1 Introduction

The purpose of this section is to give users an easy introduction, starting from simple linear regression. Based on simulations we will explain the main functions, the input and the output. First we load the LMMsolver and ggplot2 packages:

library(LMMsolver)
library(ggplot2)

1.1 Linear Regression

We will start with a simple example where the true function is linear in variable \(x\):

  f1 <- function(x) { 0.6 + 0.7*x}

Using this function we simulate data and add normal distributed noise:

set.seed(2016)
n <- 25
x <- seq(0, 1, length = n)
sigma2e <- 0.04
y <- f1(x) + rnorm(n, sd = sqrt(sigma2e))
dat1 <- data.frame(x = x, y = y)

We can fit the data using the LMMsolve function:

obj1 <- LMMsolve(fixed = y ~ x, data = dat1)

We can make predictions using the predict() function:

newdat <- data.frame(x = seq(0, 1, length = 300))
pred1 <- predict(obj1, newdata = newdat, se.fit = TRUE)
# adding the true values for comparison
pred1$y_true <- f1(pred1$x)

Note that for this linear model we could have used the standard lm() function, which will give the same result.

The following plot gives the simulated data with the predictions, and pointwise standard-error bands. The true value is plotted as dashed red line.

ggplot(data = dat1, aes(x = x, y = y)) +
  geom_point(col = "black", size = 1.5) +
  geom_line(data = pred1, aes(y=y_true), color = "red", 
            linewidth = 1, linetype = "dashed") +
  geom_line(data = pred1, aes(y = ypred), color = "blue", linewidth = 1) +
  geom_ribbon(data = pred1, aes(x=x,ymin = ypred-2*se, ymax = ypred+2*se),
              alpha = 0.2, inherit.aes = FALSE) + 
  theme_bw()

1.2 Fitting a non-linear function

In this section we will use the following non-linear function for the simulations:

f2 <- function(x) { 0.3 + 0.4*x + 0.2*sin(20*x) }

The simulated data is generated by the following code

set.seed(12)
n <- 150
x <- seq(0, 1, length = n)
sigma2e <- 0.04
y <- f2(x) + rnorm(n, sd = sqrt(sigma2e))
dat2 <- data.frame(x, y)

We can use the spline argument to fit the non-linear trend:

obj2 <- LMMsolve(fixed = y ~ 1, 
                 spline = ~spl1D(x, nseg = 50), 
                 data = dat2)

where spl1D(x, nseg = 50) defines a mixed model P-splines with 50 segments.

The model fit can be summarized in terms of effective dimensions:

summary(obj2)
#> Table with effective dimensions and penalties: 
#> 
#>         Term Effective Model Nominal Ratio Penalty
#>  (Intercept)      1.00     1       1  1.00     0.0
#>       lin(x)      1.00     1       1  1.00     0.0
#>         s(x)     11.28    53      51  0.22     0.0
#>     residual    136.72   150     148  0.92    30.3
#> 
#>  Total Effective Dimension: 150

The intercept and the slope lin(x) define the linear (or fixed) part of the model, the non-linear (or random) part is defined by s(x), with effective dimension 11.28.

Making predictions on the interval \([0,1]\) and plotting can be done in the same way as for the linear regression example:

newdat <- data.frame(x = seq(0, 1, length = 300))
pred2 <- predict(obj2, newdata = newdat, se.fit = TRUE)
pred2$y_true <- f2(pred2$x)

ggplot(data = dat2, aes(x = x, y = y)) +
  geom_point(col = "black", size = 1.5) +
  geom_line(data = pred2, aes(y = y_true), color = "red", 
            linewidth = 1, linetype ="dashed") +
  geom_line(data = pred2, aes(y = ypred), color = "blue", linewidth = 1) +
  geom_ribbon(data= pred2, aes(x=x, ymin = ypred-2*se, ymax = ypred+2*se),
              alpha=0.2, inherit.aes = FALSE) +
  theme_bw()

