Mixed Models Tutorial: Contrast Coding

Author

Reinhold Kliegl

Published

2024-09-13

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 SMLP2024: 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	2011	S105405	C051812	8.50376	4.16667
2	S20_r	Boys	2015	S111442	C056292	8.55031	4.44444
3	S20_r	Boys	2017	S111405	C061627	8.5859	3.77358
4	S20_r	Boys	2019	S105181	C105825	8.9692	4.87805
5	S20_r	Boys	2017	S104887	C012134	8.14237	3.7037
6	S20_r	Boys	2012	S100973	C013697	8.16701	4.87805
7	S20_r	Boys	2012	S100237	C007058	8.08214	4.0
8	S20_r	Boys	2016	S104991	C036718	8.37782	5.0
9	S20_r	Boys	2015	S105661	C096722	8.8898	4.87805
10	S20_r	Boys	2014	S104693	C073949	8.70363	4.44444
11	S20_r	Boys	2011	S102570	C069843	8.6653	4.7619
12	S20_r	Boys	2017	S102374	C050458	8.50103	4.7619
13	S20_r	Boys	2011	S110048	C042223	8.41889	4.65116
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
4989	Run	Girls	2012	S100122	C013571	8.16701	1012.0
4990	Run	Girls	2015	S103640	C064554	8.62149	1010.0
4991	Run	Girls	2018	S101461	C015910	8.18891	855.0
4992	Run	Girls	2014	S105790	C045328	8.45722	1104.0
4993	Run	Girls	2017	S105806	C048611	8.47912	927.0
4994	Run	Girls	2018	S111442	C117800	9.10609	1062.0
4995	Run	Girls	2017	S102222	C032543	8.33676	801.0
4996	Run	Girls	2012	S112082	C070999	8.6653	1000.0
4997	Run	Girls	2018	S113001	C053835	8.52567	783.0
4998	Run	Girls	2012	S101035	C069452	8.6653	1071.0
4999	Run	Girls	2018	S104978	C053314	8.52567	837.0
5000	Run	Girls	2019	S100596	C037264	8.38877	1008.0

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	2011	S105405	C051812	8.50376	4.16667	0.00376454	-0.828369
2	S20_r	Boys	2015	S111442	C056292	8.55031	4.44444	0.050308	-0.139897
3	S20_r	Boys	2017	S111405	C061627	8.5859	3.77358	0.0859001	-1.80262
4	S20_r	Boys	2019	S105181	C105825	8.9692	4.87805	0.469199	0.934791
5	S20_r	Boys	2017	S104887	C012134	8.14237	3.7037	-0.357632	-1.97582
6	S20_r	Boys	2012	S100973	C013697	8.16701	4.87805	-0.332991	0.934791
7	S20_r	Boys	2012	S100237	C007058	8.08214	4.0	-0.417864	-1.24145
8	S20_r	Boys	2016	S104991	C036718	8.37782	5.0	-0.122177	1.23705
9	S20_r	Boys	2015	S105661	C096722	8.8898	4.87805	0.389802	0.934791
10	S20_r	Boys	2014	S104693	C073949	8.70363	4.44444	0.203628	-0.139897
11	S20_r	Boys	2011	S102570	C069843	8.6653	4.7619	0.165298	0.646928
12	S20_r	Boys	2017	S102374	C050458	8.50103	4.7619	0.00102669	0.646928
13	S20_r	Boys	2011	S110048	C042223	8.41889	4.65116	-0.0811088	0.372455
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
4989	Run	Girls	2012	S100122	C013571	8.16701	1012.0	-0.332991	0.139241
4990	Run	Girls	2015	S103640	C064554	8.62149	1010.0	0.121492	0.124993
4991	Run	Girls	2018	S101461	C015910	8.18891	855.0	-0.311088	-0.979245
4992	Run	Girls	2014	S105790	C045328	8.45722	1104.0	-0.0427789	0.794659
4993	Run	Girls	2017	S105806	C048611	8.47912	927.0	-0.0208761	-0.466309
4994	Run	Girls	2018	S111442	C117800	9.10609	1062.0	0.606092	0.495446
4995	Run	Girls	2017	S102222	C032543	8.33676	801.0	-0.163244	-1.36395
4996	Run	Girls	2012	S112082	C070999	8.6653	1000.0	0.165298	0.0537514
4997	Run	Girls	2018	S113001	C053835	8.52567	783.0	0.0256674	-1.49218
4998	Run	Girls	2012	S101035	C069452	8.6653	1071.0	0.165298	0.559563
4999	Run	Girls	2018	S104978	C053314	8.52567	837.0	0.0256674	-1.10748
5000	Run	Girls	2019	S100596	C037264	8.38877	1008.0	-0.111225	0.110744

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	4.54162	0.425471	0.100965	1.05453
2	BPT	Boys	3.9722	0.711801	0.31007	0.992841
3	SLJ	Boys	128.926	19.0818	0.178649	0.985248
4	Star_r	Boys	2.08216	0.298869	0.0963422	1.03081
5	Run	Boys	1025.33	144.019	0.234191	1.02601
6	S20_r	Girls	4.46015	0.376233	-0.100965	0.932493
7	BPT	Girls	3.5276	0.650634	-0.31007	0.907524
8	SLJ	Girls	122.006	19.0486	-0.178649	0.98353
9	Star_r	Girls	2.0263	0.278221	-0.0963422	0.959598
10	Run	Girls	959.582	128.604	-0.234191	0.916191

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)	-0.0002	0.0142	-0.01	0.9897	0.6729
Test: Star_r	0.0051	0.0445	0.11	0.9087
Test: S20_r	-0.0039	0.0444	-0.09	0.9292
Test: SLJ	-0.0008	0.0444	-0.02	0.9861
Test: BPT	0.0045	0.0444	0.10	0.9191
Residual	0.7391

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)	-0.0211	0.0145	-1.46	0.1453	0.6651
Test: Star_r	-0.0050	0.0455	-0.11	0.9123
Test: S20_r	0.0077	0.0455	0.17	0.8662
Test: SLJ	0.0050	0.0450	0.11	0.9118
Test: BPT	-0.0283	0.0451	-0.63	0.5303
a1	0.3298	0.0492	6.70	<1e-10
Test: Star_r & a1	0.0572	0.1551	0.37	0.7125
Test: S20_r & a1	-0.1089	0.1531	-0.71	0.4771
Test: SLJ & a1	-0.0717	0.1530	-0.47	0.6396
Test: BPT & a1	0.5272	0.1541	3.42	0.0006
Residual	0.7380

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)	-0.0200	0.0142	-1.41	0.1597	0.6401
Test: Star_r	-0.0025	0.0446	-0.06	0.9551
Test: S20_r	0.0083	0.0446	0.19	0.8529
Test: SLJ	0.0030	0.0441	0.07	0.9457
Test: BPT	-0.0259	0.0442	-0.59	0.5579
a1	0.3188	0.0482	6.61	<1e-10
Sex: Boys	0.1811	0.0139	13.04	<1e-38
Test: Star_r & a1	0.0195	0.1523	0.13	0.8979
Test: S20_r & a1	-0.1020	0.1503	-0.68	0.4977
Test: SLJ & a1	-0.0577	0.1501	-0.38	0.7008
Test: BPT & a1	0.4949	0.1510	3.28	0.0011
Test: Star_r & Sex: Boys	-0.1543	0.0434	-3.56	0.0004
Test: S20_r & Sex: Boys	0.0218	0.0434	0.50	0.6147
Test: SLJ & Sex: Boys	0.0719	0.0433	1.66	0.0965
Test: BPT & Sex: Boys	0.1289	0.0433	2.98	0.0029
Residual	0.7333

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)	-0.0201	0.0142	-1.42	0.1570	0.6386
Test: Star_r	-0.0036	0.0447	-0.08	0.9361
Test: S20_r	0.0090	0.0446	0.20	0.8398
Test: SLJ	0.0034	0.0441	0.08	0.9381
Test: BPT	-0.0250	0.0442	-0.56	0.5725
a1	0.3176	0.0483	6.58	<1e-10
Sex: Boys	0.1800	0.0142	12.66	<1e-36
Test: Star_r & a1	0.0211	0.1523	0.14	0.8897
Test: S20_r & a1	-0.1051	0.1504	-0.70	0.4848
Test: SLJ & a1	-0.0556	0.1502	-0.37	0.7112
Test: BPT & a1	0.4916	0.1511	3.25	0.0011
Test: Star_r & Sex: Boys	-0.1604	0.0447	-3.59	0.0003
Test: S20_r & Sex: Boys	0.0236	0.0446	0.53	0.5972
Test: SLJ & Sex: Boys	0.0693	0.0441	1.57	0.1164
Test: BPT & Sex: Boys	0.1395	0.0442	3.15	0.0016
a1 & Sex: Boys	0.0155	0.0483	0.32	0.7483
Test: Star_r & a1 & Sex: Boys	0.0837	0.1523	0.55	0.5827
Test: S20_r & a1 & Sex: Boys	-0.0134	0.1504	-0.09	0.9288
Test: SLJ & a1 & Sex: Boys	0.0493	0.1502	0.33	0.7427
Test: BPT & a1 & Sex: Boys	-0.1781	0.1511	-1.18	0.2385
Residual	0.7344

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)	-0.0201	0.0142	-1.42	0.1570	0.6386
Test: Star_r	-0.0018	0.0223	-0.08	0.9361
Test: S20_r	0.0024	0.0128	0.19	0.8507
Test: SLJ	0.0021	0.0090	0.23	0.8192
Test: BPT	-0.0038	0.0070	-0.54	0.5918
a1	0.3176	0.0483	6.58	<1e-10
Sex: Boys	0.1800	0.0142	12.66	<1e-36
Test: Star_r & a1	0.0106	0.0761	0.14	0.8897
Test: S20_r & a1	-0.0315	0.0432	-0.73	0.4653
Test: SLJ & a1	-0.0297	0.0310	-0.96	0.3394
Test: BPT & a1	0.0805	0.0237	3.40	0.0007
Test: Star_r & Sex: Boys	-0.0802	0.0223	-3.59	0.0003
Test: S20_r & Sex: Boys	-0.0189	0.0128	-1.47	0.1408
Test: SLJ & Sex: Boys	0.0079	0.0090	0.88	0.3815
Test: BPT & Sex: Boys	0.0326	0.0070	4.66	<1e-05
a1 & Sex: Boys	0.0155	0.0483	0.32	0.7483
Test: Star_r & a1 & Sex: Boys	0.0418	0.0761	0.55	0.5827
Test: S20_r & a1 & Sex: Boys	0.0095	0.0432	0.22	0.8264
Test: SLJ & a1 & Sex: Boys	0.0171	0.0310	0.55	0.5826
Test: BPT & a1 & Sex: Boys	-0.0254	0.0237	-1.07	0.2833
Residual	0.7344

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)	-0.0201	0.0142	-1.42	0.1570	0.6386
Test: BPT-other	-0.0751	0.1400	-0.54	0.5918
Test: Star-End	-0.0036	0.0447	-0.08	0.9361
Test: S20-Star	0.0090	0.0446	0.20	0.8398
Test: SLJ-S20	0.0034	0.0441	0.08	0.9381
a1	0.3176	0.0483	6.58	<1e-10
Sex: Boys	0.1800	0.0142	12.66	<1e-36
Test: BPT-other & a1	1.6106	0.4731	3.40	0.0007
Test: Star-End & a1	0.0211	0.1523	0.14	0.8897
Test: S20-Star & a1	-0.1051	0.1504	-0.70	0.4848
Test: SLJ-S20 & a1	-0.0556	0.1502	-0.37	0.7112
Test: BPT-other & Sex: Boys	0.6527	0.1400	4.66	<1e-05
Test: Star-End & Sex: Boys	-0.1604	0.0447	-3.59	0.0003
Test: S20-Star & Sex: Boys	0.0236	0.0446	0.53	0.5972
Test: SLJ-S20 & Sex: Boys	0.0693	0.0441	1.57	0.1164
a1 & Sex: Boys	0.0155	0.0483	0.32	0.7483
Test: BPT-other & a1 & Sex: Boys	-0.5076	0.4731	-1.07	0.2833
Test: Star-End & a1 & Sex: Boys	0.0837	0.1523	0.55	0.5827
Test: S20-Star & a1 & Sex: Boys	-0.0134	0.1504	-0.09	0.9288
Test: SLJ-S20 & a1 & Sex: Boys	0.0493	0.1502	0.33	0.7427
Residual	0.7344

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)	-0.0201	0.0142	-1.42	0.1570	0.6386
Test: Star-Run	-0.0036	0.0447	-0.08	0.9361
Test: S20-Star	0.0090	0.0446	0.20	0.8398
Test: SLJ-S20	0.0034	0.0441	0.08	0.9381
Test: BPT-SLJ	-0.0250	0.0442	-0.56	0.5725
a1	0.3176	0.0483	6.58	<1e-10
Sex: Boys	0.1800	0.0142	12.66	<1e-36
Test: Star-Run & a1	0.0211	0.1523	0.14	0.8897
Test: S20-Star & a1	-0.1051	0.1504	-0.70	0.4848
Test: SLJ-S20 & a1	-0.0556	0.1502	-0.37	0.7112
Test: BPT-SLJ & a1	0.4916	0.1511	3.25	0.0011
Test: Star-Run & Sex: Boys	-0.1604	0.0447	-3.59	0.0003
Test: S20-Star & Sex: Boys	0.0236	0.0446	0.53	0.5972
Test: SLJ-S20 & Sex: Boys	0.0693	0.0441	1.57	0.1164
Test: BPT-SLJ & Sex: Boys	0.1395	0.0442	3.15	0.0016
a1 & Sex: Boys	0.0155	0.0483	0.32	0.7483
Test: Star-Run & a1 & Sex: Boys	0.0837	0.1523	0.55	0.5827
Test: S20-Star & a1 & Sex: Boys	-0.0134	0.1504	-0.09	0.9288
Test: SLJ-S20 & a1 & Sex: Boys	0.0493	0.1502	0.33	0.7427
Test: BPT-SLJ & a1 & Sex: Boys	-0.1781	0.1511	-1.18	0.2385
Residual	0.7344

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

Minimizing 11    Time: 0:00:00 ( 9.34 ms/it)
  objective:  13903.836458448579
Minimizing 111    Time: 0:00:00 ( 5.83 ms/it)
  objective:  13834.93947049731
Minimizing 141    Time: 0:00:00 ( 5.52 ms/it)
  objective:  13803.180322444721
Minimizing 164    Time: 0:00:00 ( 5.37 ms/it)
  objective:  13793.40652495341
Minimizing 187    Time: 0:00:00 ( 5.25 ms/it)
  objective:  13780.503256735621
Minimizing 210    Time: 0:00:01 ( 5.16 ms/it)
  objective:  13768.591835734904
Minimizing 233    Time: 0:00:01 ( 5.09 ms/it)
  objective:  13765.291775342273
Minimizing 256    Time: 0:00:01 ( 5.03 ms/it)
  objective:  13758.001695218787
Minimizing 279    Time: 0:00:01 ( 4.98 ms/it)
  objective:  13746.787000956567
Minimizing 302    Time: 0:00:01 ( 4.93 ms/it)
  objective:  13742.423861195399
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  objective:  13737.682923082139
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  objective:  13729.608092977003
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  objective:  13725.669959327919
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  objective:  13720.504531767401
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  objective:  13708.781179437741
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  objective:  13699.242150113314
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  objective:  13651.37543787947
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  objective:  13643.569524615268
Minimizing 509    Time: 0:00:02 ( 4.73 ms/it)
  objective:  13635.076422195016
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  objective:  13631.81883863647
Minimizing 555    Time: 0:00:02 ( 4.70 ms/it)
  objective:  13629.006307168667
Minimizing 578    Time: 0:00:02 ( 4.69 ms/it)
  objective:  13613.850664516474
Minimizing 601    Time: 0:00:02 ( 4.68 ms/it)
  objective:  13549.103670206081
Minimizing 624    Time: 0:00:02 ( 4.67 ms/it)
  objective:  13511.861908488965
Minimizing 647    Time: 0:00:03 ( 4.66 ms/it)
  objective:  13470.875118310913
Minimizing 670    Time: 0:00:03 ( 4.65 ms/it)
  objective:  13468.246512874044
Minimizing 693    Time: 0:00:03 ( 4.65 ms/it)
  objective:  13462.957171247006
Minimizing 716    Time: 0:00:03 ( 4.64 ms/it)
  objective:  13460.579024145263
Minimizing 739    Time: 0:00:03 ( 4.63 ms/it)
  objective:  13446.434734343246
Minimizing 762    Time: 0:00:03 ( 4.63 ms/it)
  objective:  13408.566653775211
Minimizing 785    Time: 0:00:03 ( 4.62 ms/it)
  objective:  13363.939997350986
Minimizing 808    Time: 0:00:03 ( 4.61 ms/it)
  objective:  13304.133667036775
Minimizing 831    Time: 0:00:03 ( 4.61 ms/it)
  objective:  13303.560846995708
Minimizing 854    Time: 0:00:03 ( 4.60 ms/it)
  objective:  13303.626580053082
Minimizing 877    Time: 0:00:04 ( 4.62 ms/it)
  objective:  13303.72923733812
Minimizing 900    Time: 0:00:04 ( 4.62 ms/it)
  objective:  13303.356774817235
Minimizing 915    Time: 0:00:04 ( 4.61 ms/it)

