MEAD4RAN
To perform Mead’s randomization test upon a 4*4 grid of spatial count data. Mead’s randomization test is
designed to test the null hypothesis of CSR (Complete Spatial Randomness).
Calling statement
mead4ran m1 ;
nran k1 (999) ;
qstatistics c1.
Input
A 4*4 matrix of quadrat counts, which may not contain any missing values.
The ordering of the counts in the matrix should be the same as the spatial ordering in the experiment
or study, and the results obtained will be dependent upon this ordering.
Subcommands
qstatistics
Specify a column in which to store simulated Q-statistics.
Output
 Observed Q-statistic
 Associated one-sided and two-sided randomization p-values
References
MANLY, F.J. (1997) Randomization, bootstrap and Monte Carlo methods in biology,
Chapman and Hall, London (Chapter 10).
Standard procedure : None
Null hypotheses : Assume that the quadrats are labelled as follows :
1
3
9
11
2
4
10
12
5
7
13
15
6
8
14
16
The null hypothesis is that the division of the quadrats into blocks of 4 [(1) = 1,2,3,4 ; (2) = 5,6,7,8 ; (3) =
9,10,11,12 ; (4) = 13,14,15,16] is random. If the data exhibit Complete Spatial Randomness (CSR), then
this division will be random, so the null hypothesis can also be viewed as CSR.
Alternative hypotheses : Clustering or regularity at an appropriate scale, resulting in a non-random
division of the quadrats into blocks.
Test-statistic : Assume that the data are as follows, so that Ti represents the quadrat count in the ith
quadrat.
T1
T2
T5
T6
T3
T4
T7
T8
T9
T10 T13 T14
T11 T12 T15 T16
We use the test-statistic Q = BSS / TSS, where TSS is the variance of the 16 counts in the 4*4 grid, and
BSS is the variance for the 4 counts (in the 2*2 grid formed by aggregating counts as follows :
AC1 = T1 + T2 + T3 + T4
AC3 = T9 + T10 + T11 + T12
Definition
BSS 
T1  T2  T3  T4 2
2
 n 2
TSS    Ti   16T .
 i 1 
AC2 = T5 + T6 + T7 + T8
AC4 = T13 + T14
10 + T15 + T16
13
14
11
2
2
2
2
 T5  T6  T7  T8   T9  T12
10  T11  T12   T13  T14  T15  T16 
 16T ,
15
4
16
Q lies between 0 and 1. In general, unusually large values of Q imply clustering, whilst unusually small
values of Q imply some form of regularity. However, it should be noted that the test is only capable of
detecting regularity or clustering at a particular spatial scale (the scale reflected by blocks of size 4).
Mead's randomization test can either be applied to data which naturally arise as a 4*4 grid of counts, or
(more commonly) by placing a 4*4 grid over a region in which locations of points are recorded, and
counting the number of points within each section of the grid.
Randomization procedure : We randomize the allocation of counts to cells within the grid, since under
the null hypothesis of complete spatial randomness this allocation should occur at random.
WORKED EXAMPLE FOR MEAD4RAN
Name of dataset
SAPLING1
Description
The raw data describes the position of 71 Swedish pine saplings in a 10 x 10m square. In this dataset, we
divide the region in 16 squares (each 2.5m x 2.5m), and count the number of saplings within each square.
Source
MANLY, F.J. (1997) Randomization, bootstrap and Monte Carlo methods in biology,
Chapman and Hall, London.
Data
Number of observations = 16
Number of variables = 1
Counts within the 4*4 grid are shown.
6
4
3
2
2
6
4
3
5
4
5
4
4
6
6
7
Worksheet
M1
Matrix of counts
Aim of analysis
To test whether the distribution of pine saplings is random.
2
Randomization procedure
MTB > Retrieve "N:\resampling\Examples\Sapling1.MTW".
Retrieving worksheet from file: N:\resampling\Examples\Sapling1.MTW
# Worksheet was saved on 09/08/01 09:13:12
Results for: Sapling1.MTW
MTB > print m1
Data Display
Matrix M1
6
4
3
2
2
6
4
3
5
4
5
4
4
6
6
7
MTB > % N:\resampling\library\mead4ran m1 ;
SUBC> nran 999 ;
SUBC> qstatistics c1.
Executing from file: N:\resampling\library\mead4ran.MAC
Mead's randomization test for a 4*4 grid
Data Display (WRITE)
Observed Q-statistic 0.3886
One-sided randomization p-value, H1: regularity
One-sided randomization p-value, H1: clustering
Two-sided randomization p-value
0.2100
0.9010
0.1050
* NOTE * For further details, see
MANLY, F.J. (1997) Randomization, bootstrap and Monte Carlo
methods in biology, Chapman and Hall, London.
Modified worksheet
C1
A column containing 999 Q-statistics, one for each simulated dataset
Discussion
There is slight evidence of clustering (we obtain a one-sided p-value of 0.105; Manly (1997) obtains
0.111), but this cannot be regarded as statistically significant. Mead's test therefore provides no real
evidence against randomness at this scale (this qualification is important - the test is scale-dependent).
3