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simulation_tutorial.html
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simulation_tutorial.html
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<!DOCTYPE html>
<HTML lang = "en">
<HEAD>
<meta charset="UTF-8"/>
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
<title>Simulation Tutorial</title>
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<BODY>
<div class ="container">
<div class = "row">
<div class = "col-md-12 twelve columns">
<div class="title">
<h1 class="title">Simulation Tutorial</h1>
<h5>Lisa DeBruine</h5>
<h5>2020-02-17</h5>
</div>
<h2>Setup</h2>
<h3>Julia</h3>
<p>Load the packages we'll be using in Julia. In pkg run the following to get the versions we're using:</p>
<ul>
<li><p><code>add MixedModels#master</code></p>
</li>
<li><p><code>add https://github.com/RePsychLing/MixedModelsSim.jl#master</code></p>
</li>
</ul>
<pre class='hljl'>
<span class='hljl-nB'>julia> </span><span class='hljl-k'>using</span><span class='hljl-t'> </span><span class='hljl-n'>Pkg</span><span class='hljl-t'>
</span><span class='hljl-nB'>julia> </span><span class='hljl-n'>Pkg</span><span class='hljl-oB'>.</span><span class='hljl-nf'>activate</span><span class='hljl-p'>()</span><span class='hljl-t'>
Activating environment at `~/Desktop/Julia/sim-tutorial/Project.toml`
</span><span class='hljl-nB'>julia> </span><span class='hljl-n'>Pkg</span><span class='hljl-oB'>.</span><span class='hljl-nf'>instantiate</span><span class='hljl-p'>()</span><span class='hljl-t'>
</span><span class='hljl-nB'>julia> </span><span class='hljl-t'>
using MixedModels # run mixed models
</span><span class='hljl-nB'>julia> </span><span class='hljl-k'>using</span><span class='hljl-t'> </span><span class='hljl-n'>MixedModelsSim</span><span class='hljl-t'> </span><span class='hljl-cs'># simulation functions for mixed models</span><span class='hljl-t'>
</span><span class='hljl-nB'>julia> </span><span class='hljl-k'>using</span><span class='hljl-t'> </span><span class='hljl-n'>RCall</span><span class='hljl-t'> </span><span class='hljl-cs'># call R functions from inside Julia</span><span class='hljl-t'>
</span><span class='hljl-nB'>julia> </span><span class='hljl-k'>using</span><span class='hljl-t'> </span><span class='hljl-n'>DataFrames</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>Tables</span><span class='hljl-t'> </span><span class='hljl-cs'># work with data tables</span><span class='hljl-t'>
</span><span class='hljl-nB'>julia> </span><span class='hljl-k'>using</span><span class='hljl-t'> </span><span class='hljl-n'>Random</span><span class='hljl-t'> </span><span class='hljl-cs'># random number generator</span><span class='hljl-t'>
</span><span class='hljl-nB'>julia> </span><span class='hljl-k'>using</span><span class='hljl-t'> </span><span class='hljl-n'>CSV</span><span class='hljl-t'> </span><span class='hljl-cs'># write CSV files</span><span class='hljl-t'>s</span>
</pre>
<h3>R</h3>
<p>Also load any packages we'll be using in R through <code>RCall()</code>.</p>
<pre class='hljl'>
<span class='hljl-so'>R"""
require(ggplot2, quietly = TRUE) # for visualisation
require(dplyr, quietly = TRUE) # for data wrangling
require(tidyr, quietly = TRUE) # for data wrangling
"""</span><span class='hljl-p'>;</span>
</pre>
<h3>Define Custom functions</h3>
<p>It's useful to be able to weave your file quickly while you're debugging, so set the number of simulations to a relatively low number while you're setting up your script and change it to a larger number when everything is debugged.</p>
<pre class='hljl'>
<span class='hljl-n'>nsims</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-ni'>1000</span><span class='hljl-t'> </span><span class='hljl-cs'># set to a low number for test, high for production</span>
</pre>
<pre class="output">
1000
</pre>
<h4>Define: ggplot_betas</h4>
<p>This function plots the beta values returned from <code>simulate_waldtests</code> using ggplot in R. If you set a figname, it will save the plot to the specified file.</p>
<pre class='hljl'>
