Mixed Models Tutorial: Contrast Coding

Author

Reinhold Kliegl

Published

2025-05-12

This script uses a subset of data reported in Fühner et al. (2021).

To circumvent delays associated with model fitting we work with models that are less complex than those in the reference publication. All the data to reproduce the models in the publication are used here, too; the script requires only a few changes to specify the more complex models in the paper.

All children were between 6.0 and 6.99 years at legal keydate (30 September) of school enrollment, that is in their ninth year of life in the third grade. To avoid delays associated with model fitting we work with a reduced data set and less complex models than those in the reference publication. The script requires only a few changes to specify the more complex models in the paper.

The script is structured in three main sections:

Setup with reading and examining the data
Contrasts coding

Effect and sequential difference contrasts
Helmert contrast
Hypothesis contrast
PCA-based contrast

Other topics

LMM goodness of fit does not depend on contrast (i.e., reparameterization)
VCs and CPs depend on contrast
VCs and CPs depend on random factor

1 Setup

1.1 Packages and functions

Code

using AlgebraOfGraphics
using CairoMakie
using Chain
using CategoricalArrays
using DataFrames
using DataFrameMacros
using MixedModels
using ProgressMeter
using SMLP2025: dataset
using Statistics
using StatsBase

ProgressMeter.ijulia_behavior(:clear);

1.2 Readme for `dataset("fggk21")`

Number of scores: 525126

Cohort: 9 levels; 2011-2019
School: 515 levels
Child: 108295 levels; all children are between 8.0 and 8.99 years old
Sex: “Girls” (n=55,086), “Boys” (n= 53,209)
age: testdate - middle of month of birthdate
Test: 5 levels
- Endurance (Run): 6 minute endurance run [m]; to nearest 9m in 9x18m field
- Coordination (Star_r): star coordination run [m/s]; 9x9m field, 4 x diagonal = 50.912 m
- Speed(S20_r): 20-meters sprint [m/s]
- Muscle power low (SLJ): standing long jump [cm]
- Muscle power up (BPT): 1-kg medicine ball push test [m]
score - see units

1.3 Preprocessing

1.3.1 Read data

tbl = dataset(:fggk21)

Arrow.Table with 525126 rows, 7 columns, and schema:
 :Cohort  String
 :School  String
 :Child   String
 :Sex     String
 :age     Float64
 :Test    String
 :score   Float64

df = DataFrame(tbl)
describe(df)

7×7 DataFrame

Row	variable	mean	min	median	max	nmissing	eltype
	Symbol	Union…	Any	Union…	Any	Int64	DataType
1	Cohort		2011		2019	0	String
2	School		S100043		S800200	0	String
3	Child		C002352		C117966	0	String
4	Sex		female		male	0	String
5	age	8.56073	7.99452	8.55852	9.10609	0	Float64
6	Test		BPT		Star_r	0	String
7	score	226.141	1.14152	4.65116	1530.0	0	Float64

1.3.2 Extract a stratified subsample

We extract a random sample of 500 children from the Sex (2) x Test (5) cells of the design. Cohort and School are random.

dat = @chain df begin
  @transform(:Sex = :Sex == "female" ? "Girls" : "Boys")
  @groupby(:Test, :Sex)
  combine(x -> x[sample(1:nrow(x), 500), :])
end

5000×7 DataFrame

4975 rows omitted

Row	Test	Sex	Cohort	School	Child	age	score
	String	String	String	String	String	Float64	Float64
1	S20_r	Boys	2018	S104759	C043600	8.44079	4.54545
2	S20_r	Boys	2015	S111030	C093868	8.87337	4.44444
3	S20_r	Boys	2017	S103421	C067472	8.64613	4.7619
4	S20_r	Boys	2018	S104875	C082004	8.76934	4.08163
5	S20_r	Boys	2016	S101047	C036394	8.37782	3.84615
6	S20_r	Boys	2014	S100961	C009273	8.1232	4.65116
7	S20_r	Boys	2014	S102489	C063865	8.61875	4.44444
8	S20_r	Boys	2018	S100341	C116663	9.10609	5.0
9	S20_r	Boys	2014	S100067	C087870	8.83231	3.84615
10	S20_r	Boys	2015	S101552	C058691	8.57221	5.0
11	S20_r	Boys	2011	S102568	C008170	8.08487	4.08163
12	S20_r	Boys	2015	S103380	C066815	8.63518	4.7619
13	S20_r	Boys	2018	S105971	C043867	8.44079	4.87805
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
4989	Run	Girls	2015	S100845	C052334	8.52019	972.0
4990	Run	Girls	2018	S100493	C090138	8.85421	927.0
4991	Run	Girls	2016	S104759	C036669	8.37782	1107.0
4992	Run	Girls	2012	S112823	C022953	8.24914	832.0
4993	Run	Girls	2019	S105934	C047829	8.47091	972.0
4994	Run	Girls	2012	S105211	C089105	8.83231	972.0
4995	Run	Girls	2015	S102258	C067981	8.65708	927.0
4996	Run	Girls	2019	S101291	C010988	8.13689	1017.0
4997	Run	Girls	2014	S103196	C092717	8.87064	980.0
4998	Run	Girls	2018	S101679	C071973	8.69268	1035.0
4999	Run	Girls	2019	S104905	C095977	8.88433	1071.0
5000	Run	Girls	2019	S103421	C057073	8.55578	954.0

1.3.3 Transformations

transform!(dat, :age, :age => (x -> x .- 8.5) => :a1) # centered age (linear)
select!(groupby(dat, :Test), :, :score => zscore => :zScore) # z-score

5000×9 DataFrame

4975 rows omitted

Row	Test	Sex	Cohort	School	Child	age	score	a1	zScore
	String	String	String	String	String	Float64	Float64	Float64	Float64
1	S20_r	Boys	2018	S104759	C043600	8.44079	4.54545	-0.059206	0.0808388
2	S20_r	Boys	2015	S111030	C093868	8.87337	4.44444	0.373374	-0.162417
3	S20_r	Boys	2017	S103421	C067472	8.64613	4.7619	0.146133	0.602102
4	S20_r	Boys	2018	S104875	C082004	8.76934	4.08163	0.269336	-1.03615
5	S20_r	Boys	2016	S101047	C036394	8.37782	3.84615	-0.122177	-1.60324
6	S20_r	Boys	2014	S100961	C009273	8.1232	4.65116	-0.376797	0.335409
7	S20_r	Boys	2014	S102489	C063865	8.61875	4.44444	0.118754	-0.162417
8	S20_r	Boys	2018	S100341	C116663	9.10609	5.0	0.606092	1.17549
9	S20_r	Boys	2014	S100067	C087870	8.83231	3.84615	0.332307	-1.60324
10	S20_r	Boys	2015	S101552	C058691	8.57221	5.0	0.0722108	1.17549
11	S20_r	Boys	2011	S102568	C008170	8.08487	4.08163	-0.415127	-1.03615
12	S20_r	Boys	2015	S103380	C066815	8.63518	4.7619	0.135181	0.602102
13	S20_r	Boys	2018	S105971	C043867	8.44079	4.87805	-0.059206	0.881804
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
4989	Run	Girls	2015	S100845	C052334	8.52019	972.0	0.0201916	-0.23285
4990	Run	Girls	2018	S100493	C090138	8.85421	927.0	0.354209	-0.539223
4991	Run	Girls	2016	S104759	C036669	8.37782	1107.0	-0.122177	0.686268
4992	Run	Girls	2012	S112823	C022953	8.24914	832.0	-0.250856	-1.18601
4993	Run	Girls	2019	S105934	C047829	8.47091	972.0	-0.0290897	-0.23285
4994	Run	Girls	2012	S105211	C089105	8.83231	972.0	0.332307	-0.23285
4995	Run	Girls	2015	S102258	C067981	8.65708	927.0	0.157084	-0.539223
4996	Run	Girls	2019	S101291	C010988	8.13689	1017.0	-0.363107	0.0735226
4997	Run	Girls	2014	S103196	C092717	8.87064	980.0	0.370637	-0.178384
4998	Run	Girls	2018	S101679	C071973	8.69268	1035.0	0.192676	0.196072
4999	Run	Girls	2019	S104905	C095977	8.88433	1071.0	0.384326	0.44117
5000	Run	Girls	2019	S103421	C057073	8.55578	954.0	0.0557837	-0.355399

dat2 = combine(
  groupby(dat, [:Test, :Sex]),
  :score => mean,
  :score => std,
  :zScore => mean,
  :zScore => std,
)

10×6 DataFrame

Row	Test	Sex	score_mean	score_std	zScore_mean	zScore_std
	String	String	Float64	Float64	Float64	Float64
1	S20_r	Boys	4.5577	0.43865	0.110335	1.05637
2	BPT	Boys	3.9764	0.718049	0.34335	1.00588
3	SLJ	Boys	130.138	19.8777	0.240967	1.0086
4	Star_r	Boys	2.08379	0.314341	0.112349	1.09023
5	Run	Boys	1040.55	156.929	0.233844	1.06841
6	S20_r	Girls	4.46607	0.385457	-0.110335	0.928271
7	BPT	Girls	3.4862	0.619837	-0.34335	0.868303
8	SLJ	Girls	120.64	18.3652	-0.240967	0.93186
9	Star_r	Girls	2.019	0.25596	-0.112349	0.887748
10	Run	Girls	971.854	127.279	-0.233844	0.866551

1.3.4 Figure of age x Sex x Test interactions

The main results of relevance here are shown in Figure 2 of Scientific Reports 11:17566.

2 Contrast coding

Contrast coding is part of StatsModels.jl. Here is the primary author’s (i.e., Dave Kleinschmidt’s) documentation of Modeling Categorical Data.

The random factors Child, School, and Cohort are assigned a Grouping contrast. This contrast is needed when the number of groups (i.e., units, levels) is very large. This is the case for Child (i.e., the 108,925 children in the full and probably also the 11,566 children in the reduced data set). The assignment is not necessary for the typical sample size of experiments. However, we use this coding of random factors irrespective of the number of units associated with them to be transparent about the distinction between random and fixed factors.

A couple of general remarks about the following examples. First, all contrasts defined in this tutorial return an estimate of the Grand Mean (GM) in the intercept, that is they are so-called sum-to-zero contrasts. In both Julia and R the default contrast is Dummy coding which is not a sum-to-zero contrast, but returns the mean of the reference (control) group - unfortunately for (quasi-)experimentally minded scientists.

Second, The factor Sex has only two levels. We use EffectCoding (also known as Sum coding in R) to estimate the difference of the levels from the Grand Mean. Unlike in R, the default sign of the effect is for the second level (base is the first, not the last level), but this can be changed with the base kwarg in the command. Effect coding is a sum-to-zero contrast, but when applied to factors with more than two levels does not yield orthogonal contrasts.

Finally, contrasts for the five levels of the fixed factor Test represent the hypotheses about differences between them. In this tutorial, we use this factor to illustrate various options.

We (initially) include only Test as fixed factor and Child as random factor. More complex LMMs can be specified by simply adding other fixed or random factors to the formula.

2.1 SeqDiffCoding: `contr1`

SeqDiffCoding was used in the publication. This specification tests pairwise differences between the five neighboring levels of Test, that is:

SDC1: 2-1
SDC2: 3-2
SDC3: 4-3
SDC4: 5-4

The levels were sorted such that these contrasts map onto four a priori hypotheses; in other words, they are theoretically motivated pairwise comparisons. The motivation also encompasses theoretically motivated interactions with Sex. The order of levels can also be explicitly specified during contrast construction. This is very useful if levels are in a different order in the dataframe. We recommend the explicit specification to increase transparency of the code.

The statistical disadvantage of SeqDiffCoding is that the contrasts are not orthogonal, that is the contrasts are correlated. This is obvious from the fact that levels 2, 3, and 4 are all used in two contrasts. One consequence of this is that correlation parameters estimated between neighboring contrasts (e.g., 2-1 and 3-2) are difficult to interpret. Usually, they will be negative because assuming some practical limitation on the overall range (e.g., between levels 1 and 3), a small “2-1” effect “correlates” negatively with a larger “3-2” effect for mathematical reasons.

Obviously, the tradeoff between theoretical motivation and statistical purity is something that must be considered carefully when planning the analysis.

contr1 = merge(
  Dict(nm => Grouping() for nm in (:School, :Child, :Cohort)),
  Dict(
    :Sex => EffectsCoding(; levels=["Girls", "Boys"]),
    :Test => SeqDiffCoding(;
      levels=["Run", "Star_r", "S20_r", "SLJ", "BPT"]
    ),
  ),
)

Dict{Symbol, StatsModels.AbstractContrasts} with 5 entries:
  :Child  => Grouping()
  :School => Grouping()
  :Test   => SeqDiffCoding(["Run", "Star_r", "S20_r", "SLJ", "BPT"])
  :Cohort => Grouping()
  :Sex    => EffectsCoding(nothing, ["Girls", "Boys"])

f_ovi_1 = @formula zScore ~ 1 + Test + (1 | Child);

m_ovi_SeqDiff_1 = fit(MixedModel, f_ovi_1, dat; contrasts=contr1)

	Est.	SE	z	p	σ_Child
(Intercept)	0.0005	0.0143	0.04	0.9710	0.7545
Test: Star_r	-0.0046	0.0442	-0.10	0.9176
Test: S20_r	0.0048	0.0442	0.11	0.9134
Test: SLJ	-0.0034	0.0442	-0.08	0.9395
Test: BPT	0.0036	0.0442	0.08	0.9355
Residual	0.6555

In this case, any differences between tests identified by the contrasts would be spurious because each test was standardized (i.e., M=0, \(SD\)=1). The differences could also be due to an imbalance in the number of boys and girls or in the number of missing observations for each test.

The primary interest in this study related to interactions of the test contrasts with and age and Sex. We start with age (linear) and its interaction with the four test contrasts.

m_ovi_SeqDiff_2 = let
  form = @formula zScore ~ 1 + Test * a1 + (1 | Child)
  fit(MixedModel, form, dat; contrasts=contr1)
end

	Est.	SE	z	p	σ_Child
(Intercept)	-0.0190	0.0146	-1.30	0.1926	0.7520
Test: Star_r	-0.0213	0.0455	-0.47	0.6393
Test: S20_r	-0.0003	0.0454	-0.01	0.9941
Test: SLJ	0.0050	0.0451	0.11	0.9121
Test: BPT	-0.0102	0.0449	-0.23	0.8203
a1	0.3022	0.0494	6.12	<1e-09
Test: Star_r & a1	0.2034	0.1515	1.34	0.1794
Test: S20_r & a1	0.1494	0.1539	0.97	0.3318
Test: SLJ & a1	-0.1342	0.1561	-0.86	0.3900
Test: BPT & a1	0.2627	0.1521	1.73	0.0842
Residual	0.6511

The difference between older and younger children is larger for Star_r than for Run (0.2473). S20_r did not differ significantly from Star_r (-0.0377) and SLJ (-0.0113) The largest difference in developmental gain was between BPT and SLJ (0.3355).

Please note that standard errors of this LMM are anti-conservative because the LMM is missing a lot of information in the RES (e..g., contrast-related VCs snd CPs for Child, School, and Cohort.

Next we add the main effect of Sex and its interaction with the four test contrasts.

m_ovi_SeqDiff_3 = let
  form = @formula zScore ~ 1 + Test * (a1 + Sex) + (1 | Child)
  fit(MixedModel, form, dat; contrasts=contr1)
end

	Est.	SE	z	p	σ_Child
(Intercept)	-0.0187	0.0142	-1.32	0.1870	0.7122
Test: Star_r	-0.0225	0.0444	-0.51	0.6121
Test: S20_r	-0.0017	0.0443	-0.04	0.9696
Test: SLJ	0.0040	0.0440	0.09	0.9267
Test: BPT	-0.0082	0.0438	-0.19	0.8519
a1	0.2886	0.0481	6.00	<1e-08
Sex: Boys	0.2069	0.0138	14.97	<1e-49
Test: Star_r & a1	0.2489	0.1478	1.68	0.0922
Test: S20_r & a1	0.1279	0.1503	0.85	0.3946
Test: SLJ & a1	-0.1075	0.1525	-0.71	0.4807
Test: BPT & a1	0.2266	0.1484	1.53	0.1267
Test: Star_r & Sex: Boys	-0.1159	0.0430	-2.70	0.0070
Test: S20_r & Sex: Boys	-0.0160	0.0430	-0.37	0.7090
Test: SLJ & Sex: Boys	0.1336	0.0429	3.11	0.0018
Test: BPT & Sex: Boys	0.0983	0.0429	2.29	0.0220
Residual	0.6569

The significant interactions with Sex reflect mostly differences related to muscle power, where the physiological constitution gives boys an advantage. The sex difference is smaller when coordination and cognition play a role – as in the Star_r test. (Caveat: SEs are estimated with an underspecified RES.)

The final step in this first series is to add the interactions between the three covariates. A significant interaction between any of the four Test contrasts and age (linear) x Sex was hypothesized to reflect a prepubertal signal (i.e., hormones start to rise in girls’ ninth year of life). However, this hypothesis is linked to a specific shape of the interaction: Girls would need to gain more than boys in tests of muscular power.

f_ovi = @formula zScore ~ 1 + Test * a1 * Sex + (1 | Child)
m_ovi_SeqDiff = fit(MixedModel, f_ovi, dat; contrasts=contr1)

	Est.	SE	z	p	σ_Child
(Intercept)	-0.0183	0.0142	-1.29	0.1961	0.7149
Test: Star_r	-0.0241	0.0444	-0.54	0.5880
Test: S20_r	-0.0023	0.0443	-0.05	0.9592
Test: SLJ	0.0050	0.0440	0.11	0.9101
Test: BPT	-0.0085	0.0438	-0.19	0.8457
a1	0.2878	0.0481	5.99	<1e-08
Sex: Boys	0.2063	0.0142	14.54	<1e-47
Test: Star_r & a1	0.2512	0.1478	1.70	0.0892
Test: S20_r & a1	0.1282	0.1502	0.85	0.3932
Test: SLJ & a1	-0.1077	0.1524	-0.71	0.4797
Test: BPT & a1	0.2265	0.1484	1.53	0.1269
Test: Star_r & Sex: Boys	-0.1235	0.0444	-2.78	0.0054
Test: S20_r & Sex: Boys	-0.0180	0.0443	-0.41	0.6840
Test: SLJ & Sex: Boys	0.1277	0.0440	2.90	0.0037
Test: BPT & Sex: Boys	0.1078	0.0438	2.46	0.0139
a1 & Sex: Boys	0.0114	0.0481	0.24	0.8124
Test: Star_r & a1 & Sex: Boys	0.1019	0.1478	0.69	0.4906
Test: S20_r & a1 & Sex: Boys	0.0373	0.1502	0.25	0.8039
Test: SLJ & a1 & Sex: Boys	0.0898	0.1524	0.59	0.5559
Test: BPT & a1 & Sex: Boys	-0.1496	0.1484	-1.01	0.3134
Residual	0.6537

The results are very clear: Despite an abundance of statistical power there is no evidence for the differences between boys and girls in how much they gain in the ninth year of life in these five tests. The authors argue that, in this case, absence of evidence looks very much like evidence of absence of a hypothesized interaction.

In the next two sections we use different contrasts. Does this have a bearing on this result? We still ignore for now that we are looking at anti-conservative test statistics.

2.2 HelmertCoding: `contr2`

The second set of contrasts uses HelmertCoding. Helmert coding codes each level as the difference from the average of the lower levels. With the default order of Test levels we get the following test statistics which we describe in reverse order of appearance in model output

HeC4: 5 - mean(1,2,3,4)
HeC3: 4 - mean(1,2,3)
HeC2: 3 - mean(1,2)
HeC1: 2 - 1

In the model output, HeC1 will be reported first and HeC4 last.