	Est.	SE	z	p	σ_Child
(Intercept)	-0.0204	0.0144	-1.41	0.1574	0.8255
Test: Star-Run	0.0026	0.0429	0.06	0.9515	0.2611
Test: S20-Star	0.0120	0.0427	0.28	0.7781	0.7761
Test: SLJ-S20	0.0027	0.0418	0.06	0.9488	0.6457
Test: BPT-SLJ	-0.0211	0.0440	-0.48	0.6306	0.4497
a1	0.3158	0.0489	6.46	<1e-09
Sex: Boys	0.1793	0.0144	12.44	<1e-34
Test: Star-Run & a1	0.0766	0.1491	0.51	0.6073
Test: S20-Star & a1	-0.1333	0.1450	-0.92	0.3577
Test: SLJ-S20 & a1	-0.0333	0.1422	-0.23	0.8151
Test: BPT-SLJ & a1	0.4508	0.1510	2.98	0.0028
Test: Star-Run & Sex: Boys	-0.1520	0.0429	-3.54	0.0004
Test: S20-Star & Sex: Boys	0.0372	0.0427	0.87	0.3835
Test: SLJ-S20 & Sex: Boys	0.0539	0.0418	1.29	0.1972
Test: BPT-SLJ & Sex: Boys	0.1366	0.0440	3.11	0.0019
a1 & Sex: Boys	0.0152	0.0489	0.31	0.7556
Test: Star-Run & a1 & Sex: Boys	0.1884	0.1491	1.26	0.2063
Test: S20-Star & a1 & Sex: Boys	-0.1017	0.1450	-0.70	0.4828
Test: SLJ-S20 & a1 & Sex: Boys	0.0426	0.1422	0.30	0.7643
Test: BPT-SLJ & a1 & Sex: Boys	-0.0983	0.1510	-0.65	0.5151
Residual	0.0000

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

Minimizing 16    Time: 0:00:00 ( 6.31 ms/it)
  objective:  14114.255837490666
Minimizing 32    Time: 0:00:00 ( 6.32 ms/it)
  objective:  13888.040142669117
Minimizing 48    Time: 0:00:00 ( 6.35 ms/it)
  objective:  13886.793092769041
Minimizing 64    Time: 0:00:00 ( 6.39 ms/it)
  objective:  13884.605187532892
Minimizing 80    Time: 0:00:00 ( 6.38 ms/it)
  objective:  13880.885654252312
Minimizing 96    Time: 0:00:00 ( 6.38 ms/it)
  objective:  13878.640970509818
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  objective:  13873.122906189885
Minimizing 128    Time: 0:00:00 ( 6.40 ms/it)
  objective:  13864.008010163656
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  objective:  13854.633876772987
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  objective:  13848.166273271261
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  objective:  13839.122497133954
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  objective:  13833.581160156167
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  objective:  13810.681218086407
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  objective:  13808.311974604858
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  objective:  13807.45010798278
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  objective:  13806.337584823908
Minimizing 272    Time: 0:00:01 ( 6.41 ms/it)
  objective:  13805.618494179158
Minimizing 288    Time: 0:00:01 ( 6.41 ms/it)
  objective:  13803.717130106701
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  objective:  13798.941189971676
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  objective:  13796.889054957657
Minimizing 336    Time: 0:00:02 ( 6.40 ms/it)
  objective:  13794.902014742618
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  objective:  13793.759005176187
Minimizing 368    Time: 0:00:02 ( 6.41 ms/it)
  objective:  13781.055386551037
Minimizing 384    Time: 0:00:02 ( 6.41 ms/it)
  objective:  13778.581060282144
Minimizing 400    Time: 0:00:02 ( 6.40 ms/it)
  objective:  13777.023213695746
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  objective:  13774.515388679194
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  objective:  13773.827489983194
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  objective:  13772.762344621202
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  objective:  13771.460847641194
Minimizing 480    Time: 0:00:03 ( 6.39 ms/it)
  objective:  13770.473503276471
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  objective:  13770.218490772371
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  objective:  13769.505627428704
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  objective:  13768.838710695418
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  objective:  13768.001351418312
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  objective:  13764.543418804313
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  objective:  13763.359771520607
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  objective:  13758.626922832878
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  objective:  13755.684864534318
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  objective:  13747.47557309849
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  objective:  13745.412205512874
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  objective:  13741.276961650065
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  objective:  13739.028560981436
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  objective:  13733.97766404369
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  objective:  13731.72612610309
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  objective:  13729.567095502629
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  objective:  13725.435781707096
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  objective:  13719.357846948405
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  objective:  13710.477508400756
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  objective:  13709.50826953975
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  objective:  13701.553962264676
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  objective:  13700.66172113661
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  objective:  13698.069737748483
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  objective:  13617.024924271267
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  objective:  13559.685301684833
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  objective:  13522.842529561174
Minimizing 1700    Time: 0:00:10 ( 6.38 ms/it)
  objective:  13522.660715117541
Minimizing 1716    Time: 0:00:10 ( 6.38 ms/it)
  objective:  13522.42746728992
Minimizing 1732    Time: 0:00:11 ( 6.38 ms/it)
  objective:  13522.25831525668
Minimizing 1748    Time: 0:00:11 ( 6.38 ms/it)
  objective:  13522.078731766509
Minimizing 1764    Time: 0:00:11 ( 6.38 ms/it)
  objective:  13520.573975666557
Minimizing 1780    Time: 0:00:11 ( 6.38 ms/it)
  objective:  13519.409893784825
Minimizing 1796    Time: 0:00:11 ( 6.38 ms/it)
  objective:  13516.934384971355
Minimizing 1812    Time: 0:00:11 ( 6.38 ms/it)
  objective:  13513.852168933816
Minimizing 1828    Time: 0:00:11 ( 6.38 ms/it)
  objective:  13513.692885404911
Minimizing 1844    Time: 0:00:11 ( 6.38 ms/it)
  objective:  13513.359807342378
Minimizing 1860    Time: 0:00:11 ( 6.38 ms/it)
  objective:  13512.326468463609
Minimizing 1876    Time: 0:00:11 ( 6.38 ms/it)
  objective:  13511.89137955226
Minimizing 1892    Time: 0:00:12 ( 6.38 ms/it)
  objective:  13510.678276761311
Minimizing 1908    Time: 0:00:12 ( 6.38 ms/it)
  objective:  13510.597185717757
Minimizing 1924    Time: 0:00:12 ( 6.38 ms/it)
  objective:  13510.005991609025
Minimizing 1940    Time: 0:00:12 ( 6.38 ms/it)
  objective:  13509.758841077746
Minimizing 1956    Time: 0:00:12 ( 6.38 ms/it)
  objective:  13508.417505786565
Minimizing 1972    Time: 0:00:12 ( 6.38 ms/it)
  objective:  13508.139966951421
Minimizing 1988    Time: 0:00:12 ( 6.38 ms/it)
  objective:  13506.91892103714
Minimizing 2004    Time: 0:00:12 ( 6.38 ms/it)
  objective:  13504.720072781121
Minimizing 2020    Time: 0:00:12 ( 6.38 ms/it)
  objective:  13501.893917283
Minimizing 2036    Time: 0:00:12 ( 6.38 ms/it)
  objective:  13498.685340638542
Minimizing 2052    Time: 0:00:13 ( 6.38 ms/it)
  objective:  13494.388352352456
Minimizing 2068    Time: 0:00:13 ( 6.37 ms/it)
  objective:  13475.030639351375
Minimizing 2084    Time: 0:00:13 ( 6.37 ms/it)
  objective:  13471.029819515985
Minimizing 2100    Time: 0:00:13 ( 6.37 ms/it)
  objective:  13466.284410792796
Minimizing 2116    Time: 0:00:13 ( 6.37 ms/it)
  objective:  13462.856176619898
Minimizing 2132    Time: 0:00:13 ( 6.37 ms/it)
  objective:  13448.49569228779
Minimizing 2148    Time: 0:00:13 ( 6.37 ms/it)
  objective:  13443.604396257564
Minimizing 2164    Time: 0:00:13 ( 6.37 ms/it)
  objective:  13441.32989058271
Minimizing 2180    Time: 0:00:13 ( 6.37 ms/it)
  objective:  13440.759679729192
Minimizing 2196    Time: 0:00:13 ( 6.37 ms/it)
  objective:  13440.221397423258
Minimizing 2212    Time: 0:00:14 ( 6.37 ms/it)
  objective:  13439.30996229405
Minimizing 2228    Time: 0:00:14 ( 6.37 ms/it)
  objective:  13438.520200098108
Minimizing 2244    Time: 0:00:14 ( 6.37 ms/it)
  objective:  13437.585572092416
Minimizing 2260    Time: 0:00:14 ( 6.37 ms/it)
  objective:  13435.95408939659
Minimizing 2277    Time: 0:00:14 ( 6.37 ms/it)
  objective:  13435.658861035728
Minimizing 2294    Time: 0:00:14 ( 6.37 ms/it)
  objective:  13434.566598619698
Minimizing 2310    Time: 0:00:14 ( 6.37 ms/it)
  objective:  13433.205542982469
Minimizing 2327    Time: 0:00:14 ( 6.37 ms/it)
  objective:  13432.584061262838
Minimizing 2343    Time: 0:00:14 ( 6.37 ms/it)
  objective:  13432.064932606809
Minimizing 2359    Time: 0:00:15 ( 6.37 ms/it)
  objective:  13429.98632201193
Minimizing 2375    Time: 0:00:15 ( 6.37 ms/it)
  objective:  13429.46347485276
Minimizing 2391    Time: 0:00:15 ( 6.37 ms/it)
  objective:  13429.195305147601
Minimizing 2407    Time: 0:00:15 ( 6.37 ms/it)
  objective:  13428.886815961392
Minimizing 2423    Time: 0:00:15 ( 6.37 ms/it)
  objective:  13427.522406682794
Minimizing 2440    Time: 0:00:15 ( 6.37 ms/it)
  objective:  13427.389515978197
Minimizing 2456    Time: 0:00:15 ( 6.37 ms/it)
  objective:  13426.914782829
Minimizing 2472    Time: 0:00:15 ( 6.37 ms/it)
  objective:  13426.637543531018
Minimizing 2488    Time: 0:00:15 ( 6.37 ms/it)
  objective:  13425.886929683067
Minimizing 2504    Time: 0:00:15 ( 6.37 ms/it)
  objective:  13425.80330065597
Minimizing 2520    Time: 0:00:16 ( 6.37 ms/it)
  objective:  13425.065453862568
Minimizing 2536    Time: 0:00:16 ( 6.37 ms/it)
  objective:  13424.269881881773
Minimizing 2552    Time: 0:00:16 ( 6.37 ms/it)
  objective:  13423.004297054067
Minimizing 2568    Time: 0:00:16 ( 6.37 ms/it)
  objective:  13421.173571606312
Minimizing 2584    Time: 0:00:16 ( 6.37 ms/it)
  objective:  13421.048990521012
Minimizing 2600    Time: 0:00:16 ( 6.37 ms/it)
  objective:  13419.500115684685
Minimizing 2616    Time: 0:00:16 ( 6.37 ms/it)
  objective:  13419.474797695526
Minimizing 2632    Time: 0:00:16 ( 6.37 ms/it)
  objective:  13419.473698453163
Minimizing 2648    Time: 0:00:16 ( 6.37 ms/it)
  objective:  13419.47370292453
Minimizing 2664    Time: 0:00:16 ( 6.37 ms/it)
  objective:  13419.473307260952
Minimizing 2680    Time: 0:00:17 ( 6.37 ms/it)
  objective:  13419.473395809546
Minimizing 2696    Time: 0:00:17 ( 6.37 ms/it)
  objective:  13419.473187002252
Minimizing 2712    Time: 0:00:17 ( 6.37 ms/it)
  objective:  13419.473267370486
Minimizing 2728    Time: 0:00:17 ( 6.37 ms/it)
  objective:  13419.47322939962
Minimizing 2744    Time: 0:00:17 ( 6.37 ms/it)
  objective:  13419.47326359243
Minimizing 2760    Time: 0:00:17 ( 6.37 ms/it)
  objective:  13419.47337249697
Minimizing 2776    Time: 0:00:17 ( 6.37 ms/it)
  objective:  13419.473252080003
Minimizing 2792    Time: 0:00:17 ( 6.37 ms/it)
  objective:  13419.473214463724
Minimizing 2808    Time: 0:00:17 ( 6.37 ms/it)
  objective:  13419.473222240107
Minimizing 2822    Time: 0:00:17 ( 6.37 ms/it)

	Est.	SE	z	p	σ_Child
(Intercept)	-0.0202	0.0142	-1.42	0.1544
Test: Star-Run	0.0021	0.0441	0.05	0.9616
Test: S20-Star	0.0046	0.0455	0.10	0.9194
Test: SLJ-S20	0.0069	0.0446	0.16	0.8761
Test: BPT-SLJ	-0.0270	0.0433	-0.62	0.5331
a1	0.3158	0.0481	6.57	<1e-10
Sex: Boys	0.1808	0.0142	12.76	<1e-36
Test: Star-Run & a1	0.0253	0.1513	0.17	0.8670
Test: S20-Star & a1	-0.1190	0.1533	-0.78	0.4379
Test: SLJ-S20 & a1	-0.0387	0.1514	-0.26	0.7982
Test: BPT-SLJ & a1	0.4872	0.1482	3.29	0.0010
Test: Star-Run & Sex: Boys	-0.1496	0.0441	-3.39	0.0007
Test: S20-Star & Sex: Boys	0.0167	0.0455	0.37	0.7134
Test: SLJ-S20 & Sex: Boys	0.0650	0.0446	1.46	0.1445
Test: BPT-SLJ & Sex: Boys	0.1433	0.0433	3.31	0.0009
a1 & Sex: Boys	0.0141	0.0481	0.29	0.7696
Test: Star-Run & a1 & Sex: Boys	0.1078	0.1513	0.71	0.4761
Test: S20-Star & a1 & Sex: Boys	-0.0261	0.1533	-0.17	0.8648
Test: SLJ-S20 & a1 & Sex: Boys	0.0687	0.1514	0.45	0.6501
Test: BPT-SLJ & a1 & Sex: Boys	-0.1915	0.1482	-1.29	0.1962
Test: BPT					0.9251
Test: SLJ					0.9786
Test: Star_r					0.9961
Test: Run					0.9648
Test: S20_r					0.9885
Residual	0.0001

