Kliegl et al. (2015) is a follow-up to Kliegl et al. (2011) (see also script kwdyz11.qmd) from an experiment looking at a variety of effects of visual cueing under four different cue-target relations (CTRs). In this experiment two rectangles are displayed (1) in horizontal orientation , (2) in vertical orientation, (3) in left diagonal orientation, or in (4) right diagonal orientation relative to a central fixation point. Subjects react to the onset of a small or a large visual target occurring at one of the four ends of the two rectangles. The target is cued validly on 70% of trials by a brief flash of the corner of the rectangle at which it appears; it is cued invalidly at the three other locations 10% of the trials each. This implies a latent imbalance in design that is not visible in the repeated-measures ANOVA, but we will show its effect in the random-effect structure and conditional modes.
There are a couple of differences between the first and this follow-up experiment, rendering it more a conceptual than a direct replication. First, the original experiment was carried out at Peking University and this follow-up at Potsdam University. Second, diagonal orientations of rectangles and large target sizes were not part of the design of Kliegl et al. (2011).
We specify three contrasts for the four-level factor CTR that are derived from spatial, object-based, and attractor-like features of attention. They map onto sequential differences between appropriately ordered factor levels. Replicating Kliegl et al. (2011), the attraction effect was not significant as a fixed effect, but yielded a highly reliable variance component (VC; i.e., reliable individual differences in positive and negative attraction effects cancel the fixed effect). Moreover, these individual differences in the attraction effect were negatively correlated with those in the spatial effect.
This comparison is of interest because a few years after the publication of Kliegl et al. (2011), the theoretically critical correlation parameter (CP) between the spatial effect and the attraction effect was determined as the source of a non-singular LMM in that paper. The present study served the purpose to estimate this parameter with a larger sample and a wider variety of experimental conditions.
Here we also include two additional experimental manipulations of target size and orientation of cue rectangle. A similar analysis was reported in the parsimonious mixed-model paper (Bates et al., 2015); it was also used in a paper of GAMEMs (Baayen et al., 2017). Data and R scripts of those analyses are also available in R-package RePsychLing.
The analysis is based on reaction times rt to maintain compatibility with Kliegl et al. (2011).
In this vignette we focus on the reduction of model complexity. And we start with a quote:
“Neither the [maximal] nor the [minimal] linear mixed models are appropriate for most repeated measures analysis. Using the [maximal] model is generally wasteful and costly in terms of statiscal power for testing hypotheses. On the other hand, the [minimal] model fails to account for nontrivial correlation among repeated measurements. This results in inflated [T]ype I error rates when non-negligible correlation does in fact exist. We can usually find middle ground, a covariance model that adequately accounts for correlation but is more parsimonious than the [maximal] model. Doing so allows us full control over [T]ype I error rates without needlessly sacrificing power.”
Stroup, W. W. (2012, p. 185). Generalized linear mixed models: Modern concepts, methods and applica?ons. CRC Press, Boca Raton.