There is some justification for the HeC4 specification in a post-hoc manner because the fifth test (BPT) turned out to be different from the other four tests in that high performance is most likely not only related to physical fitness, but also to overweight/obesity, that is for a subset of children high scores on this test might be indicative of physical unfitness. A priori the SDC4 contrast 5-4 between BPT (5) and SLJ (4) was motivated because conceptually both are tests of the physical fitness component Muscular Power, BPT for upper limbs and SLJ for lower limbs, respectively.

One could argue that there is justification for HeC3 because Run (1), Star_r (2), and S20 (3) involve running but SLJ (4) does not. Sports scientists, however, recoil. For them it does not make much sense to average the different running tests, because they draw on completely different physiological resources; it is a variant of the old apples-and-oranges problem.

The justification for HeC3 is thatRun (1) and Star_r (2) draw more strongly on cardiosrespiratory Endurance than S20 (3) due to the longer duration of the runs compared to sprinting for 20 m which is a pure measure of the physical-fitness component Speed. Again, sports scientists are not very happy with this proposal.

Finally, HeC1 contrasts the fitness components Endurance, indicated best by Run (1), and Coordination, indicated by Star_r (2). Endurance (i.e., running for 6 minutes) is considered to be the best indicator of health-related status among the five tests because it is a rather pure measure of cardiorespiratory fitness. The Star_r test requires execution of a pre-instructed sequence of forward, sideways, and backward runs. This coordination of body movements implies a demand on working memory (i.e., remembering the order of these subruns) and executive control processes, but performats also depends on endurance. HeC1 yields a measure of Coordination “corrected” for the contribution of Endurance.

The statistical advantage of HelmertCoding is that the resulting contrasts are orthogonal (uncorrelated). This allows for optimal partitioning of variance and statistical power. It is also more efficient to estimate “orthogonal” than “non-orthogonal” random-effect structures.

contr2 = Dict(
  :School => Grouping(),
  :Child => Grouping(),
  :Cohort => Grouping(),
  :Sex => EffectsCoding(; levels=["Girls", "Boys"]),
  :Test => HelmertCoding(;
    levels=["Run", "Star_r", "S20_r", "SLJ", "BPT"],
  ),
);

m_ovi_Helmert = fit(MixedModel, f_ovi, dat; contrasts=contr2)

	Est.	SE	z	p	σ_Child
(Intercept)	-0.0183	0.0142	-1.29	0.1961	0.7149
Test: Star_r	-0.0120	0.0222	-0.54	0.5880
Test: S20_r	-0.0048	0.0127	-0.37	0.7085
Test: SLJ	-0.0011	0.0090	-0.13	0.8989
Test: BPT	-0.0024	0.0069	-0.34	0.7308
a1	0.2878	0.0481	5.99	<1e-08
Sex: Boys	0.2063	0.0142	14.54	<1e-47
Test: Star_r & a1	0.1256	0.0739	1.70	0.0892
Test: S20_r & a1	0.0846	0.0441	1.92	0.0551
Test: SLJ & a1	0.0154	0.0306	0.50	0.6148
Test: BPT & a1	0.0545	0.0233	2.34	0.0195
Test: Star_r & Sex: Boys	-0.0618	0.0222	-2.78	0.0054
Test: S20_r & Sex: Boys	-0.0266	0.0127	-2.09	0.0369
Test: SLJ & Sex: Boys	0.0186	0.0090	2.08	0.0378
Test: BPT & Sex: Boys	0.0327	0.0069	4.71	<1e-05
a1 & Sex: Boys	0.0114	0.0481	0.24	0.8124
Test: Star_r & a1 & Sex: Boys	0.0509	0.0739	0.69	0.4906
Test: S20_r & a1 & Sex: Boys	0.0294	0.0441	0.67	0.5049
Test: SLJ & a1 & Sex: Boys	0.0371	0.0306	1.22	0.2241
Test: BPT & a1 & Sex: Boys	-0.0076	0.0233	-0.33	0.7440
Residual	0.6537

We forego a detailed discussion of the effects, but note that again none of the interactions between age x Sex with the four test contrasts was significant.

The default labeling of Helmert contrasts may lead to confusions with other contrasts. Therefore, we could provide our own labels:

labels=["c2.1", "c3.12", "c4.123", "c5.1234"]

Once the order of levels is memorized the proposed labelling is very transparent.

2.3 HypothesisCoding: `contr3`

The third set of contrasts uses HypothesisCoding. Hypothesis coding allows the user to specify their own a priori contrast matrix, subject to the mathematical constraint that the matrix has full rank. For example, sport scientists agree that the first four tests can be contrasted with BPT, because the difference is akin to a correction of overall physical fitness. However, they want to keep the pairwise comparisons for the first four tests.

HyC1: BPT - mean(1,2,3,4)
HyC2: Star_r - Run_r
HyC3: Run_r - S20_r
HyC4: S20_r - SLJ

contr3 = Dict(
  :School => Grouping(),
  :Child => Grouping(),
  :Cohort => Grouping(),
  :Sex => EffectsCoding(; levels=["Girls", "Boys"]),
  :Test => HypothesisCoding(
    [
      -1 -1 -1 -1 +4
      -1 +1  0  0  0
       0 -1 +1  0  0
       0  0 -1 +1  0
    ];
    levels=["Run", "Star_r", "S20_r", "SLJ", "BPT"],
    labels=["BPT-other", "Star-End", "S20-Star", "SLJ-S20"],
  ),
);

m_ovi_Hypo = fit(MixedModel, f_ovi, dat; contrasts=contr3)

	Est.	SE	z	p	σ_Child
(Intercept)	-0.0183	0.0142	-1.29	0.1961	0.7149
Test: BPT-other	-0.0478	0.1389	-0.34	0.7308
Test: Star-End	-0.0241	0.0444	-0.54	0.5880
Test: S20-Star	-0.0023	0.0443	-0.05	0.9592
Test: SLJ-S20	0.0050	0.0440	0.11	0.9101
a1	0.2878	0.0481	5.99	<1e-08
Sex: Boys	0.2063	0.0142	14.54	<1e-47
Test: BPT-other & a1	1.0904	0.4668	2.34	0.0195
Test: Star-End & a1	0.2512	0.1478	1.70	0.0892
Test: S20-Star & a1	0.1282	0.1502	0.85	0.3932
Test: SLJ-S20 & a1	-0.1077	0.1524	-0.71	0.4797
Test: BPT-other & Sex: Boys	0.6548	0.1389	4.71	<1e-05
Test: Star-End & Sex: Boys	-0.1235	0.0444	-2.78	0.0054
Test: S20-Star & Sex: Boys	-0.0180	0.0443	-0.41	0.6840
Test: SLJ-S20 & Sex: Boys	0.1277	0.0440	2.90	0.0037
a1 & Sex: Boys	0.0114	0.0481	0.24	0.8124
Test: BPT-other & a1 & Sex: Boys	-0.1524	0.4668	-0.33	0.7440
Test: Star-End & a1 & Sex: Boys	0.1019	0.1478	0.69	0.4906
Test: S20-Star & a1 & Sex: Boys	0.0373	0.1502	0.25	0.8039
Test: SLJ-S20 & a1 & Sex: Boys	0.0898	0.1524	0.59	0.5559
Residual	0.6537

With HypothesisCoding we must generate our own labels for the contrasts. The default labeling of contrasts is usually not interpretable. Therefore, we provide our own.

Anyway, none of the interactions between age x Sex with the four Test contrasts was significant for these contrasts.

contr1b = Dict(
  :School => Grouping(),
  :Child => Grouping(),
  :Cohort => Grouping(),
  :Sex => EffectsCoding(; levels=["Girls", "Boys"]),
  :Test => HypothesisCoding(
    [
      -1 +1  0  0  0
       0 -1 +1  0  0
       0  0 -1 +1  0
       0  0  0 -1 +1
    ];
    levels=["Run", "Star_r", "S20_r", "SLJ", "BPT"],
    labels=["Star-Run", "S20-Star", "SLJ-S20", "BPT-SLJ"],
  ),
);

m_ovi_SeqDiff_v2 = fit(MixedModel, f_ovi, dat; contrasts=contr1b)

	Est.	SE	z	p	σ_Child
(Intercept)	-0.0183	0.0142	-1.29	0.1961	0.7149
Test: Star-Run	-0.0241	0.0444	-0.54	0.5880
Test: S20-Star	-0.0023	0.0443	-0.05	0.9592
Test: SLJ-S20	0.0050	0.0440	0.11	0.9101
Test: BPT-SLJ	-0.0085	0.0438	-0.19	0.8457
a1	0.2878	0.0481	5.99	<1e-08
Sex: Boys	0.2063	0.0142	14.54	<1e-47
Test: Star-Run & a1	0.2512	0.1478	1.70	0.0892
Test: S20-Star & a1	0.1282	0.1502	0.85	0.3932
Test: SLJ-S20 & a1	-0.1077	0.1524	-0.71	0.4797
Test: BPT-SLJ & a1	0.2265	0.1484	1.53	0.1269
Test: Star-Run & Sex: Boys	-0.1235	0.0444	-2.78	0.0054
Test: S20-Star & Sex: Boys	-0.0180	0.0443	-0.41	0.6840
Test: SLJ-S20 & Sex: Boys	0.1277	0.0440	2.90	0.0037
Test: BPT-SLJ & Sex: Boys	0.1078	0.0438	2.46	0.0139
a1 & Sex: Boys	0.0114	0.0481	0.24	0.8124
Test: Star-Run & a1 & Sex: Boys	0.1019	0.1478	0.69	0.4906
Test: S20-Star & a1 & Sex: Boys	0.0373	0.1502	0.25	0.8039
Test: SLJ-S20 & a1 & Sex: Boys	0.0898	0.1524	0.59	0.5559
Test: BPT-SLJ & a1 & Sex: Boys	-0.1496	0.1484	-1.01	0.3134
Residual	0.6537

m_zcp_SeqD = let
  form = @formula(
    zScore ~ 1 + Test * a1 * Sex + zerocorr(1 + Test | Child)
  )
  fit(MixedModel, form, dat; contrasts=contr1b)
end

Minimizing 21    Time: 0:00:00 ( 4.89 ms/it)
   objective: 13817.378395667522


Minimizing 121    Time: 0:00:00 ( 6.23 ms/it)
   objective: 13816.039723146534


Minimizing 130    Time: 0:00:00 ( 6.15 ms/it)

	Est.	SE	z	p	σ_Child
(Intercept)	-0.0181	0.0142	-1.28	0.2022	0.7541
Test: Star-Run	-0.0250	0.0452	-0.55	0.5810	0.0000
Test: S20-Star	-0.0031	0.0441	-0.07	0.9436	0.7011
Test: SLJ-S20	0.0041	0.0430	0.10	0.9234	0.4777
Test: BPT-SLJ	-0.0031	0.0431	-0.07	0.9436	0.0011
a1	0.2879	0.0481	5.98	<1e-08
Sex: Boys	0.2064	0.0142	14.52	<1e-47
Test: Star-Run & a1	0.2123	0.1501	1.41	0.1574
Test: S20-Star & a1	0.1564	0.1491	1.05	0.2944
Test: SLJ-S20 & a1	-0.1201	0.1489	-0.81	0.4197
Test: BPT-SLJ & a1	0.2164	0.1462	1.48	0.1387
Test: Star-Run & Sex: Boys	-0.1160	0.0452	-2.57	0.0103
Test: S20-Star & Sex: Boys	-0.0189	0.0441	-0.43	0.6686
Test: SLJ-S20 & Sex: Boys	0.1239	0.0430	2.88	0.0040
Test: BPT-SLJ & Sex: Boys	0.1102	0.0431	2.56	0.0106
a1 & Sex: Boys	0.0100	0.0481	0.21	0.8349
Test: Star-Run & a1 & Sex: Boys	0.1217	0.1501	0.81	0.4175
Test: S20-Star & a1 & Sex: Boys	0.0166	0.1491	0.11	0.9113
Test: SLJ-S20 & a1 & Sex: Boys	0.0901	0.1489	0.61	0.5448
Test: BPT-SLJ & a1 & Sex: Boys	-0.1419	0.1462	-0.97	0.3315
Residual	0.4464

m_zcp_SeqD_2 = let
  form = @formula(
    zScore ~ 1 + Test * a1 * Sex + (0 + Test | Child)
  )
  fit(MixedModel, form, dat; contrasts=contr1b)
end

Minimizing 14    Time: 0:00:00 ( 7.19 ms/it)
   objective: 14022.966416593923


Minimizing 28    Time: 0:00:00 ( 7.41 ms/it)
   objective: 13940.138774912537


Minimizing 41    Time: 0:00:00 ( 7.57 ms/it)
   objective: 13840.10805608038


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   objective: 13835.861514408982


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   objective: 13831.042249772268


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   objective: 13823.530870357423


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   objective: 13814.290135784948


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   objective: 13800.537779730774


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   objective: 13794.803364359257


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   objective: 13791.834841434651


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   objective: 13788.415341010952


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   objective: 13783.44943225446


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   objective: 13781.400794885605


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   objective: 13779.596612596404


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   objective: 13771.704882211163


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   objective: 13765.585416332608


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   objective: 13759.311825371758


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   objective: 13745.907717455135


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   objective: 13745.387183864088


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   objective: 13741.331227999086


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   objective: 13738.228370264742


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   objective: 13736.92725534451


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   objective: 13733.867524039513


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   objective: 13731.566984651461


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   objective: 13729.44339004364


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   objective: 13726.897612237119


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   objective: 13722.777331838672


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   objective: 13720.633788645693


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   objective: 13718.86876307672


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   objective: 13716.818457720892


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   objective: 13712.372533651667


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   objective: 13707.141680352135


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   objective: 13701.143813416867


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   objective: 13685.275721394155


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   objective: 13680.73306182947


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   objective: 13676.824720164052


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   objective: 13665.549845649879


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   objective: 13663.546755651056


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   objective: 13661.946080316018


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   objective: 13661.206919044173


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   objective: 13658.024168540225


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   objective: 13656.82595802941


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   objective: 13639.56779410836


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   objective: 13637.042674491757


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   objective: 13627.954857697416


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   objective: 13529.001125385788


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   objective: 13527.531287358404


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   objective: 13522.051697545976


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   objective: 13519.793725965763


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   objective: 13517.365579953715


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   objective: 13514.75830789686


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   objective: 13507.402504119425


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   objective: 13500.647966940109


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   objective: 13499.587482625706


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   objective: 13494.05278235842


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   objective: 13492.949026653689


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   objective: 13490.357784765409


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   objective: 13477.079910934335


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   objective: 13472.228484947358


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   objective: 13469.810258904232


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   objective: 13466.728324297721


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   objective: 13464.252638920625


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   objective: 13461.622589715


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   objective: 13460.105298637427


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   objective: 13454.081080587333


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   objective: 13449.696121495821


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   objective: 13449.178274240017


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   objective: 13445.248910311748


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   objective: 13445.086290844069


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   objective: 13438.283016105255


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   objective: 13435.842852455607


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   objective: 13431.686367807866


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   objective: 13424.725442714785


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   objective: 13424.212087363718


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   objective: 13416.49274566937


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   objective: 13416.02467145014


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   objective: 13415.22277047584


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   objective: 13414.619120260017


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   objective: 13411.283499775353


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   objective: 13411.159341114791


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   objective: 13410.585607528046


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   objective: 13410.20914021229


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   objective: 13408.274487209274


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   objective: 13403.437287122622


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Minimizing 2116    Time: 0:00:15 ( 7.11 ms/it)
   objective: 13375.87735416711


Minimizing 2131    Time: 0:00:15 ( 7.11 ms/it)
   objective: 13372.584975778358


Minimizing 2146    Time: 0:00:15 ( 7.11 ms/it)
   objective: 13364.864463501057


Minimizing 2160    Time: 0:00:15 ( 7.11 ms/it)
   objective: 13358.745464241365


Minimizing 2175    Time: 0:00:15 ( 7.11 ms/it)
   objective: 13350.016380424728


Minimizing 2190    Time: 0:00:15 ( 7.10 ms/it)
   objective: 13349.034019571103


Minimizing 2205    Time: 0:00:15 ( 7.10 ms/it)
   objective: 13347.222547880083


Minimizing 2220    Time: 0:00:15 ( 7.10 ms/it)
   objective: 13340.817192993782


Minimizing 2235    Time: 0:00:15 ( 7.10 ms/it)
   objective: 13339.122779128287


Minimizing 2250    Time: 0:00:15 ( 7.10 ms/it)
   objective: 13338.906303990298


Minimizing 2265    Time: 0:00:16 ( 7.10 ms/it)
   objective: 13337.820854305042


Minimizing 2280    Time: 0:00:16 ( 7.10 ms/it)
   objective: 13336.377120100355


Minimizing 2295    Time: 0:00:16 ( 7.10 ms/it)
   objective: 13335.305396709562


Minimizing 2310    Time: 0:00:16 ( 7.10 ms/it)
   objective: 13330.254774353845


Minimizing 2325    Time: 0:00:16 ( 7.10 ms/it)
   objective: 13328.636023339437


Minimizing 2340    Time: 0:00:16 ( 7.10 ms/it)
   objective: 13326.52008436805


Minimizing 2355    Time: 0:00:16 ( 7.10 ms/it)
   objective: 13320.603376217725


Minimizing 2370    Time: 0:00:16 ( 7.09 ms/it)
   objective: 13317.795014853094


Minimizing 2385    Time: 0:00:16 ( 7.09 ms/it)
   objective: 13314.94550205965


Minimizing 2400    Time: 0:00:17 ( 7.09 ms/it)
   objective: 13314.17793159747


Minimizing 2415    Time: 0:00:17 ( 7.09 ms/it)
   objective: 13311.702053191024


Minimizing 2430    Time: 0:00:17 ( 7.09 ms/it)
   objective: 13310.280093945286


Minimizing 2445    Time: 0:00:17 ( 7.09 ms/it)
   objective: 13305.169094006167


Minimizing 2460    Time: 0:00:17 ( 7.09 ms/it)
   objective: 13305.13613984178


Minimizing 2475    Time: 0:00:17 ( 7.09 ms/it)
   objective: 13305.12988663392


Minimizing 2490    Time: 0:00:17 ( 7.09 ms/it)
   objective: 13305.128143352049


Minimizing 2505    Time: 0:00:17 ( 7.09 ms/it)
   objective: 13305.135148237896


Minimizing 2520    Time: 0:00:17 ( 7.09 ms/it)
   objective: 13305.127497414142


Minimizing 2535    Time: 0:00:17 ( 7.09 ms/it)
   objective: 13305.126250529938


Minimizing 2550    Time: 0:00:18 ( 7.09 ms/it)
   objective: 13305.128338735522


Minimizing 2565    Time: 0:00:18 ( 7.09 ms/it)
   objective: 13305.12579731876


Minimizing 2580    Time: 0:00:18 ( 7.09 ms/it)
   objective: 13305.127014898666


Minimizing 2595    Time: 0:00:18 ( 7.09 ms/it)
   objective: 13305.128986387106


Minimizing 2610    Time: 0:00:18 ( 7.08 ms/it)
   objective: 13305.125648041372


Minimizing 2625    Time: 0:00:18 ( 7.08 ms/it)
   objective: 13305.127349291346


Minimizing 2640    Time: 0:00:18 ( 7.08 ms/it)
   objective: 13305.124787675813


Minimizing 2655    Time: 0:00:18 ( 7.08 ms/it)
   objective: 13305.124425413378


Minimizing 2670    Time: 0:00:18 ( 7.08 ms/it)
   objective: 13305.126250520756


Minimizing 2685    Time: 0:00:19 ( 7.08 ms/it)
   objective: 13305.12789582186


Minimizing 2700    Time: 0:00:19 ( 7.08 ms/it)
   objective: 13305.124086117401


Minimizing 2715    Time: 0:00:19 ( 7.08 ms/it)
   objective: 13305.124151415657


Minimizing 2716    Time: 0:00:19 ( 7.08 ms/it)

