Hierarchical Linear Modeling

1 Preamble

1.1 Install Libraries

#install.packages("remotes")
#remotes::install_github("DevPsyLab/petersenlab")

1.2 Load Libraries

library("petersenlab")
library("lme4")
library("nlme")
library("lmerTest")
library("MASS")
library("MCMCglmm")
library("performance")
library("ggplot2")

1.3 Import Data

mydata <- read.csv("https://osf.io/cqn3d/download")

1.4 Simulate Data

set.seed(52242)

mydata$outcome <- rpois(nrow(mydata), 4)

2 Terms

These models go by a variety of different terms:

• hierarchical linear model (HLM)
• multilevel model (MLM)
• mixed effects model
• mixed model

3 Overview

https://isaactpetersen.github.io/Principles-Psychological-Assessment/reliability.html#mixedModels

4 Pre-Model Computation

It can be helpful to center the age/time variable so that the intercept in a growth curve model has meaning. For instance, we can subtract the youngest participant age to set the intercepts to be the earliest age in the sample.

mydata$ageYears <- mydata$age / 12
mydata$ageMonthsCentered <- mydata$age - min(mydata$age, na.rm = TRUE)

mydata$ageYearsCentered <- mydata$ageMonthsCentered / 12
mydata$ageYearsCenteredSquared <- mydata$ageYearsCentered ^ 2

5 Estimator: ML or REML

For small sample sizes, restricted maximum likelihood (REML) is preferred over maximum likelihood (ML). ML preferred when there is a small number (< 4) of fixed effects; REML is preferred when there are more (> 4) fixed effects. The greater the number of fixed effects, the greater the difference between REML and ML estimates. Likelihood ratio (LR) tests for REML require exactly the same fixed effects specification in both models. So, to compare models with different fixed effects with an LR test (to determine whether to include a particular fixed effect), ML must be used. In contrast to the maximum likelihood estimation, REML can produce unbiased estimates of variance and covariance parameters, variance estimates are larger in REML than ML. To compare whether an effect should be fixed or random, use REML. To simultaneously compare fixed and random effects, use ML.

6 Linear Mixed Models

The following models are models that are fit in a linear mixed modeling framework.

6.1 Growth Curve Models

6.1.1 Linear Growth Curve Model

6.1.1.1 Plot Observed Growth Curves

ggplot(
  data = mydata,
  mapping = aes(
    x = ageYears,
    y = math,
    group = id)) +
  geom_line() +
  scale_x_continuous(
    name = "Age (Years)") +
  scale_y_continuous(
    name = "Math Score")

6.1.1.2 lme4

linearMixedModel <- lmer(
  math ~ female + ageYearsCentered + female:ageYearsCentered + (1 + ageYearsCentered | id), # random intercepts and slopes; sex as a fixed-effect predictor of the intercepts and slopes
  data = mydata,
  REML = FALSE, #for ML
  na.action = na.exclude,
  control = lmerControl(optimizer = "bobyqa"))

summary(linearMixedModel)

Linear mixed model fit by maximum likelihood . t-tests use Satterthwaite's
  method [lmerModLmerTest]
Formula: math ~ female + ageYearsCentered + female:ageYearsCentered +  
    (1 + ageYearsCentered | id)
   Data: mydata
Control: lmerControl(optimizer = "bobyqa")

      AIC       BIC    logLik -2*log(L)  df.resid 
  15857.9   15903.5   -7920.9   15841.9      2213 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.3750 -0.5174  0.0051  0.5239  2.6396 

Random effects:
 Groups   Name             Variance Std.Dev. Corr
 id       (Intercept)      62.5365  7.9080       
          ageYearsCentered  0.6767  0.8226   0.08
 Residual                  32.1505  5.6701       
Number of obs: 2221, groups:  id, 932

Fixed effects:
                         Estimate Std. Error        df t value Pr(>|t|)    
(Intercept)              30.51401    0.56142 752.48747  54.352   <2e-16 ***
female                   -0.61290    0.79482 736.39886  -0.771    0.441    
ageYearsCentered          4.26792    0.11253 610.09410  37.925   <2e-16 ***
female:ageYearsCentered  -0.02558    0.16092 598.89155  -0.159    0.874    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
            (Intr) female agYrsC
female      -0.706              
ageYrsCntrd -0.635  0.448       
fml:gYrsCnt  0.444 -0.631 -0.699

6.1.1.2.1 Protoypical Growth Curve

newData <- expand.grid(
  female = c(0, 1),
  ageYears = c(
    min(mydata$ageYears, na.rm = TRUE),
    max(mydata$ageYears, na.rm = TRUE))
)

newData$ageYearsCentered <- newData$ageYears - min(newData$ageYears)

newData$sex <- NA
newData$sex[which(newData$female == 0)] <- "male"
newData$sex[which(newData$female == 1)] <- "female"
newData$sex <- as.factor(newData$sex)

newData$predictedValue <- predict( # predict.merMod
  linearMixedModel,
  newdata = newData,
  re.form = NA
)

ggplot(
  data = newData,
  mapping = aes(x = ageYears, y = predictedValue, color = sex)) +
  xlab("Age (Years)") +
  ylab("Math Score") +
  geom_line()

6.1.1.2.2 Individuals’ Growth Curves

mydata$predictedValue <- predict(
  linearMixedModel,
  newdata = mydata,
  re.form = NULL
)

ggplot(
  data = mydata,
  mapping = aes(x = ageYears, y = predictedValue, group = factor(id))) +
  xlab("Age (Years)") +
  ylab("Math Score") +
  geom_line()

6.1.1.2.3 Individuals’ Trajectories Overlaid with Prototypical Trajectory

ggplot(
  data = mydata,
  mapping = aes(x = ageYears, y = predictedValue, group = factor(id))) +
  xlab("Age (Years)") +
  ylab("Math Score") +
  geom_line() +
  geom_line(
    data = newData,
    mapping = aes(x = ageYears, y = predictedValue, group = sex, color = sex),
    linewidth = 2)

6.1.1.2.4 Extract Random Effects

ranef(linearMixedModel)