using AlgebraOfGraphics
using AlgebraOfGraphics: density
using BoxCox
using CairoMakie
using CategoricalArrays
using Chain
using DataFrameMacros
using DataFrames
using MixedModels
using MixedModelsMakie
using Random
using SMLP2025: dataset
using StatsBasedat = DataFrame(dataset(:kkl15))
describe(dat)| Row | variable | mean | min | median | max | nmissing | eltype |
|---|---|---|---|---|---|---|---|
| Symbol | Union… | Any | Union… | Any | Int64 | DataType | |
| 1 | Subj | S001 | S147 | 0 | String | ||
| 2 | CTR | dod | val | 0 | String | ||
| 3 | rt | 293.147 | 150.22 | 276.594 | 749.481 | 0 | Float32 |
| 4 | cardinal | cardinal | diagonal | 0 | String | ||
| 5 | size | big | small | 0 | String |
dat_subj = combine(
groupby(dat, [:Subj, :CTR]),
nrow => :n,
:rt => mean => :rt_m,
:rt => (c -> mean(log, c)) => :lrt_m,
)
dat_subj.CTR = categorical(dat_subj.CTR, levels=levels(dat.CTR))
describe(dat_subj)| Row | variable | mean | min | median | max | nmissing | eltype |
|---|---|---|---|---|---|---|---|
| Symbol | Union… | Any | Union… | Any | Int64 | DataType | |
| 1 | Subj | S001 | S147 | 0 | String | ||
| 2 | CTR | val | dod | 0 | CategoricalValue{String, UInt32} | ||
| 3 | n | 156.294 | 49 | 64.0 | 448 | 0 | Int64 |
| 4 | rt_m | 308.223 | 208.194 | 304.862 | 584.71 | 0 | Float32 |
| 5 | lrt_m | 5.6908 | 5.33226 | 5.69848 | 6.36141 | 0 | Float32 |
boxplot(
dat_subj.CTR.refs,
dat_subj.lrt_m;
orientation=:horizontal,
show_notch=true,
axis=(;
yticks=(
1:4,
[
"valid cue",
"same obj/diff pos",
"diff obj/same pos",
"diff obj/diff pos",
]
)
),
figure=(; resolution=(800, 300)),
)┌ Warning: Found `resolution` in the theme when creating a `Scene`. The `resolution` keyword for `Scene`s and `Figure`s has been deprecated. Use `Figure(; size = ...` or `Scene(; size = ...)` instead, which better reflects that this is a unitless size and not a pixel resolution. The key could also come from `set_theme!` calls or related theming functions. └ @ Makie ~/.julia/packages/Makie/KcEWO/src/scenes.jl:238
Mean of reaction times for four cue-target relations. Targets appeared at (a) the cued position (valid) in a rectangle, (b) in the same rectangle cue, but at its other end, (c) on the second rectangle, but at a corresponding horizontal/vertical physical distance, or (d) at the other end of the second rectangle, that is \(\sqrt{2}\) of horizontal/vertical distance diagonally across from the cue, that is also at larger physical distance compared to (c).
contrasts = Dict(
:CTR => SeqDiffCoding(; levels=["val", "sod", "dos", "dod"]),
:cardinal => EffectsCoding(; levels=["cardinal", "diagonal"]),
:size => EffectsCoding(; levels=["big", "small"])
)Dict{Symbol, StatsModels.AbstractContrasts} with 3 entries:
:CTR => SeqDiffCoding(["val", "sod", "dos", "dod"])
:size => EffectsCoding(nothing, ["big", "small"])
:cardinal => EffectsCoding(nothing, ["cardinal", "diagonal"])This is the maximum LMM for the design; size is a between-subject factor, ignoring other information such as trial number, age and gender of subjects.
m_max = let
form = @formula rt ~ 1 + CTR * cardinal * size +
(1 + CTR * cardinal | Subj)
fit(MixedModel, form, dat; contrasts)
end;Minimizing 292 Time: 0:00:00 ( 0.34 ms/it) objective: 599379.688721075 Minimizing 876 Time: 0:00:00 ( 0.39 ms/it)
issingular(m_max)trueonly(MixedModels.PCA(m_max))
Principal components based on correlation matrix
(Intercept) 1.0 … . . . . .
CTR: sod 0.6 . . . . .
CTR: dos 0.44 . . . . .
CTR: dod 0.38 1.0 . . . .
cardinal: diagonal -0.03 0.15 1.0 . . .
CTR: sod & cardinal: diagonal 0.32 … 0.02 0.26 1.0 . .
CTR: dos & cardinal: diagonal 0.42 -0.02 0.24 0.43 1.0 .