	Est.	SE	z	p	σ_Child
(Intercept)	-0.0197	0.0141	-1.40	0.1625
Test: Star-Run	-0.0169	0.0451	-0.37	0.7079
Test: S20-Star	0.0004	0.0452	0.01	0.9924
Test: SLJ-S20	0.0035	0.0437	0.08	0.9366
Test: BPT-SLJ	-0.0081	0.0428	-0.19	0.8505
a1	0.2898	0.0479	6.05	<1e-08
Sex: Boys	0.2067	0.0141	14.63	<1e-47
Test: Star-Run & a1	0.2564	0.1500	1.71	0.0873
Test: S20-Star & a1	0.1515	0.1530	0.99	0.3221
Test: SLJ-S20 & a1	-0.1651	0.1508	-1.09	0.2736
Test: BPT-SLJ & a1	0.2480	0.1447	1.71	0.0866
Test: Star-Run & Sex: Boys	-0.1329	0.0451	-2.95	0.0032
Test: S20-Star & Sex: Boys	-0.0065	0.0452	-0.14	0.8851
Test: SLJ-S20 & Sex: Boys	0.1250	0.0437	2.86	0.0043
Test: BPT-SLJ & Sex: Boys	0.1142	0.0428	2.67	0.0076
a1 & Sex: Boys	0.0063	0.0479	0.13	0.8948
Test: Star-Run & a1 & Sex: Boys	0.0909	0.1500	0.61	0.5442
Test: S20-Star & a1 & Sex: Boys	0.0095	0.1530	0.06	0.9507
Test: SLJ-S20 & a1 & Sex: Boys	0.1246	0.1508	0.83	0.4086
Test: BPT-SLJ & a1 & Sex: Boys	-0.1778	0.1447	-1.23	0.2194
Test: BPT					0.9223
Test: SLJ					0.9678
Test: Star_r					0.9892
Test: Run					0.9647
Test: S20_r					0.9876
Residual	0.0000

m_cpx_0_SeqDiff = let
  f_cpx_0 = @formula(
    zScore ~ 1 + Test * a1 * Sex + (0 + Test | Child)
  )
  fit(MixedModel, f_cpx_0, dat; contrasts=contr1b)
end

Minimizing 15    Time: 0:00:00 ( 6.95 ms/it)
   objective: 13935.55002325049


Minimizing 29    Time: 0:00:00 ( 7.25 ms/it)
   objective: 14030.447721514898


Minimizing 44    Time: 0:00:00 ( 7.21 ms/it)
   objective: 13840.506558021234


Minimizing 59    Time: 0:00:00 ( 7.17 ms/it)
   objective: 13841.017998610616


Minimizing 74    Time: 0:00:00 ( 7.14 ms/it)
   objective: 13828.929727652561


Minimizing 89    Time: 0:00:00 ( 7.11 ms/it)
   objective: 13814.481698519754


Minimizing 102    Time: 0:00:00 ( 7.34 ms/it)
   objective: 13805.742704761811


Minimizing 117    Time: 0:00:00 ( 7.31 ms/it)
   objective: 13804.17024346094


Minimizing 131    Time: 0:00:00 ( 7.33 ms/it)
   objective: 13794.445035338842


Minimizing 146    Time: 0:00:01 ( 7.30 ms/it)
   objective: 13791.694000437297


Minimizing 161    Time: 0:00:01 ( 7.28 ms/it)
   objective: 13786.580107475937


Minimizing 176    Time: 0:00:01 ( 7.26 ms/it)
   objective: 13782.1327226828


Minimizing 191    Time: 0:00:01 ( 7.25 ms/it)
   objective: 13781.11827460385


Minimizing 206    Time: 0:00:01 ( 7.23 ms/it)
   objective: 13776.277827957849


Minimizing 221    Time: 0:00:01 ( 7.22 ms/it)
   objective: 13770.553958199625


Minimizing 235    Time: 0:00:01 ( 7.22 ms/it)
   objective: 13761.105996509208


Minimizing 250    Time: 0:00:01 ( 7.21 ms/it)
   objective: 13758.155800310857


Minimizing 265    Time: 0:00:01 ( 7.20 ms/it)
   objective: 13750.2436925295


Minimizing 280    Time: 0:00:02 ( 7.19 ms/it)
   objective: 13743.314032433038


Minimizing 295    Time: 0:00:02 ( 7.18 ms/it)
   objective: 13739.954161809483


Minimizing 310    Time: 0:00:02 ( 7.17 ms/it)
   objective: 13737.288388104043


Minimizing 325    Time: 0:00:02 ( 7.16 ms/it)
   objective: 13735.222010596546


Minimizing 340    Time: 0:00:02 ( 7.16 ms/it)
   objective: 13731.952994591207


Minimizing 355    Time: 0:00:02 ( 7.15 ms/it)
   objective: 13730.94172267556


Minimizing 370    Time: 0:00:02 ( 7.14 ms/it)
   objective: 13727.24099008906


Minimizing 385    Time: 0:00:02 ( 7.14 ms/it)
   objective: 13725.143219212896


Minimizing 400    Time: 0:00:02 ( 7.14 ms/it)
   objective: 13722.553182516587


Minimizing 415    Time: 0:00:02 ( 7.13 ms/it)
   objective: 13720.566573948407


Minimizing 430    Time: 0:00:03 ( 7.13 ms/it)
   objective: 13718.72042726029


Minimizing 445    Time: 0:00:03 ( 7.12 ms/it)
   objective: 13716.606993708707


Minimizing 460    Time: 0:00:03 ( 7.12 ms/it)
   objective: 13711.305651911958


Minimizing 475    Time: 0:00:03 ( 7.11 ms/it)
   objective: 13706.826431069969


Minimizing 490    Time: 0:00:03 ( 7.11 ms/it)
   objective: 13698.254777044123


Minimizing 505    Time: 0:00:03 ( 7.10 ms/it)
   objective: 13688.028459466526


Minimizing 520    Time: 0:00:03 ( 7.10 ms/it)
   objective: 13680.308661906502


Minimizing 535    Time: 0:00:03 ( 7.09 ms/it)
   objective: 13675.493643365884


Minimizing 550    Time: 0:00:03 ( 7.09 ms/it)
   objective: 13664.521007686988


Minimizing 565    Time: 0:00:04 ( 7.08 ms/it)
   objective: 13663.318194622778


Minimizing 580    Time: 0:00:04 ( 7.08 ms/it)
   objective: 13661.843452170786


Minimizing 595    Time: 0:00:04 ( 7.07 ms/it)
   objective: 13661.138122243588


Minimizing 610    Time: 0:00:04 ( 7.07 ms/it)
   objective: 13659.346452144455


Minimizing 625    Time: 0:00:04 ( 7.06 ms/it)
   objective: 13657.65330108388


Minimizing 640    Time: 0:00:04 ( 7.06 ms/it)
   objective: 13656.089401086057


Minimizing 655    Time: 0:00:04 ( 7.05 ms/it)
   objective: 13655.51325182581


Minimizing 670    Time: 0:00:04 ( 7.05 ms/it)
   objective: 13654.283281243996


Minimizing 685    Time: 0:00:04 ( 7.05 ms/it)
   objective: 13649.18969585008


Minimizing 700    Time: 0:00:04 ( 7.05 ms/it)
   objective: 13648.564501939542


Minimizing 715    Time: 0:00:05 ( 7.05 ms/it)
   objective: 13648.188583935971


Minimizing 730    Time: 0:00:05 ( 7.05 ms/it)
   objective: 13647.904159977308


Minimizing 745    Time: 0:00:05 ( 7.05 ms/it)
   objective: 13646.986872288911


Minimizing 760    Time: 0:00:05 ( 7.04 ms/it)
   objective: 13646.176310363935


Minimizing 775    Time: 0:00:05 ( 7.04 ms/it)
   objective: 13644.723986135145


Minimizing 790    Time: 0:00:05 ( 7.04 ms/it)
   objective: 13642.541169904398


Minimizing 805    Time: 0:00:05 ( 7.04 ms/it)
   objective: 13640.755018772084


Minimizing 882    Time: 0:00:06 ( 7.21 ms/it)
   objective: 13616.267870535597


Minimizing 902    Time: 0:00:06 ( 7.21 ms/it)
   objective: 13615.30667238657


Minimizing 917    Time: 0:00:06 ( 7.21 ms/it)
   objective: 13610.93500642051


Minimizing 931    Time: 0:00:06 ( 7.21 ms/it)
   objective: 13604.114560441096


Minimizing 945    Time: 0:00:06 ( 7.21 ms/it)
   objective: 13589.214491432518


Minimizing 960    Time: 0:00:06 ( 7.20 ms/it)
   objective: 13562.114516637273


Minimizing 975    Time: 0:00:07 ( 7.20 ms/it)
   objective: 13558.936455753363


Minimizing 990    Time: 0:00:07 ( 7.20 ms/it)
   objective: 13551.994553230477


Minimizing 1005    Time: 0:00:07 ( 7.20 ms/it)
   objective: 13549.857444835092


Minimizing 1020    Time: 0:00:07 ( 7.20 ms/it)
   objective: 13544.36107341863


Minimizing 1035    Time: 0:00:07 ( 7.19 ms/it)
   objective: 13536.571155794001


Minimizing 1050    Time: 0:00:07 ( 7.19 ms/it)
   objective: 13529.852338382938


Minimizing 1065    Time: 0:00:07 ( 7.19 ms/it)
   objective: 13528.17170659671


Minimizing 1079    Time: 0:00:07 ( 7.19 ms/it)
   objective: 13522.583657595962


Minimizing 1093    Time: 0:00:07 ( 7.19 ms/it)
   objective: 13521.340204864828


Minimizing 1107    Time: 0:00:07 ( 7.19 ms/it)
   objective: 13517.785375939398


Minimizing 1122    Time: 0:00:08 ( 7.19 ms/it)
   objective: 13516.911091313756


Minimizing 1137    Time: 0:00:08 ( 7.19 ms/it)
   objective: 13511.984645280405


Minimizing 1152    Time: 0:00:08 ( 7.19 ms/it)
   objective: 13505.285675196683


Minimizing 1167    Time: 0:00:08 ( 7.19 ms/it)
   objective: 13500.314544640438


Minimizing 1182    Time: 0:00:08 ( 7.18 ms/it)
   objective: 13494.268654191328


Minimizing 1197    Time: 0:00:08 ( 7.18 ms/it)
   objective: 13493.250177363341


Minimizing 1212    Time: 0:00:08 ( 7.18 ms/it)
   objective: 13490.911775550732


Minimizing 1227    Time: 0:00:08 ( 7.18 ms/it)
   objective: 13486.685677002017


Minimizing 1242    Time: 0:00:08 ( 7.18 ms/it)
   objective: 13477.934801247065


Minimizing 1256    Time: 0:00:09 ( 7.18 ms/it)
   objective: 13477.294260866976


Minimizing 1270    Time: 0:00:09 ( 7.18 ms/it)
   objective: 13474.83213116735


Minimizing 1285    Time: 0:00:09 ( 7.18 ms/it)
   objective: 13469.953336583647


Minimizing 1300    Time: 0:00:09 ( 7.18 ms/it)
   objective: 13469.2040761604


Minimizing 1315    Time: 0:00:09 ( 7.18 ms/it)
   objective: 13464.369882058818


Minimizing 1330    Time: 0:00:09 ( 7.18 ms/it)
   objective: 13463.729216045045


Minimizing 1345    Time: 0:00:09 ( 7.17 ms/it)
   objective: 13460.23152093085


Minimizing 1359    Time: 0:00:09 ( 7.18 ms/it)
   objective: 13456.570951197995


Minimizing 1374    Time: 0:00:09 ( 7.18 ms/it)
   objective: 13451.665986530359


Minimizing 1389    Time: 0:00:09 ( 7.17 ms/it)
   objective: 13449.243604606425


Minimizing 1403    Time: 0:00:10 ( 7.17 ms/it)
   objective: 13446.9617477672


Minimizing 1418    Time: 0:00:10 ( 7.17 ms/it)
   objective: 13445.103138474682


Minimizing 1433    Time: 0:00:10 ( 7.17 ms/it)
   objective: 13444.6645133743


Minimizing 1448    Time: 0:00:10 ( 7.17 ms/it)
   objective: 13442.841042366526


Minimizing 1463    Time: 0:00:10 ( 7.17 ms/it)
   objective: 13440.564903380444


Minimizing 1478    Time: 0:00:10 ( 7.17 ms/it)
   objective: 13440.450449418466


Minimizing 1493    Time: 0:00:10 ( 7.17 ms/it)
   objective: 13439.969462861423


Minimizing 1508    Time: 0:00:10 ( 7.17 ms/it)
   objective: 13438.75618399131


Minimizing 1523    Time: 0:00:10 ( 7.16 ms/it)
   objective: 13436.185709623242


Minimizing 1538    Time: 0:00:11 ( 7.16 ms/it)
   objective: 13435.683178707972


Minimizing 1553    Time: 0:00:11 ( 7.16 ms/it)
   objective: 13434.066535873833


Minimizing 1568    Time: 0:00:11 ( 7.16 ms/it)
   objective: 13433.826754660302


Minimizing 1583    Time: 0:00:11 ( 7.16 ms/it)
   objective: 13432.790192684784


Minimizing 1598    Time: 0:00:11 ( 7.16 ms/it)
   objective: 13432.003363236756


Minimizing 1613    Time: 0:00:11 ( 7.16 ms/it)
   objective: 13430.610245230666


Minimizing 1628    Time: 0:00:11 ( 7.16 ms/it)
   objective: 13429.956040779638


Minimizing 1642    Time: 0:00:11 ( 7.16 ms/it)
   objective: 13429.637110361888


Minimizing 1657    Time: 0:00:11 ( 7.16 ms/it)
   objective: 13428.924113546615


Minimizing 1672    Time: 0:00:11 ( 7.16 ms/it)
   objective: 13436.172623718856


Minimizing 1687    Time: 0:00:12 ( 7.16 ms/it)
   objective: 13425.12975516083


Minimizing 1701    Time: 0:00:12 ( 7.16 ms/it)
   objective: 13424.867892660739


Minimizing 1715    Time: 0:00:12 ( 7.16 ms/it)
   objective: 13424.48531902449


Minimizing 1729    Time: 0:00:12 ( 7.16 ms/it)
   objective: 13423.659192971347


Minimizing 1743    Time: 0:00:12 ( 7.16 ms/it)
   objective: 13423.105239815573


Minimizing 1757    Time: 0:00:12 ( 7.16 ms/it)
   objective: 13422.288113972667


Minimizing 1771    Time: 0:00:12 ( 7.16 ms/it)
   objective: 13422.269024249908


Minimizing 1785    Time: 0:00:12 ( 7.16 ms/it)
   objective: 13421.667076807338


Minimizing 1799    Time: 0:00:12 ( 7.16 ms/it)
   objective: 13420.964802871182


Minimizing 1813    Time: 0:00:12 ( 7.16 ms/it)
   objective: 13420.586852994573


Minimizing 1827    Time: 0:00:13 ( 7.17 ms/it)
   objective: 13419.7568962878


Minimizing 1841    Time: 0:00:13 ( 7.17 ms/it)
   objective: 13419.004220421979


Minimizing 1855    Time: 0:00:13 ( 7.17 ms/it)
   objective: 13417.085500095753


Minimizing 1869    Time: 0:00:13 ( 7.17 ms/it)
   objective: 13417.005639796349


Minimizing 1883    Time: 0:00:13 ( 7.17 ms/it)
   objective: 13416.195851831828


Minimizing 1897    Time: 0:00:13 ( 7.17 ms/it)
   objective: 13415.980988124706


Minimizing 1911    Time: 0:00:13 ( 7.18 ms/it)
   objective: 13415.138905550499


Minimizing 1922    Time: 0:00:13 ( 7.20 ms/it)
   objective: 13414.619120260017


Minimizing 1935    Time: 0:00:13 ( 7.21 ms/it)
   objective: 13411.420663386045


Minimizing 1948    Time: 0:00:14 ( 7.21 ms/it)
   objective: 13411.162697422988


Minimizing 1961    Time: 0:00:14 ( 7.22 ms/it)
   objective: 13410.971883830018


Minimizing 1974    Time: 0:00:14 ( 7.23 ms/it)
   objective: 13410.36474569836


Minimizing 1988    Time: 0:00:14 ( 7.23 ms/it)
   objective: 13409.93626205984


Minimizing 2002    Time: 0:00:14 ( 7.23 ms/it)
   objective: 13406.53121360422


Minimizing 2015    Time: 0:00:14 ( 7.24 ms/it)
   objective: 13402.829963156299


Minimizing 2029    Time: 0:00:14 ( 7.24 ms/it)
   objective: 13401.095272133796


Minimizing 2043    Time: 0:00:14 ( 7.24 ms/it)
   objective: 13392.924107341489


Minimizing 2057    Time: 0:00:14 ( 7.24 ms/it)
   objective: 13392.434091976917


Minimizing 2071    Time: 0:00:15 ( 7.25 ms/it)
   objective: 13390.06634391466


Minimizing 2085    Time: 0:00:15 ( 7.25 ms/it)
   objective: 13385.129407129876


Minimizing 2099    Time: 0:00:15 ( 7.25 ms/it)
   objective: 13383.560237203128


Minimizing 2113    Time: 0:00:15 ( 7.25 ms/it)
   objective: 13375.932640828309


Minimizing 2127    Time: 0:00:15 ( 7.25 ms/it)
   objective: 13375.124705737471


Minimizing 2141    Time: 0:00:15 ( 7.25 ms/it)
   objective: 13366.709378348081


Minimizing 2155    Time: 0:00:15 ( 7.25 ms/it)
   objective: 13363.399609825254


Minimizing 2169    Time: 0:00:15 ( 7.25 ms/it)
   objective: 13351.844314403192


Minimizing 2183    Time: 0:00:15 ( 7.25 ms/it)
   objective: 13349.92387181113


Minimizing 2197    Time: 0:00:15 ( 7.25 ms/it)
   objective: 13348.388429565137


Minimizing 2211    Time: 0:00:16 ( 7.26 ms/it)
   objective: 13344.554785869463


Minimizing 2225    Time: 0:00:16 ( 7.26 ms/it)
   objective: 13339.174092886635


Minimizing 2239    Time: 0:00:16 ( 7.26 ms/it)
   objective: 13339.112612183555


Minimizing 2253    Time: 0:00:16 ( 7.26 ms/it)
   objective: 13338.679676129308


Minimizing 2267    Time: 0:00:16 ( 7.26 ms/it)
   objective: 13337.86464772551


Minimizing 2281    Time: 0:00:16 ( 7.26 ms/it)
   objective: 13336.504392860457


Minimizing 2295    Time: 0:00:16 ( 7.26 ms/it)
   objective: 13335.305396709562


Minimizing 2309    Time: 0:00:16 ( 7.26 ms/it)
   objective: 13330.883293963823


Minimizing 2323    Time: 0:00:16 ( 7.26 ms/it)
   objective: 13328.715469756746


Minimizing 2337    Time: 0:00:16 ( 7.26 ms/it)
   objective: 13327.185898506694


Minimizing 2351    Time: 0:00:17 ( 7.26 ms/it)
   objective: 13320.905246089125


Minimizing 2365    Time: 0:00:17 ( 7.26 ms/it)
   objective: 13319.886653268753


Minimizing 2379    Time: 0:00:17 ( 7.26 ms/it)
   objective: 13315.049065875632


Minimizing 2393    Time: 0:00:17 ( 7.26 ms/it)
   objective: 13314.796986523681


Minimizing 2407    Time: 0:00:17 ( 7.26 ms/it)
   objective: 13313.570152806598


Minimizing 2421    Time: 0:00:17 ( 7.26 ms/it)
   objective: 13310.78158061394


Minimizing 2435    Time: 0:00:17 ( 7.26 ms/it)
   objective: 13307.278000210645


Minimizing 2449    Time: 0:00:17 ( 7.26 ms/it)
   objective: 13305.158055766587


Minimizing 2463    Time: 0:00:17 ( 7.27 ms/it)
   objective: 13305.13115266907


Minimizing 2477    Time: 0:00:17 ( 7.27 ms/it)
   objective: 13305.126180575084


Minimizing 2491    Time: 0:00:18 ( 7.27 ms/it)
   objective: 13305.127489572638


Minimizing 2505    Time: 0:00:18 ( 7.27 ms/it)
   objective: 13305.135148237896


Minimizing 2519    Time: 0:00:18 ( 7.27 ms/it)
   objective: 13305.125354738964


Minimizing 2533    Time: 0:00:18 ( 7.27 ms/it)
   objective: 13305.12630244426


Minimizing 2547    Time: 0:00:18 ( 7.27 ms/it)
   objective: 13305.126594590067


Minimizing 2561    Time: 0:00:18 ( 7.27 ms/it)
   objective: 13305.12662712003


Minimizing 2575    Time: 0:00:18 ( 7.27 ms/it)
   objective: 13305.127051751522


Minimizing 2589    Time: 0:00:18 ( 7.27 ms/it)
   objective: 13305.12543269046


Minimizing 2603    Time: 0:00:18 ( 7.27 ms/it)
   objective: 13305.127044578709


Minimizing 2617    Time: 0:00:19 ( 7.27 ms/it)
   objective: 13305.127895258702


Minimizing 2631    Time: 0:00:19 ( 7.27 ms/it)
   objective: 13305.12594905372


Minimizing 2645    Time: 0:00:19 ( 7.27 ms/it)
   objective: 13305.12403430324


Minimizing 2659    Time: 0:00:19 ( 7.27 ms/it)
   objective: 13305.125062509847


Minimizing 2673    Time: 0:00:19 ( 7.27 ms/it)
   objective: 13305.123487864665


Minimizing 2687    Time: 0:00:19 ( 7.27 ms/it)
   objective: 13305.124633351486


Minimizing 2701    Time: 0:00:19 ( 7.27 ms/it)
   objective: 13305.124169058414


Minimizing 2715    Time: 0:00:19 ( 7.27 ms/it)
   objective: 13305.124151415657


Minimizing 2716    Time: 0:00:19 ( 7.27 ms/it)

