The sommer package was developed to provide R users with a powerful and reliable multivariate mixed model solver for different genetic and non-genetic analyses in diploid and polyploid organisms. This package allows the user to estimate variance components for a mixed model with the advantages of specifying the variance-covariance structure of the random effects, specifying heterogeneous variances, and obtaining other parameters such as BLUPs, BLUEs, residuals, fitted values, variances for fixed and random effects, etc. The core algorithms of the package are coded in C++ using the Armadillo library to optimize dense matrix operations common in the derect-inversion algorithms. Although the vignette shows examples using the mmes function with the default direct inversion algorithm (formulation=“direct”) the Henderson’s approach can be faster when the number of records surpasses the number of coefficients to estimate and setting the henderson argument to TRUE can bring significant speed ups.
The purpose of this vignette is to show how to translate the syntax formula from lme4 models to sommer models. Feel free to remove the comment marks from the lme4 code so you can compare the results.
This is the simplest model people use when a random effect is desired and the levels of the random effect are considered to have the same intercept.
# install.packages("lme4")
# library(lme4)
library(sommer)
data(DT_sleepstudy, package="enhancer")
DT <- DT_sleepstudy
###########
## lme4
###########
# fm1 <- lmer(Reaction ~ Days + (1 | Subject), data=DT)
# summary(fm1) # or vc <- VarCorr(fm1); print(vc,comp=c("Variance"))
# Random effects:
# Groups Name Variance Std.Dev.
# Subject (Intercept) 1378.2 37.12
# Residual 960.5 30.99
# Number of obs: 180, groups: Subject, 18
###########
## sommer
###########
fm2 <- mmes(Reaction ~ Days,
random= ~ Subject,
data=DT,
control = mmes_control(execution = list(inverseTol = 1e-6),
diagnostics = list(verbose = FALSE)))
summary(fm2)$varcomp
## term parameter estimate StdError Zratio
## 1 vsm(ism(Subject)) sigma2 1378.2022 505.7744 2.724935
## 2 vsm(ism(units)) sigma2 960.3449 107.0296 8.972701
This is the a model where you assume that the random effect has different intercepts based on the levels of another variable. In addition the || in lme4 assumes that slopes and intercepts have no correlation.
###########
## lme4
###########
# fm1 <- lmer(Reaction ~ Days + (Days || Subject), data=DT)
# summary(fm1) # or vc <- VarCorr(fm1); print(vc,comp=c("Variance"))
# Random effects:
# Groups Name Variance Std.Dev.
# Subject (Intercept) 627.57 25.051
# Subject.1 Days 35.86 5.988
# Residual 653.58 25.565
# Number of obs: 180, groups: Subject, 18
###########
## sommer
###########
fm2 <- mmes(Reaction ~ Days,
random= ~ Subject + vsm(ism(Days), ism(Subject)),
data=DT,
control = mmes_control(execution = list(inverseTol = 1e-6),
diagnostics = list(verbose = FALSE)))
summary(fm2)$varcomp
## term parameter estimate StdError Zratio
## 1 vsm(ism(Subject)) sigma2 627.72619 283.71771 2.212503
## 2 vsm(ism(Days), ism(Subject)) sigma2 35.84895 14.52471 2.468135
## 3 vsm(ism(units)) sigma2 653.57166 76.65528 8.526114
Notice that Days is a numerical (not factor) variable.
This is the a model where you assume that the random effect has different intercepts based on the levels of another variable. In addition a single | in lme4 assumes that slopes and intercepts have a correlation to be estimated.