$id
          (Intercept) ageYearsCentered
201       1.125313761      0.214025701
303     -12.515508864     -0.661489657
2702     12.257492354      0.430762981
4303      2.727957524      0.285002116
5002      1.943700949      0.170789619
5005      4.045982598      0.120286722
5701     12.299499720      0.346624148
6102     10.023593786      0.863981478
6801     10.850796246      0.310493680
6802     11.821009201      0.402109281
6803     13.002187545      0.661133879
9603      4.893027085      0.449774173
12401     2.267934250      0.215890170
12801     3.118093119      0.121909799
13702     9.965512365      0.174385818
13801    14.860587450      0.466936045
13803    12.244139122      0.342974140
17201     8.389382622      0.541441962
18601    -9.114431785     -0.325174564
22104    -8.224888366     -0.162602823
22901    -3.619019303     -0.120180576
23602     8.688461076      0.339697533
23701     3.179383329     -0.022949766
24402     5.504641645      0.352394764
26201     9.719849574      0.702414569
28503    -8.430258804      0.354120841
29201    12.005754759      0.186774027
29202     8.417070934      0.781161659
34401    13.282239330      0.463013476
35401     6.780780604      0.582522293
35402     5.591003405     -0.049981300
36901    15.699222750      0.479509729
36902    10.244502566      0.276867605
37403    -1.843161415     -0.402326018
38202    -3.239859256     -0.028890813
38802    -5.250923211     -0.340856108
40501     3.463053677      0.153135154
40702     7.971281145      0.173100533
42504     2.888138805      0.321177217
42505     6.585722700     -0.155175248
43103     4.433636752      0.238674652
45201     8.183501056      0.541547032
45903    15.716178377      0.502448721
46201    11.011297054      0.881456660
47201    14.720202046      0.427719979
47402     0.880121442     -0.190353845
47601     1.216021904     -0.017165043
48502    -2.519553727      0.312149007
49501     1.743541361      0.182682300
53601    -2.357792771     -0.378382000
55201     8.475516854      0.442472479
55801     2.607026981      0.226992780
56301    -5.108650931      0.279722834
56302    -0.069098635      0.007631557
57604    10.137559691      0.648424109
58503    -9.038592483     -0.145716134
62102     7.594264242      0.514296379
62103   -11.659099523      0.062806196
62202    12.282894687      0.512176358
62703     0.465584787      0.534532255
64002     0.758192010     -0.318985718
65601    12.288113181      0.422395609
65701    11.634539842      0.659937401
66002     4.577549270      0.481784449
70801    -1.720978172     -0.098222417
73602     3.129767946      0.121536903
74602     4.600091569      0.228352651
75603    -8.954581426     -0.158125796
76402     0.105445380     -0.276301811
76403     4.265544047      0.284415843
80102     3.858025733      0.287026976
82901    -3.386576272     -0.285969402
83001    -1.334354997     -0.271017391
83101    -6.092041392     -0.315778466
83102    -0.060917140     -0.148750133
83602     4.330407333     -0.021683400
83603    -0.359555292      0.065349949
85402    -2.755742657     -0.098799494
87501    -1.199237512     -0.175971326
87901     3.270068645      0.148146826
88902    -5.915202379     -0.317691655
89303    -0.058364793     -0.053342839
89803     6.548041328      0.642729376
90002    10.903475521      0.450119296
92303     1.585091240     -0.179298523
93202    -3.666591112     -0.040016525
93901    -3.564923756     -0.080818748
94002     0.625658727      0.677971537
96201    -2.014940963     -0.158988595
96202    -4.068449608     -0.163941086
98102     6.960679147      0.369901362
101701    0.168658221     -0.517413498
101702   -3.629285495      0.142066301
103501   -5.892939203     -0.645193590
108201   16.298148686      0.607038953
113001    8.357403604      0.635293810
113201    5.570042678      0.382727677
114101   -8.066311147     -0.029666619
114702    4.802081813      0.209318715
115901    2.373140471      0.543510541
116002    0.862849698     -0.047444533
116004   11.272635078      0.530370201
118402   11.180501318      0.376954789
122301   -7.041208476     -0.087766848
124201   -2.235992657     -0.057998101
124402   -6.654073179     -0.360197908
125002   -2.036353581     -0.270098667
126101    3.297895109      0.262547616
129502   -6.797821803     -0.419879120
129601   -4.382931097     -0.160115222
129602    0.466740158      0.017451223
132301    9.786381148      0.576039297
132402   -8.839831108     -0.350099054
132403    0.438064947     -0.144665332
136001   -5.458706565     -0.412198348
138901    2.422614923     -0.241055794
138902   -0.353560559     -0.051066863
139902   10.080325125      0.725346721
140004   -3.496297543      0.282378348
140104   -8.512781578     -0.655296107
140901   -0.115083608      0.716800588
140902    0.707055960     -0.308476123
141001    9.865281000      0.428920620
141601    3.020939622     -0.023542778
141603    0.627584701     -0.092923215
143901    3.106422910      0.127377649
144004    0.045069645     -0.112213249
153203   16.087092196      0.356217483
153204  -17.430077261     -0.616817016
153801    0.564118516     -0.113333334
155902   -8.936419914     -0.348618461
156401   12.187261187      0.348144705
156605   10.323654494      0.486995803
158701    8.513660697      0.336830662
158702   10.497163871      0.545777637
159605  -14.147922125     -0.755640107
159606  -11.066745066     -0.540698054
159801   -9.058337641      0.180842500
165501   -8.160174645     -0.377886697
165602    4.162599565      0.224329239
165701  -10.200591739     -0.555846787
165801    7.416486078      0.158087240
166602   -1.002994170      0.072206197
166603  -11.911124448     -0.292497710
167501   11.597707011      0.876649436
167502   17.304114029      0.737624890
169003   -0.779869964     -0.052336104
170403   -3.714540532     -0.672213772
170404   -5.285749530     -0.231791835
171401   -9.836468983     -0.069591914
172502    4.642796953     -0.407352298
173001   -5.588181640     -1.754863314
173201    0.578502007      0.031869950
174001   10.240215969      0.382702555
175405  -10.792035414     -0.348151265
175801   -7.789717327     -0.412713911
176003    8.715520376     -0.004390948
178901   -8.060279252     -0.232927660
179401   -5.537013298     -0.293072902
179701   -2.918516341     -0.698092331
180301    7.530934833      0.497157407
181801   -1.508291042     -0.134899316
182903   -0.244975730     -0.013902331
185304    2.293821497     -0.050900857
187803    2.406041176      0.434335498
188002    5.547847871     -0.253696393
188502    7.326775168      0.461852110
189601    7.020803583      0.239798464
189602    2.295523328      0.565604150
190202    2.361864346      0.109192928
190501   11.272910613      0.411262167
191202    0.109786069      0.287007128
193601   -6.460102071      0.726747858
193803    5.864314159      0.349916488
196002   -6.930064877     -0.316839993
196201    3.816025703      0.426787984
196202   -0.173952604     -0.230887906
204602    9.043456136      0.353334576
205802    8.940355783      0.987958401
207102   10.906815310      0.632212233
207202    3.946083434      0.614767925
207301   -0.305794959     -0.219240732
212001    5.175627775      0.307630681
213101   -1.604988068     -0.157754955
213702    1.853348987      0.136708618
215304   -1.680247506      0.283103740
216701    7.463945088      0.259872555
217402    4.141544076      0.234240048
219003    2.764968942      0.701685107
220401    3.029593716      0.699986352
221202    8.544703486      0.357270902
223002   15.104320740      0.623124524
224901   -4.469678843     -0.379986217
226001   -8.061210006      0.098195084
226502   -7.274549310     -0.100782710
226704   -7.527709790     -0.321128624
227002    2.461705710     -0.802262248
227101  -10.912047462     -0.095120844
227102    3.993605419      0.448123850
227201   -3.041064530     -0.137936840
231502   -7.742581030      0.041937232
233301   -1.019109719     -0.378574893
233901    9.003114610      0.382666793
233902    5.981925269      0.186142904
235503   -8.586220486     -0.338828100
236002   -3.383562465     -0.415619283
236201   -6.061759642      0.038488620
236202  -10.602033124     -0.071094358
236901    5.500138415      0.331178677
237501    6.492126787      0.518736481
238301   -1.069920262     -0.345037345
239603   -4.794009822     -0.505654150
239604   -3.138157293     -0.053065007
240701   -2.633824808     -0.230222843
240702   -5.845413982      0.313454961
244201    0.821623905     -0.010520847
244702    0.386389868      0.049039506
245903   -1.389917735     -0.441635774
248402   -9.640091283     -0.385119312
250502    1.575879281      0.185416145
252902   -8.705342925     -0.428989701
253001   -3.472250915     -0.321775121
253603   -3.952048624     -0.234004326
253802    0.813068757     -0.321677569
254901    1.557440949      0.242732961
255201    4.499756962      0.033367834
255901  -12.966621777     -0.095135814
255902   -6.408643305     -0.249179468
255903   -0.068506317      0.018661532
258801   10.066745258      0.473779137
260702   10.140891704      0.574527205
261501   -6.274288059     -0.078233942
261502   -5.275763532      0.442941285
261503   -3.768648304     -0.388219736
262601    3.713484558     -0.009543953
262602   13.156712564      0.474759507
263801   11.035679006      0.567834817
265902    0.656276991     -0.696886948
268301   -7.098532980     -0.305426775
268303   -3.792822734     -0.106412940
268901   -2.171914979     -0.078913196
268902   -0.431021818     -0.352369652
269001   -6.011169857     -0.488260650
269101    1.858122812      0.001046154
269102   -2.443573441      0.415726816
269902    4.732520442      0.381227590
271001   -2.477610234     -0.146752234
272903    3.562984351     -0.084305656
273501    6.570778698      0.281221305
274401   -2.812921787     -0.263584631
277002   -1.087288775     -0.304425636
277003    2.438031125      0.050445057
277502   -0.269337940     -0.041138191
278103    4.245009083      0.708932977
281001   -0.006211057     -0.564986398
281901    0.663646457      0.008984918
282602    2.931241086      0.373471582
282603    9.565849274      0.545267076
287702   11.892041859      0.660432677
288501    5.787422166      0.216667586
290901   13.929416909      0.821404683
290902    1.730188742      0.132665289
291101    6.259580343      0.294992854
294902   12.621060269      0.244878842
294903    4.072045749      0.087980535
295901   15.894522897      0.084921874
295902   14.439897307      0.297412167
296401    5.254230369      0.531416207
297901    0.856386616     -0.254469811
297902   -0.317311625      0.087017537
298301   -0.060917140     -0.148750133
301101   12.475974495      0.832250226
301401    2.948685577     -0.147761333
302301    0.416783844      0.060061145
302702    0.821664368     -0.089719049
302703   -0.236039902      0.125432324
304103   -4.456206190      0.013561574
304605   -4.255831240     -0.130330842
307301    3.310356602     -0.241802776
307501   -6.283556136     -0.083599724
309802    9.967256858      0.457526730
309901    3.484441714     -0.116828550
310101    1.290473853      0.216435818
311301   -0.080565206     -0.101463857
311701    1.194153804      0.038496161
312102    4.729692013      0.239638219
312201    6.695393683      0.210621990
312901   -2.426393020     -0.576742792
314501   12.287217891      0.539511886
315001    4.880252353      0.291273041
315002   13.783249241      0.739403986
315602    9.531309188      0.466146167
316301    7.479274636      0.567983219
316501    3.430731315      0.177619705
316502    4.093043825      0.093542139
319002   -0.169959992     -0.006645019
319801    2.316432546     -0.055008041
320101   -0.590631906     -0.433742436
320102    5.817675834      0.281357695
322505    8.674753283      0.562890678
324401   12.299517798      0.797447725
324402    4.946384990      0.388171137
324801    8.928434794      0.521462250
325401   -1.069020836      0.078768504
325402   13.458927332      0.509744340
325904   -7.522413853     -0.525541971
326901   -1.850392851     -0.213568608
328702    5.099223572     -0.092063809
328703    4.774712475      0.172466962
330601   -2.762453096     -0.532733818
331506   -6.722315938     -0.032499717
334201    0.565220953     -0.702132818
335001    7.357640498      0.275755052
335002    6.382367575      0.176234888
335501   -2.374164673      0.235733420
336401    1.185753997     -0.636458965
336902   -2.810051956     -0.539564109
340702    5.691374442      0.431253177
340801   -1.079168948     -0.042968743
342301    8.287865554      0.418811940
342302    2.963952990      0.322217400
342402    8.445022113      0.305838605
343302    0.814012955     -0.236369473
344602    3.405934612      0.408305541
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847302    3.291690158     -0.002044838
847701    8.235072091      0.400242080
847901   -0.432079542      0.114558409
847902   -6.633722012     -0.097355701
848201    4.468851642      0.670151752
849401    0.189416375     -0.149836441
853202   -0.429397010      0.218005012
856103   -3.586214102     -0.092203244
857003   -6.720617097     -0.493204616
858302   -4.204535539     -0.609410105
861503   -3.966967748      0.293400309
861803   -3.135359757      0.264874987
862201    4.653017738      0.132082293
863101  -12.297582748     -0.439929413
863403   -6.477163797     -0.474134043
864302    2.085691014      0.074949397
866203   -9.169437523     -0.147097347
866205   -3.492643017     -0.061050645
866402   -2.782792441     -0.029554824
867903   11.942822075      0.499397590
869001   -7.702502070     -0.202251421
869501    0.806896368      0.352096898
869601  -10.485942895     -0.461798451
870803    8.928434794      0.521462250
878701  -17.568709144     -0.626957686
879403   11.121309995      0.417858606
879404    3.311343239      0.047788637
885801    3.486722522      0.081372459
885802  -16.039468436     -1.050718441
886401   11.077378767      0.671794102
891202   -1.785951488      0.135840571
892601   -6.248449211     -0.100279844
894201   -4.410948902     -0.294281539
894802   -0.313453837     -0.115560234
894803   -5.508871671     -0.307101576
894804   -4.959265629     -0.273862791
897001   -3.323415615     -0.358621058
905003   -6.766784958     -0.578860310
907001   -0.044475957     -0.120023417
908102   -0.826610098      0.037276094
910103   -9.997531168     -0.318016485
911701    1.189104202      0.248190222
914402    4.330045404      0.059557193
918002    1.312890067     -0.031666983
918201    2.010858048      0.084121892
918202    2.960751078      0.071789615
918301   -0.253494219      0.421761227
922901    4.736230982      0.019599188
925503   -2.952847689     -0.184075604
925504   -8.155463950     -0.407774825
925701   -1.560286833     -0.085956943
926102  -11.384931242     -0.440643401
928101   -2.561314916     -0.175971001
929805    2.299878514      0.120028438
931202    6.182482161      0.760747880
931703   -5.921728745      0.253419327
938001   -4.268374580     -0.150332673
938002   -5.203853762     -0.359216554
938205    1.452960485      0.154925383
940103   -2.332263976     -0.136215098
951602   -9.264136115     -0.794456772
952201   -3.740749071     -0.123357170
964101    3.547967803      0.131810129
964102    7.585006320      0.091026089
966001    9.257072985      0.418793561
968001   -0.898120469     -0.728569335
968002   -4.421070337      0.099825430
968003   -8.385332673     -0.022239581
968401    2.945108480      0.304805734
968402    7.276622407      0.604626813
976803   -9.628901937     -0.742544730
982001   -4.695771491     -0.306490550
983703    3.745534505      0.165550609
983902    2.596009547      0.125200150
983903   -3.667967858     -0.155126491
984402   -1.217021320     -0.058975181
984404   -0.789019406     -0.327117617
985802   -3.353538083     -0.397301364
985803    0.978668861     -0.062038203
986203   -2.764543447     -0.702253665
986505    0.235705952     -0.568761949
986506   -5.064852532     -0.934175547
987401   -6.339050700      0.154132170
987701   -4.792430373     -0.496558484
988701   -8.738498236     -0.368233827
989201   -5.672395817     -0.500824137
989503   -3.963618084     -0.191185511
992201    3.713250289      0.037354172
993803   -0.339041448     -0.033276656
995101  -16.855431161     -0.793196728
995102  -16.732009976     -0.816271483
995304    1.141931270      0.039760624
997002   -7.890837919     -0.120332271
998001    2.904616596     -0.099911409
999202   -4.209454375     -0.249228417
999203   -3.445087389     -0.642747361
1002601 -12.615996043     -0.681804312
1003202  -4.900766767     -0.081557288
1003203  -7.168505838     -0.337373360
1003204  -7.477961633     -0.234615583
1003601   4.142234535      0.087851091
1003802   6.694641940      0.582885762
1004802  -6.010511003     -0.127669923
1004903  -5.696529302      0.321126050
1005201  -5.061927500     -0.154374548
1005203   7.744852438      0.359834771
1007201   1.485163277     -0.137844389
1007301 -12.207661253     -0.289172581
1007302 -12.256018327     -0.141761798
1007901  -1.587140380      0.168986153
1013201  -7.449018493     -0.201116136
1017504 -15.113587952     -0.217160076
1018402  -8.195378280     -0.336645460
1018404  -6.060598198     -0.384207768
1019104  -3.898712489     -0.395718677
1019105 -10.441463111     -0.495045209
1023702   0.230071978     -0.062853597
1030801   9.614192285      0.359470695
1031001   7.309989035      0.957296774
1031002   3.338190748      0.463618589
1031302  -7.796511938     -0.032990651
1033202 -13.370171847     -0.281812065
1033601  -5.677419865     -0.172171992
1034001  -6.334557499     -0.663053901
1036001  -2.492905799      0.635495607
1036003  -1.474746601     -0.133442885
1037502  -2.665942104     -0.015951671
1040101   4.276289331      0.440820891
1042101  -2.553330890     -0.258232185
1042201  -6.059952254     -0.189338564
1044801   4.128417062      0.245711117
1047801  -2.849801051     -0.185397056
1050001  -2.717825315     -0.095089574
1050701  -5.872749060      0.457585048
1053201  -0.250978709     -0.100236571
1053202  -6.760663767     -0.293314450
1053802  -6.284836936      0.080095322
1053804  -7.223993827     -0.248573995
1056501  -3.896931891     -0.118419463
1081101  -5.400851712     -0.293731045
1081103  -4.680976218     -0.137297987
1176403  -1.958564728     -0.236731201
1176502  -4.966749948      0.109135738
1179201   5.308169099     -0.088424313
1179202   1.720621134      0.117183301
1180202  -0.431475168     -0.508419596
1181201 -11.704987647     -0.583519704
1181902  -2.349190562     -0.066365192
1181904  -6.320454193     -0.333828815
1182604  -0.232940420     -0.239191587
1185003 -13.222234389     -0.226450008
1187001   3.002185759     -0.284717029
1187303   6.481394612      0.381759227
1189901   8.842299522      0.382244575
1190501  -2.594795950     -0.242898294
1191001   3.134902432      0.262260144
1191901   5.116734779      0.307548594
1194901  -2.642814420      0.740744619
1197902  -7.270379750     -0.242315430
1198101  -1.607639092     -0.129777591
1198901  15.013179772      0.359981902
1201701  -0.835329508     -0.169484890
1201702  -1.378932676     -0.575885435
1203201  -2.032124763     -0.578525839
1205201  -4.566609451      0.130473364
1207601   4.894676196      0.201302925
1209201   0.014426741     -0.133074642
1211502   8.396956305      0.333608159
1211503   5.392613109      0.178489598
1212303  -2.575279428     -0.048814312
1213501  -5.356118638      0.908621520
1217202  -1.881659801     -0.624995211
1217204  -6.521048243     -0.094817905
1217702  -3.406869123     -0.086193891
1217704  -2.337403065     -0.153791714
1219103  -7.263847297     -0.311537184
1219105 -12.688070163     -0.592413772
1219106 -13.168088281     -0.487925838
1219108   2.752059171      0.271825037
1221003  -0.763516936     -0.102460692
1221702  -3.839403600     -0.221064459
1224001   3.233736846     -0.091309850
1225402  -3.826603058     -0.116909854
1228103  -0.096753765     -0.003285703
1230302 -10.049267721     -0.752222025
1256601  -7.299006078     -0.291423458

