lme4
package for R
¶The lme4
package for R includes several data sets, chosen to illustrate various types of models.
The simplest model fits are those with a single, simple, scalar random-effects term. That is, there is only a main random effects term.
Dyestuff
data¶using MixedModels,RDatasets # load packages
ds = dataset("lme4","Dyestuff")
Batch | Yield | |
---|---|---|
1 | A | 1545 |
2 | A | 1440 |
3 | A | 1440 |
4 | A | 1520 |
5 | A | 1580 |
6 | B | 1540 |
7 | B | 1555 |
8 | B | 1490 |
9 | B | 1560 |
10 | B | 1495 |
11 | C | 1595 |
12 | C | 1550 |
13 | C | 1605 |
14 | C | 1510 |
15 | C | 1560 |
16 | D | 1445 |
17 | D | 1440 |
18 | D | 1595 |
19 | D | 1465 |
20 | D | 1545 |
21 | E | 1595 |
22 | E | 1630 |
23 | E | 1515 |
24 | E | 1635 |
25 | E | 1625 |
26 | F | 1520 |
27 | F | 1455 |
28 | F | 1450 |
29 | F | 1480 |
30 | F | 1445 |
⋮ | ⋮ | ⋮ |
The only covariate is the categorical variable Batch
denoting the batch of an intermediate product from which the dye was created. We are not interested in the "effect" of a particular batch on the yield of dyestuff so much as we are interested in the variability due to batch.
m1 = fit(lmm(Yield ~ 1 + (1|Batch),ds))
Linear mixed model fit by maximum likelihood Formula: Yield ~ 1 + (1 | Batch) logLik: -163.663530, deviance: 327.327060 Variance components: Variance Std.Dev. Batch 1388.331690 37.260323 Residual 2451.250503 49.510105 Number of obs: 30; levels of grouping factors: 6 Fixed-effects parameters: Estimate Std.Error z value (Intercept) 1527.5 17.6945 86.326
The first time an lmm
model is fit takes longer than subsequent fits because several methods must be compiled. Subsequent fits are quite fast. There is a second argument, verbose
, to the fit
method. It defaults to false
. We can trace the progress of the iterations by setting it to true
.
@time fit(lmm(Yield ~ 1 + (1|Batch), ds),true);
f_1: 327.76702, [1.0] f_2: 328.63496, [0.428326] f_3: 327.33773, [0.787132] f_4: 328.27031, [0.472809] f_5: 327.33282, [0.727955] f_6: 327.32706, [0.752783] f_7: 327.32706, [0.752599] f_8: 327.32706, [0.752355] f_9: 327.32706, [0.752575] f_10: 327.32706, [0.75258] FTOL_REACHED elapsed time: 0.212709603 seconds (5417164 bytes allocated)
Convergence is fast, as this is a 1-dimensional optimization and the gradient of the objective function is available.
MixedModels.hasgrad(m1)
true
The parameter being optimized is the single free parameter in the relative covariance factor, $\Lambda$. This matrix is $6\times 6$ and block diagonal with each diagonal block being a copy of
m1.λ
1-element Array{Any,1}: PDScalF(0.7525801695835216,1)
fixef(m1)
1-element Array{Float64,1}: 1527.5
ranef(m1)
1-element Array{Any,1}: 1x6 Array{Float64,2}: -16.6282 0.369516 26.9747 -21.8014 53.5798 -42.4943
deviance(m1)
327.32705988113764
DyeStuff2
data showing a singular fit¶The Dyestuff2
data are simulated to show a fit on the boundary of the parameter space.
