Using simulation to estimate uncertainty and power

Or how I learned how to stop worrying and love the bootstrap

After working through this page you will be able to:

• use simulation and the parametric bootstrap to estimate power and uncertainty;
• simulate data from scratch for a planned design;
• interpret the simulated distributions to inform design decisions.

NoteBefore you start

Prerequisites: Parametric bootstrap for mixed-effects models.

Datasets used: simulated data only.

using AlgebraOfGraphics
using CairoMakie
using DataFrames
using Distributions
using MixedModels
using MixedModelsMakie
using MixedModelsSim
using StatsBase
using Random

const progress = isinteractive()

1 From bootstrap to power analysis

In the parametric bootstrap, we used the parameter estimates from a model fitted to observed data to simulate new data and quantify the uncertainty of those estimates. The same machinery – simulate, refit, repeat – also answers a design question: if the true effects had a particular size, how often would we detect them? That is a power analysis. All we need beyond the bootstrap is a way to construct a model object for a design we have not yet run and a way to specify the “true” parameter values that generate the simulated data.

Looking at the docstring for parametricbootstrap, we see that there are keyword-arguments for the various model parameters:

@doc parametricbootstrap

parametricbootstrap([rng::AbstractRNG], nsamp::Integer, m::MixedModel{T}, ftype=T;
    β = fixef(m), σ = m.σ, θ = m.θ, progress=true, optsum_overrides=(;))

Perform nsamp parametric bootstrap replication fits of m, returning a MixedModelBootstrap.

The default random number generator is Random.GLOBAL_RNG.

ftype can be used to store the computed bootstrap values in a lower precision. ftype is not a named argument because named arguments are not used in method dispatch and thus specialization. In other words, having ftype as a positional argument has some potential performance benefits.

Keyword Arguments

• β, σ, and θ are the values of m's parameters for simulating the responses.

• σ is only valid for LinearMixedModel and GeneralizedLinearMixedModel for

families with a dispersion parameter.

• progress controls whether the progress bar is shown. Note that the progress

bar is automatically disabled for non-interactive (i.e. logging) contexts.

• optsum_overrides is used to override values of OptSummary in the models

fit during the bootstrapping process. For example, optsum_overrides=(;ftol_rel=1e-08) reduces the convergence criterion, which can greatly speed up the bootstrap fits. Taking advantage of this speed up to increase n can often lead to better estimates of coverage intervals.

Note

All coefficients are bootstrapped. In the rank deficient case, the inestimatable coefficients are treated as -0.0 in the simulations underlying the bootstrap, which will generally result in their estimate from the simulated data also being being inestimable and thus set to -0.0. However this behavior may change in future releases to explicitly drop the extraneous columns before simulation and thus not include their estimates in the bootstrap result.

These keyword arguments are forward on to the simulate! function, which simulates a new dataset based on model matrices and parameter values. The model matrices are simply taken from the model at hand. By default, the parameter values are the estimated parameter values from a fitted model.

@doc simulate!

simulate!(rng::AbstractRNG, m::MixedModel{T}; β=fixef(m), σ=m.σ, θ=T[])
simulate!(m::MixedModel; β=fixef(m), σ=m.σ, θ=m.θ)

Overwrite the response (i.e. m.trms[end]) with a simulated response vector from model m.

This simulation includes sampling new values for the random effects.

β can be specified either as a pivoted, full rank coefficient vector (cf. fixef) or as an unpivoted full dimension coefficient vector (cf. coef), where the entries corresponding to redundant columns will be ignored.

Note

Note that simulate! methods with a y::AbstractVector as the first argument (besides the RNG) and simulate methods return the simulated response. This is in contrast to simulate! methods with a m::MixedModel as the first argument, which modify the model's response and return the entire modified model.

simulate!([rng::AbstractRNG,] y::AbstractVector, m::MixedModel{T}[, newdata];
                β = coef(m), σ = m.σ, θ = T[], wts=m.wts)
simulate([rng::AbstractRNG,] m::MixedModel{T}[, newdata];
                β = coef(m), σ = m.σ, θ = T[], wts=m.wts)

Simulate a new response vector, optionally overwriting a pre-allocated vector.

