Moses Test of Extreme Reactions for Comparing Extreme-Value Behaviour Between Two Groups

Performs the Moses test of extreme reactions, a distribution-free nonparametric test for the difference in extremity of scores between two independent groups. Scores from both groups are pooled and converted to ranks. The test statistic is the span of the control group's ranks (range + 1). An exact one-tailed probability is computed for the raw span, then recomputed after trimming h extreme ranks from each end of the control group.

Usage

mosesTest(x, ...)

# S3 method for class 'formula'
mosesTest(formula, data, subset, na.action = na.pass, ...)

# Default S3 method
mosesTest(
  x,
  y,
  extreme = NULL,
  ties.method = c("first", "average", "random", "max", "min"),
  ...
)

# S3 method for class 'mosesTestResult'
print(x, digits = getOption("digits"), ...)

Arguments

x: Numeric vector of observations from the control group. Non-finite values are removed.
...: further arguments passed to or from methods
formula: A formula of the form lhs ~ rhs, where lhs gives the observations and rhs the grouping variable with exactly two levels.
data: Optional data frame containing the variables in formula.
subset: Optional vector specifying a subset of observations.
na.action: Function specifying how missing values are handled.
y: Numeric vector of observations from the experiment group. Non-finite values are removed.
extreme: Non-negative integer $h$. Number of extreme ranks trimmed from each end of the control group before recomputing the span. If NULL, defaults to max(floor(0.05 * length(x)), 1).
ties.method: Character string passed to rank(). Default "first" preserves integer-valued ranks and exact combinatorial validity.
digits: number of significant digits to display

Value

An object of class "mosesTestResult" inheriting from "htest" with components:

statistic: Named vector containing sRaw and sTrimmed.
p.value: Named vector containing p_raw and p_trimmed.
extreme: Effective trimming parameter $h$.
parameter: Vector containing n_control and n_experiment.
null.value: Null hypothesis of equal extremity.
alternative: Character string: "greater".
method: Character string describing the test.
data.name: Character string describing the data.

Details

A nonparametric test comparing the spread (range) of two independent groups, assessing whether the control group shows greater variability than the experiment group.

Distributional derivation. Under the null hypothesis, any assignment of $n_k + n_e$ pooled ranks to the two groups is equally likely. The exact distribution of the span $S$ of the control group (a subset of size $n_k$ drawn without replacement from $\{1, \ldots, N\}$) is given by Moses (1952) as:

$$P(S \le s) = \frac{1}{\binom{N}{n_k}} \sum_{i=0}^{\min(s - n_k,\, n_e)} \binom{i + n_k - 2}{i}\, \binom{n_e + 1 - i}{n_e - i}$$

This is a cumulative distribution function (CDF): $P(\mathrm{span} \le s)$.

The one-tailed p-value for the observed span $S_{obs}$ is:

$$ p = P(S \ge S_{obs}) = 1 - P(S \le S_{obs} - 1) $$

This distinction matters because some textbook presentations write the summation formula as $P(S = s)$, whereas the upper-tail probability required for hypothesis testing is obtained from the cumulative form.

Generalisation with trimming ($h > 0$). After removing $h$ extreme ranks from each end of the sorted control group ranks, the effective control-group size becomes $n_k^\prime = n_k - 2h$, yielding:

$$ P(S_h \le s) = \frac{1}{\binom{N}{n_k}} \sum_{i=0}^{\min(s - n_k^\prime,\, n_e)} \binom{i + n_k^\prime - 2}{i}\, \binom{n_e + 2h + 1 - i}{n_e - i} $$

Tie handling. The exact combinatorial distribution assumes a continuous underlying distribution (no ties). By default, ties.method = "first" is passed to rank(), producing integer-valued ranks that remain compatible with the exact formula. Tied observations are ordered according to their occurrence in the pooled sample, matching SPSS behaviour.

Alternative tie methods such as "average" may produce fractional mid-ranks. In such cases the span is rounded upward via ceiling() to retain an integer-valued test statistic, but the resulting p-values should be regarded as approximate rather than exact.

Numerical stability. The exact distribution is evaluated entirely in log-space using lchoose() and a log-sum-exp transformation, avoiding overflow for large sample sizes.

References

Moses, L.E. (1952). A two-sample test. Psychometrika, 17, 239–247. doi:10.1007/BF02288735

Examples

x <- c(0.80, 0.83, 1.89, 1.04, 1.45,
       1.38, 1.91, 1.64, 0.73, 1.46)

y <- c(1.15, 0.88, 0.90, 0.74, 1.21)

mosesTest(x, y)
#> 
#> 	 Moses Test of Extreme Reactions 
#> 
#> data:  x and y 
#> n_control = 10,  n_experiment = 5
#> 
#>   Without trimming:                 span =   15,  p-value = 0.4285714
#>   After trimming 1 from each end:   span =   12,  p-value = 0.4335664
#> 
#> alternative hypothesis: extreme values are more likely in the control group
#> 

set.seed(1479)

x2 <- sample(1:20, 10, replace = TRUE)
y2 <- sample(5:25, 6, replace = TRUE)

mosesTest(x2, y2, extreme = 2)
#> 
#> 	 Moses Test of Extreme Reactions 
#> 
#> data:  x2 and y2 
#> n_control = 10,  n_experiment = 6
#> 
#>   Without trimming:                 span =   13,  p-value = 0.9642857
#>   After trimming 2 from each end:   span =    7,  p-value = 0.9423077
#> 
#> alternative hypothesis: extreme values are more likely in the control group
#> 

df <- data.frame(
  score = c(x, y),
  group = factor(rep(
    c("control", "experiment"),
    c(length(x), length(y))
  ))
)

mosesTest(score ~ group, data = df)
#> 
#> 	 Moses Test of Extreme Reactions 
#> 
#> data:  groups[[1L]] and groups[[2L]] 
#> n_control = 10,  n_experiment = 5
#> 
#>   Without trimming:                 span =   15,  p-value = 0.4285714
#>   After trimming 1 from each end:   span =   12,  p-value = 0.4335664
#> 
#> alternative hypothesis: extreme values are more likely in the control group
#>