###########
## lme4
###########
# fm1 <- lmer(Reaction ~ Days + (Days | Subject), data=DT)
# summary(fm1) # or # vc <- VarCorr(fm1); print(vc,comp=c("Variance"))
# Random effects:
# Groups Name Variance Std.Dev. Corr
# Subject (Intercept) 612.10 24.741
# Days 35.07 5.922 0.07
# Residual 654.94 25.592
# Number of obs: 180, groups: Subject, 18
###########
## sommer
###########
fm2 <- mmes(Reaction ~ Days,
# control = mmes_control(formulation="henderson"),
random= ~ vsm(usm(cbind(1,Days)), ism(Subject)),
data=DT,
control = mmes_control(convergence = list(maxIters = 200),
execution = list(inverseTol = 1e-6),
diagnostics = list(verbose = FALSE)))
summary(fm2)$varcomp
## term parameter estimate StdError
## 1 vsm(usm(cbind(1, Days)), ism(Subject)) variance[] 612.019224 288.74643
## 2 vsm(usm(cbind(1, Days)), ism(Subject)) covariance[Days,] 9.664236 46.66108
## 3 vsm(usm(cbind(1, Days)), ism(Subject)) variance[Days] 35.057071 14.77457
## 4 vsm(ism(units)) sigma2 654.941025 77.18554
## Zratio
## 1 2.1195733
## 2 0.2071156
## 3 2.3727978
## 4 8.4852814
cov2cor(fm2$theta[[1]])
## [,1] [,2]
## [1,] 1.0000000 0.0659777
## [2,] 0.0659777 1.0000000
Notice that this last model require a new function called covm() which creates the two random effects as before but now they have to be encapsulated in covm() instead of just added.
This is the a model where you assume that the random effect has different intercepts based on the levels of another variable but there’s not a main effect. The 0 in the intercept in lme4 assumes that random slopes interact with an intercept but without a main effect.
###########
## lme4
###########
# fm1 <- lmer(Reaction ~ Days + (0 + Days | Subject), data=DT)
# summary(fm1) # or vc <- VarCorr(fm1); print(vc,comp=c("Variance"))
# Random effects:
# Groups Name Variance Std.Dev.
# Subject Days 52.71 7.26
# Residual 842.03 29.02
# Number of obs: 180, groups: Subject, 18
###########
## sommer
###########
fm2 <- mmes(Reaction ~ Days,
random= ~ vsm(ism(Days), ism(Subject)),
data=DT,
control = mmes_control(execution = list(inverseTol = 1e-6),
diagnostics = list(verbose = FALSE)))
summary(fm2)$varcomp
## term parameter estimate StdError Zratio
## 1 vsm(ism(Days), ism(Subject)) sigma2 52.70579 19.09378 2.760365
## 2 vsm(ism(units)) sigma2 842.02965 93.84896 8.972179
One of the strengths of sommer is the availability of other variance covariance structures. In this section we show 4 models available in sommer that are not available in lme4 and might be useful.
library(orthopolynom)
## diagonal model
fm2 <- mmes(Reaction ~ Days,
random= ~ vsm(dsm(Daysf), ism(Subject)),
data=DT,
control = mmes_control(execution = list(inverseTol = 1e-6),
diagnostics = list(verbose = FALSE)))
summary(fm2)$varcomp
## term parameter estimate StdError Zratio
## 1 vsm(dsm(Daysf), ism(Subject)) variance[0] 656.2651 114.9900 5.707151
## 2 vsm(dsm(Daysf), ism(Subject)) variance[1] 713.5638 445.0486 1.603339
## 3 vsm(dsm(Daysf), ism(Subject)) variance[2] 516.6946 383.4608 1.347451
## 4 vsm(dsm(Daysf), ism(Subject)) variance[3] 1072.7899 556.7946 1.926725
## 5 vsm(dsm(Daysf), ism(Subject)) variance[4] 1371.9225 651.2555 2.106581
## 6 vsm(dsm(Daysf), ism(Subject)) variance[5] 2216.1361 921.8138 2.404104
## 7 vsm(dsm(Daysf), ism(Subject)) variance[6] 3427.6151 1317.2583 2.602083
## 8 vsm(dsm(Daysf), ism(Subject)) variance[7] 2056.3370 872.7117 2.356261
## 9 vsm(dsm(Daysf), ism(Subject)) variance[8] 3114.2466 1218.9363 2.554889
## 10 vsm(dsm(Daysf), ism(Subject)) variance[9] 3989.4234 1510.9733 2.640300
## 11 vsm(ism(units)) sigma2 362.9853 255.5421 1.420452
## unstructured model
fm2 <- mmes(Reaction ~ Days,