New data can be optionally provided in tabular format.

This simulation includes sampling new values for the random effects. Thus in contrast to predict, there is no distinction in between "new" and "old" / previously observed random-effects levels.

Unlike predict, there is no type parameter for GeneralizedLinearMixedModel because the noise term in the model and simulation is always on the response scale.

The wts argument is currently ignored except for GeneralizedLinearMixedModel models with a Binomial distribution.

Note

Note that simulate! methods with a y::AbstractVector as the first argument (besides the RNG) and simulate methods return the simulated response. This is in contrast to simulate! methods with a m::MixedModel as the first argument, which modify the model's response and return the entire modified model.

So now we have a way to simulate new data with new parameter values once we have a model. We just need a way to create a model with our preferred design. We’ll use the MixedModelsSim package for that.

1.1 Simulating data from scratch.

The MixedModelsSim package provides a function simdat_crossed for simulating effects from a crossed design:

@doc simdat_crossed

simdat_crossed([RNG], subj_n, item_n;
               subj_btwn=nothing, item_btwn=nothing, both_win=nothing,
               subj_prefix="S", item_prefix="I")

Return a row table with a design specified by the:

• number of subjects (subj_n),

• number of items (item_n)

• between-subject factors (subj_btwn)

• between-item factors (item_btwn)

• within-subject/item factors (both_win)

If a factor is both between-subject and between-item, put it in both subj_btwn and item_btwn with the same keys and the same levels.

Factors should be specified as dictionaries in the following format:

Dict(
    :factor1_name => ["F1_level1", "F1_level2"],
    :factor2_name => ["F2_level1", "F2_level2", "F2_level3"]
)

In addition to design, the rowtable contains a field dv pre-populated with N(0,1) noise as a basis for further simulating a design.

Note

The number of subjects/items must divide the number of combinations of between subject/item factor levels. In other words, this function assumes a balanced design and will throw an error if that is not possible.

Let’s see what that looks like in practice. We’ll look at a simple 2 x 2 design with 20 subjects and 20 items. Our first factor age will vary between subjects and have the levels old and young. Our second factor frequency will vary between items and have the levels low and high. Finally, we also need to specify a random number generator to use for seeding the data simulation.

subj_n = 20
item_n = 20
subj_btwn = Dict(:age => ["old", "young"])
item_btwn = Dict(:frequency => ["low", "high"])
const RNG = MersenneTwister(42)
dat = simdat_crossed(RNG, subj_n, item_n;
                     subj_btwn, item_btwn)
Table(dat)

Table with 5 columns and 400 rows:
      subj  age    item  frequency  dv
    ┌─────────────────────────────────────────
 1  │ S01   old    I01   low        1.21028
 2  │ S02   young  I01   low        -0.0789986
 3  │ S03   old    I01   low        0.403512
 4  │ S04   young  I01   low        0.289951
 5  │ S05   old    I01   low        -0.0670422
 6  │ S06   young  I01   low        0.61398
 7  │ S07   old    I01   low        1.01694
 8  │ S08   young  I01   low        -0.053864
 9  │ S09   old    I01   low        -0.570387
 10 │ S10   young  I01   low        -0.857737
 11 │ S11   old    I01   low        -0.211452
 12 │ S12   young  I01   low        -0.853673
 13 │ S13   old    I01   low        0.439568
 14 │ S14   young  I01   low        -1.31491
 15 │ S15   old    I01   low        0.176901
 16 │ S16   young  I01   low        -0.254805
 17 │ S17   old    I01   low        -1.77737
 ⋮  │  ⋮      ⋮     ⋮        ⋮          ⋮

We have 400 rows – 20 subjects x 20 items. Similarly, the experimental factors are expanded out to be fully crossed. Finally, we have a dependent variable dv initialized to be draws from the standard normal distribution \(N(0,1)\).