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 16    Time: 0:00:00 ( 6.42 ms/it)
  objective:  14114.255837490666
Minimizing 32    Time: 0:00:00 ( 6.35 ms/it)
  objective:  13888.040142669117
Minimizing 48    Time: 0:00:00 ( 6.34 ms/it)
  objective:  13886.793092769041
Minimizing 64    Time: 0:00:00 ( 6.32 ms/it)
  objective:  13884.605187532892
Minimizing 80    Time: 0:00:00 ( 6.31 ms/it)
  objective:  13880.885654252312
Minimizing 96    Time: 0:00:00 ( 6.31 ms/it)
  objective:  13878.640970509818
Minimizing 112    Time: 0:00:00 ( 6.31 ms/it)
  objective:  13873.122906189885
Minimizing 128    Time: 0:00:00 ( 6.31 ms/it)
  objective:  13864.008010163656
Minimizing 144    Time: 0:00:00 ( 6.31 ms/it)
  objective:  13854.633876772987
Minimizing 160    Time: 0:00:01 ( 6.33 ms/it)
  objective:  13848.166273271261
Minimizing 176    Time: 0:00:01 ( 6.35 ms/it)
  objective:  13839.122497133954
Minimizing 192    Time: 0:00:01 ( 6.35 ms/it)
  objective:  13833.581160156167
Minimizing 208    Time: 0:00:01 ( 6.36 ms/it)
  objective:  13810.681218086407
Minimizing 224    Time: 0:00:01 ( 6.36 ms/it)
  objective:  13808.311974604858
Minimizing 240    Time: 0:00:01 ( 6.36 ms/it)
  objective:  13807.45010798278
Minimizing 256    Time: 0:00:01 ( 6.36 ms/it)
  objective:  13806.337584823908
Minimizing 272    Time: 0:00:01 ( 6.36 ms/it)
  objective:  13805.618494179158
Minimizing 288    Time: 0:00:01 ( 6.36 ms/it)
  objective:  13803.717130106701
Minimizing 304    Time: 0:00:01 ( 6.36 ms/it)
  objective:  13798.941189971676
Minimizing 320    Time: 0:00:02 ( 6.36 ms/it)
  objective:  13796.889054957657
Minimizing 336    Time: 0:00:02 ( 6.36 ms/it)
  objective:  13794.902014742618
Minimizing 352    Time: 0:00:02 ( 6.36 ms/it)
  objective:  13793.759005176187
Minimizing 368    Time: 0:00:02 ( 6.37 ms/it)
  objective:  13781.055386551037
Minimizing 384    Time: 0:00:02 ( 6.37 ms/it)
  objective:  13778.581060282144
Minimizing 400    Time: 0:00:02 ( 6.37 ms/it)
  objective:  13777.023213695746
Minimizing 416    Time: 0:00:02 ( 6.37 ms/it)
  objective:  13774.515388679194
Minimizing 432    Time: 0:00:02 ( 6.37 ms/it)
  objective:  13773.827489983194
Minimizing 448    Time: 0:00:02 ( 6.37 ms/it)
  objective:  13772.762344621202
Minimizing 464    Time: 0:00:02 ( 6.37 ms/it)
  objective:  13771.460847641194
Minimizing 480    Time: 0:00:03 ( 6.37 ms/it)
  objective:  13770.473503276471
Minimizing 496    Time: 0:00:03 ( 6.37 ms/it)
  objective:  13770.218490772371
Minimizing 512    Time: 0:00:03 ( 6.36 ms/it)
  objective:  13769.505627428704
Minimizing 528    Time: 0:00:03 ( 6.36 ms/it)
  objective:  13768.838710695418
Minimizing 544    Time: 0:00:03 ( 6.36 ms/it)
  objective:  13768.001351418312
Minimizing 560    Time: 0:00:03 ( 6.36 ms/it)
  objective:  13764.543418804313
Minimizing 576    Time: 0:00:03 ( 6.36 ms/it)
  objective:  13763.359771520607
Minimizing 592    Time: 0:00:03 ( 6.36 ms/it)
  objective:  13758.626922832878
Minimizing 608    Time: 0:00:03 ( 6.36 ms/it)
  objective:  13755.684864534318
Minimizing 624    Time: 0:00:03 ( 6.36 ms/it)
  objective:  13747.47557309849
Minimizing 640    Time: 0:00:04 ( 6.36 ms/it)
  objective:  13745.412205512874
Minimizing 656    Time: 0:00:04 ( 6.36 ms/it)
  objective:  13741.276961650065
Minimizing 672    Time: 0:00:04 ( 6.36 ms/it)
  objective:  13739.028560981436
Minimizing 688    Time: 0:00:04 ( 6.36 ms/it)
  objective:  13733.97766404369
Minimizing 704    Time: 0:00:04 ( 6.36 ms/it)
  objective:  13731.72612610309
Minimizing 720    Time: 0:00:04 ( 6.36 ms/it)
  objective:  13729.567095502629
Minimizing 736    Time: 0:00:04 ( 6.36 ms/it)
  objective:  13725.435781707096
Minimizing 752    Time: 0:00:04 ( 6.36 ms/it)
  objective:  13723.326591177567
Minimizing 768    Time: 0:00:04 ( 6.36 ms/it)
  objective:  13721.156459806076
Minimizing 784    Time: 0:00:04 ( 6.36 ms/it)
  objective:  13716.546688714207
Minimizing 800    Time: 0:00:05 ( 6.36 ms/it)
  objective:  13710.557043846038
Minimizing 816    Time: 0:00:05 ( 6.36 ms/it)
  objective:  13709.692263156001
Minimizing 832    Time: 0:00:05 ( 6.36 ms/it)
  objective:  13708.695389477769
Minimizing 848    Time: 0:00:05 ( 6.36 ms/it)
  objective:  13705.581680909298
Minimizing 864    Time: 0:00:05 ( 6.36 ms/it)
  objective:  13704.515227024804
Minimizing 880    Time: 0:00:05 ( 6.36 ms/it)
  objective:  13701.611812698276
Minimizing 896    Time: 0:00:05 ( 6.36 ms/it)
  objective:  13701.199151318546
Minimizing 912    Time: 0:00:05 ( 6.36 ms/it)
  objective:  13699.45322346087
Minimizing 928    Time: 0:00:05 ( 6.36 ms/it)
  objective:  13696.485347688005
Minimizing 944    Time: 0:00:06 ( 6.36 ms/it)
  objective:  13691.1106480907
Minimizing 960    Time: 0:00:06 ( 6.36 ms/it)
  objective:  13687.842331049076
Minimizing 976    Time: 0:00:06 ( 6.37 ms/it)
  objective:  13686.544824008914
Minimizing 992    Time: 0:00:06 ( 6.37 ms/it)
  objective:  13685.246405741498
Minimizing 1008    Time: 0:00:06 ( 6.37 ms/it)
  objective:  13681.459435788569
Minimizing 1024    Time: 0:00:06 ( 6.37 ms/it)
  objective:  13677.017677295644
Minimizing 1040    Time: 0:00:06 ( 6.37 ms/it)
  objective:  13674.4261806957
Minimizing 1056    Time: 0:00:06 ( 6.37 ms/it)
  objective:  13665.888454667685
Minimizing 1072    Time: 0:00:06 ( 6.37 ms/it)
  objective:  13663.646302449019
Minimizing 1088    Time: 0:00:06 ( 6.37 ms/it)
  objective:  13654.790378548063
Minimizing 1104    Time: 0:00:07 ( 6.37 ms/it)
  objective:  13618.658190507726
Minimizing 1120    Time: 0:00:07 ( 6.37 ms/it)
  objective:  13611.502688019653
Minimizing 1136    Time: 0:00:07 ( 6.37 ms/it)
  objective:  13606.79876544921
Minimizing 1152    Time: 0:00:07 ( 6.37 ms/it)
  objective:  13603.196299598523
Minimizing 1168    Time: 0:00:07 ( 6.37 ms/it)
  objective:  13596.703681924148
Minimizing 1184    Time: 0:00:07 ( 6.37 ms/it)
  objective:  13592.649109985156
Minimizing 1200    Time: 0:00:07 ( 6.37 ms/it)
  objective:  13581.951868077143
Minimizing 1216    Time: 0:00:07 ( 6.37 ms/it)
  objective:  13576.50065673433
Minimizing 1232    Time: 0:00:07 ( 6.37 ms/it)
  objective:  13568.850083722027
Minimizing 1248    Time: 0:00:07 ( 6.37 ms/it)
  objective:  13568.6757134695
Minimizing 1264    Time: 0:00:08 ( 6.37 ms/it)
  objective:  13565.87215453633
Minimizing 1280    Time: 0:00:08 ( 6.37 ms/it)
  objective:  13564.09458651815
Minimizing 1296    Time: 0:00:08 ( 6.37 ms/it)
  objective:  13561.03877685414
Minimizing 1312    Time: 0:00:08 ( 6.37 ms/it)
  objective:  13558.919707146932
Minimizing 1328    Time: 0:00:08 ( 6.37 ms/it)
  objective:  13557.051654180155
Minimizing 1344    Time: 0:00:08 ( 6.37 ms/it)
  objective:  13555.325656141657
Minimizing 1360    Time: 0:00:08 ( 6.37 ms/it)
  objective:  13554.907089393266
Minimizing 1376    Time: 0:00:08 ( 6.37 ms/it)
  objective:  13552.724322587863
Minimizing 1392    Time: 0:00:08 ( 6.38 ms/it)
  objective:  13551.298843020311
Minimizing 1408    Time: 0:00:08 ( 6.38 ms/it)
  objective:  13544.344535074262
Minimizing 1424    Time: 0:00:09 ( 6.38 ms/it)
  objective:  13544.145539842939
Minimizing 1440    Time: 0:00:09 ( 6.38 ms/it)
  objective:  13543.28479431101
Minimizing 1456    Time: 0:00:09 ( 6.38 ms/it)
  objective:  13541.711416189828
Minimizing 1472    Time: 0:00:09 ( 6.38 ms/it)
  objective:  13541.10667712667
Minimizing 1488    Time: 0:00:09 ( 6.38 ms/it)
  objective:  13540.027778576288
Minimizing 1506    Time: 0:00:09 ( 6.38 ms/it)
  objective:  13538.026557080506
Minimizing 1524    Time: 0:00:09 ( 6.38 ms/it)
  objective:  13532.064536667676
Minimizing 1540    Time: 0:00:09 ( 6.38 ms/it)
  objective:  13531.697094523493
Minimizing 1556    Time: 0:00:09 ( 6.38 ms/it)
  objective:  13530.989595034203
Minimizing 1572    Time: 0:00:10 ( 6.38 ms/it)
  objective:  13530.24801299021
Minimizing 1588    Time: 0:00:10 ( 6.38 ms/it)
  objective:  13529.449197291753
Minimizing 1604    Time: 0:00:10 ( 6.38 ms/it)
  objective:  13526.545397726892
Minimizing 1620    Time: 0:00:10 ( 6.38 ms/it)
  objective:  13526.357584112346
Minimizing 1636    Time: 0:00:10 ( 6.38 ms/it)
  objective:  13525.82691068593
Minimizing 1652    Time: 0:00:10 ( 6.38 ms/it)
  objective:  13525.462311212847
Minimizing 1668    Time: 0:00:10 ( 6.38 ms/it)
  objective:  13525.24217557523
Minimizing 1684    Time: 0:00:10 ( 6.38 ms/it)
  objective:  13522.842529561174
Minimizing 1700    Time: 0:00:10 ( 6.38 ms/it)
  objective:  13522.660715117541
Minimizing 1716    Time: 0:00:10 ( 6.38 ms/it)
  objective:  13522.42746728992
Minimizing 1732    Time: 0:00:11 ( 6.38 ms/it)
  objective:  13522.25831525668
Minimizing 1748    Time: 0:00:11 ( 6.38 ms/it)
  objective:  13522.078731766509
Minimizing 1764    Time: 0:00:11 ( 6.38 ms/it)
  objective:  13520.573975666557
Minimizing 1780    Time: 0:00:11 ( 6.38 ms/it)
  objective:  13519.409893784825
Minimizing 1796    Time: 0:00:11 ( 6.38 ms/it)
  objective:  13516.934384971355
Minimizing 1812    Time: 0:00:11 ( 6.38 ms/it)
  objective:  13513.852168933816
Minimizing 1828    Time: 0:00:11 ( 6.38 ms/it)
  objective:  13513.692885404911
Minimizing 1844    Time: 0:00:11 ( 6.38 ms/it)
  objective:  13513.359807342378
Minimizing 1860    Time: 0:00:11 ( 6.38 ms/it)
  objective:  13512.326468463609
Minimizing 1876    Time: 0:00:11 ( 6.38 ms/it)
  objective:  13511.89137955226
Minimizing 1892    Time: 0:00:12 ( 6.38 ms/it)
  objective:  13510.678276761311
Minimizing 1908    Time: 0:00:12 ( 6.37 ms/it)
  objective:  13510.597185717757
Minimizing 1924    Time: 0:00:12 ( 6.37 ms/it)
  objective:  13510.005991609025
Minimizing 1940    Time: 0:00:12 ( 6.37 ms/it)
  objective:  13509.758841077746
Minimizing 1956    Time: 0:00:12 ( 6.37 ms/it)
  objective:  13508.417505786565
Minimizing 1972    Time: 0:00:12 ( 6.37 ms/it)
  objective:  13508.139966951421
Minimizing 1988    Time: 0:00:12 ( 6.37 ms/it)
  objective:  13506.91892103714
Minimizing 2004    Time: 0:00:12 ( 6.37 ms/it)
  objective:  13504.720072781121
Minimizing 2020    Time: 0:00:12 ( 6.37 ms/it)
  objective:  13501.893917283
Minimizing 2037    Time: 0:00:12 ( 6.37 ms/it)
  objective:  13498.55835684936
Minimizing 2053    Time: 0:00:13 ( 6.37 ms/it)
  objective:  13492.861469826428
Minimizing 2069    Time: 0:00:13 ( 6.37 ms/it)
  objective:  13472.62510417703
Minimizing 2085    Time: 0:00:13 ( 6.37 ms/it)
  objective:  13470.143488838512
Minimizing 2101    Time: 0:00:13 ( 6.37 ms/it)
  objective:  13465.90331305188
Minimizing 2117    Time: 0:00:13 ( 6.37 ms/it)
  objective:  13462.069807859254
Minimizing 2133    Time: 0:00:13 ( 6.37 ms/it)
  objective:  13456.424154740103
Minimizing 2149    Time: 0:00:13 ( 6.37 ms/it)
  objective:  13443.77215349373
Minimizing 2165    Time: 0:00:13 ( 6.37 ms/it)
  objective:  13441.170696626985
Minimizing 2181    Time: 0:00:13 ( 6.37 ms/it)
  objective:  13440.71911347247
Minimizing 2197    Time: 0:00:13 ( 6.37 ms/it)
  objective:  13440.190142923748
Minimizing 2213    Time: 0:00:14 ( 6.37 ms/it)
  objective:  13439.209282835596
Minimizing 2229    Time: 0:00:14 ( 6.37 ms/it)
  objective:  13438.494583086635
Minimizing 2245    Time: 0:00:14 ( 6.37 ms/it)
  objective:  13437.538455738104
Minimizing 2261    Time: 0:00:14 ( 6.37 ms/it)
  objective:  13435.79037532468
Minimizing 2277    Time: 0:00:14 ( 6.37 ms/it)
  objective:  13435.658861035728
Minimizing 2293    Time: 0:00:14 ( 6.37 ms/it)
  objective:  13434.640655471478
Minimizing 2309    Time: 0:00:14 ( 6.36 ms/it)
  objective:  13433.247322296462
Minimizing 2325    Time: 0:00:14 ( 6.36 ms/it)
  objective:  13432.652307338896
Minimizing 2341    Time: 0:00:14 ( 6.36 ms/it)
  objective:  13432.19915909374
Minimizing 2357    Time: 0:00:14 ( 6.36 ms/it)
  objective:  13430.253112717313
Minimizing 2373    Time: 0:00:15 ( 6.36 ms/it)
  objective:  13429.463867911545
Minimizing 2389    Time: 0:00:15 ( 6.36 ms/it)
  objective:  13429.260241662967
Minimizing 2405    Time: 0:00:15 ( 6.36 ms/it)
  objective:  13428.990209295254
Minimizing 2421    Time: 0:00:15 ( 6.36 ms/it)
  objective:  13427.60269614379
Minimizing 2437    Time: 0:00:15 ( 6.36 ms/it)
  objective:  13427.400851618091
Minimizing 2453    Time: 0:00:15 ( 6.36 ms/it)
  objective:  13426.978909664089
Minimizing 2469    Time: 0:00:15 ( 6.36 ms/it)
  objective:  13426.696145511538
Minimizing 2485    Time: 0:00:15 ( 6.36 ms/it)
  objective:  13425.917325875605
Minimizing 2501    Time: 0:00:15 ( 6.36 ms/it)
  objective:  13425.829405077267
Minimizing 2517    Time: 0:00:16 ( 6.36 ms/it)
  objective:  13425.106727021499
Minimizing 2533    Time: 0:00:16 ( 6.36 ms/it)
  objective:  13424.395279418226
Minimizing 2549    Time: 0:00:16 ( 6.36 ms/it)
  objective:  13423.030530493124
Minimizing 2566    Time: 0:00:16 ( 6.36 ms/it)
  objective:  13421.257273031442
Minimizing 2582    Time: 0:00:16 ( 6.36 ms/it)
  objective:  13421.072418729338
Minimizing 2598    Time: 0:00:16 ( 6.36 ms/it)
  objective:  13419.50594218033
Minimizing 2614    Time: 0:00:16 ( 6.36 ms/it)
  objective:  13419.475536317419
Minimizing 2630    Time: 0:00:16 ( 6.36 ms/it)
  objective:  13419.473966114005
Minimizing 2646    Time: 0:00:16 ( 6.36 ms/it)
  objective:  13419.47356471009
Minimizing 2662    Time: 0:00:16 ( 6.36 ms/it)
  objective:  13419.47334586455
Minimizing 2678    Time: 0:00:17 ( 6.36 ms/it)
  objective:  13419.47336779043
Minimizing 2694    Time: 0:00:17 ( 6.36 ms/it)
  objective:  13419.473187581025
Minimizing 2710    Time: 0:00:17 ( 6.36 ms/it)
  objective:  13419.473338800468
Minimizing 2726    Time: 0:00:17 ( 6.36 ms/it)
  objective:  13419.473208301337
Minimizing 2743    Time: 0:00:17 ( 6.36 ms/it)
  objective:  13419.473261730353
Minimizing 2759    Time: 0:00:17 ( 6.35 ms/it)
  objective:  13419.473119695744
Minimizing 2775    Time: 0:00:17 ( 6.35 ms/it)
  objective:  13419.47325857279
Minimizing 2791    Time: 0:00:17 ( 6.35 ms/it)
  objective:  13419.473309875495
Minimizing 2807    Time: 0:00:17 ( 6.35 ms/it)
  objective:  13419.473309878013
Minimizing 2822    Time: 0:00:17 ( 6.36 ms/it)

