smoothing spline for crop density
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smoothing spline for crop density
This is the continuing saga of an analysis on which Arthur Gilmour and Brian
Cullis gave some advice a week or two ago. The data are from three lupin
varieties sown at a range of target densities. Yield (tonnes/ha) (THA) and
the realised density (plants/m2) (ESTAB) are measured.
A smoothing spline is used to fit the response of THA to ESTAB. Brian
pointed out that there are too many knots (49) in the resulting spline, and
suggested that knot points should be chosen explicitly. However, I can't
see how to do precisely this: the ASREML manual describes how to set the
number of knot points (p. 51, version of 2 October 1998) but not their
actual values.
Brian recommended that Rep should not be fitted in combination with a
spatial model, unless there was clear evidence that it was necessary. But
there is: the two .asr files embedded below show an increase of max
likelihood of about 10 from including Rep. Arthur suggested fitting COL as
a random effect (the plots are laid out in two columns), and I hoped this
might pick up the same variation as Rep, but it doesn't do much at all.
Final puzzle (for the time being): the relationship between THA and ESTAB is
negative and significant. Note that tonnes/ha, not yield/plant, is
measured. A plot of the raw data is pretty flat, which is disappointing
but okay - but how does one account for a significant negative relationship?
Is it being tested against the wrong error?
Nick Galwey
_______________________
Analysis including Rep
ASREML [30 Sep 1998] Restricted branching lupin - density trials
11/05/98 12:49:15.75 8.00 Mbyte c:\DOCS\Dracup_M\96WH14\96WH14.as
QUALIFIERS: !skip 1 !maxit 20 !spline 6
Reading C:\DOCS\Dracup_M\96WH14\96WH14.dat FREE FORMAT skipping 1
lines
Univariate analysis of THA
Using 108 records [of 108 read from 108 lines of
C:\DOCS\Dracup_M\96W]
Model term Size Type COL Minimum Mean Maximum #zero #miss
1 Variety 3 Factor 1 1 1.8333 3 0 0
2 TDENS 6 Factor 2 25 63.8889 125 0 0
WARNING - More levels in factor than expected
3 Rep 5 Factor 3 1 2.6667 5 0 0
4 PLOT 90 Factor 4 1 38.0833 90 0 0
5 COL 2 Factor 5 1 1.5000 2 0 0
6 ROW 54 Factor 6 1 27.5000 54 0 0
7 THA 1 Variate 7 0.7530 1.014 1.428 0 20
8 ESTAB 1 Covariat 8 19.60 71.19 137.6 0 18
9 mu 1 Constant Term
10 mv_estimates 20 Missing value
11 c(Variety) 2 Factor 1 1 1.8333 3 0 0
12 c(Variety).E 2 Interaction 11 c(Vari: 2 8 ESTAB : 1
13 spl(ESTAB) 4 Spline 8 19.60 71.19 137.6 0 18
14 Variety.spl( 12 Interaction 1 Variet: 3 13 spl(ESTAB) : 4
54 AR=AutoR 0.10 0.10
2 identity
Forming 48 equations: 27 dense
Initial updates will be shrunk by factor 0.387
LogL= 107.577 S2= 0.14449E-01 82 df 0.10000 0.10000 0.10000
1.0000 0.10000 0.10000
LogL= 116.191 S2= 0.13137E-01 82 df 0.15861 0.12940 0.10000E-01
1.0000 0.18802 0.24897
LogL= 121.581 S2= 0.14585E-01 82 df 0.27306 0.43350E-010.10000E-02
1.0000 0.29841 0.34974
LogL= 122.831 S2= 0.17089E-01 82 df 0.42788 0.93525E-010.10000E-03
1.0000 0.37088 0.37592
LogL= 122.976 S2= 0.17904E-01 82 df 0.54763 0.74599E-010.10000E-04
1.0000 0.39757 0.36668
LogL= 122.953 S2= 0.17851E-01 82 df 0.58050 0.94118E-010.10000E-06
1.0000 0.39947 0.36545
LogL= 122.949 S2= 0.17536E-01 82 df 0.59623 0.96594E-010.10000E-06
1.0000 0.40504 0.35305
Final parameter values 0.57348 0.50383E-010.10000E-06
1.0000 0.40207 0.36372
Source Model terms Gamma Component Compnt/StndErr
Rep 5 5 0.573475 0.100567E-01 1.07 P
spl(ESTAB) 4 4 0.503830E-01 0.883540E-03 0.30 P
Variety.spl(ESTAB) 12 12 0.100000E-06 0.175365E-08 2.27 B
Variance 108 82 1.00000 0.175365E-01 2.27 P
Residual AR=AutoR 54 0.402072 0.402072 3.37 U
Residual AR=AutoR 54 0.363720 0.363720 3.03 U
WARNING: Code B - fixed at a boundary (!GP)
C - Constrained by user (!CON)
S - Singular Information matrix
S means there is no information in the data for this parameter.