	Est.	SE	z	p	σ_Child
(Intercept)	-0.0197	0.0141	-1.40	0.1625
Test: Star-Run	-0.0169	0.0451	-0.37	0.7079
Test: S20-Star	0.0004	0.0452	0.01	0.9924
Test: SLJ-S20	0.0035	0.0437	0.08	0.9366
Test: BPT-SLJ	-0.0081	0.0428	-0.19	0.8505
a1	0.2898	0.0479	6.05	<1e-08
Sex: Boys	0.2067	0.0141	14.63	<1e-47
Test: Star-Run & a1	0.2564	0.1500	1.71	0.0873
Test: S20-Star & a1	0.1515	0.1530	0.99	0.3221
Test: SLJ-S20 & a1	-0.1651	0.1508	-1.09	0.2736
Test: BPT-SLJ & a1	0.2480	0.1447	1.71	0.0866
Test: Star-Run & Sex: Boys	-0.1329	0.0451	-2.95	0.0032
Test: S20-Star & Sex: Boys	-0.0065	0.0452	-0.14	0.8851
Test: SLJ-S20 & Sex: Boys	0.1250	0.0437	2.86	0.0043
Test: BPT-SLJ & Sex: Boys	0.1142	0.0428	2.67	0.0076
a1 & Sex: Boys	0.0063	0.0479	0.13	0.8948
Test: Star-Run & a1 & Sex: Boys	0.0909	0.1500	0.61	0.5442
Test: S20-Star & a1 & Sex: Boys	0.0095	0.1530	0.06	0.9507
Test: SLJ-S20 & a1 & Sex: Boys	0.1246	0.1508	0.83	0.4086
Test: BPT-SLJ & a1 & Sex: Boys	-0.1778	0.1447	-1.23	0.2194
Test: BPT					0.9223
Test: SLJ					0.9678
Test: Star_r					0.9892
Test: Run					0.9647
Test: S20_r					0.9876
Residual	0.0000

VarCorr(m_cpx_0_SeqDiff)

	Column	Variance	Std.Dev	Corr.
Child	Test: Run	0.93064599	0.96469995
	Test: Star_r	0.97849182	0.98918746	-0.50
	Test: S20_r	0.97540959	0.98762827	+0.71	+0.22
	Test: SLJ	0.93656874	0.96776481	+0.51	+0.13	+0.84
	Test: BPT	0.85060249	0.92228113	-0.12	+0.62	+0.40	+0.39
Residual		0.00000000	0.00001898

m_cpx_0_SeqDiff.PCA

(Child = 
Principal components based on correlation matrix
 Test: Run      1.0     .      .      .      .
 Test: Star_r  -0.5    1.0     .      .      .
 Test: S20_r    0.71   0.22   1.0     .      .
 Test: SLJ      0.51   0.13   0.84   1.0     .
 Test: BPT     -0.12   0.62   0.4    0.39   1.0

Normalized cumulative variances:
[0.5054, 0.8762, 0.9466, 1.0, 1.0]

Component loadings
                 PC1    PC2    PC3    PC4    PC5
 Test: Run     -0.4    0.53  -0.18   0.45   0.57
 Test: Star_r  -0.14  -0.67   0.5    0.35   0.4
 Test: S20_r   -0.61   0.06   0.22   0.33  -0.68
 Test: SLJ     -0.57   0.03   0.23  -0.75   0.23
 Test: BPT     -0.34  -0.52  -0.79  -0.01   0.0,)

f_cpx_1 = @formula(
  zScore ~ 1 + Test * a1 * Sex + (1 + Test | Child)
)
m_cpx_1_SeqDiff =
fit(MixedModel, f_cpx_1, dat; contrasts=contr1b)

Minimizing 14    Time: 0:00:00 ( 7.51 ms/it)
   objective: 13890.385675304973


Minimizing 27    Time: 0:00:00 ( 7.86 ms/it)
   objective: 13871.76436926304


Minimizing 40    Time: 0:00:00 ( 7.86 ms/it)
   objective: 13806.3127154706


Minimizing 54    Time: 0:00:00 ( 7.80 ms/it)
   objective: 13803.647020844368


Minimizing 68    Time: 0:00:00 ( 7.77 ms/it)
   objective: 13801.168697345594


Minimizing 82    Time: 0:00:00 ( 7.75 ms/it)
   objective: 13795.897002245558


Minimizing 94    Time: 0:00:00 ( 8.16 ms/it)
   objective: 13792.961361835189


Minimizing 105    Time: 0:00:00 ( 8.34 ms/it)
   objective: 13783.632204897669


Minimizing 118    Time: 0:00:00 ( 8.29 ms/it)
   objective: 13768.679446472132


Minimizing 131    Time: 0:00:01 ( 8.24 ms/it)
   objective: 13766.636584420476


Minimizing 145    Time: 0:00:01 ( 8.16 ms/it)
   objective: 13758.058012198173


Minimizing 159    Time: 0:00:01 ( 8.11 ms/it)
   objective: 13756.278734443527


Minimizing 172    Time: 0:00:01 ( 8.10 ms/it)
   objective: 13755.390677366624


Minimizing 184    Time: 0:00:01 ( 8.17 ms/it)
   objective: 13753.58217900603


Minimizing 196    Time: 0:00:01 ( 8.20 ms/it)
   objective: 13752.303713969788


Minimizing 208    Time: 0:00:01 ( 8.22 ms/it)
   objective: 13751.698359205253


Minimizing 220    Time: 0:00:01 ( 8.24 ms/it)
   objective: 13747.029663044912


Minimizing 233    Time: 0:00:01 ( 8.21 ms/it)
   objective: 13746.335121497037


Minimizing 247    Time: 0:00:02 ( 8.16 ms/it)
   objective: 13742.530918990447


Minimizing 262    Time: 0:00:02 ( 8.10 ms/it)
   objective: 13740.565362823243


Minimizing 277    Time: 0:00:02 ( 8.05 ms/it)
   objective: 13738.649170032375


Minimizing 292    Time: 0:00:02 ( 8.00 ms/it)
   objective: 13735.627738715917


Minimizing 307    Time: 0:00:02 ( 7.94 ms/it)
   objective: 13733.29353104728


Minimizing 322    Time: 0:00:02 ( 7.98 ms/it)
   objective: 13731.306549704199


Minimizing 336    Time: 0:00:02 ( 7.96 ms/it)
   objective: 13730.10657271295


Minimizing 349    Time: 0:00:02 ( 7.96 ms/it)
   objective: 13721.756517457525


Minimizing 364    Time: 0:00:02 ( 7.93 ms/it)
   objective: 13718.694076089303


Minimizing 378    Time: 0:00:02 ( 7.91 ms/it)
   objective: 13703.287027173154


Minimizing 392    Time: 0:00:03 ( 7.89 ms/it)
   objective: 13697.100072059005


Minimizing 406    Time: 0:00:03 ( 7.86 ms/it)
   objective: 13686.753301058645


Minimizing 420    Time: 0:00:03 ( 7.85 ms/it)
   objective: 13683.024725194009


Minimizing 435    Time: 0:00:03 ( 7.82 ms/it)
   objective: 13678.245287537477


Minimizing 449    Time: 0:00:03 ( 7.80 ms/it)
   objective: 13676.93529687295


Minimizing 463    Time: 0:00:03 ( 7.79 ms/it)
   objective: 13672.72326676538


Minimizing 477    Time: 0:00:03 ( 7.80 ms/it)
   objective: 13664.3726439259


Minimizing 491    Time: 0:00:03 ( 7.80 ms/it)
   objective: 13663.623419999254


Minimizing 505    Time: 0:00:03 ( 7.78 ms/it)
   objective: 13661.321437475413


Minimizing 519    Time: 0:00:04 ( 7.77 ms/it)
   objective: 13659.527393046566


Minimizing 533    Time: 0:00:04 ( 7.77 ms/it)
   objective: 13658.957379669053


Minimizing 547    Time: 0:00:04 ( 7.75 ms/it)
   objective: 13657.301257996547


Minimizing 561    Time: 0:00:04 ( 7.77 ms/it)
   objective: 13654.945857291757


Minimizing 576    Time: 0:00:04 ( 7.74 ms/it)
   objective: 13652.118887848315


Minimizing 591    Time: 0:00:04 ( 7.72 ms/it)
   objective: 13651.76161891676


Minimizing 605    Time: 0:00:04 ( 7.72 ms/it)
   objective: 13648.945677574528


Minimizing 619    Time: 0:00:04 ( 7.72 ms/it)
   objective: 13648.218270036235


Minimizing 633    Time: 0:00:04 ( 7.71 ms/it)
   objective: 13643.588616617613


Minimizing 647    Time: 0:00:04 ( 7.71 ms/it)
   objective: 13642.884648958036


Minimizing 660    Time: 0:00:05 ( 7.71 ms/it)
   objective: 13642.347533587948


Minimizing 673    Time: 0:00:05 ( 7.72 ms/it)
   objective: 13641.252211188668


Minimizing 687    Time: 0:00:05 ( 7.71 ms/it)
   objective: 13640.720390649702


Minimizing 701    Time: 0:00:05 ( 7.71 ms/it)
   objective: 13636.731000203436


Minimizing 714    Time: 0:00:05 ( 7.72 ms/it)
   objective: 13633.645431205405


Minimizing 728    Time: 0:00:05 ( 7.71 ms/it)
   objective: 13632.89632887935


Minimizing 741    Time: 0:00:05 ( 7.72 ms/it)
   objective: 13625.94569552198


Minimizing 755    Time: 0:00:05 ( 7.72 ms/it)
   objective: 13621.200690799891


Minimizing 769    Time: 0:00:05 ( 7.71 ms/it)
   objective: 13607.866843770356


Minimizing 782    Time: 0:00:06 ( 7.71 ms/it)
   objective: 13605.657843407454


Minimizing 795    Time: 0:00:06 ( 7.73 ms/it)
   objective: 13604.201269727142


Minimizing 809    Time: 0:00:06 ( 7.73 ms/it)
   objective: 13596.43428001619


Minimizing 823    Time: 0:00:06 ( 7.72 ms/it)
   objective: 13588.860570872836


Minimizing 836    Time: 0:00:06 ( 7.73 ms/it)
   objective: 13587.580794492234


Minimizing 850    Time: 0:00:06 ( 7.73 ms/it)
   objective: 13583.23313432777


Minimizing 863    Time: 0:00:06 ( 7.73 ms/it)
   objective: 13579.772681124552


Minimizing 876    Time: 0:00:06 ( 7.73 ms/it)
   objective: 13571.476511039065


Minimizing 888    Time: 0:00:06 ( 7.75 ms/it)
   objective: 13570.568182379713


Minimizing 901    Time: 0:00:06 ( 7.76 ms/it)
   objective: 13568.61833101335


Minimizing 913    Time: 0:00:07 ( 7.77 ms/it)
   objective: 13567.118118141028


Minimizing 927    Time: 0:00:07 ( 7.77 ms/it)
   objective: 13563.2660955787


Minimizing 941    Time: 0:00:07 ( 7.77 ms/it)
   objective: 13557.522116767454


Minimizing 954    Time: 0:00:07 ( 7.77 ms/it)
   objective: 13551.246553134639


Minimizing 967    Time: 0:00:07 ( 7.76 ms/it)
   objective: 13550.63184188486


Minimizing 981    Time: 0:00:07 ( 7.76 ms/it)
   objective: 13546.811806475605


Minimizing 994    Time: 0:00:07 ( 7.76 ms/it)
   objective: 13542.775086741836


Minimizing 1008    Time: 0:00:07 ( 7.76 ms/it)
   objective: 13533.921957198167


Minimizing 1022    Time: 0:00:07 ( 7.76 ms/it)
   objective: 13532.706088600244


Minimizing 1040    Time: 0:00:08 ( 7.78 ms/it)
   objective: 13531.052884428398


Minimizing 1056    Time: 0:00:08 ( 7.78 ms/it)
   objective: 13528.904856362897


Minimizing 1069    Time: 0:00:08 ( 7.78 ms/it)
   objective: 13524.864472359288


Minimizing 1083    Time: 0:00:08 ( 7.77 ms/it)
   objective: 13524.281981040622


Minimizing 1098    Time: 0:00:08 ( 7.76 ms/it)
   objective: 13521.829930797416


Minimizing 1113    Time: 0:00:08 ( 7.76 ms/it)
   objective: 13517.935441709611


Minimizing 1127    Time: 0:00:08 ( 7.75 ms/it)
   objective: 13511.220304674396


Minimizing 1141    Time: 0:00:08 ( 7.75 ms/it)
   objective: 13507.588071068632


Minimizing 1155    Time: 0:00:08 ( 7.74 ms/it)
   objective: 13507.475440354


Minimizing 1169    Time: 0:00:09 ( 7.74 ms/it)
   objective: 13506.05135519908


Minimizing 1183    Time: 0:00:09 ( 7.73 ms/it)
   objective: 13504.66986151343


Minimizing 1197    Time: 0:00:09 ( 7.73 ms/it)
   objective: 13504.4897735649


Minimizing 1211    Time: 0:00:09 ( 7.72 ms/it)
   objective: 13501.876626284917


Minimizing 1225    Time: 0:00:09 ( 7.71 ms/it)
   objective: 13500.573579746764


Minimizing 1240    Time: 0:00:09 ( 7.71 ms/it)
   objective: 13497.35492503993


Minimizing 1254    Time: 0:00:09 ( 7.70 ms/it)
   objective: 13497.218507995029


Minimizing 1268    Time: 0:00:09 ( 7.70 ms/it)
   objective: 13495.705932523779


Minimizing 1282    Time: 0:00:09 ( 7.70 ms/it)
   objective: 13494.090588485553


Minimizing 1296    Time: 0:00:09 ( 7.69 ms/it)
   objective: 13492.692103496505


Minimizing 1310    Time: 0:00:10 ( 7.69 ms/it)
   objective: 13489.881487492916


Minimizing 1324    Time: 0:00:10 ( 7.69 ms/it)
   objective: 13488.890225824827


Minimizing 1338    Time: 0:00:10 ( 7.68 ms/it)
   objective: 13482.259913419533


Minimizing 1352    Time: 0:00:10 ( 7.68 ms/it)
   objective: 13477.496178285175


Minimizing 1366    Time: 0:00:10 ( 7.68 ms/it)
   objective: 13472.764992438722


Minimizing 1380    Time: 0:00:10 ( 7.67 ms/it)
   objective: 13468.12035663284


Minimizing 1394    Time: 0:00:10 ( 7.67 ms/it)
   objective: 13465.417413892472


Minimizing 1408    Time: 0:00:10 ( 7.66 ms/it)
   objective: 13460.148024158902


Minimizing 1423    Time: 0:00:10 ( 7.66 ms/it)
   objective: 13453.46566693447


Minimizing 1439    Time: 0:00:11 ( 7.65 ms/it)
   objective: 13449.490159718116


Minimizing 1455    Time: 0:00:11 ( 7.64 ms/it)
   objective: 13445.280043669802


Minimizing 1471    Time: 0:00:11 ( 7.62 ms/it)
   objective: 13440.67814346122


Minimizing 1487    Time: 0:00:11 ( 7.61 ms/it)
   objective: 13432.570169299535


Minimizing 1503    Time: 0:00:11 ( 7.60 ms/it)
   objective: 13422.819201992068


Minimizing 1519    Time: 0:00:11 ( 7.59 ms/it)
   objective: 13419.548259376927


Minimizing 1535    Time: 0:00:11 ( 7.58 ms/it)
   objective: 13414.155571031311


Minimizing 1550    Time: 0:00:11 ( 7.57 ms/it)
   objective: 13412.673426609006


Minimizing 1566    Time: 0:00:11 ( 7.56 ms/it)
   objective: 13410.197393465016


Minimizing 1582    Time: 0:00:11 ( 7.56 ms/it)
   objective: 13397.249287036131


Minimizing 1598    Time: 0:00:12 ( 7.55 ms/it)
   objective: 13387.400425187821


Minimizing 1614    Time: 0:00:12 ( 7.54 ms/it)
   objective: 13374.707264886994


Minimizing 1630    Time: 0:00:12 ( 7.53 ms/it)
   objective: 13366.784097443015


Minimizing 1646    Time: 0:00:12 ( 7.52 ms/it)
   objective: 13365.892028263872


Minimizing 1662    Time: 0:00:12 ( 7.51 ms/it)
   objective: 13364.70678552131


Minimizing 1678    Time: 0:00:12 ( 7.50 ms/it)
   objective: 13362.269954144955


Minimizing 1694    Time: 0:00:12 ( 7.49 ms/it)
   objective: 13356.581782696026


Minimizing 1710    Time: 0:00:12 ( 7.48 ms/it)
   objective: 13353.26191601572


Minimizing 1726    Time: 0:00:12 ( 7.47 ms/it)
   objective: 13350.219008932458


Minimizing 1741    Time: 0:00:13 ( 7.47 ms/it)
   objective: 13342.341489751634


Minimizing 1755    Time: 0:00:13 ( 7.47 ms/it)
   objective: 13341.464259968678


Minimizing 1770    Time: 0:00:13 ( 7.46 ms/it)
   objective: 13335.559489230509


Minimizing 1783    Time: 0:00:13 ( 7.47 ms/it)
   objective: 13330.721585899199


Minimizing 1799    Time: 0:00:13 ( 7.46 ms/it)
   objective: 13325.483341219166


Minimizing 1813    Time: 0:00:13 ( 7.46 ms/it)
   objective: 13319.684631871249


Minimizing 1826    Time: 0:00:13 ( 7.47 ms/it)
   objective: 13316.78925783807


Minimizing 1839    Time: 0:00:13 ( 7.47 ms/it)
   objective: 13277.226828978412


Minimizing 1852    Time: 0:00:13 ( 7.48 ms/it)
   objective: 13276.106981269128


Minimizing 1867    Time: 0:00:13 ( 7.47 ms/it)
   objective: 13276.086361017471


Minimizing 1882    Time: 0:00:14 ( 7.47 ms/it)
   objective: 13276.099227179991


Minimizing 1897    Time: 0:00:14 ( 7.46 ms/it)
   objective: 13276.111325367296


Minimizing 1912    Time: 0:00:14 ( 7.46 ms/it)
   objective: 13276.102020608072


Minimizing 1927    Time: 0:00:14 ( 7.45 ms/it)
   objective: 13276.09983385769


Minimizing 1942    Time: 0:00:14 ( 7.45 ms/it)
   objective: 13276.100076910923


Minimizing 1957    Time: 0:00:14 ( 7.44 ms/it)
   objective: 13276.10163753244


Minimizing 1972    Time: 0:00:14 ( 7.44 ms/it)
   objective: 13276.096794010213


Minimizing 1987    Time: 0:00:14 ( 7.44 ms/it)
   objective: 13276.103102046662


Minimizing 2002    Time: 0:00:14 ( 7.43 ms/it)
   objective: 13276.094146557982


Minimizing 2017    Time: 0:00:14 ( 7.43 ms/it)
   objective: 13276.103624993222


Minimizing 2032    Time: 0:00:15 ( 7.42 ms/it)
   objective: 13276.098819102408


Minimizing 2047    Time: 0:00:15 ( 7.42 ms/it)
   objective: 13276.101316274435


Minimizing 2062    Time: 0:00:15 ( 7.41 ms/it)
   objective: 13276.098422491574


Minimizing 2065    Time: 0:00:15 ( 7.41 ms/it)