	Est.	SE	z	p	σ_Child
(Intercept)	-0.0202	0.0142	-1.42	0.1544
Test: Star-Run	0.0021	0.0441	0.05	0.9616
Test: S20-Star	0.0046	0.0455	0.10	0.9194
Test: SLJ-S20	0.0069	0.0446	0.16	0.8761
Test: BPT-SLJ	-0.0270	0.0433	-0.62	0.5331
a1	0.3158	0.0481	6.57	<1e-10
Sex: Boys	0.1808	0.0142	12.76	<1e-36
Test: Star-Run & a1	0.0253	0.1513	0.17	0.8670
Test: S20-Star & a1	-0.1190	0.1533	-0.78	0.4379
Test: SLJ-S20 & a1	-0.0387	0.1514	-0.26	0.7982
Test: BPT-SLJ & a1	0.4872	0.1482	3.29	0.0010
Test: Star-Run & Sex: Boys	-0.1496	0.0441	-3.39	0.0007
Test: S20-Star & Sex: Boys	0.0167	0.0455	0.37	0.7134
Test: SLJ-S20 & Sex: Boys	0.0650	0.0446	1.46	0.1445
Test: BPT-SLJ & Sex: Boys	0.1433	0.0433	3.31	0.0009
a1 & Sex: Boys	0.0141	0.0481	0.29	0.7696
Test: Star-Run & a1 & Sex: Boys	0.1078	0.1513	0.71	0.4761
Test: S20-Star & a1 & Sex: Boys	-0.0261	0.1533	-0.17	0.8648
Test: SLJ-S20 & a1 & Sex: Boys	0.0687	0.1514	0.45	0.6501
Test: BPT-SLJ & a1 & Sex: Boys	-0.1915	0.1482	-1.29	0.1962
Test: BPT					0.9251
Test: SLJ					0.9786
Test: Star_r					0.9961
Test: Run					0.9648
Test: S20_r					0.9885
Residual	0.0001

VarCorr(m_cpx_0_SeqDiff)

	Column	Variance	Std.Dev	Corr.
Child	Test: Run	0.93090768	0.96483557
	Test: Star_r	0.99211591	0.99605015	+0.91
	Test: S20_r	0.97715468	0.98851135	+0.39	+0.14
	Test: SLJ	0.95756803	0.97855405	+0.10	+0.22	+0.58
	Test: BPT	0.85582603	0.92510866	+0.01	+0.15	+0.05	+0.45
Residual		0.00000001	0.00009214

m_cpx_0_SeqDiff.PCA

(Child = 
Principal components based on correlation matrix
 Test: Run     1.0    .     .     .     .
 Test: Star_r  0.91  1.0    .     .     .
 Test: S20_r   0.39  0.14  1.0    .     .
 Test: SLJ     0.1   0.22  0.58  1.0    .
 Test: BPT     0.01  0.15  0.05  0.45  1.0

Normalized cumulative variances:
[0.4535, 0.7404, 0.934, 1.0, 1.0]

Component loadings
                 PC1    PC2    PC3    PC4    PC5
 Test: Run     -0.55   0.45  -0.01   0.22   0.66
 Test: Star_r  -0.54   0.4    0.28  -0.3   -0.62
 Test: S20_r   -0.42  -0.28  -0.65   0.46  -0.33
 Test: SLJ     -0.41  -0.57  -0.08  -0.65   0.27
 Test: BPT     -0.23  -0.48   0.7    0.47  -0.01,)

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 16    Time: 0:00:00 ( 6.45 ms/it)
  objective:  14024.811353037934
Minimizing 32    Time: 0:00:00 ( 6.42 ms/it)
  objective:  13924.173486113275
Minimizing 48    Time: 0:00:00 ( 6.49 ms/it)
  objective:  13865.195235835401
Minimizing 64    Time: 0:00:00 ( 6.48 ms/it)
  objective:  13855.59111137833
Minimizing 80    Time: 0:00:00 ( 6.51 ms/it)
  objective:  13832.389866768177
Minimizing 96    Time: 0:00:00 ( 6.49 ms/it)
  objective:  13829.216723079711
Minimizing 112    Time: 0:00:00 ( 6.48 ms/it)
  objective:  13816.799452020403
Minimizing 128    Time: 0:00:00 ( 6.47 ms/it)
  objective:  13810.106245972118
Minimizing 144    Time: 0:00:00 ( 6.46 ms/it)
  objective:  13806.035220530648
Minimizing 160    Time: 0:00:01 ( 6.46 ms/it)
  objective:  13803.436194042464
Minimizing 176    Time: 0:00:01 ( 6.46 ms/it)
  objective:  13797.343827064724
Minimizing 192    Time: 0:00:01 ( 6.46 ms/it)
  objective:  13794.936916576891
Minimizing 208    Time: 0:00:01 ( 6.46 ms/it)
  objective:  13789.61267160035
Minimizing 224    Time: 0:00:01 ( 6.46 ms/it)
  objective:  13787.58123911373
Minimizing 240    Time: 0:00:01 ( 6.45 ms/it)
  objective:  13780.482054511822
Minimizing 256    Time: 0:00:01 ( 6.45 ms/it)
  objective:  13775.61897878881
Minimizing 272    Time: 0:00:01 ( 6.45 ms/it)
  objective:  13758.035928020026
Minimizing 288    Time: 0:00:01 ( 6.44 ms/it)
  objective:  13750.321819907771
Minimizing 304    Time: 0:00:01 ( 6.44 ms/it)
  objective:  13686.37251752197
Minimizing 320    Time: 0:00:02 ( 6.43 ms/it)
  objective:  13686.085367152304
Minimizing 336    Time: 0:00:02 ( 6.43 ms/it)
  objective:  13679.838434752895
Minimizing 352    Time: 0:00:02 ( 6.43 ms/it)
  objective:  13670.649397342138
Minimizing 368    Time: 0:00:02 ( 6.43 ms/it)
  objective:  13661.183603822774
Minimizing 384    Time: 0:00:02 ( 6.42 ms/it)
  objective:  13657.274135823904
Minimizing 400    Time: 0:00:02 ( 6.42 ms/it)
  objective:  13636.609520525068
Minimizing 416    Time: 0:00:02 ( 6.42 ms/it)
  objective:  13628.879017406369
Minimizing 432    Time: 0:00:02 ( 6.45 ms/it)
  objective:  13628.446889834559
Minimizing 448    Time: 0:00:02 ( 6.44 ms/it)
  objective:  13626.65078940594
Minimizing 464    Time: 0:00:02 ( 6.44 ms/it)
  objective:  13625.000481995841
Minimizing 480    Time: 0:00:03 ( 6.44 ms/it)
  objective:  13622.202627835897
Minimizing 496    Time: 0:00:03 ( 6.44 ms/it)
  objective:  13616.505980841372
Minimizing 512    Time: 0:00:03 ( 6.44 ms/it)
  objective:  13614.924745476375
Minimizing 528    Time: 0:00:03 ( 6.44 ms/it)
  objective:  13614.566408613056
Minimizing 544    Time: 0:00:03 ( 6.44 ms/it)
  objective:  13611.424709523642
Minimizing 560    Time: 0:00:03 ( 6.43 ms/it)
  objective:  13609.817041164817
Minimizing 576    Time: 0:00:03 ( 6.43 ms/it)
  objective:  13607.555899118772
Minimizing 592    Time: 0:00:03 ( 6.43 ms/it)
  objective:  13605.86900086105
Minimizing 608    Time: 0:00:03 ( 6.43 ms/it)
  objective:  13603.101985367328
Minimizing 624    Time: 0:00:04 ( 6.43 ms/it)
  objective:  13599.300220054662
Minimizing 640    Time: 0:00:04 ( 6.43 ms/it)
  objective:  13596.83181452316
Minimizing 656    Time: 0:00:04 ( 6.42 ms/it)
  objective:  13594.029529099833
Minimizing 672    Time: 0:00:04 ( 6.42 ms/it)
  objective:  13591.186600783338
Minimizing 688    Time: 0:00:04 ( 6.42 ms/it)
  objective:  13562.584364625342
Minimizing 704    Time: 0:00:04 ( 6.42 ms/it)
  objective:  13560.23366492967
Minimizing 720    Time: 0:00:04 ( 6.42 ms/it)
  objective:  13555.506332108118
Minimizing 736    Time: 0:00:04 ( 6.42 ms/it)
  objective:  13530.067567872342
Minimizing 752    Time: 0:00:04 ( 6.42 ms/it)
  objective:  13527.48021714609
Minimizing 768    Time: 0:00:04 ( 6.42 ms/it)
  objective:  13522.250988840453
Minimizing 784    Time: 0:00:05 ( 6.43 ms/it)
  objective:  13520.572428970583
Minimizing 800    Time: 0:00:05 ( 6.43 ms/it)
  objective:  13506.049659739452
Minimizing 816    Time: 0:00:05 ( 6.43 ms/it)
  objective:  13499.609388647797
Minimizing 832    Time: 0:00:05 ( 6.43 ms/it)
  objective:  13463.750046722329
Minimizing 848    Time: 0:00:05 ( 6.43 ms/it)
  objective:  13462.682207931095
Minimizing 864    Time: 0:00:05 ( 6.43 ms/it)
  objective:  13457.909184397606
Minimizing 880    Time: 0:00:05 ( 6.43 ms/it)
  objective:  13456.658476929544
Minimizing 896    Time: 0:00:05 ( 6.43 ms/it)
  objective:  13453.18406374006
Minimizing 912    Time: 0:00:05 ( 6.43 ms/it)
  objective:  13452.409446501726
Minimizing 928    Time: 0:00:05 ( 6.43 ms/it)
  objective:  13445.785805933134
Minimizing 944    Time: 0:00:06 ( 6.43 ms/it)
  objective:  13444.346745903269
Minimizing 960    Time: 0:00:06 ( 6.43 ms/it)
  objective:  13442.136680922893
Minimizing 976    Time: 0:00:06 ( 6.43 ms/it)
  objective:  13441.330926762268
Minimizing 992    Time: 0:00:06 ( 6.43 ms/it)
  objective:  13439.337221631431
Minimizing 1008    Time: 0:00:06 ( 6.43 ms/it)
  objective:  13438.444600099087
Minimizing 1024    Time: 0:00:06 ( 6.43 ms/it)
  objective:  13436.440559144641
Minimizing 1040    Time: 0:00:06 ( 6.43 ms/it)
  objective:  13436.287839714292
Minimizing 1056    Time: 0:00:06 ( 6.43 ms/it)
  objective:  13435.37458282859
Minimizing 1072    Time: 0:00:06 ( 6.43 ms/it)
  objective:  13434.362357292674
Minimizing 1088    Time: 0:00:06 ( 6.43 ms/it)
  objective:  13431.150323946786
Minimizing 1104    Time: 0:00:07 ( 6.43 ms/it)
  objective:  13413.006796062487
Minimizing 1120    Time: 0:00:07 ( 6.43 ms/it)
  objective:  13412.645734241029
Minimizing 1136    Time: 0:00:07 ( 6.43 ms/it)
  objective:  13405.983011820732
Minimizing 1152    Time: 0:00:07 ( 6.43 ms/it)
  objective:  13405.888199447334
Minimizing 1168    Time: 0:00:07 ( 6.43 ms/it)
  objective:  13403.973497097788
Minimizing 1184    Time: 0:00:07 ( 6.43 ms/it)
  objective:  13402.415956620724
Minimizing 1200    Time: 0:00:07 ( 6.43 ms/it)
  objective:  13402.083079450502
Minimizing 1216    Time: 0:00:07 ( 6.43 ms/it)
  objective:  13399.025604018694
Minimizing 1232    Time: 0:00:07 ( 6.43 ms/it)
  objective:  13395.960298929756
Minimizing 1248    Time: 0:00:08 ( 6.43 ms/it)
  objective:  13390.370496926247
Minimizing 1264    Time: 0:00:08 ( 6.43 ms/it)
  objective:  13381.352556218568
Minimizing 1280    Time: 0:00:08 ( 6.43 ms/it)
  objective:  13328.211142680724
Minimizing 1296    Time: 0:00:08 ( 6.43 ms/it)
  objective:  13324.157394373207
Minimizing 1312    Time: 0:00:08 ( 6.43 ms/it)
  objective:  13313.307660419814
Minimizing 1328    Time: 0:00:08 ( 6.43 ms/it)
  objective:  13310.350158876987
Minimizing 1344    Time: 0:00:08 ( 6.43 ms/it)
  objective:  13301.69292246856
Minimizing 1360    Time: 0:00:08 ( 6.43 ms/it)
  objective:  13299.084434125965
Minimizing 1376    Time: 0:00:08 ( 6.43 ms/it)
  objective:  13286.111198838626
Minimizing 1392    Time: 0:00:08 ( 6.43 ms/it)
  objective:  13285.45799696786
Minimizing 1408    Time: 0:00:09 ( 6.42 ms/it)
  objective:  13285.273344445159
Minimizing 1424    Time: 0:00:09 ( 6.42 ms/it)
  objective:  13285.269974754046
Minimizing 1440    Time: 0:00:09 ( 6.42 ms/it)
  objective:  13285.268218495155
Minimizing 1456    Time: 0:00:09 ( 6.42 ms/it)
  objective:  13285.253399284571
Minimizing 1472    Time: 0:00:09 ( 6.42 ms/it)
  objective:  13285.262490427951
Minimizing 1488    Time: 0:00:09 ( 6.42 ms/it)
  objective:  13285.243948119925
Minimizing 1504    Time: 0:00:09 ( 6.42 ms/it)
  objective:  13285.263871290139
Minimizing 1520    Time: 0:00:09 ( 6.42 ms/it)
  objective:  13285.247976401311
Minimizing 1536    Time: 0:00:09 ( 6.42 ms/it)
  objective:  13285.244367973355
Minimizing 1552    Time: 0:00:09 ( 6.42 ms/it)
  objective:  13285.251841748584
Minimizing 1568    Time: 0:00:10 ( 6.42 ms/it)
  objective:  13285.268145087
Minimizing 1584    Time: 0:00:10 ( 6.42 ms/it)
  objective:  13285.270943894531
Minimizing 1600    Time: 0:00:10 ( 6.42 ms/it)
  objective:  13285.296802239187
Minimizing 1616    Time: 0:00:10 ( 6.43 ms/it)
  objective:  13285.244580366329
Minimizing 1629    Time: 0:00:10 ( 6.43 ms/it)

