C. H. Ettema, D. C. Coleman, G. Vellidis, R. Lowrance, and S.
Rathbun. 1998. Spatiotemporal distributions of bacterivorous nematodes
and soil resources in a restored riparian wetland. Ecology 79:2721-2734.
Supplements
Supplement 1: SAS Programs for geostatistical analysis.
Ecological Archives E079-001-S1.
Copyright
Description
This document contains supplementary material for
the statistical analysis used by Ettema et al. (1998). SAS programs for geostatistical
analysis, and accompanying output files for one of the nematodes investigated
(Prismatolaimus) are presented and explained.
Author contact information
Dept. of Environmental Sciences
Sub-department of Soil Quality
Wageningen University
POB 8005
6700 EC Wageningen
The Netherlands
E-mail: christien.ettema@bb.benp.wau.nl
File list and downloads
More complete descriptions are provided in the following section
- prisnema.dat
is a datafile with abundance data for the bacterivorous nematode Prismatolaimus.
- semivari.sas
is a SAS program for estimating and modeling pooled semivariance in nonstationary
data.*
- semivari.lst
is the accompanying output file for Prismatolaimus.
- glstest.sas
is a SAS program for testing the partial slopes of x, y-coordinates
using GLS estimation.*
- glstest.lst
is the accompanying output file for Prismatolaimus.
- unikrige.sas
is a SAS program for universal kriging.*
- prism.nov
is the accompanying output file for Prismatolaimus.
- E079001.zip
is a single file containing all the components of this supplement
*NOTE: The *.sas files all were written for
SAS version 6.08. All of these programs require the use of the SAS
matrix language IML. For a short introduction toSAS-IML, click
here.
File descriptions
1. prisnema.dat
The datafile prisnema.dat contains four tab-delimited
variables: x-coordinate, y-coordinate, sampling date (t=1,
108, 172, and 290), and Prismatolaimus abundance per assembled core
(2.22 cm diameter, 15 cm depth). See Ettema et al. (1998) for details on sampling
and nematode enumeration. Note that in each SAS program, nematode abundance
is converted to a per-meter-squared basis (using factor 2583.4745).
2. semivari.sas
The program semivari.sas contains four main
parts following the data step.
The first part (1) is a simple ordinary-least-squares (OLS) regression of
nematode abundance against x-coordinates (the y-coordinate
variable is left out in the model statement as it is not significant; it may
be added in for different data sets!). This OLS regression is necessary because
there is a large-scale spatial trend in the data along the x-direction (i.e.,
the data are non-stationary). Basically, the OLS-residuals can be considered
stationary data, with large-scale spatial trends removed.
In part 2, a main IML program (vgram) is
defined in which semivariances are estimated using the OLS-residuals instead
of the original data (residuals are read into the 'z' array in part 3). First,
for every pair of points, their spatial separation distance is calculated
as Euclidean distance (dist). Secondly, their
"temporal distance" is calculated (dtime).
For pooled spatial semivariance, we are not interested in the semivariance
between points of different sampling dates; that's why the program
contains a trick to leave these semivariances out of the final calculation:
The statement
dist=dist + 10000 *dtime;
combined with
if dist <= maxdist then append;
(the value of maxdist being set in part 3)
throws out all the semivariances of pairs of points with a dtime
> 0. Next, the distances are discretized into distance classes (this is
needed as samples were taken on an irregular lattice).
The first "if-loop" in part 2 simply adds the mean-squared differences
between subsequent pairs of observations to v[n].
It only does this for a limited number of size (distance) classes (size
and nclass being defined in part 3). For a
different data set, you have to play around with the settings of size,
nclass and maxdist.
The rule of thumb is to estimate semivariance up to 3/4 of the maximum separation
distance in your data, and have at least 30 pairs of points per distance class.
The 'do i=1 to nclass' loop in part 2 sets
the value-label of the distance class halfway (such that e.g. 'dist=15'
really covers distances 10-20m). In addition, the average semivariance (vario) is calculated, per distance class i by
dividing the summed semivariances (v[i]) by
twice the number of pairs in the distance class: 2*count[i],
which is the 2Nh in equation 1 in Ettema et al. (1998):
Equation 1: 
The output of this loop contains 3 columns of data, containing the distance
class (dist, i.e. h in Equation 1 above), the average
semivariance in that class (vario, being gamma(h)),
and the number of pairs of points in each distance class (n,
being Nh).
Part 3 of the program reads the data from the data set a
(the residuals!) into arrays, sets the values of the parameters size, nclass and maxdist,
and invokes the execution of the main program (vgram).
