The aim of the `LMMsolver`

package is to provide an
efficient and flexible system to estimate variance components using
restricted maximum likelihood or REML (Patterson and Thompson 1971), for
models where the mixed model equations are sparse (Boer 2023). An
example of an application is using splines to model spatial (Rodríguez-Álvarez
et al. 2018; Boer, Piepho, and Williams
2020) or temporal (Bustos-Korts et al. 2019)
trends. Another example is mixed model Quantitative Trait Locus (QTL)
analysis for multiparental populations, allowing for heterogeneous
residual variance and design matrices with Identity-By-Descent (IBD)
probabilities (Li et al.
2021).

A Linear Mixed Model (LMM) has the form

\[ y = X \beta + Z u + e, \quad u \sim N(0,G), \quad e \sim N(0,R) \] where \(y\) is a vector of observations, \(\beta\) is a vector with the fixed effects, \(u\) is a vector with the random effects, and \(e\) a vector of random residuals. \(X\) and \(Z\) are design matrices.

The `LMMsolve`

package can fit models where the matrices
\(G^{-1}\) and \(R^{-1}\) are a linear combination of
precision matrices \(Q_{G,i}\) and
\(Q_{R,i}\): \[
G^{-1} = \sum_{i} \psi_i Q_{G,i} \;, \quad R^{-1} = \sum_{i} \phi_i
Q_{R,i}
\] where the precision parameters \(\psi_i\) and \(\phi_i\) are estimated using REML. For most
standard mixed models \(1/{\psi_i}\)
are the variance components and \(1/{\phi_i}\) the residual variances. We use
a formulation in terms of precision parameters to allow for non-standard
mixed models using tensor product splines introduced in Rodríguez-Álvarez et al. (2015).

If the matrices \(G^{-1}\) and \(R^{-1}\) are sparse, the mixed model
equations can be solved using efficient sparse matrix algebra
implemented in the `spam`

package (Furrer and Sain 2010). To calculate
the derivatives of the log-likelihood in an efficient way, the automatic
differentiation of the Cholesky matrix (Smith 1995; Boer
2023) was implemented in C++ using the `Rcpp`

package (Eddelbuettel and Balamuta
2018).

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 1
```

In the following subsections we will use two methods to correct for spatial trend, to show some of the options of the package.

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: 72
```

The 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 -0.4593992
```

The 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 0.2395091
```

The trend can then be plotted.

`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 0.06
```

as reported in Boer, Piepho, and Williams (2020), Table 1.

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 96.84971
```

The 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: 121
```

The 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`

:

For two-dimensional mixed P-splines we use the model defined in Rodríguez-Álvarez et al. (2015). As an example we use
the `USprecip`

data set in the `spam`

package
(Furrer and Sain
2010), analysed in Rodríguez-Álvarez
et al. (2015).

```
## Get precipitation data from spam
data(USprecip, package = "spam")
## Only use observed data
USprecip <- as.data.frame(USprecip)
USprecip <- USprecip[USprecip$infill == 1, ]
```

The two-dimensional P-spline can be defined with the
`spl2D()`

function, and with longitude and latitude as
covariates. The number of segments chosen here is equal to the number of
segments used in Rodríguez-Álvarez et al. (2015).

```
obj3 <- LMMsolve(fixed = anomaly ~ 1,
spline = ~spl2D(x1 = lon, x2 = lat, nseg = c(41, 41)),
data = USprecip)
```

The summary function gives a table with the effective dimensions and the penalty parameters:

```
summary(obj3)
#> Table with effective dimensions and penalties:
#>
#> Term Effective Model Nominal Ratio Penalty
#> (Intercept) 1.00 1 1 1.00 0.00
#> lin(lon, lat) 3.00 3 3 1.00 0.00
#> s(lon) 302.60 1936 1932 0.16 0.26
#> s(lat) 409.09 1936 1932 0.21 0.08
#> residual 5190.31 5906 5902 0.88 13.53
#>
#> Total Effective Dimension: 5906
```

A plot for the smooth trend can be obtained in a similar way as for the one-dimensional examples:

```
plotDat3 <- obtainSmoothTrend(obj3, grid = c(200, 300), includeIntercept = TRUE)
plotDat3 <- sf::st_as_sf(plotDat3, coords = c("lon", "lat"))
usa <- sf::st_as_sf(maps::map("usa", regions = "main", plot = FALSE))
sf::st_crs(usa) <- sf::st_crs(plotDat3)
intersection <- sf::st_intersects(plotDat3, usa)
plotDat3 <- plotDat3[!is.na(as.numeric(intersection)), ]
ggplot(usa) +
geom_sf(color = NA) +
geom_tile(data = plotDat3,
mapping = aes(geometry = geometry, fill = ypred),
linewidth = 0,
stat = "sf_coordinates") +
scale_fill_gradientn(colors = topo.colors(100))+
labs(title = "Precipitation (anomaly)",
x = "Longitude", y = "Latitude") +
coord_sf() +
theme(panel.grid = element_blank())
```

Instead of using the `grid`

argument, `newdata`

can be used to make predictions for locations specified in a
`data.frame`

:

```
## Predictions for new data, using city coordinates from maps package.
data(us.cities, package = "maps")
## Column names have to match column names used for fitting the model.
colnames(us.cities)[colnames(us.cities) == "long"] <- "lon"
## Select columns name, lat and lon
us.cities <- us.cities[, c(1,4,5)]
head(us.cities)
#> name lat lon
#> 1 Abilene TX 32.45 -99.74
#> 2 Akron OH 41.08 -81.52
#> 3 Alameda CA 37.77 -122.26
#> 4 Albany GA 31.58 -84.18
#> 5 Albany NY 42.67 -73.80
#> 6 Albany OR 44.62 -123.09
pred3 <- obtainSmoothTrend(obj3, newdata = us.cities, includeIntercept = TRUE)
head(pred3)
#> name lat lon ypred se
#> 1 Abilene TX 32.45 -99.74 -0.50934296 0.12154785
#> 2 Akron OH 41.08 -81.52 1.04632673 0.08791266
#> 3 Alameda CA 37.77 -122.26 1.15536050 0.06550927
#> 4 Albany GA 31.58 -84.18 1.00461677 0.12751835
#> 5 Albany NY 42.67 -73.80 0.09537157 0.05353887
#> 6 Albany OR 44.62 -123.09 0.79867839 0.09363536
```

In QTL-mapping for multiparental populations the Identity-By-Descent (IBD) probabilities are used as genetic predictors in the mixed model (Li et al. 2021). The following simulated example is for illustration. It consists of three parents (A, B, and C), and two crosses AxB, and AxC. AxB is a population of 100 Doubled Haploids (DH), AxC of 80 DHs. The probabilities, pA, pB, and pC, are for a position on the genome close to a simulated QTL. This simulated data is included in the package.

```
## Load data for multiparental population.
data(multipop)
head(multipop)
#> cross ind pA pB pC pheno
#> 1 AxB AxB0001 0.17258816 0.82741184 0 9.890637
#> 2 AxB AxB0002 0.82170793 0.17829207 0 6.546568
#> 3 AxB AxB0003 0.95968439 0.04031561 0 7.899249
#> 4 AxB AxB0004 0.96564081 0.03435919 0 4.462866
#> 5 AxB AxB0005 0.04838734 0.95161266 0 5.207757
#> 6 AxB AxB0006 0.95968439 0.04031561 0 5.265580
```

The residual (genetic) variances for the two populations can be
different. Therefore we need to allow for heterogeneous residual
variances, which can be defined by using the `residual`

argument in `LMMsolve`

:

```
## Fit null model.
obj4 <- LMMsolve(fixed = pheno ~ cross,
residual = ~cross,
data = multipop)
dev4 <- deviance(obj4)
```

The QTL-probabilities are defined by the columns pA, pB, pC, and can
be included in the random part of the mixed model by using the
`group`

argument:

```
## Fit alternative model - include QTL with probabilities defined in columns 3:5
lGrp <- list(QTL = 3:5)
obj5 <- LMMsolve(fixed = pheno ~ cross,
group = lGrp,
random = ~grp(QTL),
residual = ~cross,
data = multipop)
dev5 <- deviance(obj5)
```

The approximate \(-log10(p)\) value is given by

```
## Deviance difference between null and alternative model.
dev <- dev4 - dev5
## Calculate approximate p-value.
minlog10p <- -log10(0.5 * pchisq(dev, 1, lower.tail = FALSE))
round(minlog10p, 2)
#> [1] 8.76
```

The estimated QTL effects of the parents A, B, and C are given by:

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