	Est.	SE	z	p	σ_Child
(Intercept)	-0.0209	0.0142	-1.47	0.1415	0.6739
Test: Star-Run	-0.0066	0.0441	-0.15	0.8803	0.4867
Test: S20-Star	0.0169	0.0451	0.38	0.7071	0.8479
Test: SLJ-S20	0.0033	0.0443	0.08	0.9400	0.7574
Test: BPT-SLJ	-0.0277	0.0433	-0.64	0.5222	0.6869
a1	0.3184	0.0482	6.61	<1e-10
Sex: Boys	0.1801	0.0142	12.69	<1e-36
Test: Star-Run & a1	0.0285	0.1511	0.19	0.8505
Test: S20-Star & a1	-0.1345	0.1524	-0.88	0.3776
Test: SLJ-S20 & a1	-0.0281	0.1507	-0.19	0.8519
Test: BPT-SLJ & a1	0.4785	0.1481	3.23	0.0012
Test: Star-Run & Sex: Boys	-0.1609	0.0441	-3.65	0.0003
Test: S20-Star & Sex: Boys	0.0341	0.0451	0.76	0.4497
Test: SLJ-S20 & Sex: Boys	0.0557	0.0443	1.26	0.2091
Test: BPT-SLJ & Sex: Boys	0.1442	0.0433	3.33	0.0009
a1 & Sex: Boys	0.0163	0.0482	0.34	0.7352
Test: Star-Run & a1 & Sex: Boys	0.1174	0.1511	0.78	0.4371
Test: S20-Star & a1 & Sex: Boys	-0.0582	0.1524	-0.38	0.7024
Test: SLJ-S20 & a1 & Sex: Boys	0.0795	0.1507	0.53	0.5981
Test: BPT-SLJ & a1 & Sex: Boys	-0.1729	0.1481	-1.17	0.2432
Residual	0.0000

m_cpx_1_SeqDiff.PCA

(Child = 
Principal components based on correlation matrix
 (Intercept)      1.0     .      .      .      .
 Test: Star-Run   0.7    1.0     .      .      .
 Test: S20-Star  -0.05   0.68   1.0     .      .
 Test: SLJ-S20   -0.08   0.02   0.07   1.0     .
 Test: BPT-SLJ   -0.33  -0.65  -0.59   0.45   1.0

Normalized cumulative variances:
[0.5094, 0.7459, 0.9627, 1.0, 1.0]

Component loadings
                   PC1    PC2    PC3    PC4    PC5
 (Intercept)     -0.37   0.07  -0.77   0.05  -0.51
 Test: Star-Run  -0.59  -0.23  -0.19   0.28   0.7
 Test: S20-Star  -0.45  -0.36   0.54   0.36  -0.5
 Test: SLJ-S20    0.15  -0.86  -0.17  -0.46  -0.02
 Test: BPT-SLJ    0.54  -0.26  -0.24   0.76   0.0,)

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 16    Time: 0:00:00 ( 6.40 ms/it)
  objective:  13909.676505487332
Minimizing 32    Time: 0:00:00 ( 6.37 ms/it)
  objective:  13896.099587370363
Minimizing 48    Time: 0:00:00 ( 6.37 ms/it)
  objective:  13871.873758713446
Minimizing 64    Time: 0:00:00 ( 6.38 ms/it)
  objective:  13863.124023908562
Minimizing 80    Time: 0:00:00 ( 6.38 ms/it)
  objective:  13856.94189196125
Minimizing 96    Time: 0:00:00 ( 6.39 ms/it)
  objective:  13842.188181653983
Minimizing 112    Time: 0:00:00 ( 6.40 ms/it)
  objective:  13839.50726731276
Minimizing 128    Time: 0:00:00 ( 6.41 ms/it)
  objective:  13837.155257516291
Minimizing 144    Time: 0:00:00 ( 6.42 ms/it)
  objective:  13830.863664588016
Minimizing 160    Time: 0:00:01 ( 6.44 ms/it)
  objective:  13828.025775438424
Minimizing 176    Time: 0:00:01 ( 6.44 ms/it)
  objective:  13826.538205086763
Minimizing 192    Time: 0:00:01 ( 6.44 ms/it)
  objective:  13822.536410076156
Minimizing 208    Time: 0:00:01 ( 6.44 ms/it)
  objective:  13811.104891963481
Minimizing 224    Time: 0:00:01 ( 6.44 ms/it)
  objective:  13809.431573921072
Minimizing 240    Time: 0:00:01 ( 6.44 ms/it)
  objective:  13805.780234004436
Minimizing 256    Time: 0:00:01 ( 6.44 ms/it)
  objective:  13804.238444298964
Minimizing 272    Time: 0:00:01 ( 6.44 ms/it)
  objective:  13799.451666508156
Minimizing 288    Time: 0:00:01 ( 6.44 ms/it)
  objective:  13798.54861268582
Minimizing 304    Time: 0:00:01 ( 6.43 ms/it)
  objective:  13796.967524670636
Minimizing 320    Time: 0:00:02 ( 6.43 ms/it)
  objective:  13793.040169788957
Minimizing 336    Time: 0:00:02 ( 6.43 ms/it)
  objective:  13789.318278253762
Minimizing 352    Time: 0:00:02 ( 6.43 ms/it)
  objective:  13787.191461784965
Minimizing 368    Time: 0:00:02 ( 6.43 ms/it)
  objective:  13775.19007232685
Minimizing 384    Time: 0:00:02 ( 6.44 ms/it)
  objective:  13765.76661753283
Minimizing 400    Time: 0:00:02 ( 6.44 ms/it)
  objective:  13759.748828144006
Minimizing 416    Time: 0:00:02 ( 6.45 ms/it)
  objective:  13747.320679565233
Minimizing 432    Time: 0:00:02 ( 6.45 ms/it)
  objective:  13740.724267856625
Minimizing 448    Time: 0:00:02 ( 6.45 ms/it)
  objective:  13737.443564563477
Minimizing 464    Time: 0:00:02 ( 6.45 ms/it)
  objective:  13727.924233888283
Minimizing 480    Time: 0:00:03 ( 6.45 ms/it)
  objective:  13726.879469896736
Minimizing 496    Time: 0:00:03 ( 6.45 ms/it)
  objective:  13722.19023717074
Minimizing 512    Time: 0:00:03 ( 6.45 ms/it)
  objective:  13711.933706637345
Minimizing 528    Time: 0:00:03 ( 6.44 ms/it)
  objective:  13694.389525444862
Minimizing 544    Time: 0:00:03 ( 6.44 ms/it)
  objective:  13692.521430703644
Minimizing 560    Time: 0:00:03 ( 6.44 ms/it)
  objective:  13679.394858975746
Minimizing 576    Time: 0:00:03 ( 6.44 ms/it)
  objective:  13676.304503582134
Minimizing 592    Time: 0:00:03 ( 6.44 ms/it)
  objective:  13659.886812100784
Minimizing 608    Time: 0:00:03 ( 6.44 ms/it)
  objective:  13658.559701511615
Minimizing 624    Time: 0:00:04 ( 6.43 ms/it)
  objective:  13655.029188891946
Minimizing 640    Time: 0:00:04 ( 6.43 ms/it)
  objective:  13643.545772202277
Minimizing 656    Time: 0:00:04 ( 6.43 ms/it)
  objective:  13625.891090766097
Minimizing 672    Time: 0:00:04 ( 6.43 ms/it)
  objective:  13622.509652644432
Minimizing 688    Time: 0:00:04 ( 6.43 ms/it)
  objective:  13613.715312434026
Minimizing 704    Time: 0:00:04 ( 6.43 ms/it)
  objective:  13605.400696001248
Minimizing 720    Time: 0:00:04 ( 6.42 ms/it)
  objective:  13594.809499883719
Minimizing 736    Time: 0:00:04 ( 6.42 ms/it)
  objective:  13581.65345396992
Minimizing 752    Time: 0:00:04 ( 6.43 ms/it)
  objective:  13575.671219418655
Minimizing 768    Time: 0:00:04 ( 6.43 ms/it)
  objective:  13571.77375569925
Minimizing 784    Time: 0:00:05 ( 6.43 ms/it)
  objective:  13565.570682280399
Minimizing 800    Time: 0:00:05 ( 6.42 ms/it)
  objective:  13559.198452389748
Minimizing 816    Time: 0:00:05 ( 6.42 ms/it)
  objective:  13558.167822129632
Minimizing 832    Time: 0:00:05 ( 6.42 ms/it)
  objective:  13555.72605576735
Minimizing 848    Time: 0:00:05 ( 6.42 ms/it)
  objective:  13555.308333092427
Minimizing 864    Time: 0:00:05 ( 6.42 ms/it)
  objective:  13551.865202236731
Minimizing 880    Time: 0:00:05 ( 6.42 ms/it)
  objective:  13552.14792234196
Minimizing 896    Time: 0:00:05 ( 6.42 ms/it)
  objective:  13541.961395323531
Minimizing 912    Time: 0:00:05 ( 6.42 ms/it)
  objective:  13539.26085244224
Minimizing 928    Time: 0:00:05 ( 6.42 ms/it)
  objective:  13534.52634401941
Minimizing 944    Time: 0:00:06 ( 6.42 ms/it)
  objective:  13529.647294805487
Minimizing 960    Time: 0:00:06 ( 6.42 ms/it)
  objective:  13525.608123944112
Minimizing 976    Time: 0:00:06 ( 6.42 ms/it)
  objective:  13518.669648236224
Minimizing 992    Time: 0:00:06 ( 6.42 ms/it)
  objective:  13504.806312062596
Minimizing 1008    Time: 0:00:06 ( 6.42 ms/it)
  objective:  13499.202797540624
Minimizing 1024    Time: 0:00:06 ( 6.42 ms/it)
  objective:  13494.166666677644
Minimizing 1040    Time: 0:00:06 ( 6.42 ms/it)
  objective:  13492.074847633354
Minimizing 1056    Time: 0:00:06 ( 6.42 ms/it)
  objective:  13487.3912312397
Minimizing 1072    Time: 0:00:06 ( 6.42 ms/it)
  objective:  13479.115091592379
Minimizing 1088    Time: 0:00:06 ( 6.42 ms/it)
  objective:  13459.631870452198
Minimizing 1104    Time: 0:00:07 ( 6.42 ms/it)
  objective:  13456.824117122233
Minimizing 1120    Time: 0:00:07 ( 6.41 ms/it)
  objective:  13448.88037578603
Minimizing 1136    Time: 0:00:07 ( 6.41 ms/it)
  objective:  13442.1181315094
Minimizing 1152    Time: 0:00:07 ( 6.41 ms/it)
  objective:  13388.470200981115
Minimizing 1168    Time: 0:00:07 ( 6.41 ms/it)
  objective:  13387.958741912531
Minimizing 1184    Time: 0:00:07 ( 6.41 ms/it)
  objective:  13385.393972460748
Minimizing 1200    Time: 0:00:07 ( 6.41 ms/it)
  objective:  13384.26242418532
Minimizing 1216    Time: 0:00:07 ( 6.41 ms/it)
  objective:  13382.943605110835
Minimizing 1232    Time: 0:00:07 ( 6.41 ms/it)
  objective:  13380.15657700476
Minimizing 1248    Time: 0:00:07 ( 6.41 ms/it)
  objective:  13379.39877665187
Minimizing 1264    Time: 0:00:08 ( 6.41 ms/it)
  objective:  13379.399960031791
Minimizing 1280    Time: 0:00:08 ( 6.41 ms/it)
  objective:  13379.39892343599
Minimizing 1296    Time: 0:00:08 ( 6.41 ms/it)
  objective:  13379.398193288944
Minimizing 1312    Time: 0:00:08 ( 6.41 ms/it)
  objective:  13379.39843993647
Minimizing 1328    Time: 0:00:08 ( 6.41 ms/it)
  objective:  13379.398524122575
Minimizing 1344    Time: 0:00:08 ( 6.41 ms/it)
  objective:  13379.398292011872
Minimizing 1360    Time: 0:00:08 ( 6.40 ms/it)
  objective:  13379.39802987696
Minimizing 1376    Time: 0:00:08 ( 6.40 ms/it)
  objective:  13379.398624241629
Minimizing 1392    Time: 0:00:08 ( 6.40 ms/it)
  objective:  13379.398246093158
Minimizing 1408    Time: 0:00:09 ( 6.40 ms/it)
  objective:  13379.39793954909
Minimizing 1424    Time: 0:00:09 ( 6.40 ms/it)
  objective:  13379.398001567592
Minimizing 1440    Time: 0:00:09 ( 6.40 ms/it)
  objective:  13379.398565885844
Minimizing 1456    Time: 0:00:09 ( 6.40 ms/it)
  objective:  13379.397999963054
Minimizing 1472    Time: 0:00:09 ( 6.40 ms/it)
  objective:  13379.397990717946
Minimizing 1490    Time: 0:00:09 ( 6.41 ms/it)
  objective:  13379.39789871237
Minimizing 1507    Time: 0:00:09 ( 6.41 ms/it)
  objective:  13379.398790089908
Minimizing 1507    Time: 0:00:09 ( 6.41 ms/it)

	Est.	SE	z	p	σ_Child
(Intercept)	-0.0202	0.0142	-1.42	0.1557	0.7228
Test: c5.1	-0.0095	0.0431	-0.22	0.8252	1.2210
Test: c234.15	0.0709	0.1680	0.42	0.6733	0.1935
Test: c2.34	-0.0182	0.0778	-0.23	0.8147	1.6566
Test: c3.4	-0.0059	0.0448	-0.13	0.8948	1.3566
a1	0.3187	0.0483	6.60	<1e-10
Sex: Boys	0.1797	0.0142	12.64	<1e-35
Test: c5.1 & a1	0.3490	0.1471	2.37	0.0176
Test: c234.15 & a1	-1.3501	0.5739	-2.35	0.0187
Test: c2.34 & a1	0.2687	0.2650	1.01	0.3106
Test: c3.4 & a1	0.0658	0.1525	0.43	0.6662
Test: c5.1 & Sex: Boys	0.0750	0.0431	1.74	0.0821
Test: c234.15 & Sex: Boys	-0.9226	0.1680	-5.49	<1e-07
Test: c2.34 & Sex: Boys	-0.1113	0.0778	-1.43	0.1525
Test: c3.4 & Sex: Boys	-0.0715	0.0448	-1.60	0.1106
a1 & Sex: Boys	0.0158	0.0483	0.33	0.7428
Test: c5.1 & a1 & Sex: Boys	-0.0327	0.1471	-0.22	0.8241
Test: c234.15 & a1 & Sex: Boys	0.8243	0.5739	1.44	0.1509
Test: c2.34 & a1 & Sex: Boys	0.0246	0.2650	0.09	0.9259
Test: c3.4 & a1 & Sex: Boys	-0.0400	0.1525	-0.26	0.7929
Residual	0.0000