NOTE: if the OLS yields only non-significant slopes for x, y-coordinates,
the data can be considered stationary. In that case, some statements in (3)
have to be changed:
use a;
changes into
AND
read all var{resid} into z;
changes into
read all var{prism} into z; (prism
being the original data).
As mentioned earlier, the settings of size,
nclass, and maxdist
are to be adjusted to the spatial characteristics of your data set.
Part 4 of the program, following simple print
and plot statements, fits an exponentional model to the
semivariogram using SAS's non-linear regression procedure, PROC NLIN. The
exponentional model is as follows:
Equation 2:
In the parameters statement, guess values
are given for Co, Ce,
and alpha. The model is fitted using weighted least squares
(WLS) (_weight_ statement), that is, priority is given to fitting
the model best at short distances. It is important to try different guess
values to find the best-fitting parameters (see below for judging best fit).
3. semivari.lst
The output from semivari.sas, run with prisnema.dat is semivari.lst.
It contains several parts. First, it shows the OLS output, with a significant
slope for x. It also shows the estimated semivariance (vario)
for 9 distance classes (dist) and number of
observation pairs in each class (n). NOTE:
if the slope for x was not significant, you would re-estimate the
semivariances using the original data (see above). NOTE: the final, unbiased
test for x is done in glstest.sas,
described below.
Next, the output shows the semivariogram plot, revealing spatial autocorrelation
at short distances, and leveling off (near-independence) at large distances.
The plot is followed by the non-linear regression iterations and the message
that convergence was attained. The relative structure can be calculated as
(Ce/(Co + Ce)), and the range as (3/alpha).
The model R^2 can be calculated from the ANOVA table as:
R^2 = (Corrected Total Weighted SS - Residual Weighted SS) / Corrected
Total Weighted SS)
The best fitting model has the highest R^2 and the lowest Weighted Mean Square
Residual.
The remainder of the output shows the predicted values, residuals and residual
plot.
4. glstest.sas
Although OLS-regression gives unbiased estimates of partial slopes, their
standard errors, as estimated by common regression software, are biased in
the presence of spatial correlation. Thus, for a thorough evaluation of large-scale
trend in the data, the slopes are tested using Generalized-Least-Squares (GLS).
GLS estimates of regression coefficients use information from the spatial
correlation matrix (which, in our case, is derived from the model fitted to
the residuals-based semivariogram). By giving more weight to isolated sites
than sites in crowded locations which present redundant information, GLS regression
coefficient estimates are more precise than OLS estimates, and estimates of
their standard errors are unbiased (Cressie 1993:
p. 20-24). GLS estimates can be used to test effects of independent variables,
such as x,y-coordinates.
The SAS program glstest.sas has 3 main parts
following the data step. In part 1, the subroutine vmat is defined, in which the spatial correlation (not
semivariance) matrix v[i,j] is calculated.
To understand the calculations, remember the following equations for semivariance,
covariance, and correlation:
Spatial semivariance:
Equation 3:
where C(r) is the spatial covariance function and r is
distance.
(NOTE: h is distance class - see Equation 1, above)
Spatial Correlation:
Equation 4: 
Combining Equations 3 and 4:
Equation 5: 
In the exponential model,
Equation 6:
Thus, Eq. (5) can be re-written as:
Equation 7: 
IMPORTANT: The notation i^=j
(in subroutine vmat) means "for i
unequal to j " (i.e. so that r > 0). Some SAS environments
require a slightly different notation, e.g. i¬=j.
If not sure (and if the printout shows a negative sigma2),
activate the print v statement and check that the diagonal entries
of the v-matrix are all ones. Other tricky notations: Some SAS environments
use a `, not a ', to transpose matrices.
|| is used to combine two matrices (if your screen shows these bars
split in the center, it may not work properly!). // is used
to place one matrix below the other. Check SAS-IML guides for further info.
The second section (2) contains the main program, gls,
which performs the actual GLS estimation, using the subroutine vmat
to calculate the correlation matrix. Remember the multiple regression model:
Equation 8: 
where
Equation 9: 
in which V is the correlation (ro(r))
matrix vmat. The GLS estimators for beta (beta
in the SAS program) , var(beta) (varbeta)
, and sigma-squared (sigma2) are, respectively:
Equation 10: 
Equation 11: 
Equation 12: 
The last section (3) reads data from the dataset wetland
into arrays appearing in the IML program, and sets the values of program parameters
Co, Ce, alpha,
and maxdist. The final statement (run
gls) invokes the actual execution of the main program gls
and subroutine vmat.