CTR: dod & cardinal: diagonal 0.14 0.88 0.11 0.26 0.07 1.0
Normalized cumulative variances:
[0.3611, 0.5905, 0.7428, 0.8638, 0.9457, 1.0, 1.0, 1.0]
Component loadings
PC1 … PC4 PC5 PC6 PC7 PC8
(Intercept) -0.43 -0.15 -0.29 0.47 -0.19 0.46
CTR: sod -0.35 -0.41 -0.03 -0.23 -0.12 -0.59
CTR: dos -0.39 0.35 -0.1 -0.75 -0.08 0.19
CTR: dod -0.45 0.22 -0.18 0.1 0.7 0.06
cardinal: diagonal -0.12 … -0.56 -0.48 -0.16 -0.13 0.1
CTR: sod & cardinal: diagonal -0.31 -0.43 0.72 -0.01 0.26 0.14
CTR: dos & cardinal: diagonal -0.24 0.22 -0.15 0.33 0.06 -0.6
CTR: dod & cardinal: diagonal -0.41 0.3 0.31 0.17 -0.6 0.0VarCorr(m_max)| Column | Variance | Std.Dev | Corr. | |||||||
| Subj | (Intercept) | 2938.44786 | 54.20745 | |||||||
| CTR: sod | 455.44655 | 21.34119 | +0.60 | |||||||
| CTR: dos | 56.85753 | 7.54039 | +0.44 | +0.18 | ||||||
| CTR: dod | 192.63470 | 13.87929 | +0.38 | +0.54 | +0.30 | |||||
| cardinal: diagonal | 261.91287 | 16.18372 | -0.03 | -0.01 | +0.00 | +0.15 | ||||
| CTR: sod & cardinal: diagonal | 30.20533 | 5.49594 | +0.32 | +0.21 | +0.33 | +0.02 | +0.26 | |||
| CTR: dos & cardinal: diagonal | 1.64655 | 1.28318 | +0.42 | -0.37 | +0.58 | -0.02 | +0.24 | +0.43 | ||
| CTR: dod & cardinal: diagonal | 13.89368 | 3.72742 | +0.14 | +0.28 | +0.25 | +0.88 | +0.11 | +0.26 | +0.07 | |
| Residual | 3972.11165 | 63.02469 |
The LMM m_max is overparameterized but it is not immediately apparent why.
Force CPs to zero. Reduction strategy 1 is more suited for reducing model w/o theoretical expectations about CPs. The better reduction strategy for the present experiment with an a priori interest in CPs is described as Reduction strategy 2.
m_zcp1 = let
form = @formula rt ~ 1 + CTR * cardinal * size +
zerocorr(1 + CTR * cardinal | Subj)
fit(MixedModel, form, dat; contrasts)
end;issingular(m_zcp1)trueonly(MixedModels.PCA(m_zcp1))
Principal components based on correlation matrix
(Intercept) 1.0 . . . . . . .
CTR: sod 0.0 1.0 . . . . . .
CTR: dos 0.0 0.0 1.0 . . . . .
CTR: dod 0.0 0.0 0.0 1.0 . . . .
cardinal: diagonal 0.0 0.0 0.0 0.0 1.0 . . .
CTR: sod & cardinal: diagonal 0.0 0.0 0.0 0.0 0.0 1.0 . .
CTR: dos & cardinal: diagonal 0.0 0.0 0.0 0.0 0.0 0.0 0.0 .
