Mood's Median Test for Comparing Medians Across Independent Groups

Tests whether two or more independent samples come from populations with the same median. All observations are dichotomised at the pooled median and the resulting 2 x k table is tested for independence.

Usage

moodMedianTest(x, ...)

# Default S3 method
moodMedianTest(
  x,
  g,
  ties = c("below", "above", "drop"),
  method = c("chisq", "exact"),
  correct = TRUE,
  ...
)

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

Arguments

x: a numeric vector of observations, or a formula of the form lhs ~ rhs with a numeric lhs and a grouping rhs
...: further arguments, passed to the default method
g: a vector or factor giving the group for each element of x
ties: how to treat observations exactly equal to the pooled median, one of "below" (default), "above" or "drop"
method: the test applied to the 2 x k table, either "chisq" (default) or "exact"
correct: logical, whether to apply the continuity correction; ignored unless the table is 2 x 2
formula: a formula of the form lhs ~ rhs
data: an optional data frame containing the model variables
subset: an optional vector specifying a subset of observations
na.action: a function indicating what should happen when the data contain NAs

Value

An object of class "htest" with components

statistic: Pearson's chi-squared statistic, NULL for method = "exact"
parameter: the degrees of freedom, NULL for method = "exact"
p.value: the p-value
estimate: the group medians
observed: the 2 x k table of counts above and below the pooled median
expected: the expected counts under independence
grand.median: the pooled median at which the data were dichotomised
method: a character string describing the test
data.name: a character string giving the names of the data

Details

Not to be confused with mood.test(), which is Mood's two-sample test for a difference in scale. The median test described here has no base R implementation.

The procedure is the k-sample counterpart of signTest() and shares its robustness and its modest power: only the position of each observation relative to the pooled median enters the statistic, so all information about distance is discarded. Its asymptotic relative efficiency against the F test under normality is \(2/\pi\).

That low efficiency is the price of asking a narrow question, and the alternatives ask different ones rather than the same one better: brunnerMunzelTest() tests the relative effect \(P(X < Y) + \frac{1}{2}P(X = Y) = \frac{1}{2}\) and vanWaerdenTest() tests equality of the distributions against normal-score location alternatives. Neither is a test of equal medians, so they are not drop-in replacements. Use the median test when the median is genuinely the quantity of interest, or when only the side of a threshold is trustworthy, as with coarsely recorded or thresholded data.

Observations equal to the pooled median. With an even total sample size the pooled median usually falls between two observations and the question does not arise. Otherwise ties decides: "below" (default) counts them with the lower group, following Conover, "above" with the upper group, and "drop" removes them, which retains a symmetric above-versus-below classification at the cost of a smaller effective sample size. Dropping them does not systematically enlarge the p-value; it can move it either way. "below" and "above" correspond to mid.score values of "0" and "1" in coin::median_test; "drop" has no counterpart there, since mid.score = "0.5" scores the median observations rather than removing them.

method = "exact" replaces the chi-squared approximation by Fisher's exact test on the same table and is advisable when expected counts are small; no test statistic is reported in that case. The continuity correction applies only to a 2 x 2 table, as in chisq.test().

Missing values are removed casewise.

References

Mood, A. M. (1950) Introduction to the Theory of Statistics. McGraw-Hill, New York, pp. 394-399.

Conover, W. J. (1999) Practical Nonparametric Statistics, 3rd edition. Wiley, New York, pp. 218-223.

Examples

moodMedianTest(breaks ~ tension, data = warpbreaks)
#> 
#> 	Mood's median test (3 groups, chi-squared approximation)
#> 
#> data:  breaks ~ tension
#> X-squared = 7.2993, df = 2, p-value = 0.026
#> sample estimates:
#>    L    M    H 
#> 29.5 27.0 20.5 
#> 

## the pooled median at which the data were split, and the resulting table
r <- moodMedianTest(breaks ~ tension, data = warpbreaks)
r$grand.median
#> [1] 26
## [1] 26
r$observed
#>        
#>          L  M  H
#>   above 12  9  4
#>   below  6  9 14

## small tables: use the exact test on the same 2 x k table
moodMedianTest(breaks ~ tension, data = warpbreaks, method = "exact")$p.value
#> [1] 0.03153902
## [1] 0.03153902