VarCorr(m_cpx_1_PC)

	Column	Variance	Std.Dev	Corr.
Child	(Intercept)	0.52245127	0.72280791
	Test: c5.1	1.49096279	1.22104987	-0.05
	Test: c234.15	0.03744751	0.19351359	-0.58	-0.06
	Test: c2.34	2.74421486	1.65656719	+0.20	-0.81	+0.20
	Test: c3.4	1.84046443	1.35663718	+0.05	-0.06	-0.15	+0.11
Residual		0.00000000	0.00003025

m_cpx_1_PC.PCA

(Child = 
Principal components based on correlation matrix
 (Intercept)     1.0     .      .      .      .
 Test: c5.1     -0.05   1.0     .      .      .
 Test: c234.15  -0.58  -0.06   1.0     .      .
 Test: c2.34     0.2   -0.81   0.2    1.0     .
 Test: c3.4      0.05  -0.06  -0.15   0.11   1.0

Normalized cumulative variances:
[0.3747, 0.6965, 0.8904, 0.9786, 1.0]

Component loadings
                  PC1    PC2    PC3   PC4    PC5
 (Intercept)    -0.15  -0.67  -0.27  0.59  -0.33
 Test: c5.1      0.68  -0.05   0.08  0.47   0.56
 Test: c234.15  -0.09   0.71   0.04  0.61  -0.34
 Test: c2.34    -0.7    0.05  -0.06  0.23   0.67
 Test: c3.4     -0.13  -0.21   0.96  0.12  -0.09,)

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 15    Time: 0:00:00 ( 7.02 ms/it)
  objective:  13939.913625321566
Minimizing 30    Time: 0:00:00 ( 6.91 ms/it)
  objective:  13910.867319917234
Minimizing 46    Time: 0:00:00 ( 6.75 ms/it)
  objective:  13863.238227281887
Minimizing 62    Time: 0:00:00 ( 6.67 ms/it)
  objective:  13858.913591244836
Minimizing 78    Time: 0:00:00 ( 6.62 ms/it)
  objective:  13853.354875178957
Minimizing 94    Time: 0:00:00 ( 6.58 ms/it)
  objective:  13850.220009957106
Minimizing 110    Time: 0:00:00 ( 6.56 ms/it)
  objective:  13835.335549293204
Minimizing 126    Time: 0:00:00 ( 6.54 ms/it)
  objective:  13828.44923412617
Minimizing 142    Time: 0:00:00 ( 6.53 ms/it)
  objective:  13805.81570636043
Minimizing 158    Time: 0:00:01 ( 6.52 ms/it)
  objective:  13801.878717348127
Minimizing 174    Time: 0:00:01 ( 6.51 ms/it)
  objective:  13794.969379377308
Minimizing 190    Time: 0:00:01 ( 6.50 ms/it)
  objective:  13790.72132699359
Minimizing 206    Time: 0:00:01 ( 6.49 ms/it)
  objective:  13783.895103439132
Minimizing 222    Time: 0:00:01 ( 6.49 ms/it)
  objective:  13773.982825600067
Minimizing 238    Time: 0:00:01 ( 6.48 ms/it)
  objective:  13760.321802210769
Minimizing 254    Time: 0:00:01 ( 6.48 ms/it)
  objective:  13757.076142671373
Minimizing 270    Time: 0:00:01 ( 6.48 ms/it)
  objective:  13752.878501352681
Minimizing 286    Time: 0:00:01 ( 6.47 ms/it)
  objective:  13750.356167136835
Minimizing 302    Time: 0:00:01 ( 6.48 ms/it)
  objective:  13747.576623276746
Minimizing 318    Time: 0:00:02 ( 6.47 ms/it)
  objective:  13743.237283023993
Minimizing 334    Time: 0:00:02 ( 6.47 ms/it)
  objective:  13737.640585869482
Minimizing 350    Time: 0:00:02 ( 6.47 ms/it)
  objective:  13736.064610794943
Minimizing 366    Time: 0:00:02 ( 6.47 ms/it)
  objective:  13730.7311762455
Minimizing 382    Time: 0:00:02 ( 6.47 ms/it)
  objective:  13729.954647987171
Minimizing 398    Time: 0:00:02 ( 6.47 ms/it)
  objective:  13724.61811652228
Minimizing 414    Time: 0:00:02 ( 6.46 ms/it)
  objective:  13713.877557110063
Minimizing 430    Time: 0:00:02 ( 6.46 ms/it)
  objective:  13710.707283421085
Minimizing 446    Time: 0:00:02 ( 6.46 ms/it)
  objective:  13703.309980666989
Minimizing 462    Time: 0:00:02 ( 6.46 ms/it)
  objective:  13698.578637933795
Minimizing 478    Time: 0:00:03 ( 6.46 ms/it)
  objective:  13690.326985509375
Minimizing 494    Time: 0:00:03 ( 6.46 ms/it)
  objective:  13678.076979629506
Minimizing 510    Time: 0:00:03 ( 6.46 ms/it)
  objective:  13668.404221317815
Minimizing 526    Time: 0:00:03 ( 6.46 ms/it)
  objective:  13662.937628404798
Minimizing 542    Time: 0:00:03 ( 6.46 ms/it)
  objective:  13661.766860902859
Minimizing 558    Time: 0:00:03 ( 6.45 ms/it)
  objective:  13658.350606923279
Minimizing 574    Time: 0:00:03 ( 6.45 ms/it)
  objective:  13655.642160024021
Minimizing 590    Time: 0:00:03 ( 6.45 ms/it)
  objective:  13654.037447144354
Minimizing 606    Time: 0:00:03 ( 6.45 ms/it)
  objective:  13652.377786473113
Minimizing 622    Time: 0:00:04 ( 6.45 ms/it)
  objective:  13651.78315637205
Minimizing 638    Time: 0:00:04 ( 6.45 ms/it)
  objective:  13646.042233668988
Minimizing 654    Time: 0:00:04 ( 6.45 ms/it)
  objective:  13645.42883236074
Minimizing 670    Time: 0:00:04 ( 6.45 ms/it)
  objective:  13644.149199563231
Minimizing 686    Time: 0:00:04 ( 6.45 ms/it)
  objective:  13643.29088485473
Minimizing 702    Time: 0:00:04 ( 6.45 ms/it)
  objective:  13641.422186331976
Minimizing 718    Time: 0:00:04 ( 6.45 ms/it)
  objective:  13640.28312079826
Minimizing 734    Time: 0:00:04 ( 6.45 ms/it)
  objective:  13639.21903860182
Minimizing 750    Time: 0:00:04 ( 6.45 ms/it)
  objective:  13637.155493680548
Minimizing 766    Time: 0:00:04 ( 6.45 ms/it)
  objective:  13634.248030443698
Minimizing 782    Time: 0:00:05 ( 6.46 ms/it)
  objective:  13634.005196397004
Minimizing 797    Time: 0:00:05 ( 6.47 ms/it)
  objective:  13632.016301571122
Minimizing 813    Time: 0:00:05 ( 6.47 ms/it)
  objective:  13630.701618919396
Minimizing 828    Time: 0:00:05 ( 6.48 ms/it)
  objective:  13630.291222549145
Minimizing 844    Time: 0:00:05 ( 6.48 ms/it)
  objective:  13629.149668356455
Minimizing 860    Time: 0:00:05 ( 6.48 ms/it)
  objective:  13627.48138460295
Minimizing 876    Time: 0:00:05 ( 6.48 ms/it)
  objective:  13627.302868477018
Minimizing 892    Time: 0:00:05 ( 6.48 ms/it)
  objective:  13626.045248150956
Minimizing 909    Time: 0:00:05 ( 6.47 ms/it)
  objective:  13622.774641421594
Minimizing 926    Time: 0:00:05 ( 6.46 ms/it)
  objective:  13620.095597857318
Minimizing 943    Time: 0:00:06 ( 6.46 ms/it)
  objective:  13616.466082558523
Minimizing 960    Time: 0:00:06 ( 6.45 ms/it)
  objective:  13610.019765424622
Minimizing 976    Time: 0:00:06 ( 6.45 ms/it)
  objective:  13597.232399969085
Minimizing 992    Time: 0:00:06 ( 6.45 ms/it)
  objective:  13564.582162001432
Minimizing 1008    Time: 0:00:06 ( 6.45 ms/it)
  objective:  13562.203232163229
Minimizing 1024    Time: 0:00:06 ( 6.45 ms/it)
  objective:  13557.494123637189
Minimizing 1040    Time: 0:00:06 ( 6.44 ms/it)
  objective:  13550.037087341145
Minimizing 1056    Time: 0:00:06 ( 6.44 ms/it)
  objective:  13547.899219763523
Minimizing 1072    Time: 0:00:06 ( 6.44 ms/it)
  objective:  13543.966947492416
Minimizing 1088    Time: 0:00:07 ( 6.44 ms/it)
  objective:  13538.183418521061
Minimizing 1104    Time: 0:00:07 ( 6.43 ms/it)
  objective:  13534.328781730102
Minimizing 1120    Time: 0:00:07 ( 6.43 ms/it)
  objective:  13522.845718559169
Minimizing 1136    Time: 0:00:07 ( 6.43 ms/it)
  objective:  13512.483205152479
Minimizing 1152    Time: 0:00:07 ( 6.43 ms/it)
  objective:  13475.257290880341
Minimizing 1168    Time: 0:00:07 ( 6.43 ms/it)
  objective:  13472.40100562651
Minimizing 1184    Time: 0:00:07 ( 6.43 ms/it)
  objective:  13468.642113944923
Minimizing 1200    Time: 0:00:07 ( 6.43 ms/it)
  objective:  13467.559864473122
Minimizing 1216    Time: 0:00:07 ( 6.42 ms/it)
  objective:  13466.512165257329
Minimizing 1233    Time: 0:00:07 ( 6.42 ms/it)
  objective:  13464.522786371264
Minimizing 1249    Time: 0:00:08 ( 6.42 ms/it)
  objective:  13463.420995849912
Minimizing 1265    Time: 0:00:08 ( 6.42 ms/it)
  objective:  13460.151232849224
Minimizing 1281    Time: 0:00:08 ( 6.42 ms/it)
  objective:  13457.899792656637
Minimizing 1297    Time: 0:00:08 ( 6.42 ms/it)
  objective:  13451.9291469874
Minimizing 1313    Time: 0:00:08 ( 6.42 ms/it)
  objective:  13450.113151402751
Minimizing 1329    Time: 0:00:08 ( 6.42 ms/it)
  objective:  13448.101174799755
Minimizing 1345    Time: 0:00:08 ( 6.42 ms/it)
  objective:  13443.021297650354
Minimizing 1361    Time: 0:00:08 ( 6.42 ms/it)
  objective:  13431.291078742128
Minimizing 1377    Time: 0:00:08 ( 6.41 ms/it)
  objective:  13415.687844414744
Minimizing 1393    Time: 0:00:08 ( 6.41 ms/it)
  objective:  13378.40357580177
Minimizing 1409    Time: 0:00:09 ( 6.41 ms/it)
  objective:  13371.492128896265
Minimizing 1425    Time: 0:00:09 ( 6.41 ms/it)
  objective:  13365.35182421378
Minimizing 1441    Time: 0:00:09 ( 6.41 ms/it)
  objective:  13360.097938926934
Minimizing 1457    Time: 0:00:09 ( 6.41 ms/it)
  objective:  13355.506930682255
Minimizing 1473    Time: 0:00:09 ( 6.41 ms/it)
  objective:  13350.52958149038
Minimizing 1489    Time: 0:00:09 ( 6.41 ms/it)
  objective:  13345.95370122639
Minimizing 1505    Time: 0:00:09 ( 6.41 ms/it)
  objective:  13332.142658639015
Minimizing 1521    Time: 0:00:09 ( 6.41 ms/it)
  objective:  13331.448901415439
Minimizing 1537    Time: 0:00:09 ( 6.41 ms/it)
  objective:  13327.186213538735
Minimizing 1553    Time: 0:00:09 ( 6.41 ms/it)
  objective:  13319.047840409345
Minimizing 1570    Time: 0:00:10 ( 6.40 ms/it)
  objective:  13311.97163955016
Minimizing 1586    Time: 0:00:10 ( 6.40 ms/it)
  objective:  13311.892356969998
Minimizing 1602    Time: 0:00:10 ( 6.40 ms/it)
  objective:  13311.89161115636
Minimizing 1618    Time: 0:00:10 ( 6.40 ms/it)
  objective:  13311.901818680955
Minimizing 1634    Time: 0:00:10 ( 6.40 ms/it)
  objective:  13311.905827544819
Minimizing 1653    Time: 0:00:10 ( 6.41 ms/it)
  objective:  13311.890706955557
Minimizing 1671    Time: 0:00:10 ( 6.41 ms/it)
  objective:  13311.889527278618
Minimizing 1687    Time: 0:00:10 ( 6.41 ms/it)
  objective:  13311.889010928935
Minimizing 1703    Time: 0:00:10 ( 6.40 ms/it)
  objective:  13311.896606369293
Minimizing 1719    Time: 0:00:11 ( 6.40 ms/it)
  objective:  13311.894027532748
Minimizing 1735    Time: 0:00:11 ( 6.40 ms/it)
  objective:  13311.892309053714
Minimizing 1751    Time: 0:00:11 ( 6.40 ms/it)
  objective:  13311.888171878265
Minimizing 1767    Time: 0:00:11 ( 6.40 ms/it)
  objective:  13311.892208361998
Minimizing 1783    Time: 0:00:11 ( 6.40 ms/it)
  objective:  13311.895939248017
Minimizing 1799    Time: 0:00:11 ( 6.40 ms/it)
  objective:  13311.89317696885
Minimizing 1816    Time: 0:00:11 ( 6.40 ms/it)
  objective:  13311.888492061436
Minimizing 1828    Time: 0:00:11 ( 6.40 ms/it)

	Est.	SE	z	p	σ_Child
(Intercept)	-0.0195	0.0144	-1.36	0.1749	0.7056
Test: c5.1	0.0138	0.0304	0.45	0.6495	1.0261
Test: c234.15	-0.0090	0.0308	-0.29	0.7693	0.3203
Test: c12.34	-0.0109	0.0320	-0.34	0.7334	0.9587
Test: c3.4	-0.0075	0.0317	-0.24	0.8124	0.5296
a1	0.2973	0.0489	6.08	<1e-08
Sex: Boys	0.1717	0.0144	11.94	<1e-32
Test: c5.1 & a1	-0.3226	0.1033	-3.12	0.0018
Test: c234.15 & a1	0.1449	0.1056	1.37	0.1699
Test: c12.34 & a1	0.1182	0.1084	1.09	0.2753
Test: c3.4 & a1	0.0897	0.1080	0.83	0.4061
Test: c5.1 & Sex: Boys	-0.0970	0.0304	-3.19	0.0014
Test: c234.15 & Sex: Boys	0.1603	0.0308	5.20	<1e-06
Test: c12.34 & Sex: Boys	-0.0046	0.0320	-0.14	0.8854
Test: c3.4 & Sex: Boys	-0.0218	0.0317	-0.69	0.4919
a1 & Sex: Boys	0.0236	0.0489	0.48	0.6293
Test: c5.1 & a1 & Sex: Boys	0.0551	0.1033	0.53	0.5936
Test: c234.15 & a1 & Sex: Boys	-0.1258	0.1056	-1.19	0.2336
Test: c12.34 & a1 & Sex: Boys	-0.0138	0.1084	-0.13	0.8988
Test: c3.4 & a1 & Sex: Boys	-0.0725	0.1080	-0.67	0.5019
Residual	0.0000