<span class='hljl-k'>function</span><span class='hljl-t'> </span><span class='hljl-nf'>ggplot_betas</span><span class='hljl-p'>(</span><span class='hljl-n'>sim</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>figname</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-ni'>0</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>width</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-ni'>7</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>height</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-ni'>5</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-n'>beta_df</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>DataFrame</span><span class='hljl-p'>(</span><span class='hljl-nf'>columntable</span><span class='hljl-p'>(</span><span class='hljl-n'>sim</span><span class='hljl-p'>)</span><span class='hljl-oB'>.</span><span class='hljl-n'>β</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-so'>R"""
p <- $beta_df %>%
gather(var, val, 1:ncol(.)) %>%
ggplot(aes(val, color = var)) +
geom_density(show.legend = FALSE) +
facet_wrap(~var, scales = "free")
if (is.character($figname)) {
ggsave($figname, p, width = $width, height = $height)
}
p
"""</span><span class='hljl-t'>
</span><span class='hljl-k'>end</span>
</pre>
<pre class="output">
ggplot_betas (generic function with 4 methods)
</pre>
<h2>Existing Data</h2>
<p>Load existing data from this morning's tutorial. Set the contrasts and run model 4 from the tutorial.</p>
<pre class='hljl'>
<span class='hljl-cs'># load data</span><span class='hljl-t'>
</span><span class='hljl-n'>kb07</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>MixedModels</span><span class='hljl-oB'>.</span><span class='hljl-nf'>dataset</span><span class='hljl-p'>(</span><span class='hljl-s'>"kb07"</span><span class='hljl-p'>);</span><span class='hljl-t'>
</span><span class='hljl-cs'># set contrasts</span><span class='hljl-t'>
</span><span class='hljl-n'>contrasts</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>Dict</span><span class='hljl-p'>(</span><span class='hljl-sc'>:spkr</span><span class='hljl-t'> </span><span class='hljl-oB'>=></span><span class='hljl-t'> </span><span class='hljl-nf'>HelmertCoding</span><span class='hljl-p'>(),</span><span class='hljl-t'>
</span><span class='hljl-sc'>:prec</span><span class='hljl-t'> </span><span class='hljl-oB'>=></span><span class='hljl-t'> </span><span class='hljl-nf'>HelmertCoding</span><span class='hljl-p'>(),</span><span class='hljl-t'>
</span><span class='hljl-sc'>:load</span><span class='hljl-t'> </span><span class='hljl-oB'>=></span><span class='hljl-t'> </span><span class='hljl-nf'>HelmertCoding</span><span class='hljl-p'>());</span><span class='hljl-t'>
</span><span class='hljl-cs'># define formula</span><span class='hljl-t'>
</span><span class='hljl-n'>kb07_f</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nd'>@formula</span><span class='hljl-p'>(</span><span class='hljl-t'> </span><span class='hljl-n'>rt_trunc</span><span class='hljl-t'> </span><span class='hljl-oB'>~</span><span class='hljl-t'> </span><span class='hljl-ni'>1</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-n'>spkr</span><span class='hljl-oB'>+</span><span class='hljl-n'>prec</span><span class='hljl-oB'>+</span><span class='hljl-n'>load</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-p'>(</span><span class='hljl-ni'>1</span><span class='hljl-oB'>|</span><span class='hljl-n'>subj</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-p'>(</span><span class='hljl-ni'>1</span><span class='hljl-oB'>+</span><span class='hljl-n'>prec</span><span class='hljl-oB'>|</span><span class='hljl-n'>item</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-p'>);</span><span class='hljl-t'>
</span><span class='hljl-cs'># fit model</span><span class='hljl-t'>
</span><span class='hljl-n'>kb07_m</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>fit</span><span class='hljl-p'>(</span><span class='hljl-n'>MixedModel</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>kb07_f</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>kb07</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>contrasts</span><span class='hljl-oB'>=</span><span class='hljl-n'>contrasts</span><span class='hljl-p'>)</span>
</pre>
<pre class="output">
Linear mixed model fit by maximum likelihood
rt_trunc ~ 1 + spkr + prec + load + (1 | subj) + (1 + prec | item)
logLik -2 logLik AIC BIC
-1.43319251×10⁴ 2.86638501×10⁴ 2.86818501×10⁴ 2.87312548×10⁴
Variance components:
Column Variance Std.Dev. Corr.