NOTE 1: If both x- and y-
coordinates are to be tested, add y to the design matrix xmat:
read all var{x y} into xmat;
NOTE 2: For Co, Ce
and alpha fill in the best fitting values
from the output of semivari.sas, i.e. from
semivari.lst
NOTE 3: The program is set up to test the partial slope of x across
the 4 seasons. To test for individual seasons, activate the appropriate statements
in the data step (e.g. to test for the first date, t=1, drop the data of all
other dates using the statements
if time=108 then delete;
if time=172 then delete;
if time=290 then delete;
Thus limiting the test to one sampling date will make the dtime-calculation
in part (1) irrelevant (however, leave it in place - it doesn't hurt the other
calculations, and reduces the risk that later on you forget to put it back
in).
5. glstest.lst
The glstest output, glstest.lst, should be
read as follows. The column titled beta contains
the estimates for beta_0 and beta_1; beta_1 being the slope for the x-coordinate-variable
tested. The diagonal of the matrix titled varbeta
contains the variance estimates for the betas: var(beta_0) is the top-left
value in the matrix, and var(beta_1) is the bottom-right value. The t-statistic
for beta_1 is estimated as:
t = beta_1 / sqrt{var(beta_1)}
which here = -1577.945 / sqrt(763592.47) = -1.81
Compare this value to the t_(n-2) percentage point (in this particular
test, n=107, and 2 is the amount of estimated model parameters, beta_0 and
beta_1). Test double-sided.
Note that if y (the y-coordinates) is included as an explanatory variable
in the design matrix xmat(see glstest.sas,
NOTE 1), beta would
show 3 instead of 2 estimates, and varbeta
would be a 3x3, instead of a 2x2 matrix. In addition, the outcome of the t-statistic
should be compared to the t_(n-3), instead of the t_(n-2),
percentage point
6. unikrige.sas
The SAS program for universal kriging, unikrige.sas,
consists of 7 parts. For an explanation of the universal kriging equations,
click here.
Part 1 is a simple data input step, reading the measurements z(x,y) at sampled
locations. It contains statements to drop all data except those of the sampling
date for which kriging is desired (t=1 in this example).
Part 2 creates a new data set (points), the
set of spatial points s0 for which you want to predict the
value of 'z' (point coordinates are (x0,y0),
read into (xp,yp) in part 6).
Parts 3 and 4 define two subroutines (gam1
and gam2), which are called in the main program krige in part 5. gam1
calculates the semivariances between pairs of sampled locations (x,y) and
constructs the bgamma matrix (big gamma, see universal
kriging equations). gam2 calculates the
semivariances between sampled sites and the locations (xp,yp)
for which predictions are to be made. It also constructs the sgamma
matrix (small gamma, see universal kriging equations),
which is concatenated with xmatp. xmatp
contains the value(s) of the explanatory variable(s) at the locations s0
(xp,yp) where predictions are to be made (in this program,
xmatp contains the x0-coordinates, as a significant
trend along the x-direction was found in the GLS output! Note: In ordinary
kriging, no explanatory variables are used).
Part 5, the main program krige, starts with
initializing all the matrices used in the subroutines gam1
and gam2, and subsequently calls these subroutines.
In the lambda= statement the universal
kriging equations are solved. Using the resulting lambda, z is predicted
(zhatm) for the unsampled sites s0. The 'do'
loop takes care that the resulting matrix entries zhatm[i],
xp[i] and yp[i]
are output as data columns zhat, x0,
and y0 into the IML data set pred
which was initialized in the create statement right after proc
iml in part 3 of the program.
Part 6 reads data from the dataset wetland into
arrays appearing in the IML program, and sets the values of program parameters
Co, Ce, and alpha (adjust these values according to your model fit!
I.e. use the best-fitting parameters from semivari.lst).
The final statement (run krige) invokes the
actual execution of the main program krige
and subroutines gam1 and gam2.
Note that If both x- and y- coordinates are significant
explanatory variables, y has to be added to xmat
and xmatp:
read all var{x y} into xmat;
read all var{x0 y0} into xmatp;
Finally, in part 7 of the program, the predicted values stored in the IML
data set pred are written to a space-delimited ASCII file prism.nov, which
can be used in graphics software to draw maps.
Note that unikrige.sas requires considerable
memory and computation time. We used the batch facility on our mainframe system
to temporarily access more memory and increase available CPU. For a quick
check whether the program actually works for you, reduce the points data set
to just a few data points. E.g.:
do x0=0 to 48 by 48;
do y0=0 to 108 by 108;
creates a tiny data set with which the program should run without system
protests!
References
Cressie, N. A. C. 1993. Statistics for spatial data. John Wiley and
Sons, New York, USA.
Ettema, C. H., D. C. Coleman, G. Vellidis, R. Lowrance, and S. Rathbun. 1998.
Spatio-temporal distributions of bacterivorous nematodes and soil resources
in a restored riparian wetland. Ecology 79(8): 2721-2734.
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