ST 524
Plot Size and Shape
NCSU - Fall 2008
Question 1.
Data set “uniftrialdata.xls” presents yields of a uniformity trial on winter wheat (simulated data). Unit
size (1.5m wide × 4.5m long) plots are distributed in a 6 columns × 48 rows, for a total of 288 plots (size
X = 1). Interest is in exploring the relationship between plot size and the variance among plots (in unit
basis). There are four variables that identify plots according to their size: plot2, plot4, plot8 and plot16,
where the plot sizes are 2,4,6,8, and 16 units.
1.
Following the approach presented in Swallow and H a nested analysis of variance on yield is
presented that will be used in the estimation of VX, variance among plots of size x, expressed in
unitary basis.
proc glm data=b3(where=(plot1<49 and col <7));
class
plot1 plot2 plot4 plot8 plot16 ;
model newyield = plot16 plot8(plot16) plot4(plot8*plot16) plot2(plot4*plot8*plot16) ;
random plot16 plot8(plot16) plot4(plot8*plot16) plot2(plot4*plot8*plot16) /test;
output out=outglm r=resid student=sres p=pred;
run;
The GLM Procedure
Dependent Variable: newyield
Source
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
143
209919.5978
1467.9692
2.40
<.0001
Error
144
87907.8423
610.4711
Corrected Total
287
297827.4402
R-Square
Coeff Var
Root MSE
newyield Mean
0.704836
6.033877
24.70771
409.4832
Source
plot16
plot8(plot16)
plot4(plot8*plot16)
plot(plot*plot*plot)
DF
17
18
36
72
Type I SS
50233.17637
34266.59997
62756.55429
62663.26720
Mean Square
2954.89273
1903.70000
1743.23762
870.32316
F Value
4.84
3.12
2.86
1.43
Pr > F
<.0001
<.0001
<.0001
0.0370
Source
plot16
plot8(plot16)
plot4(plot8*plot16)
DF
17
18
36
Type III SS
50233.17637
34266.59997
62756.55429
Mean Square
2954.89273
1903.70000
1743.23762
F Value
4.84
3.12
2.86
Pr > F
<.0001
<.0001
<.0001
plot(plot*plot*plot)
72
62663.26720
870.32316
1.43
0.0370
Expected Mean Squares
Source
Type III Expected Mean Square
plot16
Var(Error) + 2 Var(plot(plot*plot*plot)) + 4 Var(plot4(plot8*plot16)) + 8
Var(plot8(plot16)) + 16 Var(plot16)
plot8(plot16)
Var(Error) + 2 Var(plot(plot*plot*plot)) + 4 Var(plot4(plot8*plot16)) + 8
Var(plot8(plot16))
plot4(plot8*plot16)
Var(Error) + 2 Var(plot(plot*plot*plot)) + 4 Var(plot4(plot8*plot16))
plot(plot*plot*plot)
Var(Error) + 2 Var(plot(plot*plot*plot))
Tuesday November 25, 2008
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ST 524
Plot Size and Shape
NCSU - Fall 2008
Plot size
MS
VX
1
610.4711
V1 = 610.4711
2
870.32316
V2 
1743.23762
4
V4 
1903.70000
8
V8 
2954.89273
16
V16 
870.32316  610.4711
= 129.9261
2
1743.23762  870.32316 
218.2286
=
4
1903.70000  1743.23762 
=
20.0578
8
 2954.89273 1903.70000 
=
65.69954
16
Variance components may be obtained directly with PROC MIXED,
proc mixed data=b3(where=(plot1<49 and col<7));
class
plot1 plot2 plot4 plot8 plot16 ;
model newyield= / outp=predds ;
random plot16 plot8(plot16) plot4(plot8*plot16) plot2(plot4*plot8*plot16) ;
run;
Variance components estimates from PROC MIXED
Plot Size
Estimate
1
610.47
2
129.93
4
218.23
8
20.0578
16
65.6995
log Vx   6.0354  0.9127log  X 
The Mixed Procedure
Covariance Parameter Estimates
Cov Parm
plot16
plot8(plot16)
plot4(plot8*plot16)
plot(plot*plot*plot)
Residual
2.
Estimate
65.6995
20.0578
218.23
129.93
610.47
Size
16
8
4
16
1
Next, a regression of Vx on X, in a log scale, is used to get a raw estimate of the coefficient of soil
heterogeneity b, Smith’s b.
The REG Procedure
Model: MODEL1
Tuesday November 25, 2008
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ST 524
Plot Size and Shape
NCSU - Fall 2008
Dependent Variable: log_vx
Analysis of Variance
DF
Sum of
Squares
Mean
Square
1
3
4
4.00265
2.56906
6.57171
4.00265
0.85635
Root MSE
Dependent Mean
Coeff Var
0.92539
4.77010
19.39988
Source
Model
Error
Corrected Total
R-Square
Adj R-Sq
F Value
Pr > F
4.67
0.1194
0.6091
0.4788
Parameter Estimates
Variable
DF
Parameter
Estimate
Standard
Error
t Value
Pr > |t|
Intercept
1
6.03543
0.71681
8.42
0.0035
log_x
1
-0.91274
0.42218
-2.16
0.1194
Regression equation:
log Vx   6.0354  0.9127log  X 
Smith’s b = 0.9127
Values closer to 1 indicates increasing homogeneity of the soil. A plot size between 2 and 8 seems
adequate since for X=16 the variance among plots of size 16 is greater.
3. Additionally, we can analyze the residuals for the plot size X = 1, X = 8 and see whether the
use of a larger plot reduces the residual variation.
Check residual distribution on field

*** fit just an intercept in the model
yij     ij ,
coordinates of each plot, i = 1, 2, . . ., 48
is the yield in (i, j) plot,

where i and j are the
row and j = 1,2,3,4,5,6 column,
is the overall mean, and
 ij is
yij
the residual value
in (i, j) plot.
proc glm data = newtrial;
model newyield =
;
output out = outglm r = resid student = sres p = pred;
run;
The GLM Procedure
Dependent Variable: newyield
Sum of
Source
DF
Squares
Mean Square
Model
Tuesday November 25, 2008
1
48290839.62
48290839.62
F Value
Pr > F
46535.2
<.0001
3
ST 524
Plot Size and Shape
NCSU - Fall 2008
Error
287
297827.44
Uncorrected Total
288
48588667.06
1037.73
R-Square
Coeff Var
Root MSE
newyield Mean
0.000000
7.866930
32.21376
409.4832
Source
Intercept
DF
Type I SS
Mean Square
F Value
Pr > F
1
48290839.62
48290839.62
46535.2
<.0001
Parameter
Estimate
Standard
Error
t Value
Pr > |t|
Intercept
409.4832432
1.89821396
215.72
<.0001
*** residual plot on the field ***;
Residual plot

Standardized Residual plot
*** graph a contour plot for residuals on the field ***;
proc g3grid data=outglm out=out2;
grid row*col = sres ;
run;
proc gcontour data=out2;
plot row*col=sres/ levels= -4 -3 -2 -1 0 1 2 3 4;* pattern join;
run;
Tuesday November 25, 2008
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ST 524
Plot Size and Shape
Tuesday November 25, 2008
NCSU - Fall 2008
5