item (Intercept) 133015.240 364.71254
prec: maintain 63766.936 252.52116 -0.70
subj (Intercept) 88819.437 298.02590
Residual 462443.388 680.03190
Number of obs: 1789; levels of grouping factors: 32, 56
Fixed-effects parameters:
──────────────────────────────────────────────────────
Estimate Std.Error z value P(>|z|)
──────────────────────────────────────────────────────
(Intercept) 2181.85 77.4681 28.16 <1e-99
spkr: old 67.879 16.0785 4.22 <1e-4
prec: maintain -333.791 47.4472 -7.03 <1e-11
load: yes 78.5904 16.0785 4.89 <1e-5
──────────────────────────────────────────────────────
</pre>
<h3>Simulate data with same parameters</h3>
<p>Use the <code>simulate_waldtests()</code> function to run 1000 iterations of data sampled using the parameters from <code>m4</code>. Set up a random seed to make the simulation reproducible. You can use your favourite number.</p>
<p>To use multithreading, you need to set the number of cores you want to use. In Visual Studio Code, open the settings (gear icon in the lower left corner or cmd-,) and search for "thread". Set <code>julia.NumThreads</code> to the number of cores you want to use (at least 1 less than your total number).</p>
<pre class='hljl'>
<span class='hljl-cs'># set seed for reproducibility</span><span class='hljl-t'>
</span><span class='hljl-n'>rng</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>MersenneTwister</span><span class='hljl-p'>(</span><span class='hljl-ni'>8675309</span><span class='hljl-p'>);</span><span class='hljl-t'>
</span><span class='hljl-cs'># run nsims iterations</span><span class='hljl-t'>
</span><span class='hljl-n'>kb07_sim</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>simulate_waldtests</span><span class='hljl-p'>(</span><span class='hljl-n'>rng</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>nsims</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>kb07_m</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>use_threads</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-kc'>true</span><span class='hljl-p'>);</span>
</pre>
<p><strong>Try</strong>: Run the code above with and without <code>use_threads</code>.</p>
<p>Save all data to a csv file.</p>
<pre class='hljl'>
<span class='hljl-n'>kb07_sim_df</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>sim_to_df</span><span class='hljl-p'>(</span><span class='hljl-n'>kb07_sim</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-n'>CSV</span><span class='hljl-oB'>.</span><span class='hljl-nf'>write</span><span class='hljl-p'>(</span><span class='hljl-s'>"sim/kb07_sim.csv"</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>kb07_sim_df</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-nf'>first</span><span class='hljl-p'>(</span><span class='hljl-n'>kb07_sim_df</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>8</span><span class='hljl-p'>)</span>
</pre>
<table class="data-frame"><thead><tr><th></th><th>iteration</th><th>coefname</th><th>beta</th><th>se</th><th>z</th><th>p</th></tr><tr><th></th><th>Int64</th><th>Symbol</th><th>Float64⍰</th><th>Float64⍰</th><th>Float64⍰</th><th>Float64⍰</th></tr></thead><tbody><p>8 rows × 6 columns</p><tr><th>1</th><td>1</td><td>(Intercept)</td><td>2248.0</td><td>84.8585</td><td>26.4912</td><td>1.22408e-154</td></tr><tr><th>2</th><td>1</td><td>load: yes</td><td>48.9212</td><td>15.8225</td><td>3.09187</td><td>0.00198898</td></tr><tr><th>3</th><td>1</td><td>prec: maintain</td><td>-320.632</td><td>45.4898</td><td>-7.04844</td><td>1.8093e-12</td></tr><tr><th>4</th><td>1</td><td>spkr: old</td><td>62.8771</td><td>15.8225</td><td>3.9739</td><td>7.07065e-5</td></tr><tr><th>5</th><td>2</td><td>(Intercept)</td><td>2165.36</td><td>79.8084</td><td>27.132</td><td>4.12634e-162</td></tr><tr><th>6</th><td>2</td><td>load: yes</td><td>91.9312</td><td>16.2066</td><td>5.67246</td><td>1.40758e-8</td></tr><tr><th>7</th><td>2</td><td>prec: maintain</td><td>-353.079</td><td>37.4427</td><td>-9.42985</td><td>4.10688e-21</td></tr><tr><th>8</th><td>2</td><td>spkr: old</td><td>46.542</td><td>16.2066</td><td>2.87179</td><td>0.00408153</td></tr></tbody></table>
