Widmann, A. & Schröger E. (1999).
Bootstrapping the distribution of the city-block distance between two repeated measures [Online].
Available http://www.uni-leipzig.de/~biocog/widmann/minkowski.html
UNIVERSITÄT LEIPZIG
 

Bootstrapping the Distribution of the City-Block Distance between Two Repeated Measures

Andreas Widmann, & Erich Schröger
University of Leipzig, Germany
E-mail: widmann@rz.uni-leipzig.de

The city-block distance

      n
   C=SUM(|Y(1)i-Y(2)i|)
     i=1

is a special case of the «Minkowski-distances» representing the distances between two vectors in an n-dimensional space spanned by n orthogonal axes. It can be used as a measure of the similarity of the distributions/means of two repeated measures. An example is given in Schröger (1998) were it has been used to determine the replicability of a particular component of the event-related brain potential. Since in the case of small sample sizes or unknown distribution functions of Y(1) and Y(2) by which the probability function of C could be developed, the distribution function of C, required to interpret a particular value of C, is not known, a randomization test can be used (e.g. Edgington, 1980).

It is suggested to

  1. pool the measured values from both repeated measurements, resulting in a sample size of 2n;
  2. draw a large number of random samples from this pool, dividing the pool in two groups of size n;
  3. compute C for each sample;
  4. compute the cumulative distribution functions of the values of C; and
  5. determine the critical value for a given significance level alpha.

Steps (1) to (3) are done by the provided perl script minkowski.pl. To keep the script short and to save computing time the probability for a city-block distance equal or smaller than the empirically measured one is computed rather than the cumulative distribution function.

References:
Edgington, E. S. (1980). Randomization tests. New York: Dekker.
Schröger, E. (1998). Measurement and interpretation of the mismatch negativity. Behavior Research Methods, Instruments and Computers, 30, 131-145.


Download the perl script and documentation:
  • minkowski.zip (for dos/win32s)
  • minkowski.tar.gz (for linux/unix)
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    04.10.2001, Andreas Widmann, widmann at uni-leipzig dot de