Very small components with Comp/SE ratios of zero sometimes indicate poor
scaling. Consider rescaling the design matrix in such cases
Fitted Spline 6 (X) for spl(ESTAB)
27.040 51.208 70.956 93.893 116.031 131.900
Fitted Spline 6 (Y) for spl(ESTAB) 1
0.028978 -0.017173 -0.023585 -0.008760 0.007397 0.013143
' 0.03
0.02
,' 0.01
0.00
' -0.01
' -0.02
' -0.03
-0.04
Fitted Spline 6 (Y) for Variety.spl(ESTAB) 1
-0.74E-08 -0.29E-09 0.10E-07 0.48E-08 -0.21E-09 -0.71E-08
'''''' 0.00
0.00
0.00
-0.01
-0.01
-0.01
-0.01
-0.01
Fitted Spline 6 (Y) for Variety.spl(ESTAB) 2
0.42E-07 -0.18E-07 -0.39E-07 -0.18E-07 0.84E-08 0.25E-07
'''''' 0.00
0.00
0.00
-0.01
-0.01
-0.01
-0.01
-0.01
Fitted Spline 6 (Y) for Variety.spl(ESTAB) 3
-0.44E-08 0.26E-09 0.47E-08 0.43E-08 -0.51E-09 -0.43E-08
'''''' 0.00
0.00
0.00
-0.01
-0.01
-0.01
-0.01
-0.01
Solution Standard Error T-value T-prev
12 c(Variety).ESTAB 2 3.41 3.41 [DF
F_inc F_all]
1 0.571064E-03 0.386200E-03 1.48
2 -0.101010E-02 0.388844E-03 -2.60 -2.37
8 ESTAB 1 20.29 21.59 [DF
F_inc F_all]
3 -0.190629E-02 0.410273E-03 -4.65
11 c(Variety) 2 0.49 1.90 0.5110E-01 [DF F_i
F_a SED]
MERRIT -0.291534E-01 0.301931E-01 -0.97
85SO46-37 0.580733E-01 0.298602E-01 1.94 1.65
10 mv_estimates 20 16.08 14.48 [DF
F_inc F_all]
6 -0.899080 0.932293E-01 -9.64
7 -0.645952 0.948447E-01 -6.81 2.08
8 -1.10387 0.118815 -9.29 -2.92
9 -1.13570 0.124359 -9.13 -0.28
10 -1.13103 0.132515 -8.54 0.04
11 -1.14038 0.136347 -8.36 -0.08
12 -1.14252 0.139542 -8.19 -0.02
13 -1.14668 0.141591 -8.10 -0.04
14 -1.14913 0.143109 -8.03 -0.02
15 -1.15159 0.144174 -7.99 -0.02
16 -1.15344 0.144951 -7.96 -0.02
17 -1.15507 0.145517 -7.94 -0.01
18 -1.15638 0.145934 -7.92 -0.01
19 -1.15748 0.146244 -7.91 -0.01
20 -1.15839 0.146477 -7.91 -0.01
21 -1.15915 0.146653 -7.90 -0.01
22 -1.15978 0.146787 -7.90 -0.01
23 -1.16030 0.146891 -7.90 0.00
24 -1.16074 0.146971 -7.90 0.00
25 -1.16110 0.147035 -7.90 0.00
9 mu 1 210.45 296.72 [DF
F_inc F_all]
26 1.15333 0.669547E-01 17.23
3 Rep 5 effects fitted
13 spl(ESTAB) 4 effects fitted
14 Variety.spl(ESTAB) 12 effects fitted
1 possible outliers: see .res file
Finished: 12:49:24.69 LogL Converged