ds2 = dataset("lme4","Dyestuff2");
@time m2 = fit(lmm(Yield ~ 1 + (1|Batch), ds2),true)
f_1: 170.9011, [1.0] f_2: 163.97764, [0.281722] f_3: 162.87304, [0.0] f_4: 162.87304, [0.0] f_5: 162.87304, [0.0] XTOL_REACHED elapsed time: 0.087887694 seconds (1538732 bytes allocated)
Linear mixed model fit by maximum likelihood Formula: Yield ~ 1 + (1 | Batch) logLik: -81.436518, deviance: 162.873037 Variance components: Variance Std.Dev. Batch 0.000000 0.000000 Residual 13.346099 3.653231 Number of obs: 30; levels of grouping factors: 6 Fixed-effects parameters: Estimate Std.Error z value (Intercept) 5.6656 0.666986 8.49433
slp = dataset("lme4","sleepstudy");
m3 = fit(lmm(Reaction ~ 1+Days + (1+Days|Subject),slp),true)
f_1: 1784.6423, [1.0,0.0,1.0] f_2: 1792.09158, [1.04647,-0.384052,0.159046] f_3: 1759.76629, [1.00506,-0.0847897,0.418298] f_4: 1787.91236, [1.26209,0.662287,0.0] f_5: 1770.2265, [1.04773,0.323752,0.0] f_6: 1755.6188, [1.00967,0.0107469,0.150327] f_7: 1762.85008, [0.991808,0.14307,0.446863] f_8: 1753.29754, [1.0048,0.0534958,0.272807] f_9: 1752.5881, [1.00312,0.0418443,0.252944] f_10: 1767.54407, [0.99451,0.00122224,0.0940196] f_11: 1752.21061, [1.00224,0.0373251,0.232541] f_12: 1758.83812, [0.988744,0.0206109,0.123804] f_13: 1752.13481, [1.00085,0.0355465,0.220038] f_14: 1752.02982, [0.980566,0.015964,0.234045] f_15: 1759.88963, [0.971299,0.0118275,0.120811] f_16: 1751.98896, [0.979624,0.0155451,0.221269] f_17: 1751.98436, [0.97888,0.015535,0.224081] f_18: 1751.96796, [0.968696,0.0144745,0.223608] f_19: 1752.08826, [0.867754,0.0226905,0.23397] f_20: 1751.9463, [0.943112,0.0163709,0.226009] f_21: 1754.13022, [0.834535,0.0100178,0.166328] f_22: 1751.94908, [0.930934,0.0157232,0.218808] f_23: 1751.94123, [0.938201,0.0161115,0.223087] f_24: 1751.96529, [0.894427,0.0196256,0.223832] f_25: 1751.93978, [0.930555,0.0167127,0.223213] f_26: 1751.93974, [0.930506,0.019329,0.222173] f_27: 1751.94419, [0.913941,0.018183,0.222681] f_28: 1751.93955, [0.927971,0.0191544,0.22225] f_29: 1751.93979, [0.933502,0.0174017,0.2225] f_30: 1751.93942, [0.92986,0.0185533,0.222336] f_31: 1751.9544, [0.903287,0.0173483,0.222935] f_32: 1751.93944, [0.927141,0.0184317,0.222396] f_33: 1751.93939, [0.928786,0.0185053,0.222359] f_34: 1751.93935, [0.929171,0.0180663,0.222656] f_35: 1751.93935, [0.929702,0.0182395,0.222667] f_36: 1751.93934, [0.929337,0.0181204,0.222659] f_37: 1751.93935, [0.928905,0.0183732,0.222647] f_38: 1751.93934, [0.929269,0.0181603,0.222657] f_39: 1751.93935, [0.929106,0.0181461,0.222618] f_40: 1751.93934, [0.929236,0.0181575,0.22265] f_41: 1751.93934, [0.929197,0.0181927,0.222625] f_42: 1751.93934, [0.929229,0.018164,0.222645] f_43: 1751.93934, [0.929146,0.0181729,0.22267] f_44: 1751.93934, [0.929221,0.0181649,0.222647] f_45: 1751.93934, [0.929226,0.0181643,0.222646] FTOL_REACHED
Linear mixed model fit by maximum likelihood Formula: Reaction ~ 1 + Days + ((1 + Days) | Subject) logLik: -875.969672, deviance: 1751.939344 Variance components: Variance Std.Dev. Corr. Subject 565.516376 23.780588 32.682265 5.716840 0.08 Residual 654.940901 25.591813 Number of obs: 180; levels of grouping factors: 18 Fixed-effects parameters: Estimate Std.Error z value (Intercept) 251.405 6.63228 37.9063 Days 10.4673 1.50224 6.96779
For this model $\Lambda$ is $36\times 36$ and consists of 18 diagonal blocks of the form
m3.λ
1-element Array{Any,1}: PDLCholF(Cholesky{Float64} with factor: 2x2 Triangular{Float64,Array{Float64,2},:L,false}: 0.929226 0.0 0.0181643 0.222646)
This model also provides the gradient for the optimization
MixedModels.hasgrad(m3)
true
If we specify more than one random effects term for the same grouping factor, these are amalgamated into a single term when fitting.