NoteLatin squares, partial crossing, and continuous covariates

simdat_crossed is designed to simulate a fully crossed factorial design. If you have a partially crossed or Latin squares design, then you could delete the “extra” cells to reduce the fully crossed data here to be partially crossed. For continuous covariates, we need to separately construct the covariates and then use a tabular join to create the design. We’ll examine an example of this later.

simmod = fit(MixedModel,
             @formula(dv ~ 1 + age * frequency +
                          (1 + frequency | subj) +
                          (1 + age | item)), dat; progress)
println(simmod)

Linear mixed model fit by maximum likelihood
 dv ~ 1 + age + frequency + age & frequency + (1 + frequency | subj) + (1 + age | item)
   logLik   -2 logLik     AIC       AICc        BIC    
  -591.8575  1183.7149  1205.7149  1206.3953  1249.6210

Variance components:
             Column      Variance  Std.Dev.   Corr.
subj     (Intercept)     0.0733656 0.2708608
         frequency: low  0.0335313 0.1831157 -1.00
item     (Intercept)     0.0132766 0.1152241
         age: young      0.0005200 0.0228031 -1.00
Residual                 1.0883091 1.0432205
 Number of obs: 400; levels of grouping factors: 20, 20

  Fixed-effects parameters:
─────────────────────────────────────────────────────────────────────
                                   Coef.  Std. Error      z  Pr(>|z|)
─────────────────────────────────────────────────────────────────────
(Intercept)                   0.0816111     0.139812   0.58    0.5594
age: young                   -0.0349801     0.191027  -0.18    0.8547
frequency: low               -0.0638793     0.166657  -0.38    0.7015
age: young & frequency: low   0.00879841    0.224372   0.04    0.9687
─────────────────────────────────────────────────────────────────────

β = [250.0, -25.0, 10, 0.0]
simulate!(RNG, simmod; β)

	Est.	SE	z	p	σ_subj	σ_item
(Intercept)	0.0816	0.1398	0.58	0.5594	0.2709	0.1152
age: young	-0.0350	0.1910	-0.18	0.8547		0.0228
frequency: low	-0.0639	0.1667	-0.38	0.7015	0.1831
age: young & frequency: low	0.0088	0.2244	0.04	0.9687
Residual	1.0432

fit!(simmod)

	Est.	SE	z	p	σ_subj	σ_item
(Intercept)	249.9796	0.1267	1973.16	<1e-99	0.1641	0.1703
age: young	-25.0472	0.1655	-151.36	<1e-99		0.1041
frequency: low	9.7387	0.1640	59.39	<1e-99	0.0422
age: young & frequency: low	0.1164	0.2106	0.55	0.5804
Residual	1.0225

σ = 25.0
fit!(simulate!(RNG, simmod; β, σ))

	Est.	SE	z	p	σ_subj	σ_item
(Intercept)	256.2394	3.9124	65.49	<1e-99	4.5684	8.6530
age: young	-33.8588	4.6177	-7.33	<1e-12		7.5397
frequency: low	5.5490	5.2227	1.06	0.2880	2.8944
age: young & frequency: low	5.1727	5.9978	0.86	0.3884
Residual	23.9420

# relative to σ!
subj_re = create_re(2.0, 1.3)

2×2 LinearAlgebra.LowerTriangular{Float64, LinearAlgebra.Diagonal{Float64, Vector{Float64}}}:
 2.0   ⋅ 
 0.0  1.3

item_re = create_re(1.3, 2.0)

2×2 LinearAlgebra.LowerTriangular{Float64, LinearAlgebra.Diagonal{Float64, Vector{Float64}}}:
 1.3   ⋅ 
 0.0  2.0

θ = createθ(simmod; subj=subj_re, item=item_re)

6-element Vector{Float64}:
 2.0
 0.0
 1.3
 1.3
 0.0
 2.0

fit!(simulate!(RNG, simmod; β, σ, θ))