with conditional variances for "id" 

6.1.1.3 nlme

linearMixedModel_nlme <- lme(
  math ~ female + ageYearsCentered + female:ageYearsCentered, # sex as a fixed-effect predictor of the intercepts and slopes
  random = ~ 1 + ageYearsCentered|id, # random intercepts and slopes
  data = mydata,
  method = "ML",
  na.action = na.exclude)

summary(linearMixedModel_nlme)

Linear mixed-effects model fit by maximum likelihood
  Data: mydata 
       AIC     BIC    logLik
  15857.85 15903.5 -7920.926

Random effects:
 Formula: ~1 + ageYearsCentered | id
 Structure: General positive-definite, Log-Cholesky parametrization
                 StdDev    Corr  
(Intercept)      7.9079987 (Intr)
ageYearsCentered 0.8225933 0.082 
Residual         5.6701380       

Fixed effects:  math ~ female + ageYearsCentered + female:ageYearsCentered 
                            Value Std.Error   DF  t-value p-value
(Intercept)             30.514011 0.5619217 1287 54.30296  0.0000
female                  -0.612896 0.7955333  930 -0.77042  0.4412
ageYearsCentered         4.267923 0.1126360 1287 37.89130  0.0000
female:ageYearsCentered -0.025585 0.1610671 1287 -0.15885  0.8738
 Correlation: 
                        (Intr) female agYrsC
female                  -0.706              
ageYearsCentered        -0.635  0.448       
female:ageYearsCentered  0.444 -0.631 -0.699

Standardized Within-Group Residuals:
         Min           Q1          Med           Q3          Max 
-3.375034869 -0.517409797  0.005105047  0.523910718  2.639557775 

Number of Observations: 2221
Number of Groups: 932 

6.1.1.4 Intraclass Correlation Coefficent

icc(linearMixedModel)

icc(linearMixedModel_nlme)

6.1.2 Growth Curve Model with Timepoint-Specific Errors

Adapted from Usami & Murayama (2018):

timepointSpecificErrorsMixedModel <- lmer(
  math ~ female + ageYearsCentered + female:ageYearsCentered + (1 | id) + (1 | ageYearsCentered), # timepoint-specific errors: observations are cross-classified with person and timepoint; sex as a fixed-effect predictor of the intercepts and slopes
  data = mydata,
  REML = FALSE, #for ML
  na.action = na.exclude,
  control = lmerControl(optimizer = "bobyqa"))

summary(timepointSpecificErrorsMixedModel)

Linear mixed model fit by maximum likelihood . t-tests use Satterthwaite's
  method [lmerModLmerTest]
Formula: math ~ female + ageYearsCentered + female:ageYearsCentered +  
    (1 | id) + (1 | ageYearsCentered)
   Data: mydata
Control: lmerControl(optimizer = "bobyqa")

      AIC       BIC    logLik -2*log(L)  df.resid 
  15808.6   15848.5   -7897.3   15794.6      2214 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.5409 -0.5191  0.0051  0.5093  2.7345 

Random effects:
 Groups           Name        Variance Std.Dev.
 id               (Intercept) 76.219   8.730   
 ageYearsCentered (Intercept)  4.503   2.122   
 Residual                     30.842   5.554   
Number of obs: 2221, groups:  id, 932; ageYearsCentered, 94

Fixed effects:
                          Estimate Std. Error         df t value Pr(>|t|)    
(Intercept)               29.87438    0.75758  211.81838  39.434   <2e-16 ***
female                    -0.38123    0.82243 1960.43213  -0.464    0.643    
ageYearsCentered           4.26419    0.14813  132.95739  28.786   <2e-16 ***
female:ageYearsCentered   -0.08143    0.14596 1364.56012  -0.558    0.577    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
            (Intr) female agYrsC
female      -0.545              
ageYrsCntrd -0.758  0.321       
fml:gYrsCnt  0.356 -0.652 -0.482

6.1.3 Quadratic Growth Curve Model

When using higher-order polynomials, we could specify contrast codes for time to reduce multicollinearity between the linear and quadratic growth factors: https://tdjorgensen.github.io/SEM-in-Ed-compendium/ch27.html#saturated-growth-model

factorLoadings <- poly(
  x = c(0,1,2,3), # times (can allow unequal spacing)
  degree = 2)

factorLoadings

              1    2
[1,] -0.6708204  0.5
[2,] -0.2236068 -0.5
[3,]  0.2236068 -0.5
[4,]  0.6708204  0.5
attr(,"coefs")
attr(,"coefs")$alpha
[1] 1.5 1.5

attr(,"coefs")$norm2
[1] 1 4 5 4

attr(,"degree")
[1] 1 2
attr(,"class")
[1] "poly"   "matrix"

linearLoadings <- factorLoadings[,1]
quadraticLoadings <- factorLoadings[,2]

linearLoadings

[1] -0.6708204 -0.2236068  0.2236068  0.6708204

quadraticLoadings

[1]  0.5 -0.5 -0.5  0.5

6.1.3.1 Fit Model

quadraticGCM <- lmer(
  math ~ female + ageYearsCentered + ageYearsCenteredSquared + female:ageYearsCentered + female:ageYearsCenteredSquared + (1 + ageYearsCentered | id), # random intercepts and linear slopes; fixed quadratic slopes; sex as a fixed-effect predictor of the intercepts and slopes
  data = mydata,
  REML = FALSE, #for ML
  na.action = na.exclude,
  control = lmerControl(optimizer = "bobyqa"))

summary(quadraticGCM)