	Est.	SE	z	p	σ_Child
(Intercept)	-0.0185	0.0142	-1.30	0.1936	0.7794
Test: Star-Run	-0.0150	0.0444	-0.34	0.7346	0.6839
Test: S20-Star	0.0050	0.0451	0.11	0.9115	0.9589
Test: SLJ-S20	-0.0022	0.0439	-0.05	0.9596	0.5170
Test: BPT-SLJ	-0.0076	0.0421	-0.18	0.8573	0.7982
a1	0.2914	0.0482	6.04	<1e-08
Sex: Boys	0.2065	0.0142	14.53	<1e-47
Test: Star-Run & a1	0.2186	0.1477	1.48	0.1389
Test: S20-Star & a1	0.1980	0.1529	1.29	0.1955
Test: SLJ-S20 & a1	-0.1985	0.1514	-1.31	0.1896
Test: BPT-SLJ & a1	0.2576	0.1424	1.81	0.0705
Test: Star-Run & Sex: Boys	-0.1228	0.0444	-2.77	0.0057
Test: S20-Star & Sex: Boys	-0.0176	0.0451	-0.39	0.6955
Test: SLJ-S20 & Sex: Boys	0.1248	0.0439	2.85	0.0044
Test: BPT-SLJ & Sex: Boys	0.1129	0.0421	2.68	0.0073
a1 & Sex: Boys	0.0084	0.0482	0.17	0.8625
Test: Star-Run & a1 & Sex: Boys	0.1271	0.1477	0.86	0.3894
Test: S20-Star & a1 & Sex: Boys	0.0007	0.1529	0.00	0.9966
Test: SLJ-S20 & a1 & Sex: Boys	0.1252	0.1514	0.83	0.4081
Test: BPT-SLJ & a1 & Sex: Boys	-0.1719	0.1424	-1.21	0.2274
Residual	0.0000

m_cpx_1_SeqDiff.PCA

(Child = 
Principal components based on correlation matrix
 (Intercept)      1.0     .      .      .      .
 Test: Star-Run  -0.22   1.0     .      .      .
 Test: S20-Star   0.3   -0.85   1.0     .      .
 Test: SLJ-S20   -0.43  -0.01   0.42   1.0     .
 Test: BPT-SLJ   -0.15   0.83  -0.75  -0.05   1.0

Normalized cumulative variances:
[0.5452, 0.8344, 0.9641, 1.0, 1.0]

Component loadings
                   PC1    PC2    PC3    PC4    PC5
 (Intercept)     -0.19  -0.65  -0.66  -0.17   0.27
 Test: Star-Run   0.57   0.06  -0.26  -0.6   -0.49
 Test: S20-Star  -0.58   0.14  -0.31   0.19  -0.72
 Test: SLJ-S20   -0.11   0.75  -0.5   -0.15   0.41
 Test: BPT-SLJ    0.54   0.01  -0.38   0.74  -0.07,)

2.4 PCA-based HypothesisCoding: `contr4`

The fourth set of contrasts uses HypothesisCoding to specify the set of contrasts implementing the loadings of the four principle components of the published LMM based on test scores, not test effects (contrasts) - coarse-grained, that is roughly according to their signs. This is actually a very interesting and plausible solution nobody had proposed a priori.

PC1: BPT - Run_r
PC2: (Star_r + S20_r + SLJ) - (BPT + Run_r)
PC3: Star_r - (S20_r + SLJ)
PC4: S20_r - SLJ

PC1 contrasts the worst and the best indicator of physical health; PC2 contrasts these two against the core indicators of physical fitness; PC3 contrasts the cognitive and the physical tests within the narrow set of physical fitness components; and PC4, finally, contrasts two types of lower muscular fitness differing in speed and power.

contr4 = Dict(
  :School => Grouping(),
  :Child => Grouping(),
  :Cohort => Grouping(),
  :Sex => EffectsCoding(; levels=["Girls", "Boys"]),
  :Test => HypothesisCoding(
    [
      -1  0  0  0 +1
      -3 +2 +2 +2 -3
       0 +2 -1 -1  0
       0  0 +1 -1  0
    ];
    levels=["Run", "Star_r", "S20_r", "SLJ", "BPT"],
    labels=["c5.1", "c234.15", "c2.34", "c3.4"],
  ),
);

m_cpx_1_PC = fit(MixedModel, f_cpx_1, dat; contrasts=contr4)

Minimizing 13    Time: 0:00:00 ( 7.82 ms/it)
   objective: 13839.654648668518


Minimizing 25    Time: 0:00:00 ( 8.92 ms/it)
   objective: 13838.225382447163


Minimizing 36    Time: 0:00:00 ( 8.98 ms/it)
   objective: 13818.56330121988


Minimizing 49    Time: 0:00:00 ( 8.81 ms/it)
   objective: 13809.404680254927


Minimizing 64    Time: 0:00:00 ( 8.40 ms/it)
   objective: 13796.624458710023


Minimizing 80    Time: 0:00:00 ( 8.05 ms/it)
   objective: 13781.415882889587


Minimizing 95    Time: 0:00:00 ( 7.84 ms/it)
   objective: 13773.703605410166


Minimizing 110    Time: 0:00:00 ( 7.69 ms/it)
   objective: 13770.292016758283


Minimizing 125    Time: 0:00:00 ( 7.58 ms/it)
   objective: 13766.646473485298


Minimizing 140    Time: 0:00:01 ( 7.52 ms/it)
   objective: 13763.980022398298


Minimizing 155    Time: 0:00:01 ( 7.46 ms/it)
   objective: 13756.867441428556


Minimizing 170    Time: 0:00:01 ( 7.41 ms/it)
   objective: 13754.370242393128


Minimizing 185    Time: 0:00:01 ( 7.36 ms/it)
   objective: 13750.26631203554


Minimizing 201    Time: 0:00:01 ( 7.30 ms/it)
   objective: 13743.852039724994


Minimizing 217    Time: 0:00:01 ( 7.25 ms/it)
   objective: 13741.959671350836


Minimizing 233    Time: 0:00:01 ( 7.20 ms/it)
   objective: 13737.669671620159


Minimizing 249    Time: 0:00:01 ( 7.16 ms/it)
   objective: 13736.91137816252


Minimizing 265    Time: 0:00:01 ( 7.13 ms/it)
   objective: 13734.323030845298


Minimizing 281    Time: 0:00:01 ( 7.10 ms/it)
   objective: 13733.293745248924


Minimizing 297    Time: 0:00:02 ( 7.07 ms/it)
   objective: 13730.216231061073


Minimizing 313    Time: 0:00:02 ( 7.05 ms/it)
   objective: 13727.95395190292


Minimizing 328    Time: 0:00:02 ( 7.04 ms/it)
   objective: 13724.87736960789


Minimizing 344    Time: 0:00:02 ( 7.02 ms/it)
   objective: 13723.126195853016


Minimizing 360    Time: 0:00:02 ( 7.01 ms/it)
   objective: 13720.151337257728


Minimizing 376    Time: 0:00:02 ( 6.99 ms/it)
   objective: 13716.9417206952


Minimizing 392    Time: 0:00:02 ( 6.98 ms/it)
   objective: 13708.964599448069


Minimizing 408    Time: 0:00:02 ( 6.96 ms/it)
   objective: 13697.598658236455


Minimizing 424    Time: 0:00:02 ( 6.95 ms/it)
   objective: 13691.02385419921


Minimizing 440    Time: 0:00:03 ( 6.94 ms/it)
   objective: 13685.870252443798


Minimizing 456    Time: 0:00:03 ( 6.92 ms/it)
   objective: 13679.793721235503


Minimizing 472    Time: 0:00:03 ( 6.92 ms/it)
   objective: 13675.253432344645


Minimizing 487    Time: 0:00:03 ( 6.92 ms/it)
   objective: 13674.651015741005


Minimizing 503    Time: 0:00:03 ( 6.91 ms/it)
   objective: 13672.543631690507


Minimizing 519    Time: 0:00:03 ( 6.90 ms/it)
   objective: 13669.84812479547


Minimizing 535    Time: 0:00:03 ( 6.89 ms/it)
   objective: 13668.625441621796


Minimizing 551    Time: 0:00:03 ( 6.89 ms/it)
   objective: 13667.073952411592


Minimizing 567    Time: 0:00:03 ( 6.88 ms/it)
   objective: 13664.870316104269


Minimizing 583    Time: 0:00:04 ( 6.87 ms/it)
   objective: 13660.746490569614


Minimizing 599    Time: 0:00:04 ( 6.86 ms/it)
   objective: 13658.048230843


Minimizing 615    Time: 0:00:04 ( 6.85 ms/it)
   objective: 13657.119248768246


Minimizing 630    Time: 0:00:04 ( 6.85 ms/it)
   objective: 13656.35089113473


Minimizing 646    Time: 0:00:04 ( 6.84 ms/it)
   objective: 13652.806485525394


Minimizing 662    Time: 0:00:04 ( 6.84 ms/it)
   objective: 13652.34990733592


Minimizing 678    Time: 0:00:04 ( 6.83 ms/it)
   objective: 13650.647817353809


Minimizing 694    Time: 0:00:04 ( 6.82 ms/it)
   objective: 13649.27457414725


Minimizing 710    Time: 0:00:04 ( 6.82 ms/it)
   objective: 13647.38685172056


Minimizing 726    Time: 0:00:04 ( 6.81 ms/it)
   objective: 13646.758730487454


Minimizing 742    Time: 0:00:05 ( 6.81 ms/it)
   objective: 13642.526457257696


Minimizing 758    Time: 0:00:05 ( 6.80 ms/it)
   objective: 13636.169537371577


Minimizing 774    Time: 0:00:05 ( 6.80 ms/it)
   objective: 13632.96124910012


Minimizing 791    Time: 0:00:05 ( 6.81 ms/it)
   objective: 13628.349101218413


Minimizing 808    Time: 0:00:05 ( 6.81 ms/it)
   objective: 13624.52121566957


Minimizing 824    Time: 0:00:05 ( 6.81 ms/it)
   objective: 13611.270038491384


Minimizing 840    Time: 0:00:05 ( 6.80 ms/it)
   objective: 13599.823422297384


Minimizing 855    Time: 0:00:05 ( 6.80 ms/it)
   objective: 13596.05941426169


Minimizing 871    Time: 0:00:05 ( 6.80 ms/it)
   objective: 13592.459926267766


Minimizing 886    Time: 0:00:06 ( 6.80 ms/it)
   objective: 13582.919532892025


Minimizing 902    Time: 0:00:06 ( 6.80 ms/it)
   objective: 13581.967316555165


Minimizing 918    Time: 0:00:06 ( 6.79 ms/it)
   objective: 13581.104955048198


Minimizing 933    Time: 0:00:06 ( 6.79 ms/it)
   objective: 13579.539353518754


Minimizing 949    Time: 0:00:06 ( 6.79 ms/it)
   objective: 13577.219410013815


Minimizing 965    Time: 0:00:06 ( 6.79 ms/it)
   objective: 13574.156142222659


Minimizing 981    Time: 0:00:06 ( 6.79 ms/it)
   objective: 13570.489538518552


Minimizing 997    Time: 0:00:06 ( 6.78 ms/it)
   objective: 13564.629568649718


Minimizing 1013    Time: 0:00:06 ( 6.78 ms/it)
   objective: 13557.011200247478


Minimizing 1029    Time: 0:00:06 ( 6.78 ms/it)
   objective: 13548.983009770644


Minimizing 1045    Time: 0:00:07 ( 6.78 ms/it)
   objective: 13545.415704364146


Minimizing 1061    Time: 0:00:07 ( 6.77 ms/it)
   objective: 13534.274473336729


Minimizing 1076    Time: 0:00:07 ( 6.77 ms/it)
   objective: 13526.397855764677


Minimizing 1092    Time: 0:00:07 ( 6.77 ms/it)
   objective: 13526.02910731043


Minimizing 1108    Time: 0:00:07 ( 6.77 ms/it)
   objective: 13524.596784797177


Minimizing 1124    Time: 0:00:07 ( 6.77 ms/it)
   objective: 13523.35844618937


Minimizing 1140    Time: 0:00:07 ( 6.76 ms/it)
   objective: 13521.704159749635


Minimizing 1156    Time: 0:00:07 ( 6.76 ms/it)
   objective: 13520.38247034073


Minimizing 1172    Time: 0:00:07 ( 6.76 ms/it)
   objective: 13514.119059277567


Minimizing 1188    Time: 0:00:08 ( 6.76 ms/it)
   objective: 13513.635005657728


Minimizing 1204    Time: 0:00:08 ( 6.75 ms/it)
   objective: 13511.364085370478


Minimizing 1220    Time: 0:00:08 ( 6.75 ms/it)
   objective: 13507.368166847984


Minimizing 1235    Time: 0:00:08 ( 6.75 ms/it)
   objective: 13504.388680717886


Minimizing 1251    Time: 0:00:08 ( 6.75 ms/it)
   objective: 13501.61071496601


Minimizing 1267    Time: 0:00:08 ( 6.74 ms/it)
   objective: 13501.048878192742


Minimizing 1283    Time: 0:00:08 ( 6.74 ms/it)
   objective: 13499.573251460744


Minimizing 1299    Time: 0:00:08 ( 6.74 ms/it)
   objective: 13495.754107029294


Minimizing 1315    Time: 0:00:08 ( 6.73 ms/it)
   objective: 13495.334647044321


Minimizing 1331    Time: 0:00:08 ( 6.73 ms/it)
   objective: 13493.327089545004


Minimizing 1347    Time: 0:00:09 ( 6.73 ms/it)
   objective: 13491.96486475534


Minimizing 1363    Time: 0:00:09 ( 6.73 ms/it)
   objective: 13489.546055261773


Minimizing 1379    Time: 0:00:09 ( 6.72 ms/it)
   objective: 13483.471472059959


Minimizing 1395    Time: 0:00:09 ( 6.72 ms/it)
   objective: 13483.308337595285


Minimizing 1411    Time: 0:00:09 ( 6.72 ms/it)
   objective: 13482.795226083581


Minimizing 1427    Time: 0:00:09 ( 6.72 ms/it)
   objective: 13482.015232146601


Minimizing 1443    Time: 0:00:09 ( 6.72 ms/it)
   objective: 13481.367826951297


Minimizing 1459    Time: 0:00:09 ( 6.71 ms/it)
   objective: 13480.93880845447


Minimizing 1475    Time: 0:00:09 ( 6.71 ms/it)
   objective: 13479.85794782424


Minimizing 1491    Time: 0:00:10 ( 6.71 ms/it)
   objective: 13478.872858813185


Minimizing 1507    Time: 0:00:10 ( 6.71 ms/it)
   objective: 13474.658059804737


Minimizing 1523    Time: 0:00:10 ( 6.71 ms/it)
   objective: 13471.995945995164


Minimizing 1539    Time: 0:00:10 ( 6.71 ms/it)
   objective: 13464.616888973353


Minimizing 1555    Time: 0:00:10 ( 6.70 ms/it)
   objective: 13460.404362411784


Minimizing 1571    Time: 0:00:10 ( 6.70 ms/it)
   objective: 13457.644731633838


Minimizing 1587    Time: 0:00:10 ( 6.70 ms/it)
   objective: 13451.613302300073


Minimizing 1603    Time: 0:00:10 ( 6.70 ms/it)
   objective: 13449.419403628992


Minimizing 1619    Time: 0:00:10 ( 6.70 ms/it)
   objective: 13434.052644139214


Minimizing 1635    Time: 0:00:10 ( 6.70 ms/it)
   objective: 13431.4147641186


Minimizing 1651    Time: 0:00:11 ( 6.69 ms/it)
   objective: 13427.39952251139


Minimizing 1667    Time: 0:00:11 ( 6.69 ms/it)
   objective: 13424.18761021561


Minimizing 1683    Time: 0:00:11 ( 6.69 ms/it)
   objective: 13421.430115308001


Minimizing 1699    Time: 0:00:11 ( 6.69 ms/it)
   objective: 13417.59793613547


Minimizing 1715    Time: 0:00:11 ( 6.69 ms/it)
   objective: 13413.943964048318


Minimizing 1731    Time: 0:00:11 ( 6.69 ms/it)
   objective: 13402.807021841538


Minimizing 1747    Time: 0:00:11 ( 6.69 ms/it)
   objective: 13397.894619318948


Minimizing 1763    Time: 0:00:11 ( 6.68 ms/it)
   objective: 13377.638572248237


Minimizing 1779    Time: 0:00:11 ( 6.68 ms/it)
   objective: 13367.450521093939


Minimizing 1795    Time: 0:00:11 ( 6.68 ms/it)
   objective: 13366.676395136543


Minimizing 1811    Time: 0:00:12 ( 6.68 ms/it)
   objective: 13364.590315003239


Minimizing 1827    Time: 0:00:12 ( 6.68 ms/it)
   objective: 13364.266697261628


Minimizing 1843    Time: 0:00:12 ( 6.68 ms/it)
   objective: 13360.805357428791


Minimizing 1859    Time: 0:00:12 ( 6.68 ms/it)
   objective: 13358.895665122516


Minimizing 1875    Time: 0:00:12 ( 6.68 ms/it)
   objective: 13357.317918166838


Minimizing 1891    Time: 0:00:12 ( 6.68 ms/it)
   objective: 13351.137611861457


Minimizing 1907    Time: 0:00:12 ( 6.68 ms/it)
   objective: 13350.758849798833


Minimizing 1923    Time: 0:00:12 ( 6.68 ms/it)
   objective: 13350.757151971935


Minimizing 1939    Time: 0:00:12 ( 6.67 ms/it)
   objective: 13350.756717085955


Minimizing 1955    Time: 0:00:13 ( 6.67 ms/it)
   objective: 13350.755359990333


Minimizing 1971    Time: 0:00:13 ( 6.67 ms/it)
   objective: 13350.755347646089


Minimizing 1987    Time: 0:00:13 ( 6.67 ms/it)
   objective: 13350.754836405147


Minimizing 2003    Time: 0:00:13 ( 6.67 ms/it)
   objective: 13350.75546586253


Minimizing 2019    Time: 0:00:13 ( 6.67 ms/it)
   objective: 13350.75482834666


Minimizing 2035    Time: 0:00:13 ( 6.67 ms/it)
   objective: 13350.75492902426


Minimizing 2051    Time: 0:00:13 ( 6.67 ms/it)
   objective: 13350.755272084003


Minimizing 2067    Time: 0:00:13 ( 6.67 ms/it)
   objective: 13350.754842789378


Minimizing 2083    Time: 0:00:13 ( 6.66 ms/it)
   objective: 13350.754727388354


Minimizing 2099    Time: 0:00:13 ( 6.66 ms/it)
   objective: 13350.755164903618


Minimizing 2115    Time: 0:00:14 ( 6.66 ms/it)
   objective: 13350.754937030855


Minimizing 2131    Time: 0:00:14 ( 6.66 ms/it)
   objective: 13350.755019308723


Minimizing 2134    Time: 0:00:14 ( 6.66 ms/it)

