Median Absolute Deviation

Compute the median absolute deviation, i.e., the (lo-/hi-) median of the absolute deviations from the median, and (by default) adjust by a factor for asymptotically normal consistency. This function wraps the specific base R function mad() and extends it for the use of weights.

Usage

madX(
  x,
  weights = NULL,
  center = medianX,
  constant = 1.4826,
  medianType = c("standard", "low", "high"),
  na.rm = FALSE
)

Arguments

x

a numeric vector

weights

a numerical vector of weights the same length as x giving the weights to use for elements of x

center

a numeric center or a function applied to x. When weights are supplied, the function must support a weights argument. Defaults to medianX.

constant

scale factor (default is 1.4826)

medianType

character string selecting the "standard", "low", or "high" median for even sample sizes

na.rm

if TRUE then NA values are stripped from x before computation takes place

Value

a numeric scalar containing the scaled median absolute deviation

Details

The actual value calculated is constant * cMedian(abs(x - center)) with the default value of center being median(x), and cMedian being the usual, the ‘low’ or ‘high’ median, see the arguments description for low and high above.

The default constant = 1.4826 (approximately \(1/\Phi^{-1}(\frac 3 4)\) = 1/qnorm(3/4)) ensures consistency, i.e., $$E[mad(X_1,\dots,X_n)] = \sigma$$ for \(X_i\) distributed as \(N(\mu, \sigma^2)\) and large \(n\).

If na.rm is TRUE then NA values are stripped from x before computation takes place. If this is not done then an NA value in x will cause madX to return NA.

Confidence intervals are provided by lumen::madCI().

See also

IQR() which is simpler but less robust, iqrX() for the same using weights,
mad(), median(), var() the base R equivalents
lumen::madCI() (confidence intervals).

Other dispersion: coefVar(), iqrX(), meanAD(), meanSE(), rangeX(), varX()

Examples

madX(c(1:9))
#> [1] 2.9652
print(madX(c(1:9),     constant = 1)) ==
      madX(c(1:8, 100), constant = 1)       # = 2 ; TRUE
#> [1] 2
#> [1] TRUE
x <- c(1,2,3,5,7,8)
sort(abs(x - median(x)))
#> [1] 1 1 2 3 3 4
c(madX(x, constant = 1, medianType="standard"),
  madX(x, constant = 1, medianType="low"),
  madX(x, constant = 1, medianType="high"))
#> [1] 2.5 2.0 3.0

# use weights
x <- sample(20, 30, replace = TRUE)
z <- as.numeric(names(w <- table(x)))

(m1 <- madX(z, weights=w))
#> [1] 6.6717
(m2 <- madX(x))
#> [1] 6.6717
stopifnot(identical(m1, m2))