Linear mixed model fit by maximum likelihood . t-tests use Satterthwaite's
  method [lmerModLmerTest]
Formula: math ~ female + ageYearsCentered + ageYearsCenteredSquared +  
    female:ageYearsCentered + female:ageYearsCenteredSquared +  
    (1 + ageYearsCentered | id)
   Data: mydata
Control: lmerControl(optimizer = "bobyqa")

      AIC       BIC    logLik -2*log(L)  df.resid 
  15666.9   15724.0   -7823.5   15646.9      2211 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.3660 -0.4945  0.0048  0.5085  2.4377 

Random effects:
 Groups   Name             Variance Std.Dev. Corr 
 id       (Intercept)      69.8860  8.3598        
          ageYearsCentered  0.7099  0.8426   -0.05
 Residual                  27.1959  5.2150        
Number of obs: 2221, groups:  id, 932

Fixed effects:
                                 Estimate Std. Error         df t value
(Intercept)                      23.43112    0.84914 1380.85816  27.594
female                            0.61167    1.18734 1370.91791   0.515
ageYearsCentered                  8.79334    0.42555 1381.21301  20.664
ageYearsCenteredSquared          -0.57623    0.05255 1382.10819 -10.966
female:ageYearsCentered          -0.62379    0.60301 1386.34429  -1.034
female:ageYearsCenteredSquared    0.05700    0.07582 1393.17373   0.752
                               Pr(>|t|)    
(Intercept)                      <2e-16 ***
female                            0.607    
ageYearsCentered                 <2e-16 ***
ageYearsCenteredSquared          <2e-16 ***
female:ageYearsCentered           0.301    
female:ageYearsCenteredSquared    0.452    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
            (Intr) female agYrsC agYrCS fml:YC
female      -0.715                            
ageYrsCntrd -0.836  0.598                     
agYrsCntrdS  0.759 -0.543 -0.969              
fml:gYrsCnt  0.590 -0.829 -0.706  0.683       
fml:gYrsCnS -0.526  0.751  0.671 -0.693 -0.968

This is equivalent to:

quadraticGCM <- lmer(
  math ~ female + ageYearsCentered + I(ageYearsCentered^2) + female:ageYearsCentered + female:I(ageYearsCentered^2) + (1 + ageYearsCentered | id), # random intercepts and slopes; sex as a fixed-effect predictor of the intercepts and slopes
  data = mydata,
  REML = FALSE, #for ML
  na.action = na.exclude,
  control = lmerControl(optimizer = "bobyqa"))

summary(quadraticGCM)

Linear mixed model fit by maximum likelihood . t-tests use Satterthwaite's
  method [lmerModLmerTest]
Formula: 
math ~ female + ageYearsCentered + I(ageYearsCentered^2) + female:ageYearsCentered +  
    female:I(ageYearsCentered^2) + (1 + ageYearsCentered | id)
   Data: mydata
Control: lmerControl(optimizer = "bobyqa")

      AIC       BIC    logLik -2*log(L)  df.resid 
  15666.9   15724.0   -7823.5   15646.9      2211 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.3660 -0.4945  0.0048  0.5085  2.4377 

Random effects:
 Groups   Name             Variance Std.Dev. Corr 
 id       (Intercept)      69.8860  8.3598        
          ageYearsCentered  0.7099  0.8426   -0.05
 Residual                  27.1959  5.2150        
Number of obs: 2221, groups:  id, 932

Fixed effects:
                               Estimate Std. Error         df t value Pr(>|t|)
(Intercept)                    23.43112    0.84914 1380.85816  27.594   <2e-16
female                          0.61167    1.18734 1370.91791   0.515    0.607
ageYearsCentered                8.79334    0.42555 1381.21301  20.664   <2e-16
I(ageYearsCentered^2)          -0.57623    0.05255 1382.10819 -10.966   <2e-16
female:ageYearsCentered        -0.62379    0.60301 1386.34429  -1.034    0.301
female:I(ageYearsCentered^2)    0.05700    0.07582 1393.17373   0.752    0.452
                                
(Intercept)                  ***
female                          
ageYearsCentered             ***
I(ageYearsCentered^2)        ***
female:ageYearsCentered         
female:I(ageYearsCentered^2)    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
            (Intr) female agYrsC I(YC^2 fml:YC
female      -0.715                            
ageYrsCntrd -0.836  0.598                     
I(gYrsCn^2)  0.759 -0.543 -0.969              
fml:gYrsCnt  0.590 -0.829 -0.706  0.683       
fml:I(YC^2) -0.526  0.751  0.671 -0.693 -0.968

6.1.3.2 Protoypical Growth Curve

newData <- expand.grid(
  female = c(0, 1),
  ageYears = seq(from = min(mydata$ageYears, na.rm = TRUE), to = max(mydata$ageYears, na.rm = TRUE), length.out = 10000))

newData$ageYearsCentered <- newData$ageYears - min(newData$ageYears)
newData$ageYearsCenteredSquared <- newData$ageYearsCentered ^ 2

newData$sex <- NA
newData$sex[which(newData$female == 0)] <- "male"
newData$sex[which(newData$female == 1)] <- "female"
newData$sex <- as.factor(newData$sex)

newData$predictedValue <- predict( # predict.merMod
  quadraticGCM,
  newdata = newData,
  re.form = NA
)

ggplot(
  data = newData,
  mapping = aes(
    x = ageYears,
    y = predictedValue,
    color = sex)) +
  xlab("Age (Years)") +
  ylab("Math Score") +
  geom_line()

6.1.3.3 Individuals’ Growth Curves

mydata$predictedValue <- predict(
  quadraticGCM,
  newdata = mydata,
  re.form = NULL
)

ggplot(
  data = mydata,
  mapping = aes(
    x = ageYears,
    y = predictedValue,
    group = factor(id))) +
  xlab("Age (Years)") +
  ylab("Math Score") +
  geom_line()

6.1.3.4 Individuals’ Trajectories Overlaid with Prototypical Trajectory

ggplot(
  data = mydata,
  mapping = aes(
    x = ageYears,
    y = predictedValue,
    group = factor(id))) +
  xlab("Age (Years)") +
  ylab("Math Score") +
  geom_line() +
  geom_line(
    data = newData,
    mapping = aes(
      x = ageYears,
      y = predictedValue,
      group = sex,
      color = sex),
    linewidth = 2)

6.1.3.5 Extract Random Effects

ranef(quadraticGCM)

$id
         (Intercept) ageYearsCentered
201       0.33928713     0.2048916235
303     -14.23955413    -0.5404361816
2702     13.67892361     0.6053891399
4303      1.87926497     0.2622509407
5002      1.61867520     0.3766872283
5005      5.00342166     0.1103061842
5701     12.64428032     0.1076555275
6102      9.72171668     1.0058459560
6801     11.48936045     0.1758389252
6802     12.40495519     0.2481464174
6803     13.12879251     0.5013588794
9603      5.52979990     0.3824649037
12401     2.66417687     0.1687337604
12801     2.60493276     0.0651466650
13702    10.61884268     0.0773190112
13801    15.36440208     0.2021123643
13803    12.80038202     0.1110485222
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1031001   8.41926674     0.8884365398
1031002   2.61726444     0.5687764738
1031302  -7.49864845     0.0238281057
1033202 -15.27241276    -0.1013013000
1033601  -6.77615959    -0.1356527064
1034001  -6.38457498    -0.3774152235
1036001  -2.90674838     0.7579398885
1036003  -2.49687709    -0.1787402824
1037502  -1.21200719    -0.1915285497
1040101   3.61694408     0.4188760000
1042101  -2.47889816    -0.3250838342
1042201  -7.07864202    -0.1667731918
1044801   5.27999007     0.1352621731
1047801  -2.78469529    -0.2438990700
1050001  -3.00945189    -0.1172746551
1050701  -5.81647206     0.6866632213
1053201  -0.95441977    -0.1619816337
1053202  -6.95363077    -0.3185416524
1053802  -7.07885643     0.1638374049
1053804  -7.76244443    -0.2296186628
1056501  -5.04202222    -0.0791672561
1081101  -6.75642144    -0.2620547492
1081103  -4.96763880    -0.0609534768
1176403  -2.20154088    -0.3199205933
1176502  -5.18399523     0.1404597238
1179201   6.26432726    -0.3416049617
1179202   3.04900080    -0.0282322304
1180202   1.57919458    -0.5766208445
1181201 -12.69246491    -0.5442199223
1181902  -2.90077365    -0.1014802659
1181904  -5.61383445    -0.1279697556
1182604   0.53424136    -0.3697945245
1185003 -13.13911291    -0.0029829473
1187001   3.32587038    -0.3349836126
1187303   5.94996716     0.2982764506
1189901   9.09614222     0.3593369162
1190501  -3.11456753    -0.0952460874
1191001   2.42903740     0.2403311666
1191901   4.88269987     0.2124648289
1194901  -3.08601463     1.0566024696
1197902  -5.58073826    -0.4460158896
1198101  -1.27644936    -0.2101451737
1198901  16.65324847    -0.0262390833
1201701  -1.67105619    -0.2423445259
1201702   0.08009328    -0.5883490291
1203201  -2.06266202    -0.4507707221
1205201  -6.00924682     0.2217953590
1207601   4.48135562     0.1113411087
1209201  -0.62585073    -0.2031446927
1211502   9.32789455     0.1114924859
1211503   5.69457884     0.1078567048
1212303  -3.26708662     0.0656369113
1213501  -3.82772221     1.0464054797
1217202  -1.94939839    -0.5953455294
1217204  -7.84971939    -0.0171599235
1217702  -2.77265776    -0.0310256094
1217704  -2.68889407    -0.1430860568
1219103  -8.14296645    -0.0699275642
1219105 -14.48626571    -0.4039643861
1219106 -15.39021023    -0.3394775206
1219108   2.23229848     0.3042720787
1221003  -0.11492997    -0.1629173604
1221702  -5.09500316    -0.2257966798
1224001   2.98624921    -0.2283459623
1225402  -4.74670456    -0.1094826571
1228103  -0.52257207    -0.0103495485
1230302  -8.82049460    -0.6378126834
1256601  -8.41003160    -0.1124932283

with conditional variances for "id" 

6.1.4 Spline Growth Curve Model

6.1.4.1 Create Knot

mydata$knot <- NA
mydata$knot[which(mydata$ageYears <= 11)] <- 0
mydata$knot[which(mydata$ageYears > 11)] <- 1

6.1.4.2 Fit Model

splineGCM <- lmer(
  math ~ female + ageYearsCentered + female:ageYearsCentered + knot + knot:ageYearsCentered + female:knot + female:knot:ageYearsCentered + (1 + ageYearsCentered | id), # random intercepts and linear slopes; fixed quadratic slopes; sex as a fixed-effect predictor of the intercepts and slopes
  data = mydata,
  REML = FALSE, #for ML
  na.action = na.exclude,
  control = lmerControl(optimizer = "bobyqa"))

summary(splineGCM)