CTR: dod & cardinal: diagonal 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0
Normalized cumulative variances:
[0.1429, 0.2857, 0.4286, 0.5714, 0.7143, 0.8571, 1.0, 1.0]
Component loadings
PC1 PC2 … PC4 PC5 PC6 PC7 PC8
(Intercept) 1.0 0.0 0.0 0.0 0.0 0.0 0.0
CTR: sod 0.0 1.0 0.0 0.0 0.0 0.0 0.0
CTR: dos 0.0 0.0 0.0 0.0 0.0 0.0 0.0
CTR: dod 0.0 0.0 1.0 0.0 0.0 0.0 0.0
cardinal: diagonal 0.0 0.0 … 0.0 1.0 0.0 0.0 0.0
CTR: sod & cardinal: diagonal 0.0 0.0 0.0 0.0 1.0 0.0 0.0
CTR: dos & cardinal: diagonal 0.0 0.0 0.0 0.0 0.0 0.0 NaN
CTR: dod & cardinal: diagonal 0.0 0.0 0.0 0.0 0.0 1.0 0.0VarCorr(m_zcp1)| Column | Variance | Std.Dev | Corr. | |||||||
| Subj | (Intercept) | 2875.43244 | 53.62306 | |||||||
| CTR: sod | 483.09089 | 21.97933 | . | |||||||
| CTR: dos | 79.93243 | 8.94049 | . | . | ||||||
| CTR: dod | 216.91700 | 14.72810 | . | . | . | |||||
| cardinal: diagonal | 250.32171 | 15.82156 | . | . | . | . | ||||
| CTR: sod & cardinal: diagonal | 35.91784 | 5.99315 | . | . | . | . | . | |||
| CTR: dos & cardinal: diagonal | 0.00000 | 0.00000 | . | . | . | . | . | . | ||
| CTR: dod & cardinal: diagonal | 6.88389 | 2.62372 | . | . | . | . | . | . | . | |
| Residual | 3972.38114 | 63.02683 |
The LMM m_zcp1 is also overparameterized, but now there is clear evidence for absence of evidence for the VC of one of the interactions and the other two interaction-based VCs are also very small.
Take out VCs for interactions.
m_zcp1_rdc = let
form = @formula rt ~ 1 + CTR * cardinal * size +
zerocorr(1 + CTR + cardinal | Subj)
fit(MixedModel, form, dat; contrasts)
end;issingular(m_zcp1_rdc)falseonly(MixedModels.PCA(m_zcp1_rdc))
Principal components based on correlation matrix
(Intercept) 1.0 . . . .
CTR: sod 0.0 1.0 . . .
CTR: dos 0.0 0.0 1.0 . .
CTR: dod 0.0 0.0 0.0 1.0 .
cardinal: diagonal 0.0 0.0 0.0 0.0 1.0
Normalized cumulative variances:
[0.2, 0.4, 0.6, 0.8, 1.0]
Component loadings
PC1 PC2 PC3 PC4 PC5
(Intercept) 0.0 0.0 0.0 0.0 1.0
CTR: sod 1.0 0.0 0.0 0.0 0.0
CTR: dos 0.0 1.0 0.0 0.0 0.0
CTR: dod 0.0 0.0 1.0 0.0 0.0
cardinal: diagonal 0.0 0.0 0.0 1.0 0.0VarCorr(m_zcp1_rdc)| Column | Variance | Std.Dev | Corr. | ||||
| Subj | (Intercept) | 2881.45052 | 53.67914 | ||||
| CTR: sod | 484.31679 | 22.00720 | . | ||||
| CTR: dos | 79.54961 | 8.91906 | . | . | |||
| CTR: dod | 216.59820 | 14.71728 | . | . | . | ||
| cardinal: diagonal | 244.93991 | 15.65056 | . | . | . | . | |
| Residual | 3980.87777 | 63.09420 |
LMM m_zcp_rdc is ok . We add in CPs.
Extend zcp-reduced LMM with CPs
m_prm1 = let
form = @formula rt ~ 1 + CTR * cardinal * size +
(1 + CTR + cardinal | Subj)
fit(MixedModel, form, dat; contrasts)
end;issingular(m_prm1)falseonly(MixedModels.PCA(m_prm1))
Principal components based on correlation matrix
(Intercept) 1.0 . . . .
CTR: sod 0.6 1.0 . . .
CTR: dos 0.45 0.19 1.0 . .
CTR: dod 0.37 0.54 0.29 1.0 .
cardinal: diagonal -0.1 -0.05 -0.09 0.13 1.0
Normalized cumulative variances:
[0.4488, 0.6668, 0.8307, 0.9437, 1.0]
Component loadings
PC1 PC2 PC3 PC4 PC5
(Intercept) -0.55 -0.16 0.01 0.58 -0.58
CTR: sod -0.54 0.09 -0.49 0.17 0.66
CTR: dos -0.4 -0.25 0.8 -0.13 0.34
CTR: dod -0.49 0.36 -0.1 -0.71 -0.35
cardinal: diagonal 0.05 0.88 0.32 0.34 0.07LMM m_zcp_rdc is ok . We add in CPs.