<p>Plot betas in ggplot. In the code editor or Jupyter notebooks, you can omit the file name to just display the figure in an external window. </p>
<pre class='hljl'>
<span class='hljl-cs'># just display the image</span><span class='hljl-t'>
</span><span class='hljl-cs'># ggplot_betas(kb07_sim) </span><span class='hljl-t'>
</span><span class='hljl-cs'># save the image to a file and display (display doesn't work in weave)</span><span class='hljl-t'>
</span><span class='hljl-nf'>ggplot_betas</span><span class='hljl-p'>(</span><span class='hljl-n'>kb07_sim</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-s'>"fig/kb07_betas.png"</span><span class='hljl-p'>);</span>
</pre>
<p>In documents you want to weave, save the image to a file and use markdown to display the file. Add a semicolon to the end of the function to suppress creating the images in new windows during weaving.</p>
<p><img src="fig/kb07_betas.png" alt="" /></p>
<h3>Power calculation</h3>
<p>The function <code>power_table()</code> from <code>MixedModelsSim</code> takes the output of <code>simulate_waldtests()</code> and calculates the proportion of simulations where the p-value is less than alpha for each coefficient. You can set the <code>alpha</code> argument to change the default value of 0.05 (justify your alpha ;).</p>
<pre class='hljl'>
<span class='hljl-nf'>power_table</span><span class='hljl-p'>(</span><span class='hljl-n'>kb07_sim</span><span class='hljl-p'>)</span>
</pre>
<table class="data-frame"><thead><tr><th></th><th>coefname</th><th>power</th></tr><tr><th></th><th>Symbol</th><th>Float64</th></tr></thead><tbody><p>4 rows × 2 columns</p><tr><th>1</th><td>(Intercept)</td><td>1.0</td></tr><tr><th>2</th><td>spkr: old</td><td>0.991</td></tr><tr><th>3</th><td>prec: maintain</td><td>1.0</td></tr><tr><th>4</th><td>load: yes</td><td>0.999</td></tr></tbody></table>
<h3>Change parameters</h3>
<p>Let's say we want to check our power to detect effects of spkr, prec, and load that are half the size of our pilot data. We can set a new vector of beta values with the <code>β</code> argument to <code>simulate_waldtests</code>.</p>
<pre class='hljl'>
<span class='hljl-n'>newβ</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>kb07_m</span><span class='hljl-oB'>.</span><span class='hljl-n'>β</span><span class='hljl-t'>
</span><span class='hljl-n'>newβ</span><span class='hljl-p'>[</span><span class='hljl-ni'>2</span><span class='hljl-oB'>:</span><span class='hljl-ni'>4</span><span class='hljl-p'>]</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>kb07_m</span><span class='hljl-oB'>.</span><span class='hljl-n'>β</span><span class='hljl-p'>[</span><span class='hljl-ni'>2</span><span class='hljl-oB'>:</span><span class='hljl-ni'>4</span><span class='hljl-p'>]</span><span class='hljl-oB'>/</span><span class='hljl-ni'>2</span><span class='hljl-t'>
</span><span class='hljl-n'>kb07_sim_half</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>simulate_waldtests</span><span class='hljl-p'>(</span><span class='hljl-n'>rng</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>nsims</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>kb07_m</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>β</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>newβ</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>use_threads</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-kc'>true</span><span class='hljl-p'>);</span><span class='hljl-t'>
</span><span class='hljl-nf'>power_table</span><span class='hljl-p'>(</span><span class='hljl-n'>kb07_sim_half</span><span class='hljl-p'>)</span>
</pre>