m4 = fit(lmm(Reaction ~ 1+Days + (1|Subject) + (0+Days|Subject), slp),true)
f_1: 1784.6423, [1.0,1.0] f_2: 1755.16173, [1.04647,0.159046] f_3: 1804.58124, [0.192941,1.05905] f_4: 1795.3958, [0.437654,1.05905] f_5: 1781.60206, [0.812816,0.891382] f_6: 1765.95434, [1.00957,0.544762] f_7: 1754.16459, [1.03851,0.30914] f_8: 1752.98638, [1.03629,0.276642] f_9: 1753.26874, [1.02498,0.18224] f_10: 1752.17734, [1.03219,0.236848] f_11: 1752.66367, [0.976269,0.192399] f_12: 1752.10695, [1.01955,0.226392] f_13: 1752.26292, [0.891397,0.248476] f_14: 1752.03612, [0.977546,0.232736] f_15: 1756.75776, [0.777162,0.149161] f_16: 1752.02373, [0.950054,0.220357] f_17: 1752.00507, [0.947215,0.228948] f_18: 1752.00331, [0.946874,0.227205] f_19: 1752.00389, [0.941836,0.226055] f_20: 1752.00326, [0.945839,0.226969] f_21: 1752.00336, [0.945545,0.226456] f_22: 1752.00326, [0.94581,0.226917] f_23: 1752.00326, [0.945808,0.226927] f_24: 1752.00326, [0.945842,0.226939] f_25: 1752.00326, [0.945812,0.226928] f_26: 1752.00326, [0.945811,0.226928] FTOL_REACHED
Linear mixed model fit by maximum likelihood Formula: Reaction ~ 1 + Days + (1 | Subject) + ((0 + Days) | Subject) logLik: -876.001628, deviance: 1752.003255 Variance components: Variance Std.Dev. Corr. Subject 584.250055 24.171265 33.633148 5.799409 0.00 Residual 653.116005 25.556134 Number of obs: 180; levels of grouping factors: 18 Fixed-effects parameters: Estimate Std.Error z value (Intercept) 251.405 6.70767 37.4802 Days 10.4673 1.51931 6.88948
@time fit(lmm(Reaction ~ 1+Days + (1|Subject) + (0+Days|Subject), slp));
elapsed time: 0.005619764 seconds (702960 bytes allocated)
MixedModels.hasgrad(m4)
true
m4.λ
1-element Array{Any,1}: PDDiagF(2x2 Diagonal{Float64}: 0.945811 0.0 0.0 0.226928)
The Penicillin
data are from an assay of the potency of penicillin batches using an old method in which samples from the batch were applied to several plates of agar spiked with a bacterium and the diameter of the cleared region measured. Each of the 6 samples was applied to each of the 24 plates.
