mrrTest function

Madhava Rao-Raghunath Test for Testing Treatment vs. Control

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 $A: F(x,y) != F(y,x)$.

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").

Returns

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.

Details

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.

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.

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)

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. tools:::Rd_expr_doi("10.18642/jsata_7100121717")