<table class="data-frame"><thead><tr><th></th><th>coefname</th><th>power</th></tr><tr><th></th><th>Symbol</th><th>Float64</th></tr></thead><tbody><p>4 rows × 2 columns</p><tr><th>1</th><td>(Intercept)</td><td>1.0</td></tr><tr><th>2</th><td>spkr: old</td><td>0.529</td></tr><tr><th>3</th><td>prec: maintain</td><td>0.928</td></tr><tr><th>4</th><td>load: yes</td><td>0.692</td></tr></tbody></table>
<h1>Simulating Data from Scratch</h1>
<h2>simdat_crossed</h2>
<p>The <code>simdat_crossed()</code> function from <code>MixedModelsSim</code> lets you set up a data frame with a specified experimental design. For now, it only makes fully balanced crossed designs, but you can generate an unbalanced design by simulating data for the largest cell and deleting extra rows. </p>
<p>We will set a design where <code>subj_n</code> subjects per <code>age</code> group (O or Y) respond to <code>item_n</code> items in each of two <code>condition</code>s (A or B).</p>
<p>Your factors need to be specified separately for between-subject, between-item, and within-subject/item factors using <code>Dict</code> with the name of each factor as the keys and vectors with the names of the levels as values.</p>
<pre class='hljl'>
<span class='hljl-cs'># put between-subject factors in a Dict</span><span class='hljl-t'>
</span><span class='hljl-n'>subj_btwn</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>Dict</span><span class='hljl-p'>(</span><span class='hljl-s'>"age"</span><span class='hljl-t'> </span><span class='hljl-oB'>=></span><span class='hljl-t'> </span><span class='hljl-p'>[</span><span class='hljl-s'>"O"</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-s'>"Y"</span><span class='hljl-p'>])</span><span class='hljl-t'>
</span><span class='hljl-cs'># there are no between-item factors in this design so you can omit it or set it to nothing</span><span class='hljl-t'>
</span><span class='hljl-n'>item_btwn</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>nothing</span><span class='hljl-t'>
</span><span class='hljl-cs'># put within-subject/item factors in a Dict</span><span class='hljl-t'>
</span><span class='hljl-n'>both_win</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>Dict</span><span class='hljl-p'>(</span><span class='hljl-s'>"condition"</span><span class='hljl-t'> </span><span class='hljl-oB'>=></span><span class='hljl-t'> </span><span class='hljl-p'>[</span><span class='hljl-s'>"A"</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-s'>"B"</span><span class='hljl-p'>])</span><span class='hljl-t'>
</span><span class='hljl-cs'># simulate data</span><span class='hljl-t'>
</span><span class='hljl-n'>dat</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>simdat_crossed</span><span class='hljl-p'>(</span><span class='hljl-ni'>10</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>30</span><span class='hljl-p'>,</span><span class='hljl-t'>
</span><span class='hljl-n'>subj_btwn</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>subj_btwn</span><span class='hljl-p'>,</span><span class='hljl-t'>
</span><span class='hljl-n'>item_btwn</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>item_btwn</span><span class='hljl-p'>,</span><span class='hljl-t'>
</span><span class='hljl-n'>both_win</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>both_win</span><span class='hljl-p'>);</span>
</pre>
<h2>Fit a model</h2>
<p>Now you need to fit a model to your simulated data. Because the <code>dv</code> is just random numbers from N(0,1), there will be basically no subject or item random variance, residual variance will be near 1.0, and the estimates for all effects should be small. Don't worry, we'll specify fixed and random effects directly in <code>simulate_waldtests</code>. </p>
<pre class='hljl'>
<span class='hljl-cs'># set contrasts</span><span class='hljl-t'>
</span><span class='hljl-n'>contrasts</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>Dict</span><span class='hljl-p'>(</span><span class='hljl-sc'>:age</span><span class='hljl-t'> </span><span class='hljl-oB'>=></span><span class='hljl-t'> </span><span class='hljl-nf'>HelmertCoding</span><span class='hljl-p'>(),</span><span class='hljl-t'>