	Est.	SE	z	p	σ_Child
(Intercept)	-0.0187	0.0142	-1.32	0.1881	0.7796
Test: c5.1	-0.0195	0.0429	-0.45	0.6503	1.2413
Test: c234.15	-0.0253	0.1675	-0.15	0.8800	0.7376
Test: c2.34	-0.0086	0.0780	-0.11	0.9121	1.6770
Test: c3.4	0.0005	0.0441	0.01	0.9918	0.8814
a1	0.2896	0.0481	6.02	<1e-08
Sex: Boys	0.2065	0.0142	14.55	<1e-47
Test: c5.1 & a1	0.4780	0.1451	3.29	0.0010
Test: c234.15 & a1	0.3154	0.5682	0.55	0.5789
Test: c2.34 & a1	-0.1610	0.2600	-0.62	0.5358
Test: c3.4 & a1	0.1216	0.1528	0.80	0.4262
Test: c5.1 & Sex: Boys	0.1000	0.0429	2.33	0.0199
Test: c234.15 & Sex: Boys	-0.8634	0.1675	-5.15	<1e-06
Test: c2.34 & Sex: Boys	-0.0876	0.0780	-1.12	0.2617
Test: c3.4 & Sex: Boys	-0.1354	0.0441	-3.07	0.0021
a1 & Sex: Boys	0.0103	0.0481	0.22	0.8297
Test: c5.1 & a1 & Sex: Boys	0.0625	0.1451	0.43	0.6668
Test: c234.15 & a1 & Sex: Boys	0.7550	0.5682	1.33	0.1839
Test: c2.34 & a1 & Sex: Boys	-0.1627	0.2600	-0.63	0.5315
Test: c3.4 & a1 & Sex: Boys	-0.0711	0.1528	-0.47	0.6416
Residual	0.0000

VarCorr(m_cpx_1_PC)

	Column	Variance	Std.Dev	Corr.
Child	(Intercept)	0.60773937	0.77957641
	Test: c5.1	1.54071063	1.24125365	-0.09
	Test: c234.15	0.54401627	0.73757459	+0.90	-0.51
	Test: c2.34	2.81236949	1.67701207	+0.00	+0.22	-0.09
	Test: c3.4	0.77692881	0.88143565	+0.14	-0.26	+0.23	+0.34
Residual		0.00000000	0.00004549

m_cpx_1_PC.PCA

(Child = 
Principal components based on correlation matrix
 (Intercept)     1.0     .      .      .      .
 Test: c5.1     -0.09   1.0     .      .      .
 Test: c234.15   0.9   -0.51   1.0     .      .
 Test: c2.34     0.0    0.22  -0.09   1.0     .
 Test: c3.4      0.14  -0.26   0.23   0.34   1.0

Normalized cumulative variances:
[0.4355, 0.7003, 0.9066, 1.0, 1.0]

Component loadings
                  PC1    PC2    PC3    PC4    PC5
 (Intercept)    -0.57  -0.01   0.53   0.11  -0.62
 Test: c5.1      0.4    0.18   0.71   0.46   0.31
 Test: c234.15  -0.66  -0.08   0.15  -0.11   0.72
 Test: c2.34     0.04   0.77   0.18  -0.61  -0.0
 Test: c3.4     -0.27   0.6   -0.41   0.63  -0.0,)

There is a numerical interaction with a z-value > 2.0 for the first PCA (i.e., BPT - Run_r). This interaction would really need to be replicated to be taken seriously. It is probably due to larger “unfitness” gains in boys than girls (i.e., in BPT) relative to the slightly larger health-related “fitness” gains of girls than boys (i.e., in Run_r).

contr4b = merge(
  Dict(nm => Grouping() for nm in (:School, :Child, :Cohort)),
  Dict(
    :Sex => EffectsCoding(; levels=["Girls", "Boys"]),
    :Test => HypothesisCoding(
      [
        0.49 -0.04  0.20  0.03 -0.85
        0.70 -0.56 -0.21 -0.13  0.37
        0.31  0.68 -0.56 -0.35  0.00
        0.04  0.08  0.61 -0.78  0.13
      ];
      levels=["Run", "Star_r", "S20_r", "SLJ", "BPT"],
      labels=["c5.1", "c234.15", "c12.34", "c3.4"],
    ),
  ),
);

m_cpx_1_PC_2 = fit(MixedModel, f_cpx_1, dat; contrasts=contr4b)

Minimizing 16    Time: 0:00:00 ( 6.53 ms/it)
   objective: 13871.59355238144


Minimizing 31    Time: 0:00:00 ( 6.81 ms/it)
   objective: 13857.997566614958


Minimizing 47    Time: 0:00:00 ( 6.75 ms/it)
   objective: 13810.230650449239


Minimizing 63    Time: 0:00:00 ( 6.69 ms/it)
   objective: 13806.477999415674


Minimizing 79    Time: 0:00:00 ( 6.65 ms/it)
   objective: 13787.723296902446


Minimizing 95    Time: 0:00:00 ( 6.62 ms/it)
   objective: 13783.203560285068


Minimizing 111    Time: 0:00:00 ( 6.60 ms/it)
   objective: 13751.897308622543


Minimizing 127    Time: 0:00:00 ( 6.58 ms/it)
   objective: 13747.537182678308


Minimizing 143    Time: 0:00:00 ( 6.57 ms/it)
   objective: 13743.778347363237


Minimizing 159    Time: 0:00:01 ( 6.57 ms/it)
   objective: 13740.371215527754


Minimizing 175    Time: 0:00:01 ( 6.56 ms/it)
   objective: 13736.785634964419


Minimizing 191    Time: 0:00:01 ( 6.57 ms/it)
   objective: 13733.178324318476


Minimizing 207    Time: 0:00:01 ( 6.56 ms/it)
   objective: 13730.553422482215


Minimizing 223    Time: 0:00:01 ( 6.56 ms/it)
   objective: 13720.626095885193


Minimizing 239    Time: 0:00:01 ( 6.56 ms/it)
   objective: 13716.64395484471


Minimizing 255    Time: 0:00:01 ( 6.55 ms/it)
   objective: 13704.071555841352


Minimizing 271    Time: 0:00:01 ( 6.55 ms/it)
   objective: 13699.748881561849


Minimizing 287    Time: 0:00:01 ( 6.54 ms/it)
   objective: 13691.41695364464


Minimizing 303    Time: 0:00:01 ( 6.54 ms/it)
   objective: 13688.293740009878


Minimizing 319    Time: 0:00:02 ( 6.54 ms/it)
   objective: 13682.917231318701


Minimizing 335    Time: 0:00:02 ( 6.54 ms/it)
   objective: 13682.068943163653


Minimizing 350    Time: 0:00:02 ( 6.55 ms/it)
   objective: 13681.148421684167


Minimizing 366    Time: 0:00:02 ( 6.55 ms/it)
   objective: 13679.82627754362


Minimizing 382    Time: 0:00:02 ( 6.55 ms/it)
   objective: 13678.328394545264


Minimizing 398    Time: 0:00:02 ( 6.55 ms/it)
   objective: 13675.241080354164


Minimizing 414    Time: 0:00:02 ( 6.55 ms/it)
   objective: 13672.765742525022


Minimizing 430    Time: 0:00:02 ( 6.54 ms/it)
   objective: 13670.845569135363


Minimizing 446    Time: 0:00:02 ( 6.55 ms/it)
   objective: 13668.635444222287


Minimizing 462    Time: 0:00:03 ( 6.54 ms/it)
   objective: 13667.378963459403


Minimizing 478    Time: 0:00:03 ( 6.54 ms/it)
   objective: 13664.451218892347


Minimizing 494    Time: 0:00:03 ( 6.55 ms/it)
   objective: 13663.545820791143


Minimizing 513    Time: 0:00:03 ( 6.58 ms/it)
   objective: 13661.717227102847


Minimizing 531    Time: 0:00:03 ( 6.58 ms/it)
   objective: 13660.593083770265


Minimizing 547    Time: 0:00:03 ( 6.58 ms/it)
   objective: 13659.189737346496


Minimizing 563    Time: 0:00:03 ( 6.57 ms/it)
   objective: 13658.457858499973


Minimizing 579    Time: 0:00:03 ( 6.57 ms/it)
   objective: 13656.614258337071


Minimizing 595    Time: 0:00:03 ( 6.57 ms/it)
   objective: 13653.962785164524


Minimizing 611    Time: 0:00:04 ( 6.57 ms/it)
   objective: 13652.607580251617


Minimizing 627    Time: 0:00:04 ( 6.57 ms/it)
   objective: 13643.373120305176


Minimizing 643    Time: 0:00:04 ( 6.57 ms/it)
   objective: 13629.454646979779


Minimizing 659    Time: 0:00:04 ( 6.57 ms/it)
   objective: 13626.96180632586


Minimizing 675    Time: 0:00:04 ( 6.57 ms/it)
   objective: 13620.019630228413


Minimizing 691    Time: 0:00:04 ( 6.57 ms/it)
   objective: 13616.950880695502


Minimizing 707    Time: 0:00:04 ( 6.57 ms/it)
   objective: 13614.156561542419


Minimizing 723    Time: 0:00:04 ( 6.57 ms/it)
   objective: 13613.621490998798


Minimizing 739    Time: 0:00:04 ( 6.56 ms/it)
   objective: 13611.159687773259


Minimizing 755    Time: 0:00:04 ( 6.56 ms/it)
   objective: 13609.221749127955


Minimizing 771    Time: 0:00:05 ( 6.56 ms/it)
   objective: 13602.53982305585


Minimizing 787    Time: 0:00:05 ( 6.56 ms/it)
   objective: 13600.696271311397


Minimizing 803    Time: 0:00:05 ( 6.56 ms/it)
   objective: 13563.554501554252


Minimizing 819    Time: 0:00:05 ( 6.56 ms/it)
   objective: 13557.393535004208


Minimizing 835    Time: 0:00:05 ( 6.56 ms/it)
   objective: 13552.274755806939


Minimizing 851    Time: 0:00:05 ( 6.56 ms/it)
   objective: 13546.145336666377


Minimizing 867    Time: 0:00:05 ( 6.56 ms/it)
   objective: 13539.796065419563


Minimizing 883    Time: 0:00:05 ( 6.56 ms/it)
   objective: 13532.090749132687


Minimizing 899    Time: 0:00:05 ( 6.55 ms/it)
   objective: 13515.909446846039


Minimizing 915    Time: 0:00:05 ( 6.55 ms/it)
   objective: 13515.222772843481


Minimizing 931    Time: 0:00:06 ( 6.55 ms/it)
   objective: 13507.400077208818


Minimizing 947    Time: 0:00:06 ( 6.55 ms/it)
   objective: 13503.484526898807


Minimizing 963    Time: 0:00:06 ( 6.55 ms/it)
   objective: 13498.563033576967


Minimizing 979    Time: 0:00:06 ( 6.55 ms/it)
   objective: 13496.900826162971


Minimizing 995    Time: 0:00:06 ( 6.55 ms/it)
   objective: 13487.72209276232


Minimizing 1011    Time: 0:00:06 ( 6.55 ms/it)
   objective: 13475.246843864632


Minimizing 1027    Time: 0:00:06 ( 6.55 ms/it)
   objective: 13468.873091858331


Minimizing 1043    Time: 0:00:06 ( 6.55 ms/it)
   objective: 13459.460630733549


Minimizing 1059    Time: 0:00:06 ( 6.55 ms/it)
   objective: 13458.927747906091


Minimizing 1075    Time: 0:00:07 ( 6.55 ms/it)
   objective: 13454.098566939138


Minimizing 1091    Time: 0:00:07 ( 6.55 ms/it)
   objective: 13445.156254087458


Minimizing 1107    Time: 0:00:07 ( 6.55 ms/it)
   objective: 13435.465226784378


Minimizing 1123    Time: 0:00:07 ( 6.55 ms/it)
   objective: 13434.172027030116


Minimizing 1139    Time: 0:00:07 ( 6.55 ms/it)
   objective: 13431.287982766458


Minimizing 1155    Time: 0:00:07 ( 6.55 ms/it)
   objective: 13429.874668115444


Minimizing 1171    Time: 0:00:07 ( 6.55 ms/it)
   objective: 13428.24847368196


Minimizing 1187    Time: 0:00:07 ( 6.55 ms/it)
   objective: 13426.486852918737


Minimizing 1203    Time: 0:00:07 ( 6.55 ms/it)
   objective: 13425.400156308475


Minimizing 1219    Time: 0:00:07 ( 6.55 ms/it)
   objective: 13424.954684567623


Minimizing 1235    Time: 0:00:08 ( 6.55 ms/it)
   objective: 13423.091672238239


Minimizing 1251    Time: 0:00:08 ( 6.54 ms/it)
   objective: 13422.425867024867


Minimizing 1265    Time: 0:00:08 ( 6.56 ms/it)
   objective: 13420.763405875492


Minimizing 1279    Time: 0:00:08 ( 6.57 ms/it)
   objective: 13420.219085687277


Minimizing 1292    Time: 0:00:08 ( 6.58 ms/it)
   objective: 13415.895055166387


Minimizing 1306    Time: 0:00:08 ( 6.59 ms/it)
   objective: 13413.616429760092


Minimizing 1320    Time: 0:00:08 ( 6.60 ms/it)
   objective: 13413.540349707764


Minimizing 1334    Time: 0:00:08 ( 6.61 ms/it)
   objective: 13412.924624733001


Minimizing 1348    Time: 0:00:08 ( 6.62 ms/it)
   objective: 13411.64374108793


Minimizing 1362    Time: 0:00:09 ( 6.63 ms/it)
   objective: 13411.071869883337


Minimizing 1376    Time: 0:00:09 ( 6.64 ms/it)
   objective: 13408.639461200422


Minimizing 1389    Time: 0:00:09 ( 6.66 ms/it)
   objective: 13407.825019077223


Minimizing 1402    Time: 0:00:09 ( 6.67 ms/it)
   objective: 13402.61857793358


Minimizing 1416    Time: 0:00:09 ( 6.67 ms/it)
   objective: 13401.059068046452


Minimizing 1430    Time: 0:00:09 ( 6.68 ms/it)
   objective: 13396.464699359436


Minimizing 1444    Time: 0:00:09 ( 6.69 ms/it)
   objective: 13395.625291995151


Minimizing 1458    Time: 0:00:09 ( 6.70 ms/it)
   objective: 13389.575288060427


Minimizing 1472    Time: 0:00:09 ( 6.71 ms/it)
   objective: 13373.864892104495


Minimizing 1486    Time: 0:00:09 ( 6.71 ms/it)
   objective: 13351.688990531518


Minimizing 1500    Time: 0:00:10 ( 6.72 ms/it)
   objective: 13349.073531325921


Minimizing 1514    Time: 0:00:10 ( 6.73 ms/it)
   objective: 13344.13589087117


Minimizing 1530    Time: 0:00:10 ( 6.75 ms/it)
   objective: 13343.823474922232


Minimizing 1545    Time: 0:00:10 ( 6.76 ms/it)
   objective: 13342.396426692896


Minimizing 1559    Time: 0:00:10 ( 6.77 ms/it)
   objective: 13341.62127812038


Minimizing 1573    Time: 0:00:10 ( 6.77 ms/it)
   objective: 13339.984077722169


Minimizing 1587    Time: 0:00:10 ( 6.78 ms/it)
   objective: 13339.337764040029


Minimizing 1601    Time: 0:00:10 ( 6.79 ms/it)
   objective: 13333.994834169367


Minimizing 1615    Time: 0:00:10 ( 6.79 ms/it)
   objective: 13333.665904271096


Minimizing 1629    Time: 0:00:11 ( 6.80 ms/it)
   objective: 13329.17381773991


Minimizing 1643    Time: 0:00:11 ( 6.80 ms/it)
   objective: 13328.063312558006


Minimizing 1657    Time: 0:00:11 ( 6.81 ms/it)
   objective: 13322.098466451847


Minimizing 1671    Time: 0:00:11 ( 6.82 ms/it)
   objective: 13319.702330228087


Minimizing 1685    Time: 0:00:11 ( 6.82 ms/it)
   objective: 13302.316848352391


Minimizing 1699    Time: 0:00:11 ( 6.83 ms/it)
   objective: 13292.203654767698


Minimizing 1713    Time: 0:00:11 ( 6.83 ms/it)
   objective: 13291.479513186292


Minimizing 1727    Time: 0:00:11 ( 6.84 ms/it)
   objective: 13290.174124483048


Minimizing 1741    Time: 0:00:11 ( 6.84 ms/it)
   objective: 13290.018311803535


Minimizing 1755    Time: 0:00:12 ( 6.85 ms/it)
   objective: 13290.013038508274


Minimizing 1769    Time: 0:00:12 ( 6.85 ms/it)
   objective: 13290.011380709111


Minimizing 1783    Time: 0:00:12 ( 6.86 ms/it)
   objective: 13290.00871956181


Minimizing 1797    Time: 0:00:12 ( 6.86 ms/it)
   objective: 13290.017026267407


Minimizing 1811    Time: 0:00:12 ( 6.87 ms/it)
   objective: 13290.017662886064


Minimizing 1825    Time: 0:00:12 ( 6.87 ms/it)
   objective: 13290.012290001177


Minimizing 1839    Time: 0:00:12 ( 6.88 ms/it)
   objective: 13290.018819103207


Minimizing 1853    Time: 0:00:12 ( 6.88 ms/it)
   objective: 13290.011515248218


Minimizing 1867    Time: 0:00:12 ( 6.88 ms/it)
   objective: 13290.022128581128


Minimizing 1881    Time: 0:00:12 ( 6.89 ms/it)
   objective: 13290.008507330625


Minimizing 1895    Time: 0:00:13 ( 6.89 ms/it)
   objective: 13290.014108805932


Minimizing 1909    Time: 0:00:13 ( 6.89 ms/it)
   objective: 13290.005732615347


Minimizing 1923    Time: 0:00:13 ( 6.90 ms/it)
   objective: 13290.013272337645


Minimizing 1937    Time: 0:00:13 ( 6.90 ms/it)
   objective: 13290.019807812801


Minimizing 1951    Time: 0:00:13 ( 6.91 ms/it)
   objective: 13290.01612398091


Minimizing 1965    Time: 0:00:13 ( 6.91 ms/it)
   objective: 13290.014322023533


Minimizing 1970    Time: 0:00:13 ( 6.91 ms/it)