random= ~ vsm(usm(Daysf), ism(Subject)),
data=DT,
control = mmes_control(execution = list(inverseTol = 1e-6),
diagnostics = list(verbose = FALSE)))
summary(fm2)$varcomp
## term parameter estimate StdError
## 1 vsm(usm(Daysf), ism(Subject)) variance[0] 1268.77852 453.444065
## 2 vsm(usm(Daysf), ism(Subject)) covariance[1,0] 1025.26011 412.018184
## 3 vsm(usm(Daysf), ism(Subject)) covariance[2,0] 542.32445 302.772674
## 4 vsm(usm(Daysf), ism(Subject)) covariance[3,0] 800.96870 418.325945
## 5 vsm(usm(Daysf), ism(Subject)) covariance[4,0] 760.09680 430.117020
## 6 vsm(usm(Daysf), ism(Subject)) covariance[5,0] 935.56751 543.891217
## 7 vsm(usm(Daysf), ism(Subject)) covariance[6,0] 668.59057 587.829326
## 8 vsm(usm(Daysf), ism(Subject)) covariance[7,0] 934.99295 516.331803
## 9 vsm(usm(Daysf), ism(Subject)) covariance[8,0] 923.15021 609.787646
## 10 vsm(usm(Daysf), ism(Subject)) covariance[9,0] 1443.87797 724.815191
## 11 vsm(usm(Daysf), ism(Subject)) variance[1] 1283.40078 456.728924
## 12 vsm(usm(Daysf), ism(Subject)) covariance[2,1] 830.27716 337.251946
## 13 vsm(usm(Daysf), ism(Subject)) covariance[3,1] 1139.78810 461.506048
## 14 vsm(usm(Daysf), ism(Subject)) covariance[4,1] 1041.70165 462.612795
## 15 vsm(usm(Daysf), ism(Subject)) covariance[5,1] 1182.37196 572.665606
## 16 vsm(usm(Daysf), ism(Subject)) covariance[6,1] 853.23543 602.848268
## 17 vsm(usm(Daysf), ism(Subject)) covariance[7,1] 929.33238 515.203643
## 18 vsm(usm(Daysf), ism(Subject)) covariance[8,1] 1047.58355 622.380620
## 19 vsm(usm(Daysf), ism(Subject)) covariance[9,1] 1517.96895 735.589826
## 20 vsm(usm(Daysf), ism(Subject)) variance[2] 864.18179 305.515212
## 21 vsm(usm(Daysf), ism(Subject)) covariance[3,2] 1058.69346 395.030295
## 22 vsm(usm(Daysf), ism(Subject)) covariance[4,2] 912.38538 383.888145
## 23 vsm(usm(Daysf), ism(Subject)) covariance[5,2] 861.24333 456.261634
## 24 vsm(usm(Daysf), ism(Subject)) covariance[6,2] 917.77390 511.519355
## 25 vsm(usm(Daysf), ism(Subject)) covariance[7,2] 925.94761 434.135584
## 26 vsm(usm(Daysf), ism(Subject)) covariance[8,2] 833.40150 502.396582
## 27 vsm(usm(Daysf), ism(Subject)) covariance[9,2] 970.10478 569.046106
## 28 vsm(usm(Daysf), ism(Subject)) variance[3] 1625.70109 573.919675
## 29 vsm(usm(Daysf), ism(Subject)) covariance[4,3] 1592.55514 578.038076
## 30 vsm(usm(Daysf), ism(Subject)) covariance[5,3] 1675.69555 687.760610
## 31 vsm(usm(Daysf), ism(Subject)) covariance[6,3] 1787.99565 765.354119
## 32 vsm(usm(Daysf), ism(Subject)) covariance[7,3] 1284.71638 602.347291
## 33 vsm(usm(Daysf), ism(Subject)) covariance[8,3] 1609.52031 744.950281
## 34 vsm(usm(Daysf), ism(Subject)) covariance[9,3] 1745.53527 830.511762
## 35 vsm(usm(Daysf), ism(Subject)) variance[4] 1821.88734 632.943941
## 36 vsm(usm(Daysf), ism(Subject)) covariance[5,4] 2005.33942 755.257543
## 37 vsm(usm(Daysf), ism(Subject)) covariance[6,4] 2079.29219 829.907539
## 38 vsm(usm(Daysf), ism(Subject)) covariance[7,4] 1551.39672 653.864246
## 39 vsm(usm(Daysf), ism(Subject)) covariance[8,4] 