</span><span class='hljl-sc'>:condition</span><span class='hljl-t'> </span><span class='hljl-oB'>=></span><span class='hljl-t'> </span><span class='hljl-nf'>HelmertCoding</span><span class='hljl-p'>());</span><span class='hljl-t'>
</span><span class='hljl-n'>f1</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nd'>@formula</span><span class='hljl-t'> </span><span class='hljl-n'>dv</span><span class='hljl-t'> </span><span class='hljl-oB'>~</span><span class='hljl-t'> </span><span class='hljl-ni'>1</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-n'>age</span><span class='hljl-t'> </span><span class='hljl-oB'>*</span><span class='hljl-t'> </span><span class='hljl-n'>condition</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-p'>(</span><span class='hljl-ni'>1</span><span class='hljl-oB'>|</span><span class='hljl-n'>item</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-p'>(</span><span class='hljl-ni'>1</span><span class='hljl-oB'>|</span><span class='hljl-n'>subj</span><span class='hljl-p'>);</span><span class='hljl-t'>
</span><span class='hljl-n'>m1</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>fit</span><span class='hljl-p'>(</span><span class='hljl-n'>MixedModel</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>f1</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>dat</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>contrasts</span><span class='hljl-oB'>=</span><span class='hljl-n'>contrasts</span><span class='hljl-p'>)</span>
</pre>
<pre class="output">
Linear mixed model fit by maximum likelihood
dv ~ 1 + age + condition + age & condition + (1 | item) + (1 | subj)
logLik -2 logLik AIC BIC
-1681.5786 3363.1571 3377.1571 3412.7877
Variance components:
Column Variance Std.Dev.
item (Intercept) 0.0000000 0.0000000
subj (Intercept) 0.0000000 0.0000000
Residual 0.9653678 0.9825313
Number of obs: 1200; levels of grouping factors: 30, 20
Fixed-effects parameters:
───────────────────────────────────────────────────────────────
Estimate Std.Error z value P(>|z|)
───────────────────────────────────────────────────────────────
(Intercept) -0.00526093 0.0283632 -0.19 0.8528
age: Y 0.00751336 0.0283632 0.26 0.7911
condition: B 0.0185408 0.0283632 0.65 0.5133
age: Y & condition: B 0.0445935 0.0283632 1.57 0.1159
───────────────────────────────────────────────────────────────
</pre>
<h2>Simulate</h2>
<p>Set a seed for reproducibility and specify β, σ, and θ.</p>
<pre class='hljl'>
<span class='hljl-n'>rng</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>MersenneTwister</span><span class='hljl-p'>(</span><span class='hljl-ni'>8675309</span><span class='hljl-p'>);</span><span class='hljl-t'>
</span><span class='hljl-n'>new_beta</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-p'>[</span><span class='hljl-ni'>0</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-nfB'>0.25</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-nfB'>0.25</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>0</span><span class='hljl-p'>]</span><span class='hljl-t'>
</span><span class='hljl-n'>new_sigma</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nfB'>2.0</span><span class='hljl-t'>
</span><span class='hljl-n'>new_theta</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-p'>[</span><span class='hljl-nfB'>1.0</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-nfB'>1.0</span><span class='hljl-p'>]</span><span class='hljl-t'>
</span><span class='hljl-n'>sim1</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>simulate_waldtests</span><span class='hljl-p'>(</span><span class='hljl-n'>rng</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>nsims</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>m1</span><span class='hljl-p'>,</span><span class='hljl-t'>
</span><span class='hljl-n'>β</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>new_beta</span><span class='hljl-p'>,</span><span class='hljl-t'>