VarCorr(m_prm1)| Column | Variance | Std.Dev | Corr. | ||||
| Subj | (Intercept) | 2926.39374 | 54.09615 | ||||
| CTR: sod | 454.10921 | 21.30984 | +0.60 | ||||
| CTR: dos | 56.75233 | 7.53341 | +0.45 | +0.19 | |||
| CTR: dod | 188.74621 | 13.73849 | +0.37 | +0.54 | +0.29 | ||
| cardinal: diagonal | 245.27143 | 15.66114 | -0.10 | -0.05 | -0.09 | +0.13 | |
| Residual | 3981.71306 | 63.10082 |
We note that the critical correlation parameter between spatial (sod) and attraction (dod) is now estimated at .54 – not that close to the 1.0 boundary that caused singularity in Kliegl et al. (2011).
gof_summary = let
nms = [:m_zcp1_rdc, :m_prm1, :m_max]
mods = eval.(nms)
lrt = MixedModels.likelihoodratiotest(m_zcp1_rdc, m_prm1, m_max)
DataFrame(;
name = nms,
dof=dof.(mods),
deviance=round.(deviance.(mods), digits=0),
AIC=round.(aic.(mods),digits=0),
AICc=round.(aicc.(mods),digits=0),
BIC=round.(bic.(mods),digits=0),
χ²=vcat(:., round.(lrt.tests.deviancediff, digits=0)),
χ²_dof=vcat(:., round.(lrt.tests.dofdiff, digits=0)),
pvalue=vcat(:., round.(lrt.tests.pvalues, digits=3))
)
end| Row | name | dof | deviance | AIC | AICc | BIC | χ² | χ²_dof | pvalue |
|---|---|---|---|---|---|---|---|---|---|
| Symbol | Int64 | Float64 | Float64 | Float64 | Float64 | Any | Any | Any | |
| 1 | m_zcp1_rdc | 22 | 599486.0 | 599530.0 | 599530.0 | 599725.0 | . | . | . |
| 2 | m_prm1 | 32 | 599418.0 | 599482.0 | 599482.0 | 599766.0 | 68.0 | 10.0 | 0.0 |
| 3 | m_max | 53 | 599359.0 | 599465.0 | 599465.0 | 599936.0 | 59.0 | 21.0 | 0.0 |
AIC prefers LMM m_prm1 over m_zcp1_rdc; BIC LMM m_zcp1_rdc. As the CPs were one reason for conducting this experiment, AIC is the criterion of choice.
Relative to LMM m_max, first we take out interaction VCs and associated CPs, because these VCs are very small. This is the same as LMM m_prm1 above.
m_cpx = let
form = @formula rt ~ 1 + CTR * cardinal * size +
(1 + CTR + cardinal | Subj)
fit(MixedModel, form, dat; contrasts)
end;Now we check the significance of ensemble of CPs.
m_zcp2 = let
form = @formula rt ~ 1 + CTR * cardinal * size +
zerocorr(1 + CTR + cardinal | Subj)
fit(MixedModel, form, dat; contrasts)
end;VarCorr(m_zcp2)| Column | Variance | Std.Dev | Corr. | ||||
| Subj | (Intercept) | 2881.45052 | 53.67914 | ||||
| CTR: sod | 484.31679 | 22.00720 | . | ||||
| CTR: dos | 79.54961 | 8.91906 | . | . | |||
| CTR: dod | 216.59820 | 14.71728 | . | . | . | ||
| cardinal: diagonal | 244.93991 | 15.65056 | . | . | . | . | |
| Residual | 3980.87777 | 63.09420 |
The cardinal-related CPs are quite small. Do we need them?