Linear mixed model fit by maximum likelihood . t-tests use Satterthwaite's
  method [lmerModLmerTest]
Formula: math ~ female + ageYearsCentered + female:ageYearsCentered +  
    knot + knot:ageYearsCentered + female:knot + female:knot:ageYearsCentered +  
    (1 + ageYearsCentered | id)
   Data: mydata
Control: lmerControl(optimizer = "bobyqa")

      AIC       BIC    logLik -2*log(L)  df.resid 
  15691.2   15759.6   -7833.6   15667.2      2209 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.3041 -0.5021 -0.0075  0.5034  2.4850 

Random effects:
 Groups   Name             Variance Std.Dev. Corr 
 id       (Intercept)      68.3568  8.2678        
          ageYearsCentered  0.7018  0.8377   -0.03
 Residual                  27.7772  5.2704        
Number of obs: 2221, groups:  id, 932

Fixed effects:
                              Estimate Std. Error        df t value Pr(>|t|)
(Intercept)                    25.9318     0.7222 1223.0139  35.909  < 2e-16
female                          0.3473     1.0060 1208.6807   0.345    0.730
ageYearsCentered                6.2092     0.2328 1374.3269  26.674  < 2e-16
knot                           12.6515     1.8238 1241.0685   6.937 6.43e-12
female:ageYearsCentered        -0.3785     0.3240 1373.1043  -1.168    0.243
ageYearsCentered:knot          -3.4908     0.3746 1283.6771  -9.318  < 2e-16
female:knot                    -1.2105     2.6098 1234.2550  -0.464    0.643
female:ageYearsCentered:knot    0.3530     0.5383 1279.7386   0.656    0.512
                                
(Intercept)                  ***
female                          
ageYearsCentered             ***
knot                         ***
female:ageYearsCentered         
ageYearsCentered:knot        ***
female:knot                     
female:ageYearsCentered:knot    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
            (Intr) female agYrsC knot   fml:YC agYrC: fml:kn
female      -0.718                                          
ageYrsCntrd -0.786  0.564                                   
knot        -0.267  0.192  0.266                            
fml:gYrsCnt  0.565 -0.776 -0.718 -0.191                     
agYrsCntrd:  0.470 -0.337 -0.568 -0.925  0.408              
female:knot  0.187 -0.259 -0.186 -0.699  0.262  0.646       
fml:gYrsCn: -0.327  0.455  0.395  0.644 -0.559 -0.696 -0.927

6.1.4.3 Protoypical Growth Curve

newData <- expand.grid(
  female = c(0, 1),
  ageYears = seq(from = min(mydata$ageYears, na.rm = TRUE), to = max(mydata$ageYears, na.rm = TRUE), length.out = 10000))

newData$ageYearsCentered <- newData$ageYears - min(newData$ageYears)

newData$knot <- NA
newData$knot[which(newData$ageYears <= 11)] <- 0
newData$knot[which(newData$ageYears > 11)] <- 1

newData$sex <- NA
newData$sex[which(newData$female == 0)] <- "male"
newData$sex[which(newData$female == 1)] <- "female"
newData$sex <- as.factor(newData$sex)

newData$predictedValue <- predict( # predict.merMod
  splineGCM,
  newdata = newData,
  re.form = NA
)

ggplot(
  data = newData,
  mapping = aes(
    x = ageYears,
    y = predictedValue,
    color = sex)) +
  xlab("Age (Years)") +
  ylab("Math Score") +
  geom_line()

6.1.4.4 Individuals’ Growth Curves

mydata$predictedValue <- predict(
  splineGCM,
  newdata = mydata,
  re.form = NULL
)

ggplot(
  data = mydata,
  mapping = aes(
    x = ageYears,
    y = predictedValue,
    group = factor(id))) +
  xlab("Age (Years)") +
  ylab("Math Score") +
  geom_line()

6.1.4.5 Individuals’ Trajectories Overlaid with Prototypical Trajectory

ggplot(
  data = mydata,
  mapping = aes(
    x = ageYears,
    y = predictedValue,
    group = factor(id))) +
  xlab("Age (Years)") +
  ylab("Math Score") +
  geom_line() +
  geom_line(
    data = newData,
    mapping = aes(
      x = ageYears,
      y = predictedValue,
      group = sex,
      color = sex),
    linewidth = 2)

6.1.4.6 Extract Random Effects

ranef(splineGCM)

