Multi-environment trial of lupin, multiple varieties and densities

Yield of 9 varieties of lupin at different planting densities across 2 years and multiple locations.

Format

gen: genotype, 9 varieties
site: site, 11 levels
rep: rep, 2-3 levels
rate: seeding rate in plants/m^2
row: row
col: column
serp: factor of 4 levels for serpentine seeding effect
linrow: centered row position as a numeric variate (row-8.5)/10
lincol: centered column position as a numeric variate (col-3.5)
linrate: linear effect of seedrate, scaled (seedrate-41.92958)/10
yield: yield in tons/hectare
year: year, 1991-1992
loc: location

Details

Nine varieties of lupin were tested for yield response to plant density at 11 sites. The target density in 1991 was 10, 20, ..., 60 plants per m^2, and in 1992 was 20, 30, ..., 70 plants per m^2.

Plot dimensions are not given.

The variety Myallie was grown only in 1992.

Each site had 2 reps in 1991 and 3 reps in 1992. Each rep was laid out as a factorial RCB design; one randomization was used for all sites in 1991 and one (different) randomization was used for all sites in 1992. (This was confirmed with the principal investigator.)

In 1991 at the Mt. Barker location, the data for columns 5 and 6 was discarded due to problems with weeds.

Variety 'Myallie' was called '84L:439' in Verbyla 1997.

The year of release for the varieties is

Unicrop	1973
Illyarrie	1979
Yandee	1980
Danja	1986
Gungurru	1988
Yorrel	1989
Warrah	1989
Merrit	1991
Myallie	1995

Data retrieved Oct 2010 from https://www.blackwellpublishers.co.uk/rss. (No longer available).

Used with permission of Blackwell Publishing.

Source

Arunas P. Verbyla and Brian R. Cullis and Michael G. Kenward and Sue J. Welham, (1999). The analysis of designed experiments and longitudinal data by using smoothing splines. Appl. Statist., 48, 269--311. https://doi.org/10.1111/1467-9876.00154

Arunas P. Verbyla and Brian R. Cullis and Michael G. Kenward and Sue J. Welham, (1997). The analysis of designed experiments and longitudinal data by using smoothing splines. University of Adelaide, Department of Statistics, Research Report 97/4. https://https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.56.808

Examples

# \dontrun{
  
  library(agridat)
  data(verbyla.lupin)
  dat <- verbyla.lupin

  # The same RCB randomization was used at all sites in each year
  libs(desplot)
  desplot(dat, gen~col+row|site,
          out1=rep, num=rate,
          # aspect unknown
          main="verbyla.lupin - experiment design")


  # Figure 3 of Verbyla
  libs(lattice)
  foo <- xyplot(yield ~ rate|loc*gen, data=dat, subset=year==92,
                type=c('p','smooth'), cex=.5,
                main="verbyla.lupin: 1992 yield response curves",
                xlab="Seed rate (plants/m^2)",
                ylab="Yield (tons/ha)",
                strip=strip.custom(par.strip.text=list(cex=.7)))

  libs(latticeExtra) # for useOuterStrips
  useOuterStrips(foo,
                 strip=strip.custom(par.strip.text=list(cex=.7)),
                 strip.left=strip.custom(par.strip.text=list(cex=.7)))


  # ----------

  libs(asreml,lucid)
  # asreml 4
    
  # We try to reproduce the analysis of Verbyla 1999.
  # May not be exactly the same, but is pretty close.
  
  # Check nlevels for size of random-coefficient structures
  # length(with(dat, table(gen)))      # 9 varieties for RC1
  # length(with(dat, table(gen,site))) # 99 site:gen combinations for RC2
  
  # Make row and col into factors
  dat <- transform(dat, colf=factor(col), rowf=factor(row))
  # sort for asreml
  dat <- dat[order(dat$site, dat$rowf, dat$colf),]
    
