observations to be used. Defaults
to NULL.
an optional formula of the form ~ w defining integer valued case
weights for each observation. Defaults to NULL, implying equal
weight for all observations.
a character, the type of test statistic to be applied: either a maximum
statistic ("maximum", default), a quadratic form ("quadratic")
or a standardized scalar test statistic ("scalar").
a character, the conditional null distribution of the test statistic can be
approximated by its asymptotic distribution ("asymptotic", default)
or via Monte Carlo resampling ("approximate"). Alternatively, the
functions asymptotic or approximate can be used.
For univariate two-sample problems, "exact" or use of the function
exact computes the exact distribution. Computation of the
null distribution can be suppressed by specifying "none". It is also
possible to specify a function with one argument (an object inheriting from
"'>IndependenceTestStatistic") that returns an object of
class "'>NullDistribution".
a character, the alternative hypothesis: either "two.sided"
(default), "greater" or "less".
a function of transformations to be applied to the factor x supplied
in formula; see ‘Details’. Defaults to trafo.
a function of transformations to be applied to the variables y1,
…, yq supplied in formula; see ‘Details’. Defaults
to trafo.
a named list of scores to be attached to ordered factors; see
‘Details’. Defaults to NULL, implying equally spaced scores.
a logical, indicating that paired data have been transformed in such a way
that the (unstandardized) linear statistic is the sum of the absolute values
of the positive differences between the paired observations. Defaults to
FALSE.
further arguments to be passed to or from other methods (currently ignored).
symmetry_test provides a general symmetry test for a set of variables
measured on arbitrary scales. This function is based on the general framework
for conditional inference procedures proposed by Strasser and Weber (1999).
The salient parts of the Strasser-Weber framework are elucidated by Hothorn
et al. (2006) and a thorough description of the software implementation
is given by Hothorn et al. (2008).
The null hypothesis of symmetry is tested. The response variables and the
measurement conditions are given by y1, …, yq and x,
respectively, and block is a factor where each level corresponds to
exactly one subject with repeated measurements.
A vector of case weights, e.g., observation counts, can be supplied through
the weights argument and the type of test statistic is specified by
the teststat argument. Influence and regression functions, i.e.,
transformations of y1, …, yq and x, are specified by
the ytrafo and xtrafo arguments respectively; see
trafo for the collection of transformation functions currently
available. This allows for implementation of both novel and familiar test
statistics, e.g., the McNemar test, the Cochran
If, say, y1 and/or x are ordered factors, the default scores,
1:nlevels(y1) and 1:nlevels(x) respectively, can be altered
using the scores argument; this argument can also be used to coerce
nominal factors to class "ordered". For example, when y1 is an
ordered factor with four levels and x is a nominal factor with three
levels, scores = list(y1 = c(1, 3:5), x = c(1:2, 4)) supplies the
scores to be used. For ordered alternatives the scores must be monotonic, but
non-montonic scores are also allowed for testing against, e.g., umbrella
alternatives. The length of the score vector must be equal to the number of
factor levels.
The conditional null distribution of the test statistic is used to obtain
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.
Gerig, T. (1969). A multivariate extension of Friedman's
Hothorn, T., Hornik, K., van de Wiel, M. A. and Zeileis, A. (2006). A Lego system for conditional inference. The American Statistician 60(3), 257--263. 10.1198/000313006X118430
Hothorn, T., Hornik, K., van de Wiel, M. A. and Zeileis, A. (2008). Implementing a class of permutation tests: The coin package. Journal of Statistical Software 28(8), 1--23. 10.18637/jss.v028.i08
Puri, M. L. and Sen, P. K. (1971). Nonparametric Methods in Multivariate Analysis. New York: John Wiley & Sons.
Strasser, H. and Weber, C. (1999). On the asymptotic theory of permutation statistics. Mathematical Methods of Statistics 8(2), 220--250.
# NOT RUN {
## One-sided exact Fisher-Pitman test for paired observations
y1 <- c(1.83, 0.50, 1.62, 2.48, 1.68, 1.88, 1.55, 3.06, 1.30)
y2 <- c(0.878, 0.647, 0.598, 2.05, 1.06, 1.29, 1.06, 3.14, 1.29)
dta <- data.frame(
y = c(y1, y2),
x = gl(2, length(y1)),
block = factor(rep(seq_along(y1), 2))
)
symmetry_test(y ~ x | block, data = dta,
distribution = "exact", alternative = "greater")
## Alternatively: transform data and set 'paired = TRUE'
delta <- y1 - y2
y <- as.vector(rbind(abs(delta) * (delta >= 0), abs(delta) * (delta < 0)))
x <- factor(rep(0:1, length(delta)), labels = c("pos", "neg"))
block <- gl(length(delta), 2)
symmetry_test(y ~ x | block,
distribution = "exact", alternative = "greater",
paired = TRUE)
### Example data
### Gerig (1969, p. 1597)
gerig <- data.frame(
y1 = c( 0.547, 1.811, 2.561,
1.706, 2.509, 1.414,
-0.288, 2.524, 3.310,
1.417, 0.703, 0.961,
0.878, 0.094, 1.682,
-0.680, 2.077, 3.181,
0.056, 0.542, 2.983,
0.711, 0.269, 1.662,
-1.335, 1.545, 2.920,
1.635, 0.200, 2.065),
y2 = c(-0.575, 1.840, 2.399,
1.252, 1.574, 3.059,
-0.310, 1.553, 0.560,
0.932, 1.390, 3.083,
0.819, 0.045, 3.348,
0.497, 1.747, 1.355,
-0.285, 0.760, 2.332,
0.089, 1.076, 0.960,
-0.349, 1.471, 4.121,
0.845, 1.480, 3.391),
x = factor(rep(1:3, 10)),
b = factor(rep(1:10, each = 3))
)
### Asymptotic multivariate Friedman test
### Gerig (1969, p. 1599)
symmetry_test(y1 + y2 ~ x | b, data = gerig, teststat = "quadratic",
ytrafo = function(data)
trafo(data, numeric_trafo = rank_trafo,
block = gerig$b)) # L_n = 17.238
### Asymptotic multivariate Page test
(st <- symmetry_test(y1 + y2 ~ x | b, data = gerig,
ytrafo = function(data)
trafo(data, numeric_trafo = rank_trafo,
block = gerig$b),
scores = list(x = 1:3)))
pvalue(st, method = "step-down")
# }
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