$id
          (Intercept) ageYearsCentered
201       0.652144960     0.2083234957
303     -13.900382644    -0.5384100866
2702     13.248702169     0.4885445614
4303      2.123617918     0.2695865314
5002      1.813504926     0.3311534903
5005      4.914571158     0.1306294817
5701     12.746328285     0.1492250422
6102      9.811998768     0.9898072080
6801     11.251157093     0.2057798471
6802     12.123413767     0.2804666424
6803     13.281644740     0.5360890015
9603      5.389868582     0.3847449591
12401     2.656010922     0.1765637817
12801     2.656929651     0.0729177050
13702    10.715721324     0.1073154154
13801    15.450806205     0.2520724528
13803    12.401308398     0.1001032177
17201     7.861599067     0.4026244286
18601   -10.177265186    -0.2439801021
22104    -9.772847154    -0.1501052395
22901    -3.034416924    -0.1081377380
23602     8.698629194     0.2387282168
23701     3.154766594    -0.1140889363
24402     4.792965940     0.2287609053
26201    10.809083835     0.6206665402
28503    -8.129799847     0.3794778473
29201    13.049426803     0.2092913485
29202     8.242040820     0.7597709985
34401    14.369766601     0.5254802987
35401     6.833553015     0.6305626334
35402     5.815713216    -0.1275532970
36901    16.472051954     0.3285090538
36902    11.202203620     0.2812051478
37403    -0.583891740    -0.5036468086
38202    -2.437779179    -0.0475118466
38802    -6.196997784    -0.2974576620
40501     3.268217007     0.0214128468
40702     9.112695574    -0.0492314988
42504     1.995724022     0.3717940459
42505     6.863666740    -0.2638968855
43103     5.821996552     0.3275967123
45201     8.831273789     0.4316009873
45903    15.771000534     0.2051521382
46201    11.264243763     0.9108447155
47201    15.668737876     0.1788208935
47402     1.397962004    -0.1572621684
47601     2.316161114     0.0508547425
48502    -2.010293653     0.3936983795
49501     2.160142219     0.1134996285
53601    -1.775617993    -0.3861595134
55201     9.157903534     0.4402902346
55801     2.566584709     0.3197868128
56301    -6.033375659     0.4309999924
56302    -0.497820148    -0.0766013567
57604    10.920240580     0.5373312559
58503    -9.809490058     0.1142761543
62102     8.723136893     0.5120606501
62103   -12.441656968     0.2256406920
62202    12.496248891     0.3601452142
62703     1.085446072     0.6091981739
64002     1.922686288    -0.3572227883
65601    12.556337701     0.2046818628
65701    12.034497734     0.5863928139
66002     5.629706249     0.4789736360
70801    -2.802991869    -0.0481438997
73602     2.928465530    -0.0111033594
74602     5.341015494     0.0845985843
75603    -9.063079470    -0.1325943166
76402    -0.382474144    -0.3180360760
76403     4.039892562     0.2205538357
80102     4.373011570     0.4045018855
82901    -3.998049469    -0.2277049608
83001    -0.771642029    -0.2655275958
83101    -5.938951649    -0.1721878170
83102    -0.335988597    -0.2163531218
83602     4.768937922     0.0307470347
83603    -0.704466068     0.0518760872
85402    -4.069035989    -0.0549955050
87501    -2.216913886    -0.1313351231
87901     3.713742743     0.0534331179
88902    -5.217618285    -0.2443179929
89303    -0.549963171    -0.1108078629
89803     6.315856211     0.6272311822
90002    11.200283209     0.3496123205
92303     1.591599351    -0.2536957823
93202    -4.272559716    -0.0686060268
93901    -4.258376290    -0.0789601908
94002     1.347898206     0.8180194366
96201    -0.812094574    -0.1820390184
96202    -5.188134771    -0.1368756110
98102     7.021903082     0.3228415319
101701    0.611491352    -0.5395367529
101702   -3.309701677     0.2651586781
103501   -4.855153690    -0.5988670475
108201   17.109929489     0.3560101418
113001    8.103995908     0.6662753745
113201    5.068036082     0.3308741944
114101   -7.638376061     0.1190733697
114702    3.620812603     0.0676329525
115901    3.596298646     0.5221457642
116002    2.607162423     0.0792587285
116004   11.141442069     0.3485010978
118402   11.657468146     0.3032229400
122301   -8.237554758    -0.0444601711
124201   -3.201342516    -0.1291495001
124402   -7.411447467    -0.2726549361
125002   -2.324071622    -0.2732205004
126101    3.395328628     0.2530541153
129502   -8.109352437    -0.3358308547
129601   -3.694174646    -0.1748879841
129602   -0.004890503    -0.0001257246
132301   11.022416508     0.4585683295
132402   -9.869915706    -0.0353655374
132403    1.015949190    -0.2664092494
136001   -4.570641198    -0.3018957317
138901    3.448681327    -0.4417087283
138902   -0.659219825    -0.1648237787
139902   11.569741040     0.7989640943
140004   -4.143993718     0.2820237083
140104   -9.393058287    -0.4183388716
140901    0.506211936     0.5776220274
140902    2.008178668    -0.4332679931
141001   10.529025701     0.2681977001
141601    2.834946460    -0.1234921691
141603    1.864431525    -0.2273344283
143901    2.992285556     0.0424144798
144004   -0.604860664    -0.1055563297
153203   17.584083365     0.0812167576
153204  -19.217608036    -0.5563571110
153801    0.213455250    -0.1616756876
155902   -9.815326467    -0.2648213903
156401   13.106919035     0.1431165911
156605   10.401893628     0.3255663688
158701    8.376065438     0.2669487460
158702   11.327900531     0.6213357486
159605  -15.721145813    -0.6151441672
159606  -12.670157227    -0.4217667273
159801   -8.837673378     0.4405456251
165501   -9.556634747    -0.2746392920
165602    3.848039614     0.2136728781
165701  -10.922472572    -0.3906266026
165801    7.847561348     0.2132949362
166602   -1.772260785     0.1403003024
166603  -13.018202108    -0.2436342004
167501   11.902952461     0.8271717586
167502   18.389913126     0.5251316841
169003   -1.184804415     0.0920900276
170403   -3.640211918    -0.5911589285
170404   -5.888642474    -0.1909256162
171401   -8.774188849    -0.0289529377
172502    6.007624508    -0.4856175096
173001   -3.532202556    -1.8229571244
173201    0.100180877     0.0044099543
174001   11.496154194     0.2386557661
175405  -10.944056962    -0.3543867229
175801   -7.388428629    -0.2115091628
176003    9.279680284    -0.2422432351
178901   -7.900264290    -0.1734262827
179401   -5.569316716    -0.2015112502
179701   -1.573630049    -0.6057534998
180301    8.830015417     0.5339910070
181801   -2.130233232    -0.1011293569
182903   -0.781088208    -0.0357528437
185304    2.054010804    -0.1857004637
187803    3.237960366     0.4697704541
188002    6.682525613    -0.2336373424
188502    7.163180465     0.3866629904
189601    7.497162134     0.2684616832
189602    2.323101362     0.7762704681
190202    2.038508673    -0.0034264974
190501   11.512792021     0.3307443223
191202    0.868352422     0.3079964455
193601   -7.779654363     0.9775400072
193803    5.906687900     0.3316013332
196002   -8.362696350    -0.3529805778
196201    4.219276510     0.3626475506
196202    1.021205365    -0.2030038501
204602    8.872590054     0.1807214418
205802   10.242216349     1.0490612027
207102   10.993956855     0.5654727084
207202    3.525816664     0.6562284526
207301   -0.791545723    -0.2182437680
212001    5.056782807     0.4001862747
213101   -2.062636132    -0.2755041558
213702    1.551355094     0.2113731687
215304   -2.031118393     0.5523556292
216701    7.374241783     0.1131518312
217402    5.064895697     0.2008526677
219003    2.354076655     0.7937105244
220401    3.782415835     0.7125569436
221202   10.070996738     0.2486922321
223002   15.728425584     0.4306337911
224901   -3.013015917    -0.4251288567
226001   -8.599400863     0.1950905140
226502   -8.479978528     0.0002054359
226704   -8.653802583    -0.0481777029
227002    3.020943909    -0.7776319668
227101  -12.003922326     0.1818461513
227102    3.594571099     0.4742016447
227201   -3.471070487    -0.0158667855
231502   -8.470660512     0.1306369113
233301   -0.277743553    -0.3491408232
233901    9.142608012     0.2853892339
233902    5.977265622     0.1037109941
235503   -9.748939377    -0.2559857776
236002   -3.956374105    -0.4352266309
236201   -6.860293369     0.0635328590
236202  -11.405396328     0.0112172260
236901    6.537680492     0.3228252642
237501    6.634815535     0.6219354631
238301   -0.561639744    -0.2775688589
239603   -3.807444046    -0.4613305887
239604   -2.624217708    -0.1392523867
240701   -2.814072362    -0.2265948459
240702   -5.394431688     0.3318887771
244201    0.055103450    -0.1242383480
244702    0.003743181    -0.0272862586
245903   -1.499682835    -0.5018116428
248402  -11.221149647    -0.3177062488
250502    1.687272155     0.2908951509
252902   -9.840652753    -0.4844864290
253001   -4.097697558    -0.3324151023
253603   -4.974154728    -0.3192530906
253802    1.821490157    -0.3424047377
254901    0.933188357     0.2815704653
255201    4.481286992    -0.1425491653
255901  -14.401543986     0.1349502187
255902   -6.802601303    -0.1557736752
255903   -0.693978827     0.0204272984
258801   10.512221706     0.3394946638
260702    9.877213987     0.4397797986
261501   -5.686362487    -0.1323804043
261502   -6.663219382     0.8028337630
261503   -4.189480113    -0.4156013821
262601    3.851042222     0.0482523003
262602   13.080986760     0.2369840503
263801   11.669895820     0.4596394426
265902    1.986862239    -0.7016725848
268301   -8.241647519    -0.2575504204
268303   -4.742035217    -0.0927816751
268901   -2.954530211    -0.1571329152
268902    0.138025639    -0.3118058171
269001   -6.304862424    -0.3485069887
269101    1.861546568     0.0974767425
269102   -3.453675766     0.5784219826
269902    4.445471097     0.3364996881
271001   -1.830838376    -0.1057897562
272903    4.287569963    -0.2487570271
273501    7.635717755     0.2268629055
274401   -3.403635800    -0.2756986901
277002   -1.243933170    -0.2347542123
277003    2.418219955     0.1380275097
277502   -0.502617518    -0.0413044058
278103    3.626497373     0.8115046407
281001    1.184524548    -0.7241158255
281901    0.213884274    -0.0154378167
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752003    1.980723949    -0.5464093787
752501   -2.291324472    -0.1768991458
760102    0.002779244    -0.2678997226
763603   -9.515067832    -0.2267766664
764503   -3.529410857    -0.1965419102
765702    1.477915925    -0.1405890196
767901    3.194300097    -0.0668313731
771002   -9.628492583    -0.0331301722
775002    7.459231867     0.2618756402
778902  -13.159806097    -0.3228931091
778903   -1.398100657    -0.7319082250
783001   -8.902087517    -0.2744852246
783002   -1.867724426    -0.1984787514
783301    4.229424280     0.4597352393
783502    1.293732538     0.1286967835
783602   -1.511633114    -0.1079115367
783801    1.466023472     0.2565860748
785601    3.812960011     0.0865059612
786402    5.189301366     0.1904138171
788302  -11.728646898    -0.4109713490
788303  -11.703157030     0.1904551466
792103   -1.717768617     0.2765264126
792704   -4.263432014    -0.1608600887
793001    2.631618708     0.5444128410
794301  -10.885986014    -0.4341363188
794503   -8.046908048    -0.1400713658
795201   -4.778623885    -0.2761614132
795901    5.312868665    -0.0366368565
799803   14.440612041     0.3134571096
800602    1.575169789    -0.0671218693
804701   -1.182292795     0.1527017788
804702    0.188818697    -0.3088758177
809102   -5.605012786    -0.6008609290
809103   -9.565248243    -0.6846855454
809301   -6.198420982    -0.7978763861
810303    0.377371540     0.0057558757
811002   -8.319520622    -0.2138774760
812504    0.993614255     0.0264062237
814101    0.131310877    -0.1119582959
817402   -3.379127920    -0.3726538867
817403   -4.085829565    -0.7050319754
817404   -4.755705815    -0.5346998879
822602    7.917443103     0.1327733745
825702   -9.970917659    -0.1367166687
825902   -8.435519858    -0.1686452329
825903   -1.699014338    -0.5378369475
826503    2.977856523    -0.1537479884
826504  -20.169375616    -0.8092409240
826904  -17.714957998    -0.7663456674
826905   -8.867298417    -0.2094385227
827101    2.911580798     0.0743696646
827302   -6.515098274    -0.0472490379
828302    2.758256821     0.2710918047
828604   -0.387730778    -0.3245729275
828902   -5.870203967    -0.4487020126
828903   -5.540272045    -0.2053766389
828905  -13.835763582    -0.3159407142
829401   -8.807481226    -0.3429406667
829403    1.481529252     0.2490552553
833601   -0.624297944    -0.2652069381
834301   -0.220777081    -0.0289871697
835202    1.659102580    -0.2632755479
835703   -3.328187213     0.4162598998
837503   -6.753085789    -0.2036923842
837504  -10.706741673    -0.0458817913
837602  -14.410977124    -0.3320380379
838301   -7.498085384    -0.4155690654
838903    4.324118577     0.3380756621
841601    0.380851813     0.2425089126
841602    0.841025013    -0.0739130759
846301   12.045171447     0.3091647315
847301   -4.297862641    -0.5546112145
847302    3.482212270     0.0265452839
847701    7.917419013     0.3513418683
847901   -0.460907634     0.0713428669
847902   -7.658645851    -0.1069721236
848201    4.801092377     0.6521619294
849401    0.832776648    -0.0621926478
853202   -1.129530150     0.3017979514
856103   -2.805522684    -0.0979480708
857003   -8.202760994    -0.4038646396
858302   -2.788553034    -0.5891048005
861503   -3.786890937     0.2863354777
861803   -2.975009858     0.2570636647
862201    4.450524308     0.0632242098
863101  -13.753601142    -0.3306235274
863403   -5.858524796    -0.4329759096
864302    1.683787838     0.0675621624
866203   -9.608254207     0.0391321674
866205   -4.337558707    -0.0313018735
866402   -3.581884688    -0.0063703398
867903   12.840335929     0.3233423898
869001   -7.891305491    -0.1876723061
869501    0.086653086     0.3941996230
869601  -11.564066942    -0.3804197551
870803    9.307156689     0.4423498499
878701  -18.564325364    -0.5304544041
879403   12.691982843     0.3210558438
879404    3.242777255    -0.0845540177
885801    3.311057771     0.2319786141
885802  -16.768413564    -0.8683749095
886401   11.907542234     0.7056690429
891202   -1.081121395     0.1152864260
892601   -7.622453080    -0.1032087784
894201   -5.013783151    -0.2009735418
894802   -0.053149059    -0.0547022339
894803   -6.490546445    -0.2994334585
894804   -4.621840870    -0.1764300223
897001   -3.365401619    -0.1629674155
905003   -7.445714333    -0.3640961880
907001   -0.374151219    -0.1495011196
908102    0.219617324    -0.0345188263
910103  -11.167480720    -0.2984328178
911701    2.073122697     0.1071727942
914402    4.370790200    -0.0459269353
918002    1.007393620    -0.0978816035
918201    1.660847815     0.0600406049
918202    3.268499663    -0.0317340583
918301    0.035351706     0.5101045390
922901    4.361980088    -0.1613853887
925503   -3.438776695     0.0453044433
925504   -9.011543740    -0.3818310966
925701   -2.371167896    -0.1043786247
926102  -12.912275437    -0.3037111025
928101   -3.267917120    -0.1876491634
929805    1.834634487     0.0030545361
931202    6.403164042     0.9217389556
931703   -5.354493946     0.2360415478
938001   -3.994223109     0.0204975093
938002   -4.614140459    -0.3244595553
938205    1.567384264     0.0714191192
940103   -2.876967919    -0.1367363170
951602   -8.075174085    -0.6479917899
952201   -3.945860026    -0.1560927886
964101    3.984661661     0.0219674976
964102    8.360407930    -0.0953735583
966001    9.174016794     0.3432238340
968001   -0.629836663    -0.7223522955
968002   -4.532019378     0.1264415938
968003   -8.173387557     0.0922628561
968401    3.224844494     0.2381162652
968402    7.853238203     0.6730011095
976803   -9.046486242    -0.6135737022
982001   -5.313405819    -0.3165368330
983703    4.116384781     0.0643450735
983902    3.083564815     0.0115067859
983903   -3.849424068    -0.1295673009
984402   -1.843563677    -0.0852790816
984404    0.142760805    -0.2835791644
985802   -3.097528676    -0.2788940500
985803    0.646538579    -0.1911647301
986203   -2.560814112    -0.7966300263
986505    0.638431884    -0.6053352476
986506   -5.014895845    -1.0319359419
987401   -6.344335597     0.2169455056
987701   -3.506675824    -0.6063062528
988701   -9.319877074    -0.3575980333
989201   -5.305093824    -0.3802291161
989503   -3.932088153    -0.2082724988
992201    4.266447975    -0.0669857425
993803   -0.530178376     0.1345058204
995101  -17.804533585    -0.4776618045
995102  -17.473883352    -0.6232233901
995304    1.053291143     0.0243375727
997002   -8.979013260    -0.0885973261
998001    2.729012021    -0.2800570407
999202   -3.670524953    -0.3511758409
999203   -2.979903009    -0.5338115853
1002601 -13.034835722    -0.4012982324
1003202  -5.606812318    -0.0196977607
1003203  -7.477668046    -0.3066633100
1003204  -7.844700539    -0.1540920537
1003601   3.930180280     0.0011913967
1003802   6.540122419     0.5659007953
1004802  -5.570807619    -0.0413438230
1004903  -5.248322978     0.4111272596
1005201  -5.984591162    -0.1987134049
1005203   7.565038265     0.2057572959
1007201   1.510667986    -0.1147921220
1007301 -12.468870981    -0.0066654272
1007302 -12.326574049     0.0022638649
1007901  -2.264068422     0.1464422191
1013201  -7.986005280    -0.2003928810
1017504 -15.664769939     0.1460669934
1018402  -8.067842105    -0.2363769816
1018404  -6.508505979    -0.2698877112
1019104  -4.099081306    -0.3246502915
1019105 -11.252231849    -0.3048711496
1023702   0.234904992     0.0414158079
1030801   9.922354931     0.2550829939
1031001   8.177431667     0.9571389929
1031002   2.887020965     0.5672244473
1031302  -7.580679673     0.0151221031
1033202 -14.810454289    -0.1392893536
1033601  -6.830059011    -0.1810000462
1034001  -6.384315963    -0.4931655541
1036001  -2.463853750     0.7652192225
1036003  -2.232715330    -0.1682196086
1037502  -1.522566381    -0.1243245413
1040101   3.882500958     0.4223124057
1042101  -2.389200357    -0.2881820739
1042201  -7.337963729    -0.2381278056
1044801   5.063737729     0.1565980098
1047801  -2.760234293    -0.2101134212
1050001  -2.834390560    -0.0925525719
1050701  -5.695588916     0.6616487481
1053201  -0.599389674    -0.1378580936
1053202  -7.116932507    -0.3488730245
1053802  -7.098840842     0.1161428377
1053804  -7.518495890    -0.2127588435
1056501  -4.662066297    -0.0688020154
1081101  -6.509487554    -0.2433586628
1081103  -4.535700136    -0.0489579883
1176403  -2.478665580    -0.3619317394
1176502  -5.361130357     0.1014948455
1179201   6.169266543    -0.2715601330
1179202   2.828003388     0.0468697930
1180202   1.281663901    -0.5472646124
1181201 -12.764929271    -0.5961826909
1181902  -3.179238219    -0.1577544718
1181904  -5.521812552    -0.1832687129
1182604   0.677485736    -0.3209083737
1185003 -12.971458846    -0.0415619496
1187001   3.494246653    -0.3201390580
1187303   6.157122681     0.3331643729
1189901   9.250401396     0.3775423118
1190501  -2.887202250    -0.1068928078
1191001   2.654860996     0.2646823447
1191901   4.496785221     0.1631820463
1194901  -2.945928120     1.0021463443
1197902  -6.624966665    -0.3709318601
1198101  -1.383949304    -0.2114699808
1198901  16.464380969     0.0561320208
1201701  -1.448990580    -0.2336679925
1201702  -0.096107319    -0.5865903882
1203201  -1.895771421    -0.4878402424
1205201  -5.525566536     0.2163459801
1207601   4.719495051     0.1382756094
1209201  -0.349522573    -0.1862900381
1211502   9.264718431     0.1745673561
1211503   5.881429621     0.1257080878
1212303  -2.959730778     0.0557280182
1213501  -4.357858111     1.0978055000
1217202  -1.786493616    -0.5895225445
1217204  -7.916052388    -0.0739003214
1217702  -2.632215582    -0.0357903367
1217704  -2.184052557    -0.1210937439
1219103  -7.783664844    -0.1154947959
1219105 -14.003540434    -0.4179680812
1219106 -14.952823473    -0.3505976607
1219108   1.648006949     0.3247083801
1221003   0.152689839    -0.1226131532
1221702  -4.589571607    -0.2141049895
1224001   3.179239571    -0.2058228146
1225402  -4.958955736    -0.1677144514
1228103  -0.253089392    -0.0056286513
1230302  -9.114014727    -0.6678043203
1256601  -7.998906403    -0.1309251668