  # Make site names more useful for plots
  # dat <- transform(dat, site=factor(paste0(year,".",substring(loc,1,4))))
    
  # Initial model from top of Verbyla table 9.
  m0 <- asreml(yield ~ 1
               + site
               + linrate
               + site:linrate,
               data = dat,
               random = ~ spl(rate)
               + dev(rate)
               + site:spl(rate)
               + site:dev(rate)
               + str(~gen+gen:linrate, ~us(2):id(9)) # RC1
               + gen:spl(rate)
               + gen:dev(rate)
               + str(~site:gen+site:gen:linrate, ~us(2):id(99)) # RC2
               + site:gen:spl(rate)
               + site:gen:dev(rate),
               residual = ~ dsum( ~ ar1(rowf):ar1(colf)|site) # Spatial AR1 x AR1
               )
#> Multi-section model using the sigma parameterization.
#> ASReml 4.1.0 Fri Dec 11 17:50:41 2020
#> Spline: design points closer than 0.006 have been merged.
#> Warning: US updates modified 2 times in iteration 1 to remain positive definite.
#>           LogLik        Sigma2     DF     wall    cpu
#>  1      1035.181           1.0   1370 17:50:41    0.5 (16 restrained)
#>  2      1265.567           1.0   1370 17:50:42    0.4 (15 restrained)
#> Warning: US updates modified 2 times in iteration 3 to remain positive definite.
#>  3      1433.305           1.0   1370 17:50:42    0.3 (13 restrained)
#> Warning: US updates modified 1 times in iteration 4 to remain positive definite.
#>  4      1491.898           1.0   1370 17:50:42    0.3 (10 restrained)
#>  5      1530.115           1.0   1370 17:50:43    0.3 (5 restrained)
#>  6      1535.456           1.0   1370 17:50:43    0.3 (5 restrained)
#>  7      1537.134           1.0   1370 17:50:43    0.3 (3 restrained)
#>  8      1537.495           1.0   1370 17:50:44    0.3 (1 restrained)
#>  9      1537.559           1.0   1370 17:50:44    0.3
#> 10      1537.563           1.0   1370 17:50:44    0.3
#> Warning: US variance structures were modified in 5 instances to make them positive definite
#> Warning: Some components changed by more than 1% on the last iteration.
  m0 <- update(m0)
#> Multi-section model using the sigma parameterization.
#> ASReml 4.1.0 Fri Dec 11 17:50:44 2020
#> Spline: design points closer than 0.006 have been merged.
#>           LogLik        Sigma2     DF     wall    cpu
#>  1      1537.564           1.0   1370 17:50:45    0.4
#>  2      1537.564           1.0   1370 17:50:45    0.3
#> Warning: Some components changed by more than 1% on the last iteration.
  m0 <- update(m0)
#> Multi-section model using the sigma parameterization.
#> ASReml 4.1.0 Fri Dec 11 17:50:45 2020
#> Spline: design points closer than 0.006 have been merged.
#>           LogLik        Sigma2     DF     wall    cpu
#>  1      1537.564           1.0   1370 17:50:46    0.4
#>  2      1537.564           1.0   1370 17:50:46    0.3
#> Warning: Some components changed by more than 1% on the last iteration.
  m0 <- update(m0)
#> Multi-section model using the sigma parameterization.
#> ASReml 4.1.0 Fri Dec 11 17:50:46 2020
#> Spline: design points closer than 0.006 have been merged.
#>           LogLik        Sigma2     DF     wall    cpu
#>  1      1537.564           1.0   1370 17:50:47    0.4
#>  2      1537.564           1.0   1370 17:50:47    0.3
#> Warning: Some components changed by more than 1% on the last iteration.
  m0 <- update(m0)
#> Multi-section model using the sigma parameterization.
#> ASReml 4.1.0 Fri Dec 11 17:50:47 2020
#> Spline: design points closer than 0.006 have been merged.
#>           LogLik        Sigma2     DF     wall    cpu
#>  1      1537.564           1.0   1370 17:50:47    0.4
#>  2      1537.564           1.0   1370 17:50:48    0.3
#> Warning: Some components changed by more than 1% on the last iteration.
  m0 <- update(m0)
#> Multi-section model using the sigma parameterization.
#> ASReml 4.1.0 Fri Dec 11 17:50:48 2020
#> Spline: design points closer than 0.006 have been merged.
#>           LogLik        Sigma2     DF     wall    cpu
#>  1      1537.564           1.0   1370 17:50:48    0.4
#>  2      1537.564           1.0   1370 17:50:49    0.3

  # Variograms match Verbyla 1999 figure 7 (scale slightly different)
  plot(varioGram(m0), xlim=c(1:19), zlim=c(0,2),
       main="verbyla.lupin - variogram by site")
  