m_prm2 = let
form = @formula(rt ~ 1 + CTR * cardinal * size +
(1 + CTR | Subj) + (0 + cardinal | Subj))
fit(MixedModel, form, dat; contrasts)
end;VarCorr(m_prm2)| Column | Variance | Std.Dev | Corr. | ||||
| Subj | (Intercept) | 2923.32186 | 54.06775 | ||||
| CTR: sod | 454.38842 | 21.31639 | +0.60 | ||||
| CTR: dos | 56.46065 | 7.51403 | +0.45 | +0.19 | |||
| CTR: dod | 187.89792 | 13.70759 | +0.37 | +0.54 | +0.30 | ||
| cardinal: diagonal | 245.16943 | 15.65789 | . | . | . | . | |
| Residual | 3981.74285 | 63.10105 |
gof_summary = let
nms = [:m_zcp2, :m_prm2, :m_cpx, :m_max]
mods = eval.(nms)
lrt = MixedModels.likelihoodratiotest(m_zcp2, m_prm2, m_cpx, m_max)
DataFrame(;
name = nms,
dof=dof.(mods),
deviance=round.(deviance.(mods), digits=0),
AIC=round.(aic.(mods),digits=0),
AICc=round.(aicc.(mods),digits=0),
BIC=round.(bic.(mods),digits=0),
χ²=vcat(:., round.(lrt.tests.deviancediff, digits=0)),
χ²_dof=vcat(:., round.(lrt.tests.dofdiff, digits=0)),
pvalue=vcat(:., round.(lrt.tests.pvalues, digits=3))
)
end| Row | name | dof | deviance | AIC | AICc | BIC | χ² | χ²_dof | pvalue |
|---|---|---|---|---|---|---|---|---|---|
| Symbol | Int64 | Float64 | Float64 | Float64 | Float64 | Any | Any | Any | |
| 1 | m_zcp2 | 22 | 599486.0 | 599530.0 | 599530.0 | 599725.0 | . | . | . |
| 2 | m_prm2 | 28 | 599420.0 | 599476.0 | 599476.0 | 599725.0 | 65.0 | 6.0 | 0.0 |
| 3 | m_cpx | 32 | 599418.0 | 599482.0 | 599482.0 | 599766.0 | 2.0 | 4.0 | 0.652 |
| 4 | m_max | 53 | 599359.0 | 599465.0 | 599465.0 | 599936.0 | 59.0 | 21.0 | 0.0 |
The cardinal-related CPs could be removed w/o loss of goodness of fit. However, there is no harm in keeping them in the LMM. The data support both LMM m_prm2 and m_cpx (same as: m_prm1). We keep the slightly more complex LMM m_cpx (m_prm1).
Do model residuals meet LMM assumptions? Classic plots are
The slant in residuals show a lower and upper boundary of reaction times, that is we have have too few short and too few long residuals. Not ideal, but at least width of the residual band looks similar across the fitted values, that is there is no evidence for heteroskedasticity.
CairoMakie.activate!(; type="png")
scatter(fitted(m_prm1), residuals(m_prm1); alpha=0.3)
With many observations the scatterplot is not that informative. Contour plots or heatmaps may be an alternative.
set_aog_theme!()
draw(
data((; f=fitted(m_prm1), r=residuals(m_prm1))) *
mapping(
:f => "Fitted values from m1", :r => "Residuals from m1"
) *
density();
)
The plot of quantiles of model residuals over corresponding quantiles of the normal distribution should yield a straight line along the main diagonal.