pen = dataset("lme4","Penicillin");
m5 = fit(lmm(Diameter ~ 1 + (1|Plate) + (1|Sample),pen),true)
f_1: 364.62678, [1.0,1.0] f_2: 337.90847, [1.77431,1.88085] f_3: 333.965, [1.3028,2.43847] f_4: 333.07671, [1.71277,2.68316] f_5: 332.45223, [1.53153,2.7824] f_6: 332.40848, [1.41984,3.13078] f_7: 332.81251, [1.76305,3.15165] f_8: 332.19876, [1.54801,3.13536] f_9: 332.19519, [1.53628,3.14215] f_10: 332.19049, [1.5254,3.21805] f_11: 332.2817, [1.62101,3.21628] f_12: 332.18835, [1.53724,3.21786] f_13: 332.18857, [1.54153,3.21977] f_14: 332.18835, [1.53767,3.21805] f_15: 332.18835, [1.53754,3.2182] f_16: 332.18835, [1.53778,3.22006] f_17: 332.18839, [1.53587,3.21986] f_18: 332.18835, [1.53759,3.22004] f_19: 332.18835, [1.5376,3.22002] f_20: 332.18835, [1.5376,3.21975] f_21: 332.18835, [1.53748,3.21976] f_22: 332.18835, [1.53759,3.21975] f_23: 332.18835, [1.53759,3.21975] f_24: 332.18835, [1.53759,3.21975] f_25: 332.18835, [1.53759,3.21975] FTOL_REACHED
Linear mixed model fit by maximum likelihood Formula: Diameter ~ 1 + (1 | Plate) + (1 | Sample) logLik: -166.094174, deviance: 332.188349 Variance components: Variance Std.Dev. Plate 0.714992 0.845572 Sample 3.135184 1.770645 Residual 0.302425 0.549932 Number of obs: 144; levels of grouping factors: 24, 6 Fixed-effects parameters: Estimate Std.Error z value (Intercept) 22.9722 0.744595 30.852
@time fit(lmm(Diameter ~ 1 + (1|Plate) + (1|Sample),pen));
elapsed time: 0.028766902 seconds (7223940 bytes allocated)
MixedModels.hasgrad(m5)
true
m5.λ
2-element Array{Any,1}: PDScalF(1.53759361082603,1) PDScalF(3.219751759198035,1)
The Pastes
data provides the strength of samples of paste from each of three casks sampled from each of 10 batches.
psts = dataset("lme4","Pastes");
m6 = fit(lmm(Strength ~ 1 + (1|Sample) + (1|Batch),psts),true)
f_1: 274.8397, [1.0,1.0] f_2: 258.56865, [1.75,1.0] f_3: 277.39618, [1.0,1.75] f_4: 300.92218, [0.25,1.0] f_5: 279.7496, [1.0,0.25] f_6: 250.93944, [2.49601,1.07724] f_7: 249.5404, [2.74867,1.15646] f_8: 248.64833, [2.98474,1.63052] f_9: 248.59997, [2.99408,1.54877] f_10: 248.43446, [3.0687,1.54124] f_11: 248.29936, [3.14346,1.53526] f_12: 248.15509, [3.26562,1.58041] f_13: 248.20737, [3.38606,1.81135] f_14: 248.05426, [3.38994,1.54163] f_15: 248.05921, [3.49885,1.61304] f_16: 248.0504, [3.43672,1.57657] f_17: 248.04172, [3.43144,1.54874] f_18: 248.02318, [3.4418,1.49304] f_19: 248.01406, [3.42417,1.38112] f_20: 248.0061, [3.44879,1.38228] f_21: 247.99886, [3.47775,1.35361] f_22: 247.99762, [3.48518,1.34275] f_23: 247.99844, [3.48937,1.30221] f_24: 247.99559, [3.50242,1.3319] f_25: 247.99468, [3.51952,1.32081] f_26: 247.99472, [3.52475,1.31303] f_27: 247.9945, [3.52317,1.32736] f_28: 247.99448, [3.5253,1.33384] f_29: 247.99448, [3.52484,1.33222] f_30: 247.99447, [3.52599,1.33059] f_31: 247.99447, [3.52678,1.32968] f_32: 247.99447, [3.52749,1.32946] f_33: 247.99447, [3.52669,1.32894] f_34: 247.99447, [3.52689,1.32988] f_35: 247.99447, [3.52689,1.32992] FTOL_REACHED
Linear mixed model fit by maximum likelihood Formula: Strength ~ 1 + (1 | Sample) + (1 | Batch) logLik: -123.997233, deviance: 247.994466 Variance components: Variance Std.Dev. Sample 8.433617 2.904069 Batch 1.199179 1.095070 Residual 0.678002 0.823409 Number of obs: 60; levels of grouping factors: 30, 10 Fixed-effects parameters: Estimate Std.Error z value (Intercept) 60.0533 0.642136 93.5212
@time fit(lmm(Strength ~ 1 + (1|Sample) + (1|Batch),psts));
elapsed time: 0.015186255 seconds (2387896 bytes allocated)
typeof(m6)
LinearMixedModel{PLSNested} (constructor with 1 method)
At present the gradient is not available for models with nested grouping factors, but it will be.
