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<!DOCTYPE html>
<html>
<head><meta charset="utf-8" />
<title>IRT simulation notebook FINAL - name added</title>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">from</span> <span class="nn">pprint</span> <span class="k">import</span> <span class="n">pprint</span>
<span class="kn">import</span> <span class="nn">pystan</span>
<span class="kn">from</span> <span class="nn">scipy.special</span> <span class="k">import</span> <span class="n">expit</span>
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="k">import</span> <span class="n">pyplot</span> <span class="k">as</span> <span class="n">plt</span>
<span class="kn">from</span> <span class="nn">numpy.random</span> <span class="k">import</span> <span class="n">normal</span><span class="p">,</span> <span class="n">randint</span><span class="p">,</span> <span class="n">binomial</span><span class="p">,</span> <span class="n">choice</span>
<span class="kn">from</span> <span class="nn">numpy</span> <span class="k">import</span> <span class="n">percentile</span><span class="p">,</span> <span class="n">concatenate</span><span class="p">,</span> <span class="n">array</span><span class="p">,</span> <span class="n">linspace</span><span class="p">,</span> <span class="n">append</span>
<span class="o">%</span><span class="k">matplotlib</span> inline
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<h1 id="Teddy-Groves">Teddy Groves<a class="anchor-link" href="#Teddy-Groves">¶</a></h1><h1 id="How-to-test-IRT-models-using-simulated-data">How to test IRT models using simulated data<a class="anchor-link" href="#How-to-test-IRT-models-using-simulated-data">¶</a></h1><p>My company, <a href="https://www.footballradar.com/">Football Radar</a>, is interested in lots of tasks that can be performed better or worse, depending on how hard it is and how able the agent doing it is.</p>
<p>Specifically, we look at data encoding judgements by football scouts as to how well players perform different kinds of action, such as controlling a fast-moving ball under pressure from an opponent. Typically not all players attempt the same kinds of actions the same number of times. In these cases IRT models can be used to draw conclusions about the players' abilities and the tasks' difficulties.</p>
<p>This notebook explains how to code some IRT models using Stan and test whether they can recover input parameters when given simulated data.</p>
<h2 id="What-are-IRT-models?">What are IRT models?<a class="anchor-link" href="#What-are-IRT-models?">¶</a></h2><p>The acronym IRT stands for 'item response theory'. This kind of model and its generalisations are very well explained in Gelman and Hill (2006).</p>
<p>The main point for our purposes is that IRT models assume that some response variables - we'll call them 'grades' - depend probabilistically on unobserved features of the 'subjects' - in our application football players - and tasks, or 'items', that is that can be thought of as abilities and difficulties. In other words:</p>
$$pr(grade_{ij} = s) = f(ability_{j}, difficulty_{i})$$<p>In the case of binary data this schema is typically fleshed out as follows:</p>
$$pr(grade_{ij} = 1) = logistic(ability_{j} - difficulty_{i})$$<p>The subtraction relationship represents an assumption that the likelihood of correctness depends on the difference between the item's difficulty and the subject's ability, which is reasonable in a lot of applications. The logistic function fixes the direction of this dependence, and conveniently ensures that any possible ability/difficulty difference is mapped to a number between zero and one:</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x_axis</span> <span class="o">=</span> <span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x_axis</span><span class="p">,</span> <span class="n">expit</span><span class="p">(</span><span class="n">x_axis</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'x'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">([</span><span class="o">-</span><span class="mi">6</span><span class="p">,</span> <span class="mi">6</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'logistic(x)'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'The logistic function'</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
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<h2 id="Binary-IRT-model">Binary IRT model<a class="anchor-link" href="#Binary-IRT-model">¶</a></h2><p>Here is how to code a binary IRT model in Stan:</p>
<pre><code>data {
// numbers of things
int<lower=1> R; // responses
int<lower=1> I; // items
int<lower=1> S; // subjects
// data
int<lower=1,upper=I> item[R];
int<lower=1,upper=S> subject[R];
int<lower=0,upper=1> grade[R];
}
parameters {
// parameters
vector[S] ability;
vector[I] difficulty;
// hyperparameters
real mu_difficulty;
real<lower=0> sigma_difficulty;
real<lower=0> sigma_ability;
}
model {
// priors
ability ~ normal(0, sigma_ability);
difficulty ~ normal(0, sigma_difficulty);
mu_difficulty ~ cauchy(0, 5);
// data model
grade ~ bernoulli_logit(ability[subject] - difficulty[item] - mu_difficulty);
}</code></pre>