	Est.	SE	z	p	σ_Child
(Intercept)	-0.0185	0.0143	-1.29	0.1970	0.7140
Test: c5.1	0.0148	0.0304	0.49	0.6268	0.9333
Test: c234.15	0.0085	0.0314	0.27	0.7876	0.4130
Test: c12.34	0.0042	0.0317	0.13	0.8946	0.8652
Test: c3.4	-0.0022	0.0314	-0.07	0.9451	0.6120
a1	0.2846	0.0486	5.86	<1e-08
Sex: Boys	0.1989	0.0143	13.90	<1e-43
Test: c5.1 & a1	-0.2960	0.1028	-2.88	0.0040
Test: c234.15 & a1	-0.1117	0.1052	-1.06	0.2885
Test: c12.34 & a1	-0.1608	0.1072	-1.50	0.1338
Test: c3.4 & a1	0.0867	0.1081	0.80	0.4224
Test: c5.1 & Sex: Boys	-0.1163	0.0304	-3.82	0.0001
Test: c234.15 & Sex: Boys	0.1469	0.0314	4.68	<1e-05
Test: c12.34 & Sex: Boys	0.0012	0.0317	0.04	0.9690
Test: c3.4 & Sex: Boys	-0.0742	0.0314	-2.36	0.0181
a1 & Sex: Boys	0.0152	0.0486	0.31	0.7551
Test: c5.1 & a1 & Sex: Boys	-0.0007	0.1028	-0.01	0.9943
Test: c234.15 & a1 & Sex: Boys	-0.1098	0.1052	-1.04	0.2964
Test: c12.34 & a1 & Sex: Boys	-0.0851	0.1072	-0.79	0.4275
Test: c3.4 & a1 & Sex: Boys	-0.0764	0.1081	-0.71	0.4798
Residual	0.0000

VarCorr(m_cpx_1_PC_2)

	Column	Variance	Std.Dev	Corr.
Child	(Intercept)	0.50978242	0.71399049
	Test: c5.1	0.87111041	0.93333296	+0.21
	Test: c234.15	0.17053966	0.41296447	-0.01	+0.25
	Test: c12.34	0.74861136	0.86522330	-0.14	-0.13	-0.34
	Test: c3.4	0.37459766	0.61204384	+0.18	+0.20	-0.70	+0.74
Residual		0.00000000	0.00001471

m_cpx_1_PC_2.PCA

(Child = 
Principal components based on correlation matrix
 (Intercept)     1.0     .      .      .      .
 Test: c5.1      0.21   1.0     .      .      .
 Test: c234.15  -0.01   0.25   1.0     .      .
 Test: c12.34   -0.14  -0.13  -0.34   1.0     .
 Test: c3.4      0.18   0.2   -0.7    0.74   1.0

Normalized cumulative variances:
[0.4413, 0.7064, 0.883, 0.9978, 1.0]

Component loadings
                  PC1    PC2    PC3    PC4    PC5
 (Intercept)    -0.03   0.64   0.64  -0.4   -0.13
 Test: c5.1      0.06   0.71  -0.54   0.34  -0.29
 Test: c234.15   0.54   0.13  -0.38  -0.6    0.43
 Test: c12.34   -0.54  -0.12  -0.36  -0.6   -0.44
 Test: c3.4     -0.64   0.25  -0.09   0.05   0.72,)

f_zcp_1 = @formula(zScore ~ 1 + Test*a1*Sex + zerocorr(1 + Test | Child))
m_zcp_1_PC_2 = fit(MixedModel, f_zcp_1, dat; contrasts=contr4b)

Minimizing 20    Time: 0:00:00 ( 5.04 ms/it)
   objective: 13810.057358319566


Minimizing 39    Time: 0:00:00 ( 5.20 ms/it)
   objective: 13730.813873894405


Minimizing 58    Time: 0:00:00 ( 5.24 ms/it)
   objective: 13724.538064611235


Minimizing 77    Time: 0:00:00 ( 5.26 ms/it)
   objective: 13710.474507531992


Minimizing 96    Time: 0:00:00 ( 5.27 ms/it)
   objective: 13647.562402165444


Minimizing 115    Time: 0:00:00 ( 5.29 ms/it)
   objective: 13611.106590911026


Minimizing 134    Time: 0:00:00 ( 5.30 ms/it)
   objective: 13576.733085163753


Minimizing 153    Time: 0:00:00 ( 5.30 ms/it)
   objective: 13558.048755897857


Minimizing 172    Time: 0:00:00 ( 5.34 ms/it)
   objective: 13509.381745426232


Minimizing 191    Time: 0:00:01 ( 5.34 ms/it)
   objective: 13481.781214827017


Minimizing 210    Time: 0:00:01 ( 5.34 ms/it)
   objective: 13450.810533560798


Minimizing 229    Time: 0:00:01 ( 5.34 ms/it)
   objective: 13406.050215996322


Minimizing 248    Time: 0:00:01 ( 5.33 ms/it)
   objective: 13346.68317726448


Minimizing 267    Time: 0:00:01 ( 5.33 ms/it)
   objective: 13296.43048955404


Minimizing 286    Time: 0:00:01 ( 5.33 ms/it)
   objective: 13238.159803197195


Minimizing 305    Time: 0:00:01 ( 5.33 ms/it)
   objective: 13238.336645650183


Minimizing 324    Time: 0:00:01 ( 5.33 ms/it)
   objective: 13238.422524073874


Minimizing 344    Time: 0:00:01 ( 5.33 ms/it)
   objective: 13238.316954537775


Minimizing 363    Time: 0:00:01 ( 5.34 ms/it)
   objective: 13238.316595623823


Minimizing 382    Time: 0:00:02 ( 5.34 ms/it)
   objective: 13238.587249206437


Minimizing 388    Time: 0:00:02 ( 5.35 ms/it)

	Est.	SE	z	p	σ_Child
(Intercept)	-0.0181	0.0144	-1.25	0.2099	0.8044
Test: c5.1	0.0203	0.0301	0.68	0.4992	0.4292
Test: c234.15	0.0187	0.0323	0.58	0.5628	0.6410
Test: c12.34	0.0099	0.0324	0.30	0.7613	0.8346
Test: c3.4	-0.0040	0.0303	-0.13	0.8946	0.4587
a1	0.2823	0.0489	5.77	<1e-08
Sex: Boys	0.1993	0.0144	13.81	<1e-42
Test: c5.1 & a1	-0.3142	0.1015	-3.09	0.0020
Test: c234.15 & a1	-0.1317	0.1080	-1.22	0.2224
Test: c12.34 & a1	-0.1737	0.1096	-1.59	0.1129
Test: c3.4 & a1	0.1107	0.1043	1.06	0.2887
Test: c5.1 & Sex: Boys	-0.1019	0.0301	-3.39	0.0007
Test: c234.15 & Sex: Boys	0.1459	0.0323	4.52	<1e-05
Test: c12.34 & Sex: Boys	0.0024	0.0324	0.07	0.9404
Test: c3.4 & Sex: Boys	-0.0659	0.0303	-2.18	0.0296
a1 & Sex: Boys	0.0138	0.0489	0.28	0.7785
Test: c5.1 & a1 & Sex: Boys	-0.0543	0.1015	-0.53	0.5930
Test: c234.15 & a1 & Sex: Boys	-0.1159	0.1080	-1.07	0.2830
Test: c12.34 & a1 & Sex: Boys	-0.0909	0.1096	-0.83	0.4069
Test: c3.4 & a1 & Sex: Boys	-0.0966	0.1043	-0.93	0.3546
Residual	0.0000

VarCorr(m_zcp_1_PC_2)

	Column	Variance	Std.Dev	Corr.
Child	(Intercept)	0.64710343	0.80442740
	Test: c5.1	0.18418780	0.42917106	.
	Test: c234.15	0.41085521	0.64097988	.	.
	Test: c12.34	0.69663707	0.83464787	.	.	.
	Test: c3.4	0.21041552	0.45871071	.	.	.	.
Residual		0.00000000	0.00000185

MixedModels.likelihoodratiotest(m_zcp_1_PC_2, m_cpx_1_PC_2)

	model-dof	deviance	χ²	χ²-dof	P(>χ²)
zScore ~ 1 + Test + a1 + Sex + Test & a1 + Test & Sex + a1 & Sex + Test & a1 & Sex + zerocorr(1 + Test \| Child)	26	13238
zScore ~ 1 + Test + a1 + Sex + Test & a1 + Test & Sex + a1 & Sex + Test & a1 & Sex + (1 + Test \| Child)	36	13290	-52	10	NaN

3 Other topics

3.1 Contrasts are re-parameterizations of the same model

The choice of contrast does not affect the model objective, in other words, they all yield the same goodness of fit. It does not matter whether a contrast is orthogonal or not.

[
  objective(m_ovi_SeqDiff),
  objective(m_ovi_Helmert),
  objective(m_ovi_Hypo),
]

3-element Vector{Float64}:
 13831.346285093714
 13831.346285093721
 13831.346285093721

3.2 VCs and CPs depend on contrast coding

Trivially, the meaning of a contrast depends on its definition. Consequently, the contrast specification has a big effect on the random-effect structure. As an illustration, we refit the LMMs with variance components (VCs) and correlation parameters (CPs) for Child-related contrasts of Test. Unfortunately, it is not easy, actually rather quite difficult, to grasp the meaning of correlations of contrast-based effects; they represent two-way interactions.

begin
  f_Child = @formula zScore ~
    1 + Test * a1 * Sex + (1 + Test | Child)
  m_Child_SDC = fit(MixedModel, f_Child, dat; contrasts=contr1)
  m_Child_HeC = fit(MixedModel, f_Child, dat; contrasts=contr2)
  m_Child_HyC = fit(MixedModel, f_Child, dat; contrasts=contr3)
  m_Child_PCA = fit(MixedModel, f_Child, dat; contrasts=contr4)
end

Minimizing 14    Time: 0:00:00 ( 7.39 ms/it)
   objective: 13890.385675304977


Minimizing 27    Time: 0:00:00 ( 7.67 ms/it)
   objective: 13871.76436926304


Minimizing 40    Time: 0:00:00 ( 7.69 ms/it)
   objective: 13806.312715469485


Minimizing 54    Time: 0:00:00 ( 7.65 ms/it)
   objective: 13803.64702088428


Minimizing 68    Time: 0:00:00 ( 7.59 ms/it)
   objective: 13801.168697482783


Minimizing 82    Time: 0:00:00 ( 7.54 ms/it)
   objective: 13795.896993058695


Minimizing 96    Time: 0:00:00 ( 7.50 ms/it)
   objective: 13792.040475330967


Minimizing 110    Time: 0:00:00 ( 7.48 ms/it)
   objective: 13776.871409524276


Minimizing 124    Time: 0:00:00 ( 7.45 ms/it)
   objective: 13771.929540173987


Minimizing 138    Time: 0:00:01 ( 7.44 ms/it)
   objective: 13764.788934173048


Minimizing 152    Time: 0:00:01 ( 7.43 ms/it)
   objective: 13759.583772965314


Minimizing 166    Time: 0:00:01 ( 7.42 ms/it)
   objective: 13757.51751541219


Minimizing 180    Time: 0:00:01 ( 7.43 ms/it)
   objective: 13751.992620923807


Minimizing 194    Time: 0:00:01 ( 7.42 ms/it)
   objective: 13749.637258091547


Minimizing 208    Time: 0:00:01 ( 7.42 ms/it)
   objective: 13747.66741289875


Minimizing 222    Time: 0:00:01 ( 7.41 ms/it)
   objective: 13746.689597125267


Minimizing 236    Time: 0:00:01 ( 7.40 ms/it)
   objective: 13745.668209902695


Minimizing 250    Time: 0:00:01 ( 7.40 ms/it)
   objective: 13742.172718282018


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Minimizing 1904    Time: 0:00:12 ( 6.56 ms/it)
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Minimizing 1918    Time: 0:00:12 ( 6.57 ms/it)
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Minimizing 1932    Time: 0:00:12 ( 6.58 ms/it)
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Minimizing 1946    Time: 0:00:12 ( 6.58 ms/it)
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Minimizing 1960    Time: 0:00:12 ( 6.59 ms/it)
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Minimizing 1974    Time: 0:00:13 ( 6.60 ms/it)
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Minimizing 1988    Time: 0:00:13 ( 6.60 ms/it)
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Minimizing 2002    Time: 0:00:13 ( 6.61 ms/it)
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Minimizing 2016    Time: 0:00:13 ( 6.62 ms/it)
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   objective: 13493.327089545004


Minimizing 1345    Time: 0:00:09 ( 7.35 ms/it)
   objective: 13492.036011094118


Minimizing 1359    Time: 0:00:09 ( 7.35 ms/it)
   objective: 13490.635112589967


Minimizing 1373    Time: 0:00:10 ( 7.35 ms/it)
   objective: 13485.198695314713


Minimizing 1387    Time: 0:00:10 ( 7.35 ms/it)
   objective: 13483.3166857549


Minimizing 1401    Time: 0:00:10 ( 7.35 ms/it)
   objective: 13483.205672908029


Minimizing 1415    Time: 0:00:10 ( 7.35 ms/it)
   objective: 13482.55790900805


Minimizing 1429    Time: 0:00:10 ( 7.35 ms/it)
   objective: 13481.62311884229


Minimizing 1443    Time: 0:00:10 ( 7.35 ms/it)
   objective: 13481.367826951297


Minimizing 1457    Time: 0:00:10 ( 7.35 ms/it)
   objective: 13480.968584902941


Minimizing 1471    Time: 0:00:10 ( 7.34 ms/it)
   objective: 13480.279389296404


Minimizing 1486    Time: 0:00:10 ( 7.34 ms/it)
   objective: 13478.903736104221


Minimizing 1500    Time: 0:00:11 ( 7.34 ms/it)
   objective: 13476.904573330627


Minimizing 1514    Time: 0:00:11 ( 7.34 ms/it)
   objective: 13474.297266668895


Minimizing 1529    Time: 0:00:11 ( 7.34 ms/it)
   objective: 13470.002833767147


Minimizing 1544    Time: 0:00:11 ( 7.33 ms/it)
   objective: 13462.06207514344


Minimizing 1558    Time: 0:00:11 ( 7.33 ms/it)
   objective: 13460.157778409106


Minimizing 1572    Time: 0:00:11 ( 7.33 ms/it)
   objective: 13457.770786462854


Minimizing 1587    Time: 0:00:11 ( 7.33 ms/it)
   objective: 13451.613302300073


Minimizing 1601    Time: 0:00:11 ( 7.33 ms/it)
   objective: 13449.58774241776


Minimizing 1616    Time: 0:00:11 ( 7.33 ms/it)
   objective: 13435.123563433503


Minimizing 1630    Time: 0:00:11 ( 7.33 ms/it)
   objective: 13431.518788441215


Minimizing 1644    Time: 0:00:12 ( 7.33 ms/it)
   objective: 13429.287647056568


Minimizing 1658    Time: 0:00:12 ( 7.33 ms/it)
   objective: 13425.87517054005


Minimizing 1672    Time: 0:00:12 ( 7.33 ms/it)
   objective: 13423.867507700372


Minimizing 1686    Time: 0:00:12 ( 7.33 ms/it)
   objective: 13421.30905430179


Minimizing 1700    Time: 0:00:12 ( 7.33 ms/it)
   objective: 13417.335188697267


Minimizing 1714    Time: 0:00:12 ( 7.33 ms/it)
   objective: 13414.501033505934


Minimizing 1728    Time: 0:00:12 ( 7.33 ms/it)
   objective: 13404.343132971437


Minimizing 1742    Time: 0:00:12 ( 7.33 ms/it)
   objective: 13400.651505039918


Minimizing 1756    Time: 0:00:12 ( 7.33 ms/it)
   objective: 13392.777656685605


Minimizing 1770    Time: 0:00:12 ( 7.33 ms/it)
   objective: 13370.173149351438


Minimizing 1785    Time: 0:00:13 ( 7.32 ms/it)
   objective: 13367.392684728227


Minimizing 1799    Time: 0:00:13 ( 7.32 ms/it)
   objective: 13365.677839866112


Minimizing 1813    Time: 0:00:13 ( 7.32 ms/it)
   objective: 13364.530090216183


Minimizing 1828    Time: 0:00:13 ( 7.32 ms/it)
   objective: 13364.155301184568


Minimizing 1842    Time: 0:00:13 ( 7.32 ms/it)
   objective: 13361.429482887208


Minimizing 1856    Time: 0:00:13 ( 7.32 ms/it)
   objective: 13358.921734349293


Minimizing 1870    Time: 0:00:13 ( 7.32 ms/it)
   objective: 13358.768845975777


Minimizing 1884    Time: 0:00:13 ( 7.32 ms/it)
   objective: 13352.927795224285


Minimizing 1898    Time: 0:00:13 ( 7.32 ms/it)
   objective: 13350.789998397682


Minimizing 1912    Time: 0:00:13 ( 7.32 ms/it)
   objective: 13350.75662746164


Minimizing 1926    Time: 0:00:14 ( 7.32 ms/it)
   objective: 13350.758289736434


Minimizing 1940    Time: 0:00:14 ( 7.32 ms/it)
   objective: 13350.755252319737


Minimizing 1954    Time: 0:00:14 ( 7.32 ms/it)
   objective: 13350.755989147568


Minimizing 1969    Time: 0:00:14 ( 7.31 ms/it)
   objective: 13350.755935735477


Minimizing 1983    Time: 0:00:14 ( 7.31 ms/it)
   objective: 13350.75501378013


Minimizing 1997    Time: 0:00:14 ( 7.31 ms/it)
   objective: 13350.75552872727


Minimizing 2011    Time: 0:00:14 ( 7.31 ms/it)
   objective: 13350.755187449511


Minimizing 2026    Time: 0:00:14 ( 7.31 ms/it)
   objective: 13350.755086330813


Minimizing 2040    Time: 0:00:14 ( 7.31 ms/it)
   objective: 13350.755011224697


Minimizing 2054    Time: 0:00:15 ( 7.31 ms/it)
   objective: 13350.755408414727


Minimizing 2068    Time: 0:00:15 ( 7.31 ms/it)
   objective: 13350.755042749253


Minimizing 2082    Time: 0:00:15 ( 7.31 ms/it)
   objective: 13350.754936058685


Minimizing 2097    Time: 0:00:15 ( 7.31 ms/it)
   objective: 13350.755062139142


Minimizing 2112    Time: 0:00:15 ( 7.30 ms/it)
   objective: 13350.755512846517


Minimizing 2127    Time: 0:00:15 ( 7.30 ms/it)
   objective: 13350.7553692472


Minimizing 2134    Time: 0:00:15 ( 7.30 ms/it)

	Est.	SE	z	p	σ_Child
(Intercept)	-0.0187	0.0142	-1.32	0.1881	0.7796
Test: c5.1	-0.0195	0.0429	-0.45	0.6503	1.2413
Test: c234.15	-0.0253	0.1675	-0.15	0.8800	0.7376
Test: c2.34	-0.0086	0.0780	-0.11	0.9121	1.6770
Test: c3.4	0.0005	0.0441	0.01	0.9918	0.8814
a1	0.2896	0.0481	6.02	<1e-08
Sex: Boys	0.2065	0.0142	14.55	<1e-47
Test: c5.1 & a1	0.4780	0.1451	3.29	0.0010
Test: c234.15 & a1	0.3154	0.5682	0.55	0.5789
Test: c2.34 & a1	-0.1610	0.2600	-0.62	0.5358
Test: c3.4 & a1	0.1216	0.1528	0.80	0.4262
Test: c5.1 & Sex: Boys	0.1000	0.0429	2.33	0.0199
Test: c234.15 & Sex: Boys	-0.8634	0.1675	-5.15	<1e-06
Test: c2.34 & Sex: Boys	-0.0876	0.0780	-1.12	0.2617
Test: c3.4 & Sex: Boys	-0.1354	0.0441	-3.07	0.0021
a1 & Sex: Boys	0.0103	0.0481	0.22	0.8297
Test: c5.1 & a1 & Sex: Boys	0.0625	0.1451	0.43	0.6668
Test: c234.15 & a1 & Sex: Boys	0.7550	0.5682	1.33	0.1839
Test: c2.34 & a1 & Sex: Boys	-0.1627	0.2600	-0.63	0.5315
Test: c3.4 & a1 & Sex: Boys	-0.0711	0.1528	-0.47	0.6416
Residual	0.0000

VarCorr(m_Child_SDC)

	Column	Variance	Std.Dev	Corr.
Child	(Intercept)	0.5914079	0.7690305
	Test: Star_r	0.6339085	0.7961837	+0.25
	Test: S20_r	0.4534167	0.6733622	+0.17	-0.60
	Test: SLJ	0.3369005	0.5804313	-0.40	+0.42	+0.10
	Test: BPT	0.9286516	0.9636657	-0.19	+0.25	-0.49	-0.16
Residual		0.0000000	0.0000063

VarCorr(m_Child_HeC)

	Column	Variance	Std.Dev	Corr.
Child	(Intercept)	0.35728219	0.59773087
	Test: Star_r	0.32844898	0.57310468	-0.15
	Test: S20_r	0.07994767	0.28275020	+0.38	-0.74
	Test: SLJ	0.05820973	0.24126694	+0.10	-0.22	+0.77
	Test: BPT	0.05211986	0.22829775	-0.31	+0.62	-0.09	+0.54
Residual		0.00000000	0.00002633

VarCorr(m_Child_HyC)

	Column	Variance	Std.Dev	Corr.
Child	(Intercept)	0.4923681	0.7016894
	Test: BPT-other	2.3466174	1.5318673	+0.69
	Test: Star-End	2.1868944	1.4788152	-0.12	+0.55
	Test: S20-Star	1.6424392	1.2815769	+0.34	-0.40	-0.89
	Test: SLJ-S20	0.7990362	0.8938882	-0.25	+0.15	+0.24	-0.22
Residual		0.0000000	0.0000581

VarCorr(m_Child_PCA)

	Column	Variance	Std.Dev	Corr.
Child	(Intercept)	0.60773937	0.77957641
	Test: c5.1	1.54071063	1.24125365	-0.09
	Test: c234.15	0.54401627	0.73757459	+0.90	-0.51
	Test: c2.34	2.81236949	1.67701207	+0.00	+0.22	-0.09
	Test: c3.4	0.77692881	0.88143565	+0.14	-0.26	+0.23	+0.34
Residual		0.00000000	0.00004549

The CPs for the various contrasts are in line with expectations. For the SDC we observe substantial negative CPs between neighboring contrasts. For the orthogonal HeC, all CPs are small; they are uncorrelated. HyC contains some of the SDC contrasts and we observe again the negative CPs. The (roughly) PCA-based contrasts are small with one exception; there is a sizeable CP of +.41 between GM and the core of adjusted physical fitness (c234.15).