    # Sequence of models in Verbyla 1999 table 10
  m1 <- update(m0, fixed= ~ .
               + at(site, c(2,5,6,8,9,10)):lincol
               + at(site, c(3,5,7,8)):linrow
               + at(site, c(2,3,5,7,8,9,11)):serp
               , random = ~ .
               + at(site, c(3,6,7,9)):rowf
               + at(site, c(1,2,3,9,10)):colf
               + at(site, c(5,7,8,10)):units)
#> Multi-section model using the sigma parameterization.
#> ASReml 4.1.0 Fri Dec 11 17:50:49 2020
#> Spline: design points closer than 0.006 have been merged.
#> Warning: US updates modified 1 times in iteration 1 to remain positive definite.
#>           LogLik        Sigma2     DF     wall    cpu
#>  1      1300.766           1.0   1339 17:50:50    0.8 (15 restrained)
#>  2      1373.661           1.0   1339 17:50:51    0.6 (11 restrained)
#> Warning: US updates modified 1 times in iteration 3 to remain positive definite.
#>  3      1521.164           1.0   1339 17:50:51    0.6 (12 restrained)
#> Warning: US updates modified 1 times in iteration 4 to remain positive definite.
#>  4      1550.613           1.0   1339 17:50:52    0.7 (7 restrained)
#>  5      1571.582           1.0   1339 17:50:53    0.7 (3 restrained)
#>  6      1578.652           1.0   1339 17:50:53    0.6 (2 restrained)
#>  7      1582.262           1.0   1339 17:50:54    0.6 (1 restrained)
#>  8      1582.866           1.0   1339 17:50:54    0.6
#>  9      1582.912           1.0   1339 17:50:55    0.6
#> 10      1582.921           1.0   1339 17:50:56    0.6
#> 11      1582.925           1.0   1339 17:50:56    0.6
#> 12      1582.928           1.0   1339 17:50:57    0.6
#> 13      1582.930           1.0   1339 17:50:57    0.6
#> Warning: US variance structures were modified in 3 instances to make them positive definite
#> Warning: Log-likelihood not converged
#> Warning: Some components changed by more than 1% on the last iteration.
  m1 <- update(m1)
#> Multi-section model using the sigma parameterization.
#> ASReml 4.1.0 Fri Dec 11 17:50:58 2020
#> Spline: design points closer than 0.006 have been merged.
#>           LogLik        Sigma2     DF     wall    cpu
#>  1      1582.932           1.0   1339 17:50:58    0.8
#>  2      1582.932           1.0   1339 17:50:59    0.6
#>  3      1582.933           1.0   1339 17:51:00    0.7
#>  4      1582.933           1.0   1339 17:51:00    0.7
#>  5      1582.934           1.0   1339 17:51:01    0.7
#>  6      1582.934           1.0   1339 17:51:02    0.7
#> Warning: Some components changed by more than 1% on the last iteration.
  
  m2 <- update(m1,
               random = ~ .
               - site:gen:spl(rate) - site:gen:dev(rate))
#> Multi-section model using the sigma parameterization.
#> ASReml 4.1.0 Fri Dec 11 17:51:02 2020
#> Spline: design points closer than 0.006 have been merged.
#>           LogLik        Sigma2     DF     wall    cpu
#>  1      1582.935           1.0   1339 17:51:02    0.3 (1 restrained)
#>  2      1582.935           1.0   1339 17:51:03    0.2 (1 restrained)
#>  3      1582.935           1.0   1339 17:51:03    0.3
  
  m3 <- update(m2,
               random = ~ .
               - site:dev(rate) - gen:dev(rate))
#> Multi-section model using the sigma parameterization.
#> ASReml 4.1.0 Fri Dec 11 17:51:03 2020
#> Spline: design points closer than 0.006 have been merged.
#>           LogLik        Sigma2     DF     wall    cpu
#>  1      1582.936           1.0   1339 17:51:03    0.2 (1 restrained)
#>  2      1582.936           1.0   1339 17:51:03    0.1 (1 restrained)
#>  3      1582.936           1.0   1339 17:51:03    0.1
  
  m4 <- update(m3,
               random = ~ .
               - dev(rate))
#> Multi-section model using the sigma parameterization.
#> ASReml 4.1.0 Fri Dec 11 17:51:04 2020
#> Spline: design points closer than 0.006 have been merged.
#>           LogLik        Sigma2     DF     wall    cpu
#>  1      1582.936           1.0   1339 17:51:04    0.1 (1 restrained)
#>  2      1582.936           1.0   1339 17:51:04    0.1 (1 restrained)
#>  3      1582.936           1.0   1339 17:51:04    0.1
  
  m5 <- update(m4,
               random = ~ .
               - at(site, c(5,7,8,10)):units + at(site, c(5,7,8)):units)
#> Multi-section model using the sigma parameterization.
#> ASReml 4.1.0 Fri Dec 11 17:51:04 2020
#> Spline: design points closer than 0.006 have been merged.
#>           LogLik        Sigma2     DF     wall    cpu
#>  1      1582.936           1.0   1339 17:51:04    0.1
#>  2      1582.936           1.0   1339 17:51:04    0.1
  