2031.24336 822.544602
## 40 vsm(usm(Daysf), ism(Subject)) covariance[9,4] 2200.83568 914.359995
## 41 vsm(usm(Daysf), ism(Subject)) variance[5] 2934.25537 1029.107633
## 42 vsm(usm(Daysf), ism(Subject)) covariance[6,5] 2605.86800 1054.744915
## 43 vsm(usm(Daysf), ism(Subject)) covariance[7,5] 1943.83905 836.864790
## 44 vsm(usm(Daysf), ism(Subject)) covariance[8,5] 3061.11908 1125.462884
## 45 vsm(usm(Daysf), ism(Subject)) covariance[9,5] 3240.13035 1235.852640
## 46 vsm(usm(Daysf), ism(Subject)) variance[6] 3988.45679 1363.983743
## 47 vsm(usm(Daysf), ism(Subject)) covariance[7,6] 2307.78914 961.835533
## 48 vsm(usm(Daysf), ism(Subject)) covariance[8,6] 2930.00477 1196.080618
## 49 vsm(usm(Daysf), ism(Subject)) covariance[9,6] 2213.36722 1213.841096
## 50 vsm(usm(Daysf), ism(Subject)) variance[7] 2534.50387 883.079248
## 51 vsm(usm(Daysf), ism(Subject)) covariance[8,7] 2435.35192 980.468596
## 52 vsm(usm(Daysf), ism(Subject)) covariance[9,7] 2401.95756 1059.128113
## 53 vsm(usm(Daysf), ism(Subject)) variance[8] 3813.09647 1337.192658
## 54 vsm(usm(Daysf), ism(Subject)) covariance[9,8] 3850.07266 1432.545785
## 55 vsm(usm(Daysf), ism(Subject)) variance[9] 4812.81819 1692.833775
## 56 vsm(ism(units)) sigma2 20.04725 6.613576
## Zratio
## 1 2.798093
## 2 2.488386
## 3 1.791194
## 4 1.914700
## 5 1.767186
## 6 1.720137
## 7 1.137389
## 8 1.810837
## 9 1.513888
## 10 1.992064
## 11 2.809984
## 12 2.461890
## 13 2.469714
## 14 2.251779
## 15 2.064681
## 16 1.415340
## 17 1.803816
## 18 1.683188
## 19 2.063608
## 20 2.828605
## 21 2.680031
## 22 2.376696
## 23 1.887608
## 24 1.794211
## 25 2.132854
## 26 1.658852
## 27 1.704791
## 28 2.832628
## 29 2.755104
## 30 2.436452
## 31 2.336168
## 32 2.132850
## 33 2.160574
## 34 2.101759
## 35 2.878434
## 36 2.655173
## 37 2.505450
## 38 2.372659
## 39 2.469463
## 40 2.406968
## 41 2.851262
## 42 2.470614
## 43 2.322764
## 44 2.719876
## 45 2.621777
## 46 2.924123
## 47 2.399359
## 48 2.449672
## 49 1.823441
## 50 2.870075
## 51 2.483865
## 52 2.267863
## 53 2.851569
## 54 2.687574
## 55 2.843054
## 56 3.031227
## random regression (legendre polynomials)
fm2 <- mmes(Reaction ~ Days,
random= ~ vsm(dsm(leg(Days,1)), ism(Subject)),
data=DT,
control = mmes_control(execution = list(inverseTol = 1e-6),
diagnostics = list(verbose = FALSE)))
summary(fm2)$varcomp
## term parameter estimate StdError
## 1 vsm(dsm(leg(Days, 1)), ism(Subject)) variance[leg0] 2817.4417 1011.41776
## 2 vsm(dsm(leg(Days, 1)), ism(Subject)) variance[leg1] 473.4674 199.55780
## 3 vsm(ism(units)) sigma2 654.9410 77.18554
## Zratio
## 1 2.785636
## 2 2.372583
## 3 8.485282
## unstructured random regression (legendre)
fm2 <- mmes(Reaction ~ Days,
random= ~ vsm(usm(leg(Days,1)), ism(Subject)),
data=DT,
control = mmes_control(execution = list(inverseTol = 1e-6),
diagnostics = list(verbose = FALSE)))
summary(fm2)$varcomp
## term parameter estimate
## 1 vsm(usm(leg(Days, 1)), ism(Subject)) variance[leg0] 2817.4277
## 2 vsm(usm(leg(Days, 1)), ism(Subject)) covariance[leg1,leg0] 869.9652
## 3 vsm(usm(leg(Days, 1)), ism(Subject)) variance[leg1] 473.4643
## 4 vsm(ism(units)) sigma2 654.9410
## StdError Zratio
## 1 1011.41057 2.785642
## 2 381.08010 2.282893
## 3 199.55625 2.372586
## 4 77.18554 8.485281
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