Do these differences in CPs imply that we can move to zcpLMMs when we have orthogonal contrasts? We pursue this question with by refitting the four LMMs with zerocorr() and compare the goodness of fit.

begin
  f_Child0 = @formula zScore ~
    1 + Test * a1 * Sex + zerocorr(1 + Test | Child)
  m_Child_SDC0 = fit(MixedModel, f_Child0, dat; contrasts=contr1)
  m_Child_HeC0 = fit(MixedModel, f_Child0, dat; contrasts=contr2)
  m_Child_HyC0 = fit(MixedModel, f_Child0, dat; contrasts=contr3)
  m_Child_PCA0 = fit(MixedModel, f_Child0, dat; contrasts=contr4)
end

Minimizing 24    Time: 0:00:00 ( 4.35 ms/it)
   objective: 13816.597751965644


Minimizing 47    Time: 0:00:00 ( 4.43 ms/it)
   objective: 13816.452156025942


Minimizing 69    Time: 0:00:00 ( 4.53 ms/it)
   objective: 13816.11576494258


Minimizing 91    Time: 0:00:00 ( 4.58 ms/it)
   objective: 13816.072738053463


Minimizing 113    Time: 0:00:00 ( 4.58 ms/it)
   objective: 13816.03981854927


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   objective: 13816.039719243297


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Minimizing 23    Time: 0:00:00 ( 4.40 ms/it)
   objective: 13799.10750528917


Minimizing 45    Time: 0:00:00 ( 4.49 ms/it)
   objective: 13768.799730749375


Minimizing 71    Time: 0:00:00 ( 4.78 ms/it)
   objective: 13762.569963867749


Minimizing 96    Time: 0:00:00 ( 4.78 ms/it)
   objective: 13750.02979783131


Minimizing 118    Time: 0:00:00 ( 4.76 ms/it)
   objective: 13741.448367137931


Minimizing 140    Time: 0:00:00 ( 4.73 ms/it)
   objective: 13728.473903083777


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   objective: 13721.30532047743


Minimizing 184    Time: 0:00:00 ( 4.70 ms/it)
   objective: 13677.37932639882


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   objective: 13601.322975120376


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   objective: 13547.741712284165


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   objective: 13509.745018863847


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   objective: 13502.54654987184


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   objective: 13499.932596615763


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   objective: 13497.982841130426


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   objective: 13496.375571846496


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   objective: 13495.965794345779


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   objective: 13494.00120331494


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   objective: 13470.096761252484


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   objective: 13457.024853461058


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   objective: 13448.147686523778


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   objective: 13436.843324584595


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   objective: 13435.15031586244


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   objective: 13432.020571169473


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   objective: 13430.877550988022


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   objective: 13430.119051387694


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   objective: 13429.189395009133


Minimizing 602    Time: 0:00:02 ( 4.65 ms/it)
   objective: 13407.127364948843


Minimizing 624    Time: 0:00:02 ( 4.64 ms/it)
   objective: 13391.704669993764


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   objective: 13373.9115598395


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   objective: 13359.591070896306


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   objective: 13346.397987804274


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   objective: 13350.595585781994


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   objective: 13301.539059799368


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   objective: 13288.123543561087


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   objective: 13257.762713758086


Minimizing 800    Time: 0:00:03 ( 4.64 ms/it)
   objective: 13246.703103755295


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   objective: 13246.56117360582


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   objective: 13246.680514526044


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   objective: 13246.696569821142


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   objective: 13246.533304656783


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   objective: 13820.405037926314


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   objective: 13803.891680738147


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   objective: 13800.190492696034


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   objective: 13798.547937753887


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   objective: 13796.052000503572


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   objective: 13741.177013262572


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   objective: 13664.281738268837


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   objective: 13652.919266567435


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   objective: 13638.333783335758


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   objective: 13631.376320183277


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   objective: 13588.834563515578


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   objective: 13684.462732357351


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   objective: 13672.93285427327


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   objective: 13664.17399843769


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   objective: 13657.209823832942


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   objective: 13647.493560197338


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   objective: 13642.06863578891


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   objective: 13632.52374387877


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   objective: 13541.90676655266


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   objective: 13468.501886905193


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   objective: 13419.805263769565


Minimizing 464    Time: 0:00:02 ( 4.62 ms/it)
   objective: 13344.618731673138


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	Est.	SE	z	p	σ_Child
(Intercept)	-0.0182	0.0142	-1.29	0.1987	0.7272
Test: c5.1	-0.0265	0.0436	-0.61	0.5437	1.2382
Test: c234.15	-0.0540	0.1696	-0.32	0.7503	0.1638
Test: c2.34	-0.0007	0.0763	-0.01	0.9925	1.8128
Test: c3.4	-0.0065	0.0452	-0.14	0.8859	1.1986
a1	0.2870	0.0480	5.98	<1e-08
Sex: Boys	0.2063	0.0142	14.56	<1e-47
Test: c5.1 & a1	0.4789	0.1471	3.26	0.0011
Test: c234.15 & a1	0.2973	0.5742	0.52	0.6047
Test: c2.34 & a1	-0.1468	0.2544	-0.58	0.5639
Test: c3.4 & a1	0.0959	0.1567	0.61	0.5406
Test: c5.1 & Sex: Boys	0.0998	0.0436	2.29	0.0220
Test: c234.15 & Sex: Boys	-0.8445	0.1696	-4.98	<1e-06
Test: c2.34 & Sex: Boys	-0.1012	0.0763	-1.33	0.1849
Test: c3.4 & Sex: Boys	-0.1305	0.0452	-2.89	0.0039
a1 & Sex: Boys	0.0115	0.0480	0.24	0.8102
Test: c5.1 & a1 & Sex: Boys	0.0569	0.1471	0.39	0.6987
Test: c234.15 & a1 & Sex: Boys	0.7131	0.5742	1.24	0.2142
Test: c2.34 & a1 & Sex: Boys	-0.1392	0.2544	-0.55	0.5844
Test: c3.4 & a1 & Sex: Boys	-0.0742	0.1567	-0.47	0.6358
Residual	0.0000

MixedModels.likelihoodratiotest(m_Child_SDC0, m_Child_SDC)

	model-dof	deviance	χ²	χ²-dof	P(>χ²)
zScore ~ 1 + Test + a1 + Sex + Test & a1 + Test & Sex + a1 & Sex + Test & a1 & Sex + zerocorr(1 + Test \| Child)	26	13816
zScore ~ 1 + Test + a1 + Sex + Test & a1 + Test & Sex + a1 & Sex + Test & a1 & Sex + (1 + Test \| Child)	36	13263	553	10	<1e-99

MixedModels.likelihoodratiotest(m_Child_HeC0, m_Child_HeC)

	model-dof	deviance	χ²	χ²-dof	P(>χ²)
zScore ~ 1 + Test + a1 + Sex + Test & a1 + Test & Sex + a1 & Sex + Test & a1 & Sex + zerocorr(1 + Test \| Child)	26	13246
zScore ~ 1 + Test + a1 + Sex + Test & a1 + Test & Sex + a1 & Sex + Test & a1 & Sex + (1 + Test \| Child)	36	13317	-70	10	NaN

MixedModels.likelihoodratiotest(m_Child_HyC0, m_Child_HyC)

	model-dof	deviance	χ²	χ²-dof	P(>χ²)
zScore ~ 1 + Test + a1 + Sex + Test & a1 + Test & Sex + a1 & Sex + Test & a1 & Sex + zerocorr(1 + Test \| Child)	26	13291
zScore ~ 1 + Test + a1 + Sex + Test & a1 + Test & Sex + a1 & Sex + Test & a1 & Sex + (1 + Test \| Child)	36	13373	-82	10	NaN

MixedModels.likelihoodratiotest(m_Child_PCA0, m_Child_PCA)

	model-dof	deviance	χ²	χ²-dof	P(>χ²)
zScore ~ 1 + Test + a1 + Sex + Test & a1 + Test & Sex + a1 & Sex + Test & a1 & Sex + zerocorr(1 + Test \| Child)	26	13203
zScore ~ 1 + Test + a1 + Sex + Test & a1 + Test & Sex + a1 & Sex + Test & a1 & Sex + (1 + Test \| Child)	36	13351	-148	10	NaN

Obviously, we can not drop CPs from any of the LMMs. The full LMMs all have the same objective, but we can compare the goodness-of-fit statistics of zcpLMMs more directly.

begin
  zcpLMM = ["SDC0", "HeC0", "HyC0", "PCA0"]
  mods = [m_Child_SDC0, m_Child_HeC0, m_Child_HyC0, m_Child_PCA0]
  gof_summary = sort!(
    DataFrame(;
      zcpLMM=zcpLMM,
      dof=dof.(mods),
      deviance=deviance.(mods),
      AIC=aic.(mods),
      BIC=bic.(mods),
    ),
    :deviance,
  )
end

4×5 DataFrame

Row	zcpLMM	dof	deviance	AIC	BIC
	String	Int64	Float64	Float64	Float64
1	PCA0	26	13203.0	13255.0	13424.5
2	HeC0	26	13246.3	13298.3	13467.7
3	HyC0	26	13290.8	13342.8	13512.3
4	SDC0	26	13816.0	13868.0	14037.5

The best fit was obtained for the PCA-based zcpLMM. Somewhat surprisingly the second best fit was obtained for the SDC. The relatively poor performance of HeC-based zcpLMM is puzzling to me. I thought it might be related to imbalance in design in the present data, but this does not appear to be the case. The same comparison of SequentialDifferenceCoding and Helmert Coding also showed a worse fit for the zcp-HeC LMM than the zcp-SDC LMM.

3.3 VCs and CPs depend on random factor

VCs and CPs resulting from a set of test contrasts can also be estimated for the random factor School. Of course, these VCs and CPs may look different from the ones we just estimated for Child.

The effect of age (i.e., developmental gain) varies within School. Therefore, we also include its VCs and CPs in this model; the school-related VC for Sex was not significant.

f_School = @formula zScore ~
  1 + Test * a1 * Sex + (1 + Test + a1 | School);
m_School_SeqDiff = fit(MixedModel, f_School, dat; contrasts=contr1);
m_School_Helmert = fit(MixedModel, f_School, dat; contrasts=contr2);
m_School_Hypo = fit(MixedModel, f_School, dat; contrasts=contr3);
m_School_PCA = fit(MixedModel, f_School, dat; contrasts=contr4);

Minimizing 156    Time: 0:00:00 ( 0.64 ms/it)
   objective: 13758.66414070674


Minimizing 316    Time: 0:00:00 ( 0.64 ms/it)
   objective: 13755.493017954828


Minimizing 474    Time: 0:00:00 ( 0.64 ms/it)
   objective: 13755.244497496962


Minimizing 632    Time: 0:00:00 ( 0.64 ms/it)
   objective: 13755.242491858691


Minimizing 667    Time: 0:00:00 ( 0.64 ms/it)

Minimizing 155    Time: 0:00:00 ( 0.65 ms/it)
   objective: 13788.852361436366


Minimizing 312    Time: 0:00:00 ( 0.64 ms/it)
   objective: 13765.421204869956


Minimizing 469    Time: 0:00:00 ( 0.64 ms/it)
   objective: 13757.912823438124


Minimizing 627    Time: 0:00:00 ( 0.64 ms/it)
   objective: 13755.578232256667


Minimizing 786    Time: 0:00:00 ( 0.64 ms/it)
   objective: 13755.284033742779


Minimizing 945    Time: 0:00:00 ( 0.64 ms/it)
   objective: 13755.251526817008


Minimizing 1104    Time: 0:00:00 ( 0.64 ms/it)
   objective: 13755.244424622788


Minimizing 1263    Time: 0:00:00 ( 0.64 ms/it)
   objective: 13755.242837795895


Minimizing 1399    Time: 0:00:00 ( 0.64 ms/it)

Minimizing 157    Time: 0:00:00 ( 0.64 ms/it)
   objective: 13763.773664301352


Minimizing 317    Time: 0:00:00 ( 0.63 ms/it)
   objective: 13755.62375023458


Minimizing 477    Time: 0:00:00 ( 0.63 ms/it)
   objective: 13755.247816056839


Minimizing 637    Time: 0:00:00 ( 0.63 ms/it)
   objective: 13755.242545093663


Minimizing 776    Time: 0:00:00 ( 0.64 ms/it)

Minimizing 157    Time: 0:00:00 ( 0.64 ms/it)
   objective: 13765.860078727028


Minimizing 316    Time: 0:00:00 ( 0.64 ms/it)
   objective: 13756.905922266638


Minimizing 476    Time: 0:00:00 ( 0.63 ms/it)
   objective: 13755.402770428072


Minimizing 634    Time: 0:00:00 ( 0.63 ms/it)
   objective: 13755.243922014259


Minimizing 757    Time: 0:00:00 ( 0.63 ms/it)

VarCorr(m_School_SeqDiff)

	Column	Variance	Std.Dev	Corr.
School	(Intercept)	0.037029	0.192430
	Test: Star_r	0.270040	0.519654	-0.11
	Test: S20_r	0.259852	0.509757	-0.20	-0.62
	Test: SLJ	0.218225	0.467146	+0.17	+0.20	-0.79
	Test: BPT	0.120254	0.346776	-0.41	+0.08	+0.04	-0.40
	a1	0.158829	0.398533	+0.10	+0.27	+0.01	+0.13	-0.12
Residual		0.815336	0.902960

VarCorr(m_School_Helmert)

	Column	Variance	Std.Dev	Corr.
School	(Intercept)	0.0370359	0.1924472
	Test: Star_r	0.0675297	0.2598647	-0.11
	Test: S20_r	0.0182484	0.1350868	-0.32	-0.13
	Test: SLJ	0.0045444	0.0674123	-0.03	+0.22	-0.50
	Test: BPT	0.0031070	0.0557406	-0.53	+0.26	-0.23	-0.01
	a1	0.1587905	0.3984853	+0.10	+0.27	+0.19	+0.40	+0.15
Residual		0.8153303	0.9029564

VarCorr(m_School_Hypo)

	Column	Variance	Std.Dev	Corr.
School	(Intercept)	0.037031	0.192434
	Test: BPT-other	1.242925	1.114865	-0.53
	Test: Star-End	0.270094	0.519706	-0.11	+0.26
	Test: S20-Star	0.259874	0.509778	-0.20	-0.32	-0.62
	Test: SLJ-S20	0.218230	0.467151	+0.17	+0.13	+0.20	-0.79
	a1	0.158860	0.398572	+0.10	+0.15	+0.27	+0.01	+0.13
Residual		0.815332	0.902957

VarCorr(m_School_PCA)

	Column	Variance	Std.Dev	Corr.
School	(Intercept)	0.037030	0.192433
	Test: c5.1	0.177228	0.420984	-0.53
	Test: c234.15	2.386718	1.544901	+0.05	+0.28
	Test: c2.34	0.503218	0.709379	+0.17	+0.52	+0.17
	Test: c3.4	0.218196	0.467115	-0.17	-0.07	+0.09	-0.48
	a1	0.158941	0.398674	+0.10	+0.39	+0.32	-0.10	-0.13
Residual		0.815338	0.902961

We compare again how much of the fit resides in the CPs.

begin
  f_School0 = @formula zScore ~
    1 + Test * a1 * Sex + zerocorr(1 + Test + a1 | School)
  m_School_SDC0 = fit(MixedModel, f_School0, dat; contrasts=contr1)
  m_School_HeC0 = fit(MixedModel, f_School0, dat; contrasts=contr2)
  m_School_HyC0 = fit(MixedModel, f_School0, dat; contrasts=contr3)
  m_School_PCA0 = fit(MixedModel, f_School0, dat; contrasts=contr4)
  #
  zcpLMM2 = ["SDC0", "HeC0", "HyC0", "PCA0"]
  mods2 = [
    m_School_SDC0, m_School_HeC0, m_School_HyC0, m_School_PCA0
  ]
  gof_summary2 = sort!(
    DataFrame(;
      zcpLMM=zcpLMM2,
      dof=dof.(mods2),
      deviance=deviance.(mods2),
      AIC=aic.(mods2),
      BIC=bic.(mods2),
    ),
    :deviance,
  )
end

Minimizing 201    Time: 0:00:00 ( 0.50 ms/it)
   objective: 13810.858868448784


Minimizing 261    Time: 0:00:00 ( 0.50 ms/it)

Minimizing 202    Time: 0:00:00 ( 0.50 ms/it)
   objective: 13788.245579942606


Minimizing 208    Time: 0:00:00 ( 0.50 ms/it)

4×5 DataFrame

Row	zcpLMM	dof	deviance	AIC	BIC
	String	Int64	Float64	Float64	Float64
1	HeC0	27	13782.4	13836.4	14012.3
2	PCA0	27	13788.2	13842.2	14018.2
3	HyC0	27	13807.6	13861.6	14037.5
4	SDC0	27	13810.9	13864.9	14040.8

For the random factor School the Helmert contrast, followed by PCA-based contrasts have least information in the CPs; SDC has the largest contribution from CPs. Interesting.

4 That’s it

That’s it for this tutorial. It is time to try your own contrast coding. You can use these data; there are many alternatives to set up hypotheses for the five tests. Of course and even better, code up some contrasts for data of your own.

Have fun!

Fühner, T., Granacher, U., Golle, K., & Kliegl, R. (2021). Age and sex effects in physical fitness components of 108,295 third graders including 515 primary schools and 9 cohorts. Scientific Reports, 11(1). https://doi.org/10.1038/s41598-021-97000-4

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