1 Smooth trend in one dimension.
1.1 Oats field trial
As a first example we will use an oats field trial from the
agridat package. There were 24 varieties in 3 replicates,
each consisting of 6 incomplete blocks of 4 plots. The plots were laid
out in a single row.
## Load data.
data(john.alpha, package = "agridat")
head(john.alpha)
#> plot rep block gen yield row col
#> 1 1 R1 B1 G11 4.1172 1 1
#> 2 2 R1 B1 G04 4.4461 2 1
#> 3 3 R1 B1 G05 5.8757 3 1
#> 4 4 R1 B1 G22 4.5784 4 1
#> 5 5 R1 B2 G21 4.6540 5 1
#> 6 6 R1 B2 G10 4.1736 6 1In the following subsections we will use two methods to correct for spatial trend, to show some of the options of the package.
1.1.1 Modelling spatial trend using P-splines
In this subsection we will illustrate how the package can be used to fit mixed model P-splines, for details see Boer, Piepho, and Williams (2020).
In the following mixed model we include rep and gen as fixed effect,
and we use spl1D to model a one dimensional P-spline (Eilers and Marx
1996) with 100 segments, and the default choice of cubical
B-splines and second order differences:
## Fit mixed model with fixed and spline part.
obj1 <- LMMsolve(fixed = yield ~ rep + gen,
spline = ~spl1D(x = plot, nseg = 100),
data = john.alpha)A high number of segments can be used for splines in one dimension, as the corresponding mixed model equations are sparse, and therefore can be solved fast (Smith 1995; Furrer and Sain 2010).
We can obtain a table with effective dimensions (see e.g. Rodríguez-Álvarez et al. (2018)) and penalties (the
precision parameters \(\psi_i\) and
\(\phi_i\)) using the
summary function:
summary(obj1)
#> Table with effective dimensions and penalties:
#>
#> Term Effective Model Nominal Ratio Penalty
#> (Intercept) 1.00 1 1 1.00 0.00
#> rep 2.00 2 2 1.00 0.00
#> gen 23.00 23 23 1.00 0.00
#> lin(plot) 1.00 1 1 1.00 0.00
#> s(plot) 4.18 103 45 0.09 3431.87
#> residual 40.82 72 45 0.91 13.28
#>
#> Total Effective Dimension: 72The effective dimension gives a good balance between model complexity
and fit to the data for the random terms in the model. In the table
above the first four terms are fixed effects and not penalized, and
therefore the effective dimension is equal to the number of parameters
in the model. The splF is the fixed part of the spline, the
linear trend. The term s(plot) is a random effect, with
effective dimension 4.2, indicating that is important to correct for
spatial trend.
The estimated genetic effects are given by the coef
function:
genEff <- coef(obj1)$gen
head(genEff, 4)
#> gen_G01 gen_G02 gen_G03 gen_G04
#> 0.0000000 -0.5699756 -1.5231789 -0.4593992The first genotype (G01) is the reference, as genotypes were modelled as fixed effect in the model.
The smooth trend with the standard errors along the field on a dense grid of 1000 points can be obtained as follows:
## Extract smooth trend from mixed model.
plotDat1 <- obtainSmoothTrend(obj1, grid = 1000, includeIntercept = TRUE)
head(plotDat1)
#> plot ypred se
#> 1 1.000000 5.036391 0.2462874
#> 2 1.071071 5.035374 0.2448789
#> 3 1.142142 5.034355 0.2434968
#> 4 1.213213 5.033335 0.2421411
#> 5 1.284284 5.032314 0.2408118
#> 6 1.355355 5.031290 0.2395091The trend can then be plotted.
## Plot smooth trend.
ggplot(data = plotDat1, aes(x = plot, y = ypred)) +
geom_line(color = "red", linewidth = 1) +
labs(title = "Smooth spatial trend oats data", x = "plotnr", y = "yield") +
theme(panel.grid = element_blank())
1.1.2 Modelling spatial
trend using random and ginverse
arguments.