CairoMakie.activate!(; type="png")
qqnorm(
residuals(m_prm1);
qqline=:none,
axis=(;
xlabel="Standard normal quantiles",
ylabel="Quantiles of the residuals from model m1",
),
)
cm1 = only(ranefinfo(m_prm1))
caterpillar!(Figure(; resolution=(800, 1200)), cm1; orderby=2)┌ Warning: Found `resolution` in the theme when creating a `Scene`. The `resolution` keyword for `Scene`s and `Figure`s has been deprecated. Use `Figure(; size = ...` or `Scene(; size = ...)` instead, which better reflects that this is a unitless size and not a pixel resolution. The key could also come from `set_theme!` calls or related theming functions. └ @ Makie ~/.julia/packages/Makie/KcEWO/src/scenes.jl:238
shrinkageplot!(Figure(; resolution=(1000, 1200)), m_prm1)┌ Warning: Found `resolution` in the theme when creating a `Scene`. The `resolution` keyword for `Scene`s and `Figure`s has been deprecated. Use `Figure(; size = ...` or `Scene(; size = ...)` instead, which better reflects that this is a unitless size and not a pixel resolution. The key could also come from `set_theme!` calls or related theming functions. └ @ Makie ~/.julia/packages/Makie/KcEWO/src/scenes.jl:238
Here we
We generate 2500 samples for the 15 model parameters (4 fixed effect, 7 VCs, 15 CPs, and 1 residual).
samp = parametricbootstrap(MersenneTwister(1234321), 2500, m_prm1;
optsum_overrides=(; ftol_rel=1e-8));tbl = samp.tblTable with 48 columns and 2500 rows:
obj β01 β02 β03 β04 β05 β06 ⋯
┌────────────────────────────────────────────────────────────────────
1 │ 5.99123e5 318.146 26.2903 13.9174 6.74944 5.97167 27.2175 ⋯
2 │ 5.99228e5 315.166 21.7222 13.8924 1.41165 4.28871 26.216 ⋯
3 │ 5.99391e5 304.787 22.8413 14.4555 1.30937 5.97979 28.3124 ⋯
4 │ 5.98758e5 304.983 25.7334 14.2529 4.46301 7.3687 25.5255 ⋯
5 │ 5.99417e5 305.661 20.3663 11.7683 1.95922 7.55567 22.6423 ⋯
6 │ 5.99825e5 303.204 20.6372 11.9589 3.01109 3.64718 14.051 ⋯
7 │ 5.9885e5 308.687 26.3962 14.9357 4.30657 9.44674 28.0379 ⋯
8 │ 5.99054e5 308.726 22.0484 14.0206 2.04468 2.35748 21.458 ⋯
9 │ 5.99529e5 300.775 22.2847 10.537 4.48452 4.40397 21.3034 ⋯
10 │ 5.99096e5 314.717 27.6003 12.8212 1.9483 5.6583 24.3917 ⋯
11 │ 5.99564e5 313.363 25.41 11.5791 4.62007 9.92006 28.7788 ⋯
12 │ 5.99323e5 312.305 22.9754 15.0654 0.256928 8.85752 25.8754 ⋯
13 │ 5.99502e5 310.986 23.8323 14.729 1.69965 9.73568 25.5329 ⋯
14 │ 5.997e5 317.858 25.1774 13.9608 4.48191 8.0803 36.8897 ⋯
15 │ 6.00337e5 309.607 24.4717 13.2423 2.80069 3.8812 24.2719 ⋯
16 │ 600049.0 308.469 24.842 10.4018 3.39803 6.35019 28.8359 ⋯
17 │ 5.98992e5 313.968 24.6616 12.7233 3.90727 7.20239 27.1067 ⋯
⋮ │ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱confint(samp)DictTable with 2 columns and 32 rows:
par lower upper
────┬────────────────────
β01 │ 297.126 319.558
β02 │ 18.785 28.3171
β03 │ 10.2014 15.9642
β04 │ -1.03419 6.55744
β05 │ 3.25478 9.7972
β06 │ 14.051 37.0229
β07 │ 1.86817 5.3913
β08 │ -0.975109 3.7586
β09 │ -2.88731 1.77475
β10 │ 3.98408 13.7493
β11 │ -3.31111 2.47015
β12 │ 3.9137 11.1273
β13 │ -1.07411 5.60924
β14 │ -2.49725 1.0596
β15 │ -2.50563 2.27366
β16 │ 1.76716 6.47652
ρ01 │ 0.343104 0.785229
⋮ │ ⋮ ⋮We can also visualize the shortest coverage intervals for fixed effects with the ridgeplot() command:
ridgeplot(samp; show_intercept=false)
draw(
data(tbl) *
mapping(:σ => "Residual") *
density();
figure=(; resolution=(800, 400)),
)┌ Warning: Found `resolution` in the theme when creating a `Scene`. The `resolution` keyword for `Scene`s and `Figure`s has been deprecated. Use `Figure(; size = ...` or `Scene(; size = ...)` instead, which better reflects that this is a unitless size and not a pixel resolution. The key could also come from `set_theme!` calls or related theming functions. └ @ Makie ~/.julia/packages/Makie/KcEWO/src/scenes.jl:238
The shortest coverage interval for the GM ranges from x to x ms and the associate variance component from .x to .x. To keep the plot range small we do not include their densities here.