1.3 Non-Gaussian data

The LMMsolver package can also be used for non-gaussian data, using the family argument, with default family = gaussian. As an example we use count data using the Poisson distribution, defined by \[ Pr(X=k) = \frac{\lambda^k e^{-\lambda}}{k!} \;, \] with parameter \(\lambda > 0\) and \(k\) is the number of occurrences. More general, the value of the parameter \(lambda\) can depend on another variable time \(x\), for example time. Here we will assume that \(x\) is defined on the interval \([0,1]\) and defined by:

\[ \lambda(x) = 4 + 3x + 4 \sin(7 x) \] Using this function we simulate the following data

set.seed(1234)
n <- 150
x <- seq(0, 1, length=n)
fun_lambda <- function(x) { 4 + 3*x + 4*sin(7*x) }
x <- seq(0, 1, length = n)
y <- rpois(n = n, lambda = fun_lambda(x)) 
dat3 <- data.frame(x = x, y = y)

Now we fit the data with the argument family = poisson():

obj3 <- LMMsolve(fixed = y ~ 1, 
                spline = ~spl1D(x, nseg = 50), 
                family = poisson(), 
                data = dat3)
summary(obj3)
#> Table with effective dimensions and penalties: 
#> 
#>         Term Effective Model Nominal Ratio Penalty
#>  (Intercept)      1.00     1       1  1.00       0
#>       lin(x)      1.00     1       1  1.00       0
#>         s(x)      6.54    53      51  0.13       0
#>     residual    141.46   150     148  0.96       1
#> 
#>  Total Effective Dimension: 150

Making predictions and plotting the data is similar to the Gaussian data we showed before:

newdat <- data.frame(x = seq(0, 1, length = 300))
pred3 <- predict(obj3, newdata = newdat, se.fit = TRUE)
pred3$y_true <- fun_lambda(pred3$x)

ggplot(data = dat3, aes(x = x, y = y)) +
  geom_point(col = "black", size = 1.5) +
  geom_line(data = pred3, aes(y = y_true), color = "red", 
            linewidth = 1, linetype ="dashed") +
  geom_line(data = pred3, aes(y = ypred), color = "blue", linewidth = 1) +
  geom_ribbon(data= pred3, aes(x=x, ymin = ypred-2*se, ymax = ypred+2*se),
              alpha=0.2, inherit.aes = FALSE) +
  theme_bw()

1.4 Smoothing combining two experiments

In this section we will give a bit more complicated example, to show some further options of LMMsolver. Suppose there are two experiments, A and B, with the same true unobserved non-linear function f2(x) as defined before.

The simulated data is given by the following code:

set.seed(1234)
nA <-  50
nB <- 100
mu_A <-  0.10
mu_B <- -0.10
sigma2e_A <- 0.04
sigma2e_B <- 0.10

x1 <- runif(n = nA)
x2 <- runif(n = nB)
y1 <- f2(x1) + rnorm(nA, sd = sqrt(sigma2e_A)) + mu_A
y2 <- f2(x2) + rnorm(nB, sd = sqrt(sigma2e_B)) + mu_B
Experiment <- as.factor(c(rep("A", nA), rep("B", nB)))
dat4 <- data.frame(x = c(x1, x2), y = c(y1,y2), Experiment = Experiment)

Before analyzing the data in further detail a boxplot gives some insight:

ggplot(dat4, aes(x = Experiment, y = y, fill = Experiment)) +  
  geom_boxplot() + 
  geom_point(position = position_jitterdodge(), alpha = 0.3)

Comparing the two experiments we can see that:

There is a clear difference in mean/median between the two experiments. This can be corrected for by adding the argument random = ~Experiment.
The variance in experiment A is smaller than in B. This implies that is important to allow for heterogeneous variances which can be modelled by defining residual = ~Experiment.