_______________________________________________________________________________
Analysis without Rep
ASREML [30 Sep 1998] Restricted branching lupin - density trials
11/05/98 13:00:38.59 8.00 Mbyte c:\DOCS\Dracup_M\96WH14\96WH14.as
QUALIFIERS: !skip 1 !maxit 20 !spline 6
Reading C:\DOCS\Dracup_M\96WH14\96WH14.dat FREE FORMAT skipping 1
lines
Univariate analysis of THA
Using 108 records [of 108 read from 108 lines of
C:\DOCS\Dracup_M\96W]
Model term Size Type COL Minimum Mean Maximum #zero #miss
1 Variety 3 Factor 1 1 1.8333 3 0 0
2 TDENS 6 Factor 2 25 63.8889 125 0 0
WARNING - More levels in factor than expected
3 Rep 5 Factor 3 1 2.6667 5 0 0
4 PLOT 90 Factor 4 1 38.0833 90 0 0
5 COL 2 Factor 5 1 1.5000 2 0 0
6 ROW 54 Factor 6 1 27.5000 54 0 0
7 THA 1 Variate 7 0.7530 1.014 1.428 0 20
8 ESTAB 1 Covariat 8 19.60 71.19 137.6 0 18
9 mu 1 Constant Term
10 mv_estimates 20 Missing value
11 c(Variety) 2 Factor 1 1 1.8333 3 0 0
12 c(Variety).E 2 Interaction 11 c(Vari: 2 8 ESTAB : 1
13 spl(ESTAB) 4 Spline 8 19.60 71.19 137.6 0 18
14 Variety.spl( 12 Interaction 1 Variet: 3 13 spl(ESTAB) : 4
54 AR=AutoR 0.10 0.10
2 identity
Forming 43 equations: 31 dense
Initial updates will be shrunk by factor 0.387
LogL= 96.2577 S2= 0.19830E-01 82 df 0.10000 0.10000 1.0000
0.10000 0.10000
LogL= 107.380 S2= 0.17230E-01 82 df 0.21167 0.10000E-01 1.0000
0.18847 0.27718
LogL= 112.476 S2= 0.19496E-01 82 df 0.12292 0.10000E-02 1.0000
0.29283 0.37269
LogL= 113.324 S2= 0.23731E-01 82 df 0.10749 0.10000E-03 1.0000
0.35410 0.40540
LogL= 113.357 S2= 0.25198E-01 82 df 0.10322 0.10000E-04 1.0000
0.36859 0.41021
LogL= 113.357 S2= 0.25306E-01 82 df 0.10422 0.10000E-06 1.0000
0.36964 0.41051
Final parameter values 0.10382 0.10000E-06 1.0000
0.37045 0.41085
Source Model terms Gamma Component Compnt/StndErr
spl(ESTAB) 4 4 0.103817 0.262719E-02 0.64 P
Variety.spl(ESTAB) 12 12 0.100000E-06 0.253061E-08 3.11 B
Variance 108 82 1.00000 0.253061E-01 3.11 P
Residual AR=AutoR 54 0.370449 0.370449 3.65 U
Residual AR=AutoR 54 0.410850 0.410850 4.05 U
WARNING: Code B - fixed at a boundary (!GP)
C - Constrained by user (!CON)
S - Singular Information matrix
S means there is no information in the data for this parameter.
