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