	Est.	SE	z	p	σ_subj	σ_item
(Intercept)	238.6765	15.4590	15.44	<1e-53	39.9114	27.0075
age: young	-21.2541	21.2976	-1.00	0.3183		34.8575
frequency: low	-0.3036	16.0265	-0.02	0.9849	31.2202
age: young & frequency: low	16.0378	21.5626	0.74	0.4570
Residual	25.9787

samp = parametricbootstrap(RNG, 1000, simmod; β, σ, θ, progress)

MixedModelBootstrap with 1000 samples
      parameter  min        q25        median      mean        q75       ⋯
    ┌─────────────────────────────────────────────────────────────────────
 1  │ β1         187.504    236.563    250.196     249.917     263.367   ⋯
 2  │ β2         -118.074   -45.386    -24.2367    -25.4062    -6.48009  ⋯
 3  │ β3         -57.7974   -2.0594    8.93123     9.78849     22.555    ⋯
 4  │ β4         -77.082    -17.5927   0.253178    -0.0776182  17.2086   ⋯
 5  │ σ          22.204     24.3262    24.9756     24.9547     25.5221   ⋯
 6  │ σ1         25.6549    42.027     47.4457     47.7116     53.3682   ⋯
 7  │ σ2         13.7514    26.8112    31.2347     30.9942     34.9932   ⋯
 8  │ ρ1         -0.808928  -0.18075   -0.0242876  -0.0201442  0.151477  ⋯
 9  │ σ3         14.4239    27.508     31.2277     31.2821     35.1992   ⋯
 10 │ σ4         17.287     42.1595    47.4245     47.5599     52.6321   ⋯
 11 │ ρ2         -0.696028  -0.203852  -0.0184107  -0.0147826  0.156928  ⋯
 12 │ θ1         1.01087    1.68976    1.90887     1.91418     2.13819   ⋯
 13 │ θ2         -1.19827   -0.227775  -0.0266265  -0.0274248  0.185927  ⋯
 14 │ θ3         0.369048   1.03083    1.20271     1.20325     1.37222   ⋯
 15 │ θ4         0.570767   1.09377    1.24564     1.25551     1.40709   ⋯
 16 │ θ5         -1.32032   -0.386843  -0.0346073  -0.0343756  0.302657  ⋯
 17 │ θ6         0.683923   1.62089    1.84137     1.84508     2.0708    ⋯

ridgeplot(samp)

let f = Figure()
    ax = Axis(f[1, 1])
    coefplot!(ax, samp;
              conf_level=0.8,
              vline_at_zero=true,
              show_intercept=true)
    ridgeplot!(ax, samp;
               conf_level=0.8,
               vline_at_zero=true,
               show_intercept=true,
#               xlabel="Normalized density and 80% range")
    )
    scatter!(ax, β, length(β):-1:1;
             marker=:x,
             markersize=20,
             color=:red)
    f
end

coefpvalues = DataFrame()
# @showprogress 
for subj_n in [20, 40, 60, 80, 100, 120, 140],  item_n in [40, 60, 80, 100, 120, 140]
    dat = simdat_crossed(RNG, subj_n, item_n;
                         subj_btwn, item_btwn)
    simmod = MixedModel(@formula(dv ~ 1 + age * frequency +
                                     (1 + frequency | subj) +
                                     (1 + age | item)),
                        dat)

    θ = createθ(simmod; subj=subj_re, item=item_re)
    # optsum_overrides speeds up the replicate fits;
    # see the reduced-precision note on the bootstrap page
    simboot = parametricbootstrap(RNG, 100, simmod;
                                  β, σ, θ,
                                  optsum_overrides=(;ftol_rel=1e-8),
                                  progress)
    df = DataFrame(simboot.coefpvalues)
    df[!, :subj_n] .= subj_n
    df[!, :item_n] .= item_n
    append!(coefpvalues, df)
end

power = combine(groupby(coefpvalues, [:coefname, :subj_n, :item_n]),
                :p => (p -> mean(p .< 0.05)) => :power)

