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Richard McGarvey, Paul Burch, and Janet M. Matthews. 2015. Precision of systematic
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organisms. Ecological Applications
APPENDIX A. Summary and mathematical formulas for three proposed covered grid sample
variance estimators.
In this Appendix we describe three ‘covered grid’ methods proposed in this study to estimate the
variance of the sample mean from a systematic survey. These covered grid estimators extend the
balanced difference method proposed by Yates (1960), and more specifically extend the linear
v4-v6 balanced difference estimators given by Wolter (2007).
The three covered grid estimators were written for use with the tested (n = 100) systematic
survey design of a 10 x 10 square grid of sample locations within a 1-km2 square study region. R
code for these versions are found in online Supplement 2. These can be generalized for use with
irregular shaped study regions, in particular, for application to a grid of systematic sample
locations where the number of transects (or quadrats) in each row and column can vary as
needed. The generalized formulas are presented in the second subsection below.
In Yates’ balanced difference estimators, (1) each square difference term is written as sums
and differences of individual sampled values (as in the v4-v6 estimators of Wolter), rather than as
differences of each sampled value from the mean; (2) for balanced difference estimators (v4-v6),
these squared differences are computed among more than two neighboring sample values.
To construct covered grid estimators, we adopted four additional rules to extend the balanced
difference method. (3) The values included in each squared balanced difference term include
only sample values within a single row or column. (4) Each balanced difference includes all the
values in each row or column. (5) All values in each multiple-value balanced difference will be
weighted equally (unlike, e.g., v4-v6). (6) For one of the three covered grid estimators, the
twice-covered grid (TCG) variant, both row and column squared balanced differences are
included in the overall square difference sum. The row covered and column covered grid
estimators, unlike twice-covered grid and other balanced difference estimators of Yates and
Wolter, use each transect sample value in only one squared balanced difference term.
To construct the covered grid estimators, we generally followed Wolter’s (2007:301)
procedure for constructing variants of the balanced difference method, but we did not place
higher weighting on the more central values in each balanced difference term.
Covered grid variance estimators used in the simulation variance estimator comparisons
Defining mathematical notation, let lower case subscripts r and c denote rows and columns
respectively in the presumed two-dimensional systematic 10 x 10 grid of sample locations.
vTCG = the estimated twice-covered grid variance.
R = the total number of rows in the systematic grid of sample locations, and thus the numbers
of balanced differences included in the row square difference sum = 10.
C = the total number of columns in the systematic grid of sample locations, and thus the
numbers of balanced differences included in the column square difference sum = 10.
nD 2 R = The sum of squared weighting coefficients assumed for each balanced difference row
= 12   1  12   1  12   1  12   1  12   1  10.
2
2
2
2
2
nD 2C = The sum of squared weighting coefficients assumed for each balanced difference
column = 12   1  12   1  12   1  12   1  12   1  10.
2
2
2
2
2
n = total number of sampling locations, where here the number of transects = 100.
x
r ,c
; r  1, R; c  1, C = survey data counts of organisms in each transect, specified by row r
and column c.
Dr = the computed balanced difference for each row, r.
Dc = the computed balanced difference for each column, c.
f = standard finite correction ratio (here, the fraction of the study region area covered by
transects).
With this notation, the formula for the twice-covered grid estimator becomes
vTCG 
(1  f )(1/ n) 
 1

(R  C) 
 nD2 R
R
D
r
r 1
2

1
nD2C
C
D
c 1
c
2


.


(A.1)
The balanced differences for each row r and column c are written:
Dr  xr ,1  xr ,2  xr ,3  xr ,4  xr ,5  xr ,6  xr ,7  xr ,8  xr ,9  xr ,10
Dc  x1,c  x2,c  x3,c  x4,c  x5,c  x6,c  x7,c  x8,c  x9,c  x10,c .
(A.2)
(A.3)
The versions of the covered grid estimators using only rows or only columns, called row-covered
grid (vRCG) and column-covered grid (vCCG) are written by simply omitting either columns or
rows, respectively, from the twice-covered grid estimator above, that is:
vRCG 
vCCG 
(1  f )(1/ n)  R
2
 Dr 
( R  nD2 R )  r 1

(1  f )(1/ n)  C
2
 Dc 
(C  nD2C )  c 1

(A.4)
.
(A.5)
Generalized covered grid variance estimators: variable numbers of sample locations in rows or
columns
In the denominators above, nD 2 R and nD 2C normalize the sum of balanced differences
squared, to account for the number of sample values used in each term and their weightings.
More generally, if the number of sample locations, and thus of measured sample values,
varies from row to row and column to column, these covered grid estimators can be written:
vTCG 
2
C
D2 
(1  f )(1/ n) 
 R Dr

 c 

 R  C   r 1 nD2 ,r c1 nD2 ,c 
2

1 R D 

vRCG  (1  f )(1/ n)   r 
 R r 1 nD2 ,r 


vCCG
2

 1 C Dc 

 (1  f )(1/ n)  
.
C
c 1 nD2 , c 



(A.6)
(A.7)
(A.8)
The quantities nD 2 R and nD 2C above are constants (= 10 for both row and column) because the
number of sample values in each row and column of the perfectly square systematic transect
sample array we simulated is 10 for all rows and columns. But if the number of sample locations
should vary among rows or columns, the quantities of nD2 R,r and nD2C ,c will depend on the row,
r, or column, c. These quantities are still written as the sum of the weightings squared, that is as
the sum of the squared coefficients in front of each element in each balanced difference sum
(Wolter 2007:301),
nr
nD2 ,r    wi ,r 
2
(A.9)
i 1
nc
nD2 ,c    wi ,c 
2
(A.10)
i 1
Each generalized balanced difference sum, for each row, r, and column, c, individually is
written as
nr
Dr   wi ,r  xi ,r
(A.11)
i 1
nc
Dc   wi ,c  xi ,c
(A.12)
i 1
One requirement of the balanced difference method (in order that these differences continue
to quantify the difference from the mean though written as differences among multiple individual
sample values) is that the sum of the weightings must always be zero:
nr
w
i 1
i ,r
nc
w
i 1
i ,c
0
 0.
(A.13)
=
(A.14)
The sum of these covered grid weightings for the tested systematic survey design of 10
sample locations in each row and each column was 1  1  1  1  1  1  1  1  1  1 , satisfying this
requirement that they sum to zero. For the generalized case, if the number of transects is odd,
then the weightings must be adjusted, for example as in Wolter’s v4-v6, so that the sum of
weightings (for each row and column) is always zero.
This formulation produces balanced difference sum terms, from each sampled row or column,
of approximately equal weighting. When applied to irregular shaped study regions, additional
weightings can be added which account for less or more of the study region being covered by
each row or column. These additional weightings would ideally be scaled in proportion to the
measured area of the survey region covered by each row or column, or for evenly spaced rows or
columns of an irregular study region, in proportion to the width or height of the study region
covered by each row or column of the systematic sample.
LITERATURE CITED
Wolter, K. M. 2007. Introduction to variance estimation, second edition. Springer Science +
Business Media, LLC, New York.
Yates, F. 1960. Sampling methods for censuses and surveys, third edition. Charles Griffin & Co
Ltd, London.