MixedModels.hasgrad(m6)
false
The InstEval
data are several years of instructor evaluations at ETH-Zurich. The student id is S
and the instructor is D
.
inst = dataset("lme4","InstEval");
m7 = fit(lmm(Y ~ Dept*Service + (1|S) + (1|D), inst))
Linear mixed model fit by maximum likelihood Formula: Y ~ Dept * Service + (1 | S) + (1 | D) logLik: -118792.776708, deviance: 237585.553415 Variance components: Variance Std.Dev. S 0.105418 0.324681 D 0.258416 0.508347 Residual 1.384728 1.176745 Number of obs: 73421; levels of grouping factors: 2972, 1128 Fixed-effects parameters: Estimate Std.Error z value (Intercept) 3.22961 0.064053 50.4209 Dept5 0.129536 0.101294 1.27882 Dept10 -0.176751 0.0881352 -2.00545 Dept12 0.0517102 0.0817524 0.632522 Dept6 0.0347319 0.085621 0.405647 Dept7 0.14594 0.0997984 1.46235 Dept4 0.151689 0.0816897 1.85689 Dept8 0.104206 0.118751 0.877517 Dept9 0.0440401 0.0962985 0.457329 Dept14 0.0517546 0.0986029 0.524879 Dept1 0.0466719 0.101942 0.457828 Dept3 0.0563461 0.0977925 0.57618 Dept11 0.0596536 0.100233 0.59515 Dept2 0.00556281 0.110867 0.0501756 Service1 0.252025 0.0686507 3.67112 Dept5&Service1 -0.180757 0.123179 -1.46744 Dept10&Service1 0.0186492 0.110017 0.169512 Dept12&Service1 -0.282269 0.0792937 -3.55979 Dept6&Service1 -0.494464 0.0790278 -6.25683 Dept7&Service1 -0.392054 0.110313 -3.55403 Dept4&Service1 -0.278547 0.0823727 -3.38154 Dept8&Service1 -0.189526 0.111449 -1.70056 Dept9&Service1 -0.499868 0.0885423 -5.64553 Dept14&Service1 -0.497162 0.0917162 -5.42065 Dept1&Service1 -0.24042 0.0982071 -2.4481 Dept3&Service1 -0.223013 0.0890548 -2.50422 Dept11&Service1 -0.516997 0.0809077 -6.38997 Dept2&Service1 -0.384773 0.091843 -4.18946
Not too long ago this would have been considered a "large" model to fit.
@time fit(lmm(Y ~ Dept*Service + (1|S) + (1|D), inst));
elapsed time: 5.873610568 seconds (333351324 bytes allocated, 8.24% gc time)
typeof(m7)
LinearMixedModel{PLSDiag{Int32}} (constructor with 1 method)
This is the first model for which the sparse matrix representation of $\Lambda'Z'Z\Lambda$ and the sparse Cholesky factor is used. At present SuiteSparse is used for the factorization but we are considering alternatives. It is challenging to reduce the amount of copying done by SuiteSparse
In some disciplines models with random effects are all called Multilevel or "Hierarchical Linear" models. Most of the software to fit such models assumes that grouping factors are nested. Some of the data sets used in a review of software for fitting such models is available in the mlmRev
package for R.