The model in LMMsolve() is given by:

obj4 <- LMMsolve(fixed= y ~ 1, 
                 spline = ~spl1D(x, nseg = 50),
                 random = ~Experiment,
                 residual = ~Experiment,
                 data = dat4)

The table of effective dimensions is given by:

summary(obj4)
#> Table with effective dimensions and penalties: 
#> 
#>            Term Effective Model Nominal Ratio Penalty
#>     (Intercept)      1.00     1       1  1.00    0.00
#>          lin(x)      1.00     1       1  1.00    0.00
#>      Experiment      0.93     2       1  0.93   77.98
#>            s(x)      7.89    53      51  0.15    0.00
#>  Experiment_A!R     43.65    50      50  0.87   32.15
#>  Experiment_B!R     95.52   100     100  0.96    9.01
#> 
#>  Total Effective Dimension: 150

And making the predictions:

newdat <- data.frame(x=seq(0, 1, length = 300))
pred4 <- predict(obj4, newdata = newdat, se.fit = TRUE)
pred4$y_true <- f2(pred4$x)
ggplot(data = dat4, aes(x = x, y = y, colour = Experiment)) +
  geom_point(size = 1.5) +
  geom_line(data = pred4, aes(y = y_true), color="red", 
            linewidth = 1, linetype = "dashed") +
  geom_line(data = pred4, aes(y = ypred), color = "blue", linewidth = 1) +
  geom_ribbon(data = pred4, aes(x = x,ymin = ypred-2*se, ymax = ypred+2*se),
              alpha = 0.2, inherit.aes = FALSE) + 
  theme_bw()

The estimated random effects for Experiment can be obtained using the coef() function:

coef(obj4)$Experiment
#> Experiment_A Experiment_B 
#>   0.07719336  -0.07719336

The sum of the effects is equal to zero, as expected for a standard random term.

2 Smooth trends in two dimensions

For two-dimensional mixed P-splines as defined in Boer (2023) we will use two examples. The first example is US precipitation data. The second example models a data set for Sea Surface Temperature (SST) described in Cressie, Sainsbury-Dale, and Zammit-Mangion (2022).

2.1 US precipitation example

As a first 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).

obj5 <- 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(obj5)
#> 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, using the predict() function. First we make predictions on a regular two-dimensional grid:

lon_range <- range(USprecip$lon)
lat_range <- range(USprecip$lat)
newdat <- expand.grid(lon = seq(lon_range[1], lon_range[2], length = 200),
                      lat = seq(lat_range[1], lat_range[2], length = 300))
plotDat5 <- predict(obj5, newdata = newdat)

For plotting the predictions for USA main land we use the maps and sf packages:

plotDat5 <- sf::st_as_sf(plotDat5, coords = c("lon", "lat"))
usa <- sf::st_as_sf(maps::map("usa", regions = "main", plot = FALSE))
sf::st_crs(usa) <- sf::st_crs(plotDat5)
intersection <- sf::st_intersects(plotDat5, usa)
plotDat5 <- plotDat5[!is.na(as.numeric(intersection)), ]

ggplot(usa) + 
  geom_sf(color = NA) +
  geom_tile(data = plotDat5, 
            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())

2.2 Sea Surface Temperatures

The second example using two-dimensional P-splines is for Sea Surface Temperatures (SST) data (Cressie, Sainsbury-Dale, and Zammit-Mangion 2022). In their study they compare a wide range of software packages to analyse the SST data. For the comparison they focus on a region of the ocean known as the Brazil-Malvinas confluence zone, an energetic region of the ocean just off the coast of Argentina and Uruguay, where the warm Brazil current and the cold Malvinas current meet (Cressie, Sainsbury-Dale, and Zammit-Mangion 2022).