168×4 DataFrame

143 rows omitted

Row	coefname	subj_n	item_n	power
	Symbol	Int64	Int64	Float64
1	(Intercept)	20	40	1.0
2	age: young	20	40	0.29
3	frequency: low	20	40	0.07
4	age: young & frequency: low	20	40	0.06
5	(Intercept)	20	60	1.0
6	age: young	20	60	0.21
7	frequency: low	20	60	0.17
8	age: young & frequency: low	20	60	0.04
9	(Intercept)	20	80	1.0
10	age: young	20	80	0.15
11	frequency: low	20	80	0.17
12	age: young & frequency: low	20	80	0.07
13	(Intercept)	20	100	1.0
⋮	⋮	⋮	⋮	⋮
157	(Intercept)	140	100	1.0
158	age: young	140	100	0.59
159	frequency: low	140	100	0.21
160	age: young & frequency: low	140	100	0.04
161	(Intercept)	140	120	1.0
162	age: young	140	120	0.71
163	frequency: low	140	120	0.35
164	age: young & frequency: low	140	120	0.05
165	(Intercept)	140	140	1.0
166	age: young	140	140	0.62
167	frequency: low	140	140	0.3
168	age: young & frequency: low	140	140	0.03

power = combine(groupby(coefpvalues,
                        [:coefname, :subj_n, :item_n]),
                :p => (p -> mean(p .< 0.05)) => :power,
                :p => (p -> sem(p .< 0.05)) => :power_se)

168×5 DataFrame

143 rows omitted

Row	coefname	subj_n	item_n	power	power_se
	Symbol	Int64	Int64	Float64	Float64
1	(Intercept)	20	40	1.0	0.0
2	age: young	20	40	0.29	0.0456048
3	frequency: low	20	40	0.07	0.0256432
4	age: young & frequency: low	20	40	0.06	0.0238683
5	(Intercept)	20	60	1.0	0.0
6	age: young	20	60	0.21	0.040936
7	frequency: low	20	60	0.17	0.0377525
8	age: young & frequency: low	20	60	0.04	0.0196946
9	(Intercept)	20	80	1.0	0.0
10	age: young	20	80	0.15	0.035887
11	frequency: low	20	80	0.17	0.0377525
12	age: young & frequency: low	20	80	0.07	0.0256432
13	(Intercept)	20	100	1.0	0.0
⋮	⋮	⋮	⋮	⋮	⋮
157	(Intercept)	140	100	1.0	0.0
158	age: young	140	100	0.59	0.0494311
159	frequency: low	140	100	0.21	0.040936
160	age: young & frequency: low	140	100	0.04	0.0196946
161	(Intercept)	140	120	1.0	0.0
162	age: young	140	120	0.71	0.0456048
163	frequency: low	140	120	0.35	0.0479372
164	age: young & frequency: low	140	120	0.05	0.0219043
165	(Intercept)	140	140	1.0	0.0
166	age: young	140	140	0.62	0.0487832
167	frequency: low	140	140	0.3	0.0460566
168	age: young & frequency: low	140	140	0.03	0.0171447

select!(power, :coefname, :subj_n, :item_n, :power,
        [:power, :power_se] => ByRow((p, se) -> [p - 1.96*se, p + 1.96*se]) => [:lower, :upper])

168×6 DataFrame

143 rows omitted

Row	coefname	subj_n	item_n	power	lower	upper
	Symbol	Int64	Int64	Float64	Float64	Float64
1	(Intercept)	20	40	1.0	1.0	1.0
2	age: young	20	40	0.29	0.200615	0.379385
3	frequency: low	20	40	0.07	0.0197392	0.120261
4	age: young & frequency: low	20	40	0.06	0.0132181	0.106782
5	(Intercept)	20	60	1.0	1.0	1.0
6	age: young	20	60	0.21	0.129765	0.290235
7	frequency: low	20	60	0.17	0.0960051	0.243995
8	age: young & frequency: low	20	60	0.04	0.00139851	0.0786015
9	(Intercept)	20	80	1.0	1.0	1.0
10	age: young	20	80	0.15	0.0796614	0.220339
11	frequency: low	20	80	0.17	0.0960051	0.243995
12	age: young & frequency: low	20	80	0.07	0.0197392	0.120261
13	(Intercept)	20	100	1.0	1.0	1.0
⋮	⋮	⋮	⋮	⋮	⋮	⋮
157	(Intercept)	140	100	1.0	1.0	1.0
158	age: young	140	100	0.59	0.493115	0.686885
159	frequency: low	140	100	0.21	0.129765	0.290235
160	age: young & frequency: low	140	100	0.04	0.00139851	0.0786015
161	(Intercept)	140	120	1.0	1.0	1.0
162	age: young	140	120	0.71	0.620615	0.799385
163	frequency: low	140	120	0.35	0.256043	0.443957
164	age: young & frequency: low	140	120	0.05	0.00706759	0.0929324
165	(Intercept)	140	140	1.0	1.0	1.0
166	age: young	140	140	0.62	0.524385	0.715615
167	frequency: low	140	140	0.3	0.209729	0.390271
168	age: young & frequency: low	140	140	0.03	-0.00360354	0.0636035

