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

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 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")

6.1.2 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 deviance 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

6.1.2.1 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()

6.1.2.2 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()

6.1.2.3 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)

6.1.2.4 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
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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
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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
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138902   -0.353560559     -0.051066863
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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
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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
346501    5.906782681      0.108977062
347603   -0.806916394     -0.093040005
348103    3.580506175      0.234122072
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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" 

6.1.3 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 

6.2 Intraclass Correlation Coefficent

icc(linearMixedModel)

icc(linearMixedModel_nlme)

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 deviance 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 4.4.2 (2024-10-31)
Platform: x86_64-pc-linux-gnu
Running under: Ubuntu 24.04.1 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_3.5.1      performance_0.13.0 MCMCglmm_2.36      ape_5.8-1         
 [5] coda_0.19-4.1      MASS_7.3-61        lmerTest_3.1-3     nlme_3.1-166      
 [9] lme4_1.1-36        Matrix_1.7-1       petersenlab_1.1.1 

loaded via a namespace (and not attached):
 [1] tidyselect_1.2.1    psych_2.4.12        viridisLite_0.4.2  
 [4] farver_2.1.2        dplyr_1.1.4         fastmap_1.2.0      
 [7] tensorA_0.36.2.1    digest_0.6.37       rpart_4.1.23       
[10] lifecycle_1.0.4     cluster_2.1.6       magrittr_2.0.3     
[13] compiler_4.4.2      rlang_1.1.5         Hmisc_5.2-2        
[16] sass_0.4.9          tools_4.4.2         yaml_2.3.10        
[19] data.table_1.16.4   knitr_1.49          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_0.8-87     
[28] purrr_1.0.4         numDeriv_2016.8-1.1 nnet_7.3-19        
[31] grid_4.4.2          stats4_4.4.2        lavaan_0.6-19      
[34] xtable_1.8-4        colorspace_2.1-1    scales_1.3.0       
[37] insight_1.0.2       cli_3.6.4           mvtnorm_1.3-3      
[40] rmarkdown_2.29      reformulas_0.4.0    generics_0.1.3     
[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_4.4.2       parallel_4.4.2      base64enc_0.1-3    
[52] mitools_2.4         vctrs_0.6.5         boot_1.3-31        
[55] jsonlite_1.9.0      Formula_1.2-5       htmlTable_2.4.3    
[58] jquerylib_0.1.4     glue_1.8.0          nloptr_2.1.1       
[61] cubature_2.1.1      stringi_1.8.4       gtable_0.3.6       
[64] quadprog_1.5-8      munsell_0.5.1       tibble_3.2.1       
[67] pillar_1.10.1       htmltools_0.5.8.1   R6_2.6.1           
[70] Rdpack_2.6.2        mix_1.0-13          evaluate_1.0.3     
[73] pbivnorm_0.6.0      lattice_0.22-6      rbibutils_2.3      
[76] backports_1.5.0     corpcor_1.6.10      bslib_0.9.0        
[79] Rcpp_1.0.14         gridExtra_2.3       checkmate_2.3.2    
[82] xfun_0.51           pkgconfig_2.0.3    

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