with conditional variances for "id" 

7 Generalized Linear Mixed Models

https://bbolker.github.io/mixedmodels-misc/glmmFAQ.html (archived at https://perma.cc/9RFS-BCE7; source code: https://github.com/bbolker/mixedmodels-misc/blob/master/glmmFAQ.rmd)

7.1 lmer

generalizedLinearMixedModel <- glmer(
  outcome ~ female + ageYearsCentered + (ageYearsCentered | id),
  family = poisson(link = "log"),
  data = mydata,
  na.action = na.exclude)

boundary (singular) fit: see help('isSingular')

summary(generalizedLinearMixedModel)

Generalized linear mixed model fit by maximum likelihood (Laplace
  Approximation) [glmerMod]
 Family: poisson  ( log )
Formula: outcome ~ female + ageYearsCentered + (ageYearsCentered | id)
   Data: mydata

      AIC       BIC    logLik -2*log(L)  df.resid 
   9329.7    9363.9   -4658.8    9317.7      2215 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-2.0331 -0.5546 -0.0205  0.5350  5.0222 

Random effects:
 Groups Name             Variance  Std.Dev. Corr 
 id     (Intercept)      0.0058309 0.07636       
        ageYearsCentered 0.0002845 0.01687  -1.00
Number of obs: 2221, groups:  id, 932

Fixed effects:
                 Estimate Std. Error z value Pr(>|z|)    
(Intercept)      1.343986   0.027432  48.994   <2e-16 ***
female           0.007439   0.021264   0.350   0.7265    
ageYearsCentered 0.010279   0.005877   1.749   0.0803 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
            (Intr) female
female      -0.414       
ageYrsCntrd -0.835  0.037
optimizer (Nelder_Mead) convergence code: 0 (OK)
boundary (singular) fit: see help('isSingular')

7.2 MASS

glmmPQLmodel <- glmmPQL(
  outcome ~ female + ageYearsCentered,
  random = ~ 1 + ageYearsCentered|id,
  family = poisson(link = "log"),
  data = mydata)

iteration 1

summary(glmmPQLmodel)

Linear mixed-effects model fit by maximum likelihood
  Data: mydata 
  AIC BIC logLik
   NA  NA     NA

Random effects:
 Formula: ~1 + ageYearsCentered | id
 Structure: General positive-definite, Log-Cholesky parametrization
                 StdDev       Corr  
(Intercept)      5.533290e-05 (Intr)
ageYearsCentered 9.320922e-08 0     
Residual         1.014184e+00       

Variance function:
 Structure: fixed weights
 Formula: ~invwt 
Fixed effects:  outcome ~ female + ageYearsCentered 
                     Value   Std.Error   DF  t-value p-value
(Intercept)      1.3453017 0.027100741 1288 49.64077  0.0000
female           0.0074130 0.021543030  930  0.34410  0.7308
ageYearsCentered 0.0100806 0.005850753 1288  1.72296  0.0851
 Correlation: 
                 (Intr) female
female           -0.423       
ageYearsCentered -0.829  0.036

Standardized Within-Group Residuals:
        Min          Q1         Med          Q3         Max 
-2.00970268 -0.55110605 -0.02158036  0.53179575  4.98914420 

Number of Observations: 2221
Number of Groups: 932 

7.3 MCMCglmm

MCMCglmmModel <- MCMCglmm(
  outcome ~ female + ageYearsCentered,
  random = ~ us(ageYearsCentered):id,
  family = "poisson",
  data = na.omit(mydata[,c("id","outcome","female","ageYearsCentered")]))

                       MCMC iteration = 0

 Acceptance ratio for liability set 1 = 0.000410

                       MCMC iteration = 1000

 Acceptance ratio for liability set 1 = 0.439819

                       MCMC iteration = 2000

 Acceptance ratio for liability set 1 = 0.439977

                       MCMC iteration = 3000

 Acceptance ratio for liability set 1 = 0.444608

                       MCMC iteration = 4000

 Acceptance ratio for liability set 1 = 0.496749

                       MCMC iteration = 5000

 Acceptance ratio for liability set 1 = 0.490778

                       MCMC iteration = 6000

 Acceptance ratio for liability set 1 = 0.512362

                       MCMC iteration = 7000

 Acceptance ratio for liability set 1 = 0.428231

                       MCMC iteration = 8000

 Acceptance ratio for liability set 1 = 0.412336

                       MCMC iteration = 9000

 Acceptance ratio for liability set 1 = 0.471726

                       MCMC iteration = 10000

 Acceptance ratio for liability set 1 = 0.428491

                       MCMC iteration = 11000

 Acceptance ratio for liability set 1 = 0.400078

                       MCMC iteration = 12000

 Acceptance ratio for liability set 1 = 0.346435

                       MCMC iteration = 13000

 Acceptance ratio for liability set 1 = 0.276170

summary(MCMCglmmModel)

 Iterations = 3001:12991
 Thinning interval  = 10
 Sample size  = 1000 

 DIC: 9323.524 

 G-structure:  ~us(ageYearsCentered):id

                                     post.mean  l-95% CI  u-95% CI eff.samp
ageYearsCentered:ageYearsCentered.id 6.754e-06 1.087e-08 4.053e-05    7.788

 R-structure:  ~units

      post.mean l-95% CI u-95% CI eff.samp
units   0.00877 0.001634  0.01728    7.797

 Location effects: outcome ~ female + ageYearsCentered 

                 post.mean  l-95% CI  u-95% CI eff.samp  pMCMC    
(Intercept)       1.338707  1.281096  1.387614    55.58 <0.001 ***
female            0.007688 -0.031193  0.052271    58.65  0.706    
ageYearsCentered  0.010561  0.000813  0.021420    57.32  0.046 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