VarCorr(m_cpx_1_PC_2)

	Column	Variance	Std.Dev	Corr.
Child	(Intercept)	0.49789509	0.70561681
	Test: c5.1	1.05293764	1.02612750	+0.22
	Test: c234.15	0.10260819	0.32032513	-0.73	-0.29
	Test: c12.34	0.91908429	0.95868884	-0.03	+0.29	+0.62
	Test: c3.4	0.28045222	0.52957740	+0.00	+0.67	+0.03	+0.48
Residual		0.00000000	0.00001035

m_cpx_1_PC_2.PCA

(Child = 
Principal components based on correlation matrix
 (Intercept)     1.0     .      .      .      .
 Test: c5.1      0.22   1.0     .      .      .
 Test: c234.15  -0.73  -0.29   1.0     .      .
 Test: c12.34   -0.03   0.29   0.62   1.0     .
 Test: c3.4      0.0    0.67   0.03   0.48   1.0

Normalized cumulative variances:
[0.4088, 0.8023, 0.9457, 1.0, 1.0]

Component loadings
                  PC1    PC2    PC3    PC4    PC5
 (Intercept)    -0.34   0.47   0.67   0.11  -0.45
 Test: c5.1      0.22   0.6   -0.31  -0.68  -0.18
 Test: c234.15   0.54  -0.44   0.14  -0.09  -0.7
 Test: c12.34    0.59   0.11   0.59  -0.12   0.52
 Test: c3.4      0.43   0.46  -0.29   0.71  -0.11,)

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 23    Time: 0:00:00 ( 4.36 ms/it)
  objective:  13869.894021228893
Minimizing 46    Time: 0:00:00 ( 4.37 ms/it)
  objective:  13794.040911325883
Minimizing 69    Time: 0:00:00 ( 4.39 ms/it)
  objective:  13784.970455246392
Minimizing 92    Time: 0:00:00 ( 4.40 ms/it)
  objective:  13779.471178143729
Minimizing 115    Time: 0:00:00 ( 4.40 ms/it)
  objective:  13773.025761794284
Minimizing 138    Time: 0:00:00 ( 4.41 ms/it)
  objective:  13765.4177577171
Minimizing 161    Time: 0:00:00 ( 4.42 ms/it)
  objective:  13713.394187523223
Minimizing 184    Time: 0:00:00 ( 4.42 ms/it)
  objective:  13657.975229637479
Minimizing 207    Time: 0:00:00 ( 4.42 ms/it)
  objective:  13638.33706301749
Minimizing 230    Time: 0:00:01 ( 4.43 ms/it)
  objective:  13575.369048534616
Minimizing 253    Time: 0:00:01 ( 4.43 ms/it)
  objective:  13558.416566038555
Minimizing 276    Time: 0:00:01 ( 4.43 ms/it)
  objective:  13544.235095906566
Minimizing 299    Time: 0:00:01 ( 4.43 ms/it)
  objective:  13539.218911140015
Minimizing 322    Time: 0:00:01 ( 4.43 ms/it)
  objective:  13517.446220974132
Minimizing 345    Time: 0:00:01 ( 4.44 ms/it)
  objective:  13465.568427735954
Minimizing 368    Time: 0:00:01 ( 4.44 ms/it)
  objective:  13433.176291078795
Minimizing 391    Time: 0:00:01 ( 4.44 ms/it)
  objective:  13375.006261611154
Minimizing 414    Time: 0:00:01 ( 4.44 ms/it)
  objective:  13364.373333253694
Minimizing 437    Time: 0:00:01 ( 4.44 ms/it)
  objective:  13345.454479329972
Minimizing 460    Time: 0:00:02 ( 4.44 ms/it)
  objective:  13310.22451820763
Minimizing 483    Time: 0:00:02 ( 4.44 ms/it)
  objective:  13295.862223566
Minimizing 506    Time: 0:00:02 ( 4.44 ms/it)
  objective:  13272.240414376109
Minimizing 529    Time: 0:00:02 ( 4.44 ms/it)
  objective:  13271.382023496597
Minimizing 552    Time: 0:00:02 ( 4.44 ms/it)
  objective:  13271.43403263406
Minimizing 575    Time: 0:00:02 ( 4.44 ms/it)
  objective:  13271.381324219401
Minimizing 598    Time: 0:00:02 ( 4.44 ms/it)
  objective:  13271.38895813725
Minimizing 611    Time: 0:00:02 ( 4.44 ms/it)

	Est.	SE	z	p	σ_Child
(Intercept)	-0.0188	0.0144	-1.31	0.1909	0.7062
Test: c5.1	0.0139	0.0311	0.45	0.6550	0.7390
Test: c234.15	-0.0070	0.0312	-0.22	0.8223	0.5653
Test: c12.34	-0.0068	0.0319	-0.21	0.8304	0.8891
Test: c3.4	-0.0081	0.0318	-0.26	0.7985	0.7641
a1	0.2975	0.0488	6.10	<1e-08
Sex: Boys	0.1721	0.0144	11.99	<1e-32
Test: c5.1 & a1	-0.3201	0.1053	-3.04	0.0024
Test: c234.15 & a1	0.1533	0.1061	1.44	0.1485
Test: c12.34 & a1	0.1091	0.1081	1.01	0.3131
Test: c3.4 & a1	0.1019	0.1085	0.94	0.3475
Test: c5.1 & Sex: Boys	-0.0960	0.0311	-3.09	0.0020
Test: c234.15 & Sex: Boys	0.1654	0.0312	5.31	<1e-06
Test: c12.34 & Sex: Boys	-0.0017	0.0319	-0.05	0.9576
Test: c3.4 & Sex: Boys	-0.0285	0.0318	-0.90	0.3699
a1 & Sex: Boys	0.0220	0.0488	0.45	0.6522
Test: c5.1 & a1 & Sex: Boys	0.0739	0.1053	0.70	0.4831
Test: c234.15 & a1 & Sex: Boys	-0.1072	0.1061	-1.01	0.3123
Test: c12.34 & a1 & Sex: Boys	-0.0327	0.1081	-0.30	0.7623
Test: c3.4 & a1 & Sex: Boys	-0.0582	0.1085	-0.54	0.5913
Residual	0.0000

VarCorr(m_zcp_1_PC_2)

	Column	Variance	Std.Dev	Corr.
Child	(Intercept)	0.49878776	0.70624908
	Test: c5.1	0.54613699	0.73901082	.
	Test: c234.15	0.31957127	0.56530635	.	.
	Test: c12.34	0.79053637	0.88912112	.	.	.
	Test: c3.4	0.58385462	0.76410380	.	.	.	.
Residual		0.00000000	0.00000300

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	13271
zScore ~ 1 + Test + a1 + Sex + Test & a1 + Test & Sex + a1 & Sex + Test & a1 & Sex + (1 + Test \| Child)	36	13312	-41	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}:
 13892.27976824274
 13892.279768242752
 13892.279768242775

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 16    Time: 0:00:00 ( 6.59 ms/it)
  objective:  14024.811353037938
Minimizing 32    Time: 0:00:00 ( 6.60 ms/it)
  objective:  13924.173486113337
Minimizing 48    Time: 0:00:00 ( 6.58 ms/it)
  objective:  13865.195235837524
Minimizing 64    Time: 0:00:00 ( 6.58 ms/it)
  objective:  13855.59111130773
Minimizing 80    Time: 0:00:00 ( 6.59 ms/it)
  objective:  13832.38985233817
Minimizing 95    Time: 0:00:00 ( 6.61 ms/it)
  objective:  13828.837692023975
Minimizing 111    Time: 0:00:00 ( 6.61 ms/it)
  objective:  13817.27057060633
Minimizing 127    Time: 0:00:00 ( 6.61 ms/it)
  objective:  13810.340534459952
Minimizing 143    Time: 0:00:00 ( 6.61 ms/it)
  objective:  13807.193008818398
Minimizing 159    Time: 0:00:01 ( 6.60 ms/it)
  objective:  13800.329232569777
Minimizing 175    Time: 0:00:01 ( 6.60 ms/it)
  objective:  13798.11892171307
Minimizing 191    Time: 0:00:01 ( 6.60 ms/it)
  objective:  13794.770109306663
Minimizing 207    Time: 0:00:01 ( 6.60 ms/it)
  objective:  13793.47261367708
Minimizing 223    Time: 0:00:01 ( 6.60 ms/it)
  objective:  13789.961517085456
Minimizing 239    Time: 0:00:01 ( 6.60 ms/it)
  objective:  13785.389820989043
Minimizing 255    Time: 0:00:01 ( 6.60 ms/it)
  objective:  13781.519651723462
Minimizing 271    Time: 0:00:01 ( 6.60 ms/it)
  objective:  13779.685321301631
Minimizing 285    Time: 0:00:01 ( 6.65 ms/it)
  objective:  13775.542847835895
Minimizing 300    Time: 0:00:01 ( 6.66 ms/it)
  objective:  13771.99801289468
Minimizing 316    Time: 0:00:02 ( 6.66 ms/it)
  objective:  13767.592715729963
Minimizing 332    Time: 0:00:02 ( 6.66 ms/it)
  objective:  13752.012646833819
Minimizing 348    Time: 0:00:02 ( 6.65 ms/it)
  objective:  13740.230257913372
Minimizing 364    Time: 0:00:02 ( 6.65 ms/it)
  objective:  13739.05283264295
Minimizing 380    Time: 0:00:02 ( 6.65 ms/it)
  objective:  13731.354336251472
Minimizing 396    Time: 0:00:02 ( 6.65 ms/it)
  objective:  13725.277911287452
Minimizing 412    Time: 0:00:02 ( 6.65 ms/it)
  objective:  13724.115276370874
Minimizing 428    Time: 0:00:02 ( 6.65 ms/it)
  objective:  13720.451237022338
Minimizing 444    Time: 0:00:02 ( 6.64 ms/it)
  objective:  13719.265403313108
Minimizing 460    Time: 0:00:03 ( 6.64 ms/it)
  objective:  13716.679817508077
Minimizing 476    Time: 0:00:03 ( 6.64 ms/it)
  objective:  13714.960314693108
Minimizing 492    Time: 0:00:03 ( 6.64 ms/it)
  objective:  13712.825537219964
Minimizing 508    Time: 0:00:03 ( 6.63 ms/it)
  objective:  13711.829683199994
Minimizing 524    Time: 0:00:03 ( 6.63 ms/it)
  objective:  13707.080740765665
Minimizing 540    Time: 0:00:03 ( 6.63 ms/it)
  objective:  13705.823170981836
Minimizing 556    Time: 0:00:03 ( 6.63 ms/it)
  objective:  13701.043558330006
Minimizing 572    Time: 0:00:03 ( 6.63 ms/it)
  objective:  13698.439479008874
Minimizing 588    Time: 0:00:03 ( 6.63 ms/it)
  objective:  13680.487920234744
Minimizing 604    Time: 0:00:04 ( 6.62 ms/it)
  objective:  13674.904535163594
Minimizing 620    Time: 0:00:04 ( 6.62 ms/it)
  objective:  13662.809697138626
Minimizing 636    Time: 0:00:04 ( 6.62 ms/it)
  objective:  13659.244371029155
Minimizing 652    Time: 0:00:04 ( 6.62 ms/it)
  objective:  13658.123844005077
Minimizing 668    Time: 0:00:04 ( 6.62 ms/it)
  objective:  13656.063631564168
Minimizing 684    Time: 0:00:04 ( 6.62 ms/it)
  objective:  13644.591790679318
Minimizing 700    Time: 0:00:04 ( 6.62 ms/it)
  objective:  13643.799963810103
Minimizing 716    Time: 0:00:04 ( 6.62 ms/it)
  objective:  13642.729613092371
Minimizing 732    Time: 0:00:04 ( 6.62 ms/it)
  objective:  13640.630726932439
Minimizing 748    Time: 0:00:04 ( 6.61 ms/it)
  objective:  13637.16412978623
Minimizing 764    Time: 0:00:05 ( 6.61 ms/it)
  objective:  13635.398062140914
Minimizing 814    Time: 0:00:05 ( 6.68 ms/it)
  objective:  13625.737488262392
Minimizing 833    Time: 0:00:05 ( 6.68 ms/it)
  objective:  13620.499209843903
Minimizing 849    Time: 0:00:05 ( 6.68 ms/it)
  objective:  13607.740170154379
Minimizing 865    Time: 0:00:05 ( 6.67 ms/it)
  objective:  13605.72598889748
Minimizing 881    Time: 0:00:05 ( 6.67 ms/it)
  objective:  13598.098010686896
Minimizing 897    Time: 0:00:05 ( 6.67 ms/it)
  objective:  13593.86731170665
Minimizing 913    Time: 0:00:06 ( 6.67 ms/it)
  objective:  13587.252360190098
Minimizing 929    Time: 0:00:06 ( 6.67 ms/it)
  objective:  13586.898995500334
Minimizing 945    Time: 0:00:06 ( 6.66 ms/it)
  objective:  13577.171542824144
Minimizing 961    Time: 0:00:06 ( 6.66 ms/it)
  objective:  13576.564650978704
Minimizing 977    Time: 0:00:06 ( 6.66 ms/it)
  objective:  13569.785844782797
Minimizing 993    Time: 0:00:06 ( 6.66 ms/it)
  objective:  13561.018324949204
Minimizing 1009    Time: 0:00:06 ( 6.66 ms/it)
  objective:  13558.74873250653
Minimizing 1025    Time: 0:00:06 ( 6.65 ms/it)
  objective:  13551.278762117385
Minimizing 1041    Time: 0:00:06 ( 6.65 ms/it)
  objective:  13547.621881526255
Minimizing 1057    Time: 0:00:07 ( 6.65 ms/it)
  objective:  13542.527148943802
Minimizing 1073    Time: 0:00:07 ( 6.65 ms/it)
  objective:  13539.023982444589
Minimizing 1089    Time: 0:00:07 ( 6.65 ms/it)
  objective:  13530.778042806072
Minimizing 1105    Time: 0:00:07 ( 6.65 ms/it)
  objective:  13503.571805410422
Minimizing 1121    Time: 0:00:07 ( 6.65 ms/it)
  objective:  13498.968577340813
Minimizing 1137    Time: 0:00:07 ( 6.64 ms/it)
  objective:  13489.549519655578
Minimizing 1153    Time: 0:00:07 ( 6.64 ms/it)
  objective:  13488.478217932512
Minimizing 1169    Time: 0:00:07 ( 6.64 ms/it)
  objective:  13487.359776293277
Minimizing 1185    Time: 0:00:07 ( 6.64 ms/it)
  objective:  13486.235603482884
Minimizing 1201    Time: 0:00:07 ( 6.64 ms/it)
  objective:  13483.309305434173
Minimizing 1217    Time: 0:00:08 ( 6.64 ms/it)
  objective:  13475.266108318567
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Minimizing 1905    Time: 0:00:12 ( 6.61 ms/it)
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Minimizing 1921    Time: 0:00:12 ( 6.61 ms/it)
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Minimizing 16    Time: 0:00:00 ( 6.45 ms/it)
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Minimizing 32    Time: 0:00:00 ( 6.48 ms/it)
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Minimizing 112    Time: 0:00:00 ( 6.53 ms/it)
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Minimizing 1830    Time: 0:00:12 ( 6.58 ms/it)
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Minimizing 1878    Time: 0:00:12 ( 6.58 ms/it)
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Minimizing 1894    Time: 0:00:12 ( 6.58 ms/it)
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Minimizing 1910    Time: 0:00:12 ( 6.58 ms/it)
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Minimizing 1926    Time: 0:00:12 ( 6.58 ms/it)
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Minimizing 1942    Time: 0:00:12 ( 6.58 ms/it)
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Minimizing 1958    Time: 0:00:12 ( 6.57 ms/it)
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Minimizing 1990    Time: 0:00:13 ( 6.57 ms/it)
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Minimizing 2006    Time: 0:00:13 ( 6.57 ms/it)
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Minimizing 1866    Time: 0:00:12 ( 6.75 ms/it)
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Minimizing 1898    Time: 0:00:12 ( 6.75 ms/it)
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Minimizing 1914    Time: 0:00:12 ( 6.74 ms/it)
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Minimizing 1930    Time: 0:00:13 ( 6.74 ms/it)
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Minimizing 1962    Time: 0:00:13 ( 6.74 ms/it)
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Minimizing 1978    Time: 0:00:13 ( 6.74 ms/it)
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Minimizing 2010    Time: 0:00:13 ( 6.74 ms/it)
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Minimizing 16    Time: 0:00:00 ( 6.42 ms/it)
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	Est.	SE	z	p	σ_Child
(Intercept)	-0.0202	0.0142	-1.42	0.1557	0.7228
Test: c5.1	-0.0095	0.0431	-0.22	0.8252	1.2210
Test: c234.15	0.0709	0.1680	0.42	0.6733	0.1935
Test: c2.34	-0.0182	0.0778	-0.23	0.8147	1.6566
Test: c3.4	-0.0059	0.0448	-0.13	0.8948	1.3566
a1	0.3187	0.0483	6.60	<1e-10
Sex: Boys	0.1797	0.0142	12.64	<1e-35
Test: c5.1 & a1	0.3490	0.1471	2.37	0.0176
Test: c234.15 & a1	-1.3501	0.5739	-2.35	0.0187
Test: c2.34 & a1	0.2687	0.2650	1.01	0.3106
Test: c3.4 & a1	0.0658	0.1525	0.43	0.6662
Test: c5.1 & Sex: Boys	0.0750	0.0431	1.74	0.0821
Test: c234.15 & Sex: Boys	-0.9226	0.1680	-5.49	<1e-07
Test: c2.34 & Sex: Boys	-0.1113	0.0778	-1.43	0.1525
Test: c3.4 & Sex: Boys	-0.0715	0.0448	-1.60	0.1106
a1 & Sex: Boys	0.0158	0.0483	0.33	0.7428
Test: c5.1 & a1 & Sex: Boys	-0.0327	0.1471	-0.22	0.8241
Test: c234.15 & a1 & Sex: Boys	0.8243	0.5739	1.44	0.1509
Test: c2.34 & a1 & Sex: Boys	0.0246	0.2650	0.09	0.9259
Test: c3.4 & a1 & Sex: Boys	-0.0400	0.1525	-0.26	0.7929
Residual	0.0000