chem = dataset("mlmRev","Chem97");
@time m8 = fit(lmm(Score ~ 1+GCSECnt+Gender+Age + (1|School) + (1|Lea),chem))
elapsed time: 1.459777227 seconds (203399116 bytes allocated, 21.25% gc time)
Linear mixed model fit by maximum likelihood Formula: Score ~ 1 + GCSECnt + Gender + Age + (1 | School) + (1 | Lea) logLik: -70498.755697, deviance: 140997.511394 Variance components: Variance Std.Dev. School 1.131754 1.063839 Lea 0.018988 0.137798 Residual 5.041983 2.245436 Number of obs: 31022; levels of grouping factors: 2410, 131 Fixed-effects parameters: Estimate Std.Error z value (Intercept) 5.97244 0.0348207 171.52 GCSECnt 2.56977 0.0171238 150.07 GenderF -0.744255 0.0302938 -24.5679 Age -0.0376029 0.00382113 -9.84078
In experimental psychology and especially in psycholinguistics it is common to observe characteristics, such as reaction time, on several subjects exposed to a number of items. Only recently has it been possible to fit models with random effects for Subject
and Item
. These blocking factors are crossed or nearly crossed. The experimental factors are often in a 2-level factorial design.
A recent paper indicated that it is important to use a "maximal" set of random effects for the blocking factors when analyzing such data. Such models usually end up severely overparameterized and are difficult to fit. The optimization problem has many free variables and the optimum is poorly defined and usually on the boundary of the parameter space.
bs10 = MixedModels.rdata("bs10");
names(bs10)
12-element Array{Symbol,1}: :SubjID :ItemID :Spkr :Filler :ms :d :d2 :Spkr2 :dif :SF :F :S
@time m9 = fit(lmm(dif ~ 1+S+F+SF + (1+S+F+SF|SubjID) + (1+S+F+SF|ItemID), bs10))
elapsed time: 1.691779479 seconds (243729412 bytes allocated, 22.98% gc time)
Linear mixed model fit by maximum likelihood Formula: dif ~ 1 + S + F + SF + ((1 + S + F + SF) | SubjID) + ((1 + S + F + SF) | ItemID) logLik: -515.477648, deviance: 1030.955295 Variance components: Variance Std.Dev. Corr. SubjID 0.013393 0.115730 0.011216 0.105907 -0.56 0.003373 0.058075 0.99 0.99 0.005192 0.072053 -0.13 -0.13 -0.13 ItemID 0.000305 0.017459 0.000079 0.008904 -1.00 0.000136 0.011647 1.00 1.00 0.000083 0.009125 -1.00 -1.00 -1.00 Residual 0.127779 0.357462 Number of obs: 1104; levels of grouping factors: 92, 12 Fixed-effects parameters: Estimate Std.Error z value (Intercept) 0.039221 0.0169329 2.31626 S -0.0174094 0.0156289 -1.11392 F 0.0174819 0.0127947 1.36633 SF -0.0322645 0.0133832 -2.41082
Notice that there are 10 parameters to estimate in the covariance factor for ItemID
and only 12 distict levels of that factor. The covariance matrices for the random effects are both apparently 4-dimensional but in fact are singular. The SubjID
random effects lie in a 2-dimensional subspace and the ItemID
random effects in a 1-dimensional subspace.
m9.λ
2-element Array{Any,1}: PDLCholF(Cholesky{Float64} with factor: 4x4 Triangular{Float64,Array{Float64,2},:L,false}: 0.323755 0.0 0.0 0.0 -0.166342 0.24517 0.0 0.0 0.161121 -0.0208524 1.2824e-6 0.0 -0.0265681 0.199809 -1.86487e-6 0.0) PDLCholF(Cholesky{Float64} with factor: 4x4 Triangular{Float64,Array{Float64,2},:L,false}: 0.048841 0.0 0.0 0.0 -0.0249089 9.75571e-7 0.0 0.0 0.0325814 -7.88902e-8 0.0 0.0 -0.0255267 -3.63047e-8 4.95277e-10 0.0)
This is a 20-dimensional optimization problem on the profiled log-likelihood. Other examples involve 70 or more nonlinear parameters.