</span><span class='hljl-n'>σ</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>new_sigma</span><span class='hljl-p'>,</span><span class='hljl-t'>
</span><span class='hljl-n'>θ</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>new_theta</span><span class='hljl-p'>,</span><span class='hljl-t'>
</span><span class='hljl-n'>use_threads</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-kc'>true</span><span class='hljl-p'>);</span>
</pre>
<h2>Explore simulation output</h2>
<pre class='hljl'>
<span class='hljl-nf'>ggplot_betas</span><span class='hljl-p'>(</span><span class='hljl-n'>sim1</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-s'>"fig/simbetas.png"</span><span class='hljl-p'>);</span>
</pre>
<p><img src="fig/simbetas.png" alt="" /></p>
<h2>Power</h2>
<pre class='hljl'>
<span class='hljl-nf'>power_table</span><span class='hljl-p'>(</span><span class='hljl-n'>sim1</span><span class='hljl-p'>)</span>
</pre>
<table class="data-frame"><thead><tr><th></th><th>coefname</th><th>power</th></tr><tr><th></th><th>Symbol</th><th>Float64</th></tr></thead><tbody><p>4 rows × 2 columns</p><tr><th>1</th><td>(Intercept)</td><td>0.062</td></tr><tr><th>2</th><td>age: Y</td><td>0.132</td></tr><tr><th>3</th><td>condition: B</td><td>0.989</td></tr><tr><th>4</th><td>age: Y & condition: B</td><td>0.056</td></tr></tbody></table>
<h2>Try your own design</h2>
<p>Edit <code>my_dat</code> below and make sure <code>my_f</code> is updated for your new design. Also make sure <code>my_beta</code> has the right number of elements (check <code>my_m.β</code> for the number and order). You can also change <code>my_sigma</code> and <code>my_theta</code>. Set the seed in <code>my_rng</code> to your favourite number.</p>
<pre class='hljl'>
<span class='hljl-n'>my_dat</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>simdat_crossed</span><span class='hljl-p'>(</span><span class='hljl-ni'>10</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>10</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-n'>my_f</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nd'>@formula</span><span class='hljl-t'> </span><span class='hljl-n'>dv</span><span class='hljl-t'> </span><span class='hljl-oB'>~</span><span class='hljl-t'> </span><span class='hljl-ni'>1</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-p'>(</span><span class='hljl-ni'>1</span><span class='hljl-oB'>|</span><span class='hljl-n'>item</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-p'>(</span><span class='hljl-ni'>1</span><span class='hljl-oB'>|</span><span class='hljl-n'>subj</span><span class='hljl-p'>);</span><span class='hljl-t'>
</span><span class='hljl-n'>my_m</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>fit</span><span class='hljl-p'>(</span><span class='hljl-n'>MixedModel</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>my_f</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>my_dat</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-n'>my_beta</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-p'>[</span><span class='hljl-nfB'>0.0</span><span class='hljl-p'>]</span><span class='hljl-t'>
</span><span class='hljl-n'>my_sigma</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nfB'>2.0</span><span class='hljl-t'>
</span><span class='hljl-n'>my_theta</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-p'>[</span><span class='hljl-nfB'>1.0</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-nfB'>1.0</span><span class='hljl-p'>]</span><span class='hljl-t'>
</span><span class='hljl-n'>my_rng</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>MersenneTwister</span><span class='hljl-p'>(</span><span class='hljl-ni'>8675309</span><span class='hljl-p'>);</span><span class='hljl-t'>
</span><span class='hljl-n'>my_nsims</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-ni'>1000</span><span class='hljl-t'>