data(power) * mapping(:subj_n, :item_n, :power; layout=:coefname) * visual(Heatmap) |> draw

dat = simdat_crossed(RNG, subj_n, item_n;
                     subj_btwn, item_btwn)
dat = DataFrame(dat)

400×5 DataFrame

375 rows omitted

Row	subj	age	item	frequency	dv
	String	String	String	String	Float64
1	S01	old	I01	low	-0.882085
2	S02	young	I01	low	-0.0201484
3	S03	old	I01	low	0.945484
4	S04	young	I01	low	-0.247553
5	S05	old	I01	low	0.453218
6	S06	young	I01	low	0.924281
7	S07	old	I01	low	0.220375
8	S08	young	I01	low	1.85978
9	S09	old	I01	low	-0.44618
10	S10	young	I01	low	-0.754939
11	S11	old	I01	low	0.0105376
12	S12	young	I01	low	0.929521
13	S13	old	I01	low	-0.314341
⋮	⋮	⋮	⋮	⋮	⋮
389	S09	old	I20	high	2.2703
390	S10	young	I20	high	-0.00613232
391	S11	old	I20	high	-0.923031
392	S12	young	I20	high	0.539303
393	S13	old	I20	high	-0.868366
394	S14	young	I20	high	2.10268
395	S15	old	I20	high	-1.23199
396	S16	young	I20	high	-1.02512
397	S17	old	I20	high	0.818806
398	S18	young	I20	high	1.7468
399	S19	old	I20	high	-0.327621
400	S20	young	I20	high	-0.841135

item_covariates = unique!(select(dat, :item))

20×1 DataFrame

Row	item
	String
1	I01
2	I02
3	I03
4	I04
5	I05
6	I06
7	I07
8	I08
9	I09
10	I10
11	I11
12	I12
13	I13
14	I14
15	I15
16	I16
17	I17
18	I18
19	I19
20	I20

item_covariates[!, :chaos] = rand(RNG,
                                  Normal(5, 2),
                                  nrow(item_covariates))

20-element Vector{Float64}:
 4.283869606409811
 1.766354546694887
 3.5819077117990936
 4.426082794175044
 3.979161685431401
 4.150057335954468
 5.209549095470591
 5.3931183588041325
 5.836973833266723
 6.042800312594512
 3.1607596007727814
 6.519844952694776
 9.286581793894484
 7.995006071550052
 4.189786264631877
 2.4169512074675787
 3.445736901440534
 5.123574599692554
 4.422113183811228
 1.9739443958799194

leftjoin!(dat, item_covariates; on=:item)