kb07 = MixedModels.rdata("kb07")
names(kb07)'
1x15 Array{Symbol,2}: :SubjID :ItemID :Spkr :Prec :CogLoad … :S :P :C :SP :SC :PC :SPC
@time m10 = fit(lmm(RTtrunc ~ 1+S+P+C+SP+SC+PC+SPC + (1+S+P+C+SP+SC+PC+SPC|SubjID) + (1+S+P+C+SP+SC+PC+SPC|ItemID), kb07))
elapsed time: 266.277216762 seconds (26724434372 bytes allocated, 16.81% gc time)
Linear mixed model fit by maximum likelihood Formula: RTtrunc ~ 1 + S + P + C + SP + SC + PC + SPC + ((1 + S + P + C + SP + SC + PC + SPC) | SubjID) + ((1 + S + P + C + SP + SC + PC + SPC) | ItemID) logLik: -14293.158811, deviance: 28586.317622 Variance components: Variance Std.Dev. Corr. SubjID 90769.700667 301.280103 20729.176368 143.976305 -0.43 22176.490819 148.917732 -0.47 -0.47 30348.287532 174.207599 0.21 0.21 0.21 141271.253273 375.860683 0.20 0.20 0.20 0.20 29142.540830 170.711865 0.47 0.47 0.47 0.47 0.47 118762.228836 344.618962 -0.10 -0.10 -0.10 -0.10 -0.10 -0.10 243331.145812 493.286069 -0.48 -0.48 -0.48 -0.48 -0.48 -0.48 -0.48 ItemID 129824.695548 360.311942 7421.991851 86.150983 -0.34 249574.280629 499.574099 -0.68 -0.68 11792.165997 108.591740 0.20 0.20 0.20 16683.129834 129.163191 0.57 0.57 0.57 0.57 25917.846598 160.990207 0.28 0.28 0.28 0.28 0.28 75203.218971 274.232053 0.08 0.08 0.08 0.08 0.08 0.08 308511.015871 555.437680 0.04 0.04 0.04 0.04 0.04 0.04 0.04 Residual 399612.544911 632.149148 Number of obs: 1790; levels of grouping factors: 56, 32 Fixed-effects parameters: Estimate Std.Error z value (Intercept) 2180.63 76.8192 28.3865 S -133.98 38.6681 -3.46487 P -667.763 95.3327 -7.00455 C 157.974 42.4683 3.71981 SP 88.6069 81.342 1.08931 SC -75.6977 70.02 -1.08109 PC 21.048 89.6846 0.234689 SPC -191.608 168.155 -1.13947
m10.λ
2-element Array{Any,1}: PDLCholF(Cholesky{Float64} with factor: 8x8 Triangular{Float64,Array{Float64,2},:L,false}: 0.476597 0.0 0.0 … 0.0 0.0 0.0 -0.0989296 0.205149 0.0 0.0 0.0 0.0 -0.111553 -0.0341986 0.204649 0.0 0.0 0.0 0.0590232 -0.0332038 0.156032 0.0 0.0 0.0 0.116584 -0.443693 -0.377608 0.0 0.0 0.0 0.127793 -0.097419 0.0203143 … 5.74227e-57 0.0 0.0 -0.0556877 0.0506364 -0.0536056 5.96388e-57 4.04582e-144 0.0 -0.371105 0.174335 -0.520435 -1.03749e-56 -3.00707e-144 0.0) PDLCholF(Cholesky{Float64} with factor: 8x8 Triangular{Float64,Array{Float64,2},:L,false}: 0.569979 0.0 0.0 … 0.0 0.0 0.0 -0.0467818 0.128002 0.0 0.0 0.0 0.0 -0.534095 -0.576095 0.0860161 0.0 0.0 0.0 0.0347278 0.00688954 -0.0257139 0.0 0.0 0.0 0.116132 -0.123402 -0.0684817 0.0 0.0 0.0 0.0713495 0.0186925 -0.0655391 … 4.9444e-5 0.0 0.0 0.0339327 -0.0951286 0.402184 -4.56546e-5 0.0 0.0 0.03904 -0.431245 -0.061637 -3.57477e-5 1.3111e-121 0.0)