VarCorr(m_Child_SDC)

	Column	Variance	Std.Dev	Corr.
Child	(Intercept)	0.39373499	0.62748306
	Test: Star_r	0.12276468	0.35037791	+0.78
	Test: S20_r	1.39060677	1.17923991	-0.12	+0.49
	Test: SLJ	0.43824825	0.66200321	-0.02	-0.11	+0.04
	Test: BPT	0.87203896	0.93383026	-0.12	-0.64	-0.81	+0.40
Residual		0.00000000	0.00000536

VarCorr(m_Child_HeC)

	Column	Variance	Std.Dev	Corr.
Child	(Intercept)	0.35218940	0.59345548
	Test: Star_r	0.03747918	0.19359541	+0.03
	Test: S20_r	0.13396740	0.36601557	-0.04	-0.66
	Test: SLJ	0.07924506	0.28150499	-0.05	+0.11	+0.60
	Test: BPT	0.05640052	0.23748794	-0.35	+0.74	-0.15	+0.52
Residual		0.00000000	0.00002190

VarCorr(m_Child_HyC)

	Column	Variance	Std.Dev	Corr.
Child	(Intercept)	0.42541656	0.65223965
	Test: BPT-other	1.84585859	1.35862379	+0.94
	Test: Star-End	0.25315852	0.50314861	+0.79	+0.91
	Test: S20-Star	2.19702441	1.48223629	-0.16	+0.09	+0.27
	Test: SLJ-S20	0.39267032	0.62663412	+0.21	+0.21	+0.33	-0.42
Residual		0.00000000	0.00000702

VarCorr(m_Child_PCA)

	Column	Variance	Std.Dev	Corr.
Child	(Intercept)	0.52245127	0.72280791
	Test: c5.1	1.49096279	1.22104987	-0.05
	Test: c234.15	0.03744751	0.19351359	-0.58	-0.06
	Test: c2.34	2.74421486	1.65656719	+0.20	-0.81	+0.20
	Test: c3.4	1.84046443	1.35663718	+0.05	-0.06	-0.15	+0.11
Residual		0.00000000	0.00003025

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

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Minimizing 649    Time: 0:00:02 ( 4.36 ms/it)
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Minimizing 669    Time: 0:00:02 ( 4.36 ms/it)

	Est.	SE	z	p	σ_Child
(Intercept)	-0.0197	0.0142	-1.39	0.1653	0.6961
Test: c5.1	-0.0151	0.0421	-0.36	0.7203	1.1281
Test: c234.15	0.0703	0.1689	0.42	0.6774	2.2636
Test: c2.34	-0.0176	0.0798	-0.22	0.8256	2.0754
Test: c3.4	-0.0070	0.0461	-0.15	0.8799	1.2604
a1	0.3176	0.0483	6.57	<1e-10
Sex: Boys	0.1803	0.0142	12.67	<1e-36
Test: c5.1 & a1	0.3487	0.1432	2.44	0.0149
Test: c234.15 & a1	-1.4444	0.5748	-2.51	0.0120
Test: c2.34 & a1	0.2549	0.2707	0.94	0.3464
Test: c3.4 & a1	0.0584	0.1568	0.37	0.7096
Test: c5.1 & Sex: Boys	0.0731	0.0421	1.74	0.0822
Test: c234.15 & Sex: Boys	-0.9452	0.1689	-5.60	<1e-07
Test: c2.34 & Sex: Boys	-0.1133	0.0798	-1.42	0.1559
Test: c3.4 & Sex: Boys	-0.0722	0.0461	-1.56	0.1176
a1 & Sex: Boys	0.0153	0.0483	0.32	0.7507
Test: c5.1 & a1 & Sex: Boys	-0.0609	0.1432	-0.43	0.6704
Test: c234.15 & a1 & Sex: Boys	0.7166	0.5748	1.25	0.2125
Test: c2.34 & a1 & Sex: Boys	-0.0340	0.2707	-0.13	0.9000
Test: c3.4 & a1 & Sex: Boys	-0.0453	0.1568	-0.29	0.7725
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	13287
zScore ~ 1 + Test + a1 + Sex + Test & a1 + Test & Sex + a1 & Sex + Test & a1 & Sex + (1 + Test \| Child)	36	13280	6	10	0.7898

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	13273
zScore ~ 1 + Test + a1 + Sex + Test & a1 + Test & Sex + a1 & Sex + Test & a1 & Sex + (1 + Test \| Child)	36	13352	-79	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	13274
zScore ~ 1 + Test + a1 + Sex + Test & a1 + Test & Sex + a1 & Sex + Test & a1 & Sex + (1 + Test \| Child)	36	13313	-38	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	13256
zScore ~ 1 + Test + a1 + Sex + Test & a1 + Test & Sex + a1 & Sex + Test & a1 & Sex + (1 + Test \| Child)	36	13379	-123	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	13256.2	13308.2	13477.6
2	HeC0	26	13273.1	13325.1	13494.5
3	HyC0	26	13274.2	13326.2	13495.7
4	SDC0	26	13286.7	13338.7	13508.1

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 78    Time: 0:00:00 ( 1.29 ms/it)
  objective:  13869.813318317241
Minimizing 228    Time: 0:00:00 ( 0.88 ms/it)
  objective:  13833.568709129437
Minimizing 379    Time: 0:00:00 ( 0.80 ms/it)
  objective:  13833.309562206396
Minimizing 530    Time: 0:00:00 ( 0.76 ms/it)
  objective:  13833.277408958445
Minimizing 681    Time: 0:00:00 ( 0.75 ms/it)
  objective:  13833.24130269386
Minimizing 831    Time: 0:00:00 ( 0.74 ms/it)
  objective:  13833.2230505581
Minimizing 982    Time: 0:00:00 ( 0.73 ms/it)
  objective:  13833.217785527395
Minimizing 1052    Time: 0:00:00 ( 0.72 ms/it)
Minimizing 156    Time: 0:00:00 ( 0.65 ms/it)
  objective:  13909.421882276463
Minimizing 310    Time: 0:00:00 ( 0.66 ms/it)
  objective:  13843.453787979204
Minimizing 462    Time: 0:00:00 ( 0.66 ms/it)
  objective:  13834.35116310957
Minimizing 613    Time: 0:00:00 ( 0.66 ms/it)
  objective:  13833.417878872582
Minimizing 764    Time: 0:00:00 ( 0.66 ms/it)
  objective:  13833.34551947917
Minimizing 915    Time: 0:00:00 ( 0.66 ms/it)
  objective:  13833.314887042341
Minimizing 1067    Time: 0:00:00 ( 0.66 ms/it)
  objective:  13833.312227428132
Minimizing 1218    Time: 0:00:00 ( 0.66 ms/it)
  objective:  13833.312092375516
Minimizing 1370    Time: 0:00:00 ( 0.66 ms/it)
  objective:  13833.310624735019
Minimizing 1521    Time: 0:00:01 ( 0.66 ms/it)
  objective:  13833.30259941027
Minimizing 1672    Time: 0:00:01 ( 0.66 ms/it)
  objective:  13833.295867777319
Minimizing 1823    Time: 0:00:01 ( 0.66 ms/it)
  objective:  13833.292661405161
Minimizing 1974    Time: 0:00:01 ( 0.66 ms/it)
  objective:  13833.28914915711
Minimizing 2126    Time: 0:00:01 ( 0.66 ms/it)
  objective:  13833.284707074281
Minimizing 2278    Time: 0:00:01 ( 0.66 ms/it)
  objective:  13833.262421689125
Minimizing 2430    Time: 0:00:01 ( 0.66 ms/it)
  objective:  13833.241990863187
Minimizing 2581    Time: 0:00:01 ( 0.66 ms/it)
  objective:  13833.247079755502
Minimizing 2732    Time: 0:00:01 ( 0.66 ms/it)
  objective:  13833.221055382282
Minimizing 2883    Time: 0:00:01 ( 0.66 ms/it)
  objective:  13833.218046906093
Minimizing 3035    Time: 0:00:02 ( 0.66 ms/it)
  objective:  13833.217803885223
Minimizing 3083    Time: 0:00:02 ( 0.66 ms/it)
Minimizing 154    Time: 0:00:00 ( 0.65 ms/it)
  objective:  13835.103652066451
Minimizing 306    Time: 0:00:00 ( 0.66 ms/it)
  objective:  13833.436662036229
Minimizing 458    Time: 0:00:00 ( 0.66 ms/it)
  objective:  13833.29274952111
Minimizing 610    Time: 0:00:00 ( 0.66 ms/it)
  objective:  13833.281680094417
Minimizing 762    Time: 0:00:00 ( 0.66 ms/it)
  objective:  13833.251871440329
Minimizing 914    Time: 0:00:00 ( 0.66 ms/it)
  objective:  13833.228720240853
Minimizing 1066    Time: 0:00:00 ( 0.66 ms/it)
  objective:  13833.220389796224
Minimizing 1218    Time: 0:00:00 ( 0.66 ms/it)
  objective:  13833.217827011029
Minimizing 1371    Time: 0:00:00 ( 0.66 ms/it)
  objective:  13833.217686776083
Minimizing 1425    Time: 0:00:00 ( 0.66 ms/it)
Minimizing 154    Time: 0:00:00 ( 0.65 ms/it)
  objective:  13835.864365767455
Minimizing 307    Time: 0:00:00 ( 0.68 ms/it)
  objective:  13833.492581241022
Minimizing 454    Time: 0:00:00 ( 0.68 ms/it)
  objective:  13833.290651618909
Minimizing 605    Time: 0:00:00 ( 0.68 ms/it)
  objective:  13833.236150193568
Minimizing 757    Time: 0:00:00 ( 0.68 ms/it)
  objective:  13833.22598418184
Minimizing 909    Time: 0:00:00 ( 0.67 ms/it)
  objective:  13833.218017018437
Minimizing 1059    Time: 0:00:00 ( 0.67 ms/it)
  objective:  13833.217763653089
Minimizing 1208    Time: 0:00:00 ( 0.67 ms/it)
  objective:  13833.217692227265
Minimizing 1264    Time: 0:00:00 ( 0.67 ms/it)

Minimizing 1265    Time: 0:00:00 ( 0.67 ms/it)
  objective:  13833.217685724894

VarCorr(m_School_SeqDiff)

	Column	Variance	Std.Dev	Corr.
School	(Intercept)	0.043165	0.207763
	Test: Star_r	0.114727	0.338713	+0.05
	Test: S20_r	0.044376	0.210657	+0.45	-0.46
	Test: SLJ	0.116747	0.341683	-0.54	-0.58	-0.25
	Test: BPT	0.010374	0.101855	+0.14	+0.64	+0.13	-0.43
	a1	0.117291	0.342478	+0.40	-0.38	+0.10	+0.36	-0.51
Residual		0.858846	0.926739

VarCorr(m_School_Helmert)

	Column	Variance	Std.Dev	Corr.
School	(Intercept)	0.0431606	0.2077512
	Test: Star_r	0.0286809	0.1693545	+0.05
	Test: S20_r	0.0044357	0.0666008	+0.52	+0.36
	Test: SLJ	0.0041207	0.0641930	-0.45	-0.58	-0.48
	Test: BPT	0.0015528	0.0394051	-0.37	-0.24	-0.12	+0.86
	a1	0.1173043	0.3424971	+0.40	-0.38	-0.22	+0.37	+0.10
Residual		0.8588610	0.9267476

VarCorr(m_School_Hypo)

	Column	Variance	Std.Dev	Corr.
School	(Intercept)	0.043162	0.207754
	Test: BPT-other	0.621347	0.788256	-0.37
	Test: Star-End	0.114756	0.338757	+0.05	-0.24
	Test: S20-Star	0.044361	0.210619	+0.45	+0.08	-0.46
	Test: SLJ-S20	0.116710	0.341629	-0.54	+0.70	-0.58	-0.25
	a1	0.117322	0.342523	+0.40	+0.10	-0.38	+0.10	+0.36
Residual		0.858849	0.926741

VarCorr(m_School_PCA)

	Column	Variance	Std.Dev	Corr.
School	(Intercept)	0.043161	0.207753
	Test: c5.1	0.069985	0.264547	-0.22
	Test: c234.15	1.022013	1.010947	+0.29	+0.11
	Test: c2.34	0.222635	0.471842	-0.01	-0.07	+0.80
	Test: c3.4	0.116712	0.341631	+0.54	-0.19	+0.84	+0.50
	a1	0.117327	0.342530	+0.40	-0.13	-0.33	-0.35	-0.36
Residual		0.858852	0.926742

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 214    Time: 0:00:00 ( 0.48 ms/it)
  objective:  13858.910668556311
Minimizing 352    Time: 0:00:00 ( 0.48 ms/it)

Minimizing 353    Time: 0:00:00 ( 0.49 ms/it)
  objective:  13858.882947855234
Minimizing 216    Time: 0:00:00 ( 0.47 ms/it)
  objective:  13853.611245351643
Minimizing 235    Time: 0:00:00 ( 0.48 ms/it)

4×5 DataFrame

Row	zcpLMM	dof	deviance	AIC	BIC
	String	Int64	Float64	Float64	Float64
1	SDC0	27	13853.4	13907.4	14083.4
2	PCA0	27	13853.6	13907.6	14083.6
3	HeC0	27	13854.1	13908.1	14084.0
4	HyC0	27	13858.9	13912.9	14088.8

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