8 Nonlinear Mixed Models

nonlinearModel <- nlme(
  height ~ SSasymp(age, Asym, R0, lrc),
  data = Loblolly,
  fixed = Asym + R0 + lrc ~ 1,
  random = Asym ~ 1)

summary(nonlinearModel)

Nonlinear mixed-effects model fit by maximum likelihood
  Model: height ~ SSasymp(age, Asym, R0, lrc) 
  Data: Loblolly 
      AIC      BIC   logLik
  239.486 251.6401 -114.743

Random effects:
 Formula: Asym ~ 1 | Seed
            Asym  Residual
StdDev: 3.650645 0.7188624

Fixed effects:  list(Asym ~ 1, R0 ~ 1, lrc ~ 1) 
         Value Std.Error DF   t-value p-value
Asym 101.44830 2.4616151 68  41.21209       0
R0    -8.62749 0.3179519 68 -27.13459       0
lrc   -3.23373 0.0342695 68 -94.36168       0
 Correlation: 
    Asym   R0    
R0   0.704       
lrc -0.908 -0.827

Standardized Within-Group Residuals:
        Min          Q1         Med          Q3         Max 
-2.23604174 -0.62389999  0.05912777  0.65724316  1.95790785 

Number of Observations: 84
Number of Groups: 14 

9 Robust Mixed Models

To evaluate the extent to which a finding could driven by outliers, this could be done in a number of different ways, such as:

• identifying and excluding influential observations based on DFBETAS or Cook’s distance (Nieuwenhuis, Grotenhuis, & Pelzer, 2012)
• fitting mixed models using rank-based estimation (Bilgic & Susmann, 2013; Finch, 2017) or robust estimating equations (Koller, 2016)
• estimating robust standard errors using a sandwich estimator (Wang & Merkle, 2018)

10 Assumptions

The within-group errors:

1. are independent
2. are identically normally distributed
3. have mean zero and variance sigma-squared
4. are independent of the random effects

The random effects:

5. are normally distributed
6. have mean zero and covariance matrix Psi (not depending on the group)
7. are independent for different groups

11 Examining Model Assumptions

11.1 Resources

Pinheiro and Bates (2000) book (p. 174, section 4.3.1)

https://stats.stackexchange.com/questions/77891/checking-assumptions-lmer-lme-mixed-models-in-r (archived at https://perma.cc/J5GC-PCUT)

11.2 QQ Plots

Make QQ plots for each level of the random effects. Vary the level from 0, 1, to 2 so that you can check the between- and within-subject residuals.

qqnorm(linearMixedModel_nlme,
       ~ ranef(., level = 1))

11.3 PP Plots

ppPlot(linearMixedModel)

11.4 QQ Plot of residuals

qqnorm(resid(linearMixedModel))
qqline(resid(linearMixedModel))

11.5 Plot residuals

plot(linearMixedModel)

11.6 Plot residuals by group (in the example below, level 2 represents the individual)

plot(linearMixedModel,
     as.factor(id) ~ resid(.),
     abline = 0,
     xlab = "Residuals")

11.7 Plot residuals by levels of a predictor

plot(linearMixedModel_nlme,
     resid(., type = "p") ~ fitted(.) | female) #type = "p" specifies standardized residuals

11.8 Can model heteroscedasticity of the within-group error with the weights argument

linearMixedModel_nlmeVarStructure <- lme(
  math ~ female + ageYearsCentered,
  random = ~ 1 + ageYearsCentered|id,
  weights = varIdent(form = ~ 1 | female),
  method = "ML",
  data = mydata,
  na.action = na.exclude)

summary(linearMixedModel_nlmeVarStructure)

Linear mixed-effects model fit by maximum likelihood
  Data: mydata 
       AIC      BIC    logLik
  15857.83 15903.48 -7920.915

Random effects:
 Formula: ~1 + ageYearsCentered | id
 Structure: General positive-definite, Log-Cholesky parametrization
                 StdDev    Corr  
(Intercept)      7.9177716 (Intr)
ageYearsCentered 0.8278343 0.076 
Residual         5.6410162       

Variance function:
 Structure: Different standard deviations per stratum
 Formula: ~1 | female 
 Parameter estimates:
       1        0 
1.000000 1.009161 
Fixed effects:  math ~ female + ageYearsCentered 
                     Value Std.Error   DF  t-value p-value
(Intercept)      30.554856 0.5040373 1288 60.62022  0.0000
female           -0.692653 0.6172485  930 -1.12216  0.2621
ageYearsCentered  4.255258 0.0805531 1288 52.82553  0.0000
 Correlation: 
                 (Intr) female
female           -0.614       
ageYearsCentered -0.507  0.014

Standardized Within-Group Residuals:
        Min          Q1         Med          Q3         Max 
-3.37982974 -0.51663213  0.00445497  0.52228733  2.63205084 

Number of Observations: 2221
Number of Groups: 932 

11.9 Plot observed and fitted values

plot(linearMixedModel,
     math ~ fitted(.))

11.10 Plot QQ plot of residuals by levels of a predictor

qqnorm(linearMixedModel_nlme, ~ resid(.) | female)

qqnorm(linearMixedModel_nlme, ~ resid(.))

11.11 QQ plot of random effects

Make QQ plots for each level of the random effects. Vary the level from 0, 1, to 2 so that you can check the between- and within-subject residuals.

qqnorm(linearMixedModel_nlme,
       ~ ranef(., level = 0))

Error in effects[[1L]]: subscript out of bounds

qqnorm(linearMixedModel_nlme,
       ~ ranef(., level = 1))

qqnorm(linearMixedModel_nlme,
       ~ ranef(., level = 2))

Error in eval(i, data, env): object '.y' not found

11.12 QQ plot of random effects by levels of a predictor

qqnorm(linearMixedModel_nlme, 
       ~ ranef(., level = 1) | female)

11.13 Pairs plot

pairs(linearMixedModel_nlme)

pairs(linearMixedModel_nlme,
      ~ ranef(., level = 1) | female)

11.14 Variance functions for modeling heteroscedasticity

• varFixed: fixed variance
• varIdent: different variances per stratum
• varPower: power of covariate
• varExp: exponential of covariate
• varConstPower: constant plus power of covariate
• varComb: combination of variance functions

11.15 Correlation structures for modeling dependence

• corCompSymm: compound symmetry
• corSymm: general
• corAR1: autoregressive of order 1
• corCAR1: continuous-time AR(1)
• corARMA: autoregressive-moving average
• corExp: exponential
• corGaus: Gaussian
• corLin: linear
• corRatio: rational quadratic
• corSpher: spherical

12 Power Analysis

• https://aguinis.shinyapps.io/ml_power/
• https://www.causalevaluation.org/power-analysis.html
  • https://powerupr.shinyapps.io/index/
• https://koumurayama.shinyapps.io/tmethod_mlm/
• https://webpower.psychstat.org/wiki/models/index

13 Session Info

sessionInfo()

R version 4.5.1 (2025-06-13)
Platform: x86_64-pc-linux-gnu
Running under: Ubuntu 24.04.2 LTS

Matrix products: default
BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3 
LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so;  LAPACK version 3.12.0

locale:
 [1] LC_CTYPE=C.UTF-8       LC_NUMERIC=C           LC_TIME=C.UTF-8       
 [4] LC_COLLATE=C.UTF-8     LC_MONETARY=C.UTF-8    LC_MESSAGES=C.UTF-8   
 [7] LC_PAPER=C.UTF-8       LC_NAME=C              LC_ADDRESS=C          
[10] LC_TELEPHONE=C         LC_MEASUREMENT=C.UTF-8 LC_IDENTIFICATION=C   

time zone: UTC
tzcode source: system (glibc)

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
 [1] ggplot2_3.5.2      performance_0.15.0 MCMCglmm_2.36      ape_5.8-1         
 [5] coda_0.19-4.1      MASS_7.3-65        lmerTest_3.1-3     nlme_3.1-168      
 [9] lme4_1.1-37        Matrix_1.7-3       petersenlab_1.2.0 

loaded via a namespace (and not attached):
 [1] tidyselect_1.2.1    psych_2.5.6         viridisLite_0.4.2  
 [4] dplyr_1.1.4         farver_2.1.2        fastmap_1.2.0      
 [7] tensorA_0.36.2.1    digest_0.6.37       rpart_4.1.24       
[10] lifecycle_1.0.4     cluster_2.1.8.1     magrittr_2.0.3     
[13] compiler_4.5.1      rlang_1.1.6         Hmisc_5.2-3        
[16] sass_0.4.10         tools_4.5.1         yaml_2.3.10        
[19] data.table_1.17.8   knitr_1.50          labeling_0.4.3     
[22] htmlwidgets_1.6.4   mnormt_2.1.1        plyr_1.8.9         
[25] RColorBrewer_1.1-3  withr_3.0.2         foreign_0.8-90     
[28] purrr_1.1.0         numDeriv_2016.8-1.1 nnet_7.3-20        
[31] grid_4.5.1          stats4_4.5.1        lavaan_0.6-19      
[34] xtable_1.8-4        colorspace_2.1-1    scales_1.4.0       
[37] insight_1.3.1       cli_3.6.5           mvtnorm_1.3-3      
[40] rmarkdown_2.29      reformulas_0.4.1    generics_0.1.4     
[43] rstudioapi_0.17.1   reshape2_1.4.4      minqa_1.2.8        
[46] DBI_1.2.3           cachem_1.1.0        stringr_1.5.1      
[49] splines_4.5.1       parallel_4.5.1      base64enc_0.1-3    
[52] mitools_2.4         vctrs_0.6.5         boot_1.3-31        
[55] jsonlite_2.0.0      Formula_1.2-5       htmlTable_2.4.3    
[58] jquerylib_0.1.4     glue_1.8.0          nloptr_2.2.1       
[61] cubature_2.1.4      stringi_1.8.7       gtable_0.3.6       
[64] quadprog_1.5-8      tibble_3.3.0        pillar_1.11.0      
[67] htmltools_0.5.8.1   R6_2.6.1            Rdpack_2.6.4       
[70] mix_1.0-13          evaluate_1.0.4      pbivnorm_0.6.0     
[73] lattice_0.22-7      rbibutils_2.3       backports_1.5.0    
[76] corpcor_1.6.10      bslib_0.9.0         Rcpp_1.1.0         
[79] gridExtra_2.3       checkmate_2.3.2     xfun_0.52          
[82] pkgconfig_2.0.3    

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