The function has implemented the nonparametric test of Madhava Rao and Raghunath (2016) for testing paired two-samples for symmetry. The null hypothesis \(H: F(x,y) = F(y,x)\) is tested against the alternative \(A: F(x,y) \ne F(y,x)\).
mrrTest(x, ...)# S3 method for default
mrrTest(x, y = NULL, m = NULL, ...)
# S3 method for formula
mrrTest(formula, data, subset, na.action, ...)
numeric vector of data values. Non-finite (e.g., infinite or missing) values will be omitted.
further arguments to be passed to or from methods.
an optional numeric vector of data values: as with x non-finite values will be omitted.
numeric, optional integer number, whereas \(n = k m\) needs to be full filled.
a formula of the form response ~ group where
response gives the data values and group a vector or
factor of the corresponding groups.
an optional matrix or data frame (or similar: see
model.frame) containing the variables in the
formula formula. By default the variables are taken from
environment(formula).
an optional vector specifying a subset of observations to be used.
a function which indicates what should happen when
the data contain NAs. Defaults to getOption("na.action").
A list with class "htest" containing the following components:
a character string indicating what type of test was performed.
a character string giving the name(s) of the data.
the estimated quantile of the test statistic.
the p-value for the test.
the parameters of the test statistic, if any.
a character string describing the alternative hypothesis.
the estimates, if any.
the estimate under the null hypothesis, if any.
Let \(X_i\) and \(Y_i, ~ i \le n\) denote continuous variables that were observed on the same \(i\)th test item (e.g. patient) with \(i = 1, \ldots n\). Let $$ U_i = X_i + Y_i \qquad V_i = X_i - Y_i $$
Let \(U_{(i)}\) be the \(i\)th order statistic, \(U_{(1)} \le U_{(2)} \le \ldots U_{(n)}\) and \(k\) the number of clusters, with the condition:
$$ n = k ~ m. $$
Further, let the divider denote \(d_0 = -\infty\), \(d_k = \infty\), and else $$ d_j = \frac{ U_{(jm)} + U_{(jm+1)} }{2}, ~ 1 \le j \le k -1 $$
The two counts are $$ n_j^{+} = \left\{ \begin{array}{lr} 1 & \mathrm{if}~ d_{j-1} < u_i < d_j, v_i > 0 \\ 0 & \end{array} \right. $$
and $$ n_j^{-} = \left\{ \begin{array}{lr} 1 & \mathrm{if}~ d_{j-1} < u_i < d_j, v_i \le 0 \\ 0 & \end{array} \right. $$
The test statistic is $$ M = \sum_{j = 1}^k \frac{\left(n_j^{+} - n_j^{-}\right)^2} {m} $$
The exact p-values for \(5 \le n \le 30\) are taken from an internal look-up table. The exact p-values were taken from Table 7, Appendix B of Madhava Rao and Raghunath (2016).
If m = NULL the function uses \(n = m\) for
all prime numbers, otherwise it tries to find an value for
m in such a way, that for \(k = n / m\) all variables
are integer.
Madhava Rao, K.S., Ragunath, M. (2016) A Simple Nonparametric Test for Testing Treatment Versus Control. J Stat Adv Theory Appl 16, 133<U+2013>162. 10.18642/jsata_7100121717
# NOT RUN {
## Madhava Rao and Raghunath (2016), p. 151
## Inulin clearance of living donors
## and recipients of their kidneys
x <- c(61.4, 63.3, 63.7, 80.0, 77.3, 84.0, 105.0)
y <- c(70.8, 89.2, 65.8, 67.1, 87.3, 85.1, 88.1)
mrrTest(x, y)
## formula method
## Student's Sleep Data
mrrTest(extra ~ group, data = sleep)
# }
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