They divided the data within this region into a training and a testing data set, each consisting of approximately 8,000 observations.

data(SeaSurfaceTemp)
head(SeaSurfaceTemp, 5)
#>        lon      lat    sst  type
#> 1 -51.5607 -38.2629 289.94 train
#> 2 -55.0255 -49.3163 278.60 train
#> 3 -48.4228 -35.7470 291.51 train
#> 4 -48.7221 -44.2118 282.78 train
#> 5 -54.5217 -47.3870 282.44 train
table(SeaSurfaceTemp$type)
#> 
#>  test train 
#>  7894  7713

First we convert SST from Kelvin to Celsius and split the data in the training and test set:

# convert from Kelvin to Celsius
df <- SeaSurfaceTemp
df$sst <- df$sst - 273.15
### split in training and test set
df_train <- df[df$type == "train", ]
df_test <- df[df$type == "test", ]

The next plot shows the raw data, using the same color palette as in Cressie, Sainsbury-Dale, and Zammit-Mangion (2022).

nasa_palette <- c(
  "#03006d","#02008f","#0000b6","#0001ef","#0000f6","#0428f6","#0b53f7",
  "#0f81f3","#18b1f5","#1ff0f7","#27fada","#3efaa3","#5dfc7b","#85fd4e",
  "#aefc2a","#e9fc0d","#f6da0c","#f5a009","#f6780a","#f34a09","#f2210a",
  "#f50008","#d90009","#a80109","#730005"
)

map_layer <- geom_map(
  data = map_data("world"), map = map_data("world"),
  aes(group = group, map_id = region),
  fill = "black", colour = "white", linewidth = 0.1
)

# Brazil-Malvinas confluence zone
BM_box <- cbind(lon = c(-60, -48), lat = c(-50, -35))

ggplot() +
  scale_colour_gradientn(colours = nasa_palette, name = expression(degree*C)) +
  xlab("Longitude (deg)") + ylab("Latitude (deg)") +
  map_layer + xlim(BM_box[, "lon"]) + ylim(BM_box[, "lat"]) + theme_bw() +
  coord_fixed(expand = FALSE) +
  geom_point(data = df_train, aes(lon, lat, colour = sst), size=0.5)

For this complicated data we need more segments for spl2D() as in the previous example, because of the strong local changes in Sea Surface Temperatures in this region.

obj6 <- LMMsolve(fixed = sst ~ 1, 
                 spline = ~spl2D(lon, lat, nseg = c(70, 70),
                                 x1lim = BM_box[, "lon"], x2lim = BM_box[, "lat"]),
                 data = df_train, tolerance = 1.0e-1)
summary(obj6)
#> 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)    755.49  5329    5325  0.14    0.00
#>         s(lat)    689.68  5329    5325  0.13    0.00
#>       residual   6263.84  7713    7709  0.81    6.65
#> 
#>  Total Effective Dimension: 7713

The predictions on a grid are shown in the next figure

lon_range <- BM_box[, "lon"]
lat_range <- BM_box[, "lat"]
newdat <- expand.grid(lon = seq(lon_range[1], lon_range[2], length = 200),
                      lat = seq(lat_range[1], lat_range[2], length = 200))

pred_grid <- predict(obj6, newdata = newdat, se.fit=TRUE)
pred_grid <- pred_grid[pred_grid$se<5, ]

## Plot predictions on a grid
ggplot(pred_grid) +
  geom_tile(aes(x = lon, y = lat, fill = ypred)) +
  scale_fill_gradientn(colours = nasa_palette) +
  labs(
    fill = "pred.",
    x = "Longitude (deg)", y = "Latitude (deg)"
  ) +
  map_layer +
  theme_bw() +
  coord_fixed(expand = FALSE, xlim = BM_box[, "lon"], ylim = BM_box[, "lat"]) +
  scale_x_continuous(breaks = c(-58, -54, -50))

The standard errors for the predictions are in the column se in the data frame pred_grid and can be plotted using the following code:

 ## Plot standard error
ggplot(pred_grid) +
   geom_raster(aes(x = lon, y = lat, fill = se)) +
   scale_fill_distiller(palette = "BrBG", direction = -1) +
   labs( fill = "s.e.", x = "Longitude (deg)", y = "Latitude (deg)") +
   map_layer +
   theme_bw() +
   coord_fixed(expand = FALSE, xlim = c(-60, -48), ylim = c(-50, -35)) +
   scale_x_continuous(breaks = c(-58, -54, -50))

Predictions for the test set are given by

pred_test <- predict(obj6, newdata = df_test)
ggplot(pred_test, aes(x = sst,y = ypred)) + geom_point() +
     xlab("observed SST (Celsius)") + ylab("predicted SST (Celsius)") +
     geom_abline(intercept=0,slope=1,col='red') + theme_bw()

Calculation of the root mean squared prediction error (RMSPE) for the test set:

Y <- (pred_test$sst - pred_test$ypred)^2
RMSE <- sqrt(mean(Y))
round(RMSE, 2)
#> [1] 0.45

The RMSPE is in the same range (0.44-0.46) as for the software packages used in Cressie, Sainsbury-Dale, and Zammit-Mangion (2022). On a standard desktop the calculations using LMMsolver take less than 10 seconds, taking advantage of the sparse structure of the P-splines mixed model (Boer 2023).

3 Examples from Quantitative Genetics

In this section we will show some examples from quantitative genetics, to illustrate some further options of the package.

3.1 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   1

We will use 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:

obj7 <- LMMsolve(fixed = yield ~ rep + gen,
                 random = ~plotF, 
                 ginverse = lGinv, 
                 data = john.alpha)

The absolute deviance (\(-2*logL\)) and variances for the LV-model are

round(deviance(obj7, relative = FALSE), 2)
#> [1] 54.49
summary(obj7, which = "variances")
#> Table with variances: 
#> 
#>   VarComp Variance
#>     plotF     0.01
#>  residual     0.06

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

3.2 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  96.84971

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

obj8 <- LMMsolve(fixed = biomass ~ 1,
                 spline = ~spl1D(x = das, nseg = 50), 
                 data = APSIMdat)

The effective dimensions are:

summary(obj8)
#> 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)      6.46    53      51  0.13    0.01
#>     residual    112.54   121     119  0.95    0.00
#> 
#>  Total Effective Dimension: 121

The fitted smooth trend can be obtained as explained before:

das_range <- range(APSIMdat$das)
newdat <- data.frame(das=seq(das_range[1], das_range[2], length = 300))
pred8 <- predict(obj8, newdata = newdat, se.fit = TRUE)
ggplot(data = APSIMdat, aes(x = das, y = biomass)) +
  geom_point(size = 1.0) +
  geom_line(data = pred8, aes(y = ypred), color = "blue", linewidth = 1) +
  geom_ribbon(data = pred8, aes(x = das,ymin = ypred-2*se, ymax = ypred+2*se),
              alpha = 0.2, inherit.aes = FALSE) + 
    labs(title = "APSIM biomass as function of time", 
       x = "days after sowing", y = "biomass (kg)") +
  theme_bw()

The growth rate (first derivative) as function of time can be obtained using deriv = 1 in function obtainSmoothTrend:

plotDatDt <- obtainSmoothTrend(obj8, 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_bw()

3.3 QTL mapping with IBD probabilities.

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.
obj9 <- LMMsolve(fixed = pheno ~ cross, 
                 residual = ~cross, 
                 data = multipop)
dev0 <- deviance(obj9, relative = FALSE)

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)
obj10 <- LMMsolve(fixed = pheno ~ cross, 
                 group = lGrp,
                 random = ~grp(QTL),
                 residual = ~cross,
                 data = multipop) 
dev1 <- deviance(obj10, relative = FALSE)

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

## Deviance difference between null and alternative model.
dev <- dev0 - dev1
## 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:

coef(obj10)$QTL
#>     QTL_pA     QTL_pB     QTL_pC 
#> -1.2676362  0.6829275  0.5847088

Mixed Models and Smoothing

2024-08-26

The LMMsolver package