rn = renamer([
"(Intercept)" => "GM",
"CTR: sod" => "spatial effect",
"CTR: dos" => "object effect",
"CTR: dod" => "attraction effect",
"(Intercept), CTR: sod" => "GM, spatial",
"(Intercept), CTR: dos" => "GM, object",
"CTR: sod, CTR: dos" => "spatial, object",
"(Intercept), CTR: dod" => "GM, attraction",
"CTR: sod, CTR: dod" => "spatial, attraction",
"CTR: dos, CTR: dod" => "object, attraction",
])
draw(
data(tbl) *
mapping(
[:β02, :β03, :β04] .=> "Experimental effect size [ms]";
color=dims(1) =>
renamer(["spatial effect", "object effect", "attraction effect"]) =>
"Experimental effects",
) *
density();
figure=(; resolution=(800, 350)),
)┌ Warning: Found `resolution` in the theme when creating a `Scene`. The `resolution` keyword for `Scene`s and `Figure`s has been deprecated. Use `Figure(; size = ...` or `Scene(; size = ...)` instead, which better reflects that this is a unitless size and not a pixel resolution. The key could also come from `set_theme!` calls or related theming functions. └ @ Makie ~/.julia/packages/Makie/KcEWO/src/scenes.jl:238
The densitiies correspond nicely with the shortest coverage intervals.
draw(
data(tbl) *
mapping(
[:σ2, :σ3, :σ4] .=> "Standard deviations [ms]";
color=dims(1) =>
renamer(["spatial effect", "object effect", "attraction effect"]) =>
"Variance components",
) *
density();
figure=(; resolution=(800, 350)),
)┌ Warning: Found `resolution` in the theme when creating a `Scene`. The `resolution` keyword for `Scene`s and `Figure`s has been deprecated. Use `Figure(; size = ...` or `Scene(; size = ...)` instead, which better reflects that this is a unitless size and not a pixel resolution. The key could also come from `set_theme!` calls or related theming functions. └ @ Makie ~/.julia/packages/Makie/KcEWO/src/scenes.jl:238
The VC are all very nicely defined.
draw(
data(tbl) *
mapping(
[:ρ01, :ρ02, :ρ03, :ρ04, :ρ05, :ρ06] .=> "Correlation";
color=dims(1) =>
renamer(["GM, spatial", "GM, object", "spatial, object",
"GM, attraction", "spatial, attraction", "object, attraction"]) =>
"Correlation parameters",
) *
density();
figure=(; resolution=(800, 350)),
)┌ Warning: Found `resolution` in the theme when creating a `Scene`. The `resolution` keyword for `Scene`s and `Figure`s has been deprecated. Use `Figure(; size = ...` or `Scene(; size = ...)` instead, which better reflects that this is a unitless size and not a pixel resolution. The key could also come from `set_theme!` calls or related theming functions. └ @ Makie ~/.julia/packages/Makie/KcEWO/src/scenes.jl:238
Three CPs stand out positively, the correlation between GM and the spatial effect, GM and attraction effect, and the correlation between spatial and attraction effects. The second CP was positive, but not significant in the first study. The third CP replicates a CP that was judged questionable in script kwdyz11.jl. The three remaining CPs are not well defined for reaction times.
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