400×6 DataFrame

375 rows omitted

Row	subj	age	item	frequency	dv	chaos
	String	String	String	String	Float64	Float64?
1	S01	old	I01	low	-0.882085	4.28387
2	S02	young	I01	low	-0.0201484	4.28387
3	S03	old	I01	low	0.945484	4.28387
4	S04	young	I01	low	-0.247553	4.28387
5	S05	old	I01	low	0.453218	4.28387
6	S06	young	I01	low	0.924281	4.28387
7	S07	old	I01	low	0.220375	4.28387
8	S08	young	I01	low	1.85978	4.28387
9	S09	old	I01	low	-0.44618	4.28387
10	S10	young	I01	low	-0.754939	4.28387
11	S11	old	I01	low	0.0105376	4.28387
12	S12	young	I01	low	0.929521	4.28387
13	S13	old	I01	low	-0.314341	4.28387
⋮	⋮	⋮	⋮	⋮	⋮	⋮
389	S09	old	I20	high	2.2703	1.97394
390	S10	young	I20	high	-0.00613232	1.97394
391	S11	old	I20	high	-0.923031	1.97394
392	S12	young	I20	high	0.539303	1.97394
393	S13	old	I20	high	-0.868366	1.97394
394	S14	young	I20	high	2.10268	1.97394
395	S15	old	I20	high	-1.23199	1.97394
396	S16	young	I20	high	-1.02512	1.97394
397	S17	old	I20	high	0.818806	1.97394
398	S18	young	I20	high	1.7468	1.97394
399	S19	old	I20	high	-0.327621	1.97394
400	S20	young	I20	high	-0.841135	1.97394

simmod = fit(MixedModel,
             @formula(dv ~ 1 + age * frequency * chaos +
                          (1 + frequency | subj) +
                          (1 + age | item)), dat; progress)

	Est.	SE	z	p	σ_subj	σ_item
(Intercept)	0.0560	0.2412	0.23	0.8164	0.0000	0.0000
age: young	-0.3565	0.3411	-1.05	0.2959		0.0000
frequency: low	0.0548	0.3699	0.15	0.8822	0.0000
chaos	-0.0135	0.0484	-0.28	0.7809
age: young & frequency: low	0.1030	0.5232	0.20	0.8439
age: young & chaos	0.1036	0.0685	1.51	0.1305
frequency: low & chaos	-0.0421	0.0738	-0.57	0.5683
age: young & frequency: low & chaos	-0.0317	0.1044	-0.30	0.7617
Residual	0.9456

dat[!, :dv] = rand(RNG, Bernoulli(), nrow(dat))

400-element Vector{Bool}:
 0
 1
 0
 0
 1
 1
 1
 0
 0
 0
 ⋮
 1
 1
 1
 0
 1
 1
 0
 0
 1

dat[!, :dv] = rand(RNG, Poisson(), nrow(dat))

400-element Vector{Int64}:
 2
 1
 2
 0
 0
 1
 0
 2
 0
 1
 ⋮
 0
 2
 0
 1
 1
 3
 0
 1
 1

dat[!, :n] = rand(RNG, Poisson(), nrow(dat)) .+ 3

400-element Vector{Int64}:
 5
 3
 3
 7
 4
 3
 5
 3
 3
 4
 ⋮
 4
 3
 3
 4
 4
 3
 6
 4
 3

dat[!, :dv] = rand.(RNG, Binomial.(dat[!, :n])) ./ dat[!, :n]

400-element Vector{Float64}:
 0.6
 0.3333333333333333
 0.6666666666666666
 0.42857142857142855
 0.5
 0.6666666666666666
 0.2
 0.3333333333333333
 0.6666666666666666
 0.5
 ⋮
 0.5
 0.6666666666666666
 0.3333333333333333
 0.25
 1.0
 1.0
 0.5
 0.75
 1.0

1.2 Exercises

1. Effect size and power. Re-run the simulation across a range of assumed effect sizes, holding the sample size fixed. Plot estimated power against effect size. What shape do you expect?

NoteSolution

Power increases monotonically with effect size, tracing an S-shaped curve from the nominal false-positive rate (when the true effect is zero) toward 1 (for large effects). The interesting region is the middle, where small design changes move power the most.

2. Sample size. Now hold the effect size fixed and vary the number of subjects. How does required sample size trade off against effect size for a target power (say 0.8)?

NoteSolution

For a fixed target power, required sample size grows rapidly as the effect size shrinks (roughly inversely with the square of the effect size). Simulating a grid over both lets you read off feasible design points rather than relying on a single closed-form approximation.

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