Another way to correct for spatial trend is using the Linear Variance (LV) model, which is closely connected to the P-splines model (Boer, Piepho, and Williams 2020). First we need to define the precision matrix for the LV model, see Appendix in Boer, Piepho, and Williams (2020) for details:
## Add plot as factor.
john.alpha$plotF <- as.factor(john.alpha$plot)
## Define the precision matrix, see eqn (A2) in Boer et al (2020).
N <- nrow(john.alpha)
cN <- c(1 / sqrt(N - 1), rep(0, N - 2), 1 / sqrt(N - 1))
D <- diff(diag(N), diff = 1)
Delta <- 0.5 * crossprod(D)
LVinv <- 0.5 * (2 * Delta + cN %*% t(cN))
## Add LVinv to list, with name corresponding to random term.
lGinv <- list(plotF = LVinv)Given the precision matrix for the LV model we can define the model
in LMMsolve using the random and ginverse
arguments:
## Fit mixed model with first degree B-splines and first order differences.
obj2 <- LMMsolve(fixed = yield ~ rep + gen,
random = ~plotF,
ginverse = lGinv,
data = john.alpha)The deviance for the LV-model is 54.49 and the variances
summary(obj2, which = "variances")
#> Table with variances:
#>
#> VarComp Variance
#> plotF 0.01
#> residual 0.06as reported in Boer, Piepho, and Williams (2020), Table 1.
1.2 Model biomass as function of time.
In this section we show an example of mixed model P-splines to fit biomass as function of time. As an example we use wheat data simulated with the crop growth model APSIM. This data set is included in the package. For details on this simulated data see Bustos-Korts et al. (2019).
data(APSIMdat)
head(APSIMdat)
#> env geno das biomass
#> 1 Emerald_1993 g001 20 65.57075
#> 2 Emerald_1993 g001 21 60.70499
#> 3 Emerald_1993 g001 22 74.06247
#> 4 Emerald_1993 g001 23 63.73951
#> 5 Emerald_1993 g001 24 101.88005
#> 6 Emerald_1993 g001 25 96.84971The first column is the environment, Emerald in 1993, the second column the simulated genotype (g001), the third column is days after sowing (das), and the last column is the simulated biomass with medium measurement error.
The model can be fitted with
The effective dimensions are:
summary(obj2)
#> Table with effective dimensions and penalties:
#>
#> Term Effective Model Nominal Ratio Penalty
#> (Intercept) 1.00 1 1 1.00 0.00
#> lin(das) 1.00 1 1 1.00 0.00
#> s(das) 6.56 203 119 0.06 0.01
#> residual 112.44 121 119 0.94 0.00
#>
#> Total Effective Dimension: 121The fitted smooth trend can be obtained as explained before, with standard error bands in blue:
plotDat2 <- obtainSmoothTrend(obj2, grid = 1000, includeIntercept = TRUE)
ggplot(data = APSIMdat, aes(x = das, y = biomass)) +
geom_point(size = 1.2) +
geom_line(data = plotDat2, aes(y = ypred), color = "red", linewidth = 1) +
geom_line(data = plotDat2, aes(y = ypred-2*se), col='blue', linewidth = 1) +
geom_line(data = plotDat2, aes(y = ypred+2*se), col='blue', linewidth = 1) +
labs(title = "APSIM biomass as function of time",
x = "days after sowing", y = "biomass (kg)") +
theme(panel.grid = element_blank())
The growth rate (first derivative) as function of time can be
obtained using deriv = 1 in function
obtainSmoothTrend:
plotDatDt <- obtainSmoothTrend(obj2, grid = 1000, deriv = 1)
ggplot(data = plotDatDt, aes(x = das, y = ypred)) +
geom_line(color = "red", linewidth = 1) +
labs(title = "APSIM growth rate as function of time",
x = "days after sowing", y = "growth rate (kg/day)") +
theme(panel.grid = element_blank())
