mrrTest {PMCMRplus}

Madhava Rao-Raghunath Test for Testing Treatment vs. Control

Description

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).

Usage

mrrTest(x, ...)

## Default S3 method:
mrrTest(x, y = NULL, m = NULL, ...)

## S3 method for class 'formula'
mrrTest(formula, data, subset, na.action, ...)

Arguments

x

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.

y

an optional numeric vector of data values: as with x non-finite values will be omitted.

m

numeric, optional integer number, whereas n = k m needs to be full filled.

formula

a formula of the form response ~ group where response gives the data values and group a vector or factor of the corresponding groups.

data

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).

subset

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

na.action

a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

Details

Let X_i and Y_i, ~ i \le n denote continuous variables that were observed on the same ith 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 ith 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.

Value

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

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated quantile of the test statistic.

p.value

the p-value for the test.

parameter

the parameters of the test statistic, if any.

alternative

a character string describing the alternative hypothesis.

estimates

the estimates, if any.

null.value

the estimate under the null hypothesis, if any.

Note

The function returns an error code if a value for m is provided that does not lead to an integer of the ratio k = n /m.

The function also returns an error code, if a tabulated value for given n, m and calculated M can not be found in the look-up table.

References

Madhava Rao, K.S., Ragunath, M. (2016) A Simple Nonparametric Test for Testing Treatment Versus Control. J Stat Adv Theory Appl 16, 133–162. doi:10.18642/jsata_7100121717

Examples

## 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)