  # Variance components are a pretty good match to Verbyla 1997, table 15
  libs(lucid)
  vc(m5)
#>                               effect  component  std.error z.ratio bound %ch
#>                            spl(rate)  0.001288  0.001019     1.3       P 0  
#>                       spl(rate):site  0.0001556 0.0001227    1.3       P 0  
#>            gen+gen:linrate!us(2)_1:1  0.005445  0.003733     1.5       P 0  
#>            gen+gen:linrate!us(2)_2:1  0.001004  0.0007598    1.3       P 0  
#>            gen+gen:linrate!us(2)_2:2  0.0003661 0.0002168    1.7       P 0  
#>                        spl(rate):gen  0.0002057 0.0001326    1.6       P 0  
#>  site:gen+site:gen:linrate!us(2)_1:1  0.01881   0.003402     5.5       P 0  
#>  site:gen+site:gen:linrate!us(2)_2:1  0.00175   0.0004706    3.7       P 0  
#>  site:gen+site:gen:linrate!us(2)_2:2  0.000167  0.00008968   1.9       P 0  
#>                   at(site, S03):rowf  0.002594  0.006235     0.42      P 0  
#>                   at(site, S06):rowf  0.001127  0.00226      0.5       P 0  
#>                   at(site, S07):rowf  0.00019   0.002012     0.094     P 0.1
#>                   at(site, S09):rowf  0.004526  0.002323     1.9       P 0  
#>                   at(site, S01):colf  0.01041   0.00854      1.2       P 0  
#>                   at(site, S02):colf  0.02498   0.02076      1.2       P 0  
#>                   at(site, S03):colf  0.08167   0.07359      1.1       P 0  
#>                   at(site, S09):colf  0.004083  0.003688     1.1       P 0  
#>                   at(site, S10):colf  0.0004588 0.001058     0.43      P 0.2
#>                  at(site, S05):units  0.004378  0.007786     0.56      P 0  
#>                  at(site, S07):units  0.01323   0.002712     4.9       P 0  
#>                  at(site, S08):units  0.01302   0.004202     3.1       P 0  
#>                           site_S01!R  0.03273   0.005654     5.8       P 0  
#>                    site_S01!rowf!cor  0.1852    0.1417       1.3       U 0  
#>                    site_S01!colf!cor  0.1407    0.121        1.2       U 0  
#>                           site_S02!R  0.03999   0.00753      5.3       P 0  
#>                    site_S02!rowf!cor  0.3103    0.135        2.3       U 0  
#>                    site_S02!colf!cor  0.1673    0.1369       1.2       U 0  
#>                           site_S03!R  0.0505    0.0157       3.2       P 0  
#>                    site_S03!rowf!cor  0.4707    0.179        2.6       U 0  
#>                    site_S03!colf!cor -0.01851   0.2423      -0.076     U 0  
#>                           site_S04!R  0.03301   0.005216     6.3       P 0  
#>                    site_S04!rowf!cor -0.04041   0.1159      -0.35      U 0  
#>                    site_S04!colf!cor  0.1109    0.1261       0.88      U 0  
#>                           site_S05!R  0.01772   0.008592     2.1       P 0  
#>                    site_S05!rowf!cor  0.5623    0.2644       2.1       U 0  
#>                    site_S05!colf!cor  0.08337   0.2043       0.41      U 0  
#>                           site_S06!R  0.02409   0.003996     6         P 0  
#>                    site_S06!rowf!cor  0.3216    0.08428      3.8       U 0  
#>                    site_S06!colf!cor  0.2214    0.1186       1.9       U 0  
#>                           site_S07!R  0.03658   0.02023      1.8       P 0  
#>                    site_S07!rowf!cor  0.9019    0.06323     14         U 0  
#>                    site_S07!colf!cor  0.5703    0.2383       2.4       U 0  
#>                           site_S08!R  0.01813   0.006293     2.9       P 0  
#>                    site_S08!rowf!cor  0.7429    0.1436       5.2       U 0  
#>                    site_S08!colf!cor  0.1768    0.2027       0.87      U 0  
#>                           site_S09!R  0.01725   0.002548     6.8       P 0  
#>                    site_S09!rowf!cor  0.2509    0.09548      2.6       U 0  
#>                    site_S09!colf!cor -0.03375   0.1213      -0.28      U 0  
#>                           site_S10!R  0.01035   0.001703     6.1       P 0  
#>                    site_S10!rowf!cor  0.5222    0.07746      6.7       U 0.1
#>                    site_S10!colf!cor  0.1709    0.09266      1.8       U 0  
#>                           site_S11!R  0.05398   0.006905     7.8       P 0  
#>                    site_S11!rowf!cor  0.1939    0.09649      2         U 0  
#>                    site_S11!colf!cor  0.1034    0.1002       1         U 0  
  .001004/sqrt(.005446*.0003662) # .711 correlation for RC1
#> [1] 0.7109445
  .00175/sqrt(.01881*.000167) # .987 correlation for RC2
#> [1] 0.9873825
  
  # Matches Verbyla 1999 figure 5
  plot(varioGram(m5),
       main="verbyla.lupin - final model variograms",
       xlim=c(1:19), zlim=c(0,1.5))
  
# }