Very small components with Comp/SE ratios of zero sometimes indicate poor
scaling. Consider rescaling the design matrix in such cases
Fitted Spline 6 (X) for spl(ESTAB)
27.040 51.208 70.956 93.893 116.031 131.900
Fitted Spline 6 (Y) for spl(ESTAB) 1
0.029947 -0.024106 -0.022770 0.000291 0.011445 0.005194
' 0.03
0.02
', 0.01
' 0.00
-0.01
, -0.02
' -0.03
-0.04
Fitted Spline 6 (Y) for Variety.spl(ESTAB) 1
-0.34E-07 0.20E-08 0.35E-07 0.33E-07 0.72E-09 -0.37E-07
'''''' 0.00
0.00
0.00
-0.01
-0.01
-0.01
-0.01
-0.01
Fitted Spline 6 (Y) for Variety.spl(ESTAB) 2
0.84E-07 -0.32E-07 -0.76E-07 -0.47E-07 0.13E-07 0.58E-07
'''''' 0.00
0.00
0.00
-0.01
-0.01
-0.01
-0.01
-0.01
Fitted Spline 6 (Y) for Variety.spl(ESTAB) 3
-0.22E-07 0.72E-08 0.20E-07 0.14E-07 -0.32E-08 -0.16E-07
'''''' 0.00
0.00
0.00
-0.01
-0.01
-0.01
-0.01
-0.01
Solution Standard Error T-value T-prev
12 c(Variety).ESTAB 2 1.87 1.87 [DF
F_inc F_all]
1 0.524790E-03 0.449518E-03 1.17
2 -0.860156E-03 0.450230E-03 -1.91 -1.79
8 ESTAB 1 14.23 15.25 [DF
F_inc F_all]
3 -0.182029E-02 0.466187E-03 -3.90
11 c(Variety) 2 0.29 1.28 0.5934E-01 [DF F_i
F_a SED]
MERRIT -0.338268E-01 0.349969E-01 -0.97
85SO46-37 0.554641E-01 0.347685E-01 1.60 1.45
10 mv_estimates 20 13.06 11.56 [DF
F_inc F_all]
6 -0.876066 0.107972 -8.11
7 -0.786603 0.105018 -7.49 0.64
8 -1.01553 0.124373 -8.17 -1.40
9 -1.05113 0.132126 -7.96 -0.27
10 -1.04928 0.146039 -7.18 0.01
11 -1.06321 0.152003 -6.99 -0.10
12 -1.06760 0.157684 -6.77 -0.03
13 -1.07495 0.161254 -6.67 -0.05
14 -1.07946 0.164125 -6.58 -0.03
15 -1.08415 0.166177 -6.52 -0.03
16 -1.08773 0.167764 -6.48 -0.03
17 -1.09098 0.168956 -6.46 -0.02
18 -1.09365 0.169877 -6.44 -0.02
19 -1.09597 0.170587 -6.42 -0.02
20 -1.09793 0.171141 -6.42 -0.01
21 -1.09960 0.171575 -6.41 -0.01
22 -1.10103 0.171918 -6.40 -0.01
23 -1.10224 0.172191 -6.40 -0.01
24 -1.10327 0.172409 -6.40 -0.01
25 -1.10415 0.172586 -6.40 -0.01
9 mu 1 320.97 370.71 [DF
F_inc F_all]
26 1.14303 0.593665E-01 19.25
13 spl(ESTAB) 4 1.80 1.14 [DF
F_inc F_all]
27 0.578399E-01 0.360540E-01 1.60
28 0.109743E-01 0.360288E-01 0.30 -0.75
29 -0.159594E-01 0.392685E-01 -0.41 -0.42
30 -0.206031E-01 0.474531E-01 -0.43 -0.07
14 Variety.spl(ESTAB) 12 effects fitted
1 possible outliers: see .res file
Finished: 13:00:45.39 LogL Converged
_____________________________________________________________________
N.W. Galwey,
Faculty of Agriculture,
University of Western Australia,
Nedlands, WA 6709, Australia.
Tel.: +61 9 380 1959 (direct line)
+61 9 380 2554 (switchboard)
Fax: +61 9 380 1108