<p>The data model is what makes this an IRT model, so naturally it has the form described above - the grade vector has a logistic distribution, where each repsonse's success probability depends on the difference between the ability of the subject attempting it and the difficulty of the item being attempted.</p>
<p>The difficulty parameters are given a non-centered parameterisation, so that the mean difficulty and the deviations of individual items' difficulties from this mean are defined separately as <code>mu_difficulty</code> and <code>difficulty</code> respectively. In applications where a lot of data is available, a centred parameterisation, i.e. <code>difficulty ~ normal(mu_difficulty, sigma_difficulty)</code> and <code>grade ~ bernoulli_logit(ability[subject] - difficulty[item])</code>, might be more appropriate. See the section of the Stan manual on 'Reparameterisation' for more on this topic.</p>
<p>A bit of care has to be taken with the prior distributions for the <code>difficulty</code> and <code>ability</code> parameters, as the data model is only sensitive to their differences, not their absolute values. This notebook deals with this issue using 'soft centering', that is, fixing the location of one or other parameter's prior distribution, so that, all else equal, the model will choose ability parameter values that sum to zero.</p>
<p>Alternatively, a sum-to-zero constraint could have been imposed directly using a <code>transformed parameters</code> block. See the section of the Stan manual on 'Parameterizing Centered Vectors' for details about this approach, known as 'hard centering'.</p>
<p>The scale of the <code>ability</code> and <code>difficulty</code> parameters' priors are set hierarchically, using hyperparameters <code>sigma_ability</code> and <code>sigma_difficulty</code>. In this way we avoid having to specify the scales, which is generally a good idea in the absence of any information.</p>
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<p>Once we have coded our model, we can put it in a <code>.stan</code> file and compile it using the following code:</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">binary_irt_model</span> <span class="o">=</span> <span class="n">pystan</span><span class="o">.</span><span class="n">StanModel</span><span class="p">(</span><span class="n">file</span><span class="o">=</span><span class="s2">"binary_irt.stan"</span><span class="p">)</span>
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<h3 id="Simulated-data">Simulated data<a class="anchor-link" href="#Simulated-data">¶</a></h3><p>A good way to check if a model is doing what you think it should is to test it on simulated data.</p>
<p>Here we verify that the binary IRT model we just compiled can recover some randomly generated parameters.</p>
<p>First we need to generate some random data. Here's how to do it using functions from numpy and scipy:</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">generate_binary_irt_data</span><span class="p">(</span><span class="n">sim_input</span><span class="p">):</span>
<span class="c1"># simulate abilities, difficulties, and subject/item combinations</span>
<span class="n">sim_ability</span> <span class="o">=</span> <span class="n">normal</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span>
<span class="n">scale</span><span class="o">=</span><span class="n">sim_input</span><span class="p">[</span><span class="s1">'sigma_ability'</span><span class="p">],</span>
<span class="n">size</span><span class="o">=</span><span class="n">sim_input</span><span class="p">[</span><span class="s1">'S'</span><span class="p">])</span>
<span class="n">sim_difficulty</span> <span class="o">=</span> <span class="n">normal</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="n">sim_input</span><span class="p">[</span><span class="s1">'mu_difficulty'</span><span class="p">],</span>
<span class="n">scale</span><span class="o">=</span><span class="n">sim_input</span><span class="p">[</span><span class="s1">'sigma_difficulty'</span><span class="p">],</span>
<span class="n">size</span><span class="o">=</span><span class="n">sim_input</span><span class="p">[</span><span class="s1">'I'</span><span class="p">])</span>
<span class="n">sim_subject</span> <span class="o">=</span> <span class="n">randint</span><span class="p">(</span><span class="n">low</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span>
<span class="n">high</span><span class="o">=</span><span class="n">sim_input</span><span class="p">[</span><span class="s1">'S'</span><span class="p">],</span>
<span class="n">size</span><span class="o">=</span><span class="n">sim_input</span><span class="p">[</span><span class="s1">'R'</span><span class="p">])</span>
<span class="n">sim_item</span> <span class="o">=</span> <span class="n">randint</span><span class="p">(</span><span class="n">low</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span>
<span class="n">high</span><span class="o">=</span><span class="n">sim_input</span><span class="p">[</span><span class="s1">'I'</span><span class="p">],</span>
<span class="n">size</span><span class="o">=</span><span class="n">sim_input</span><span class="p">[</span><span class="s1">'R'</span><span class="p">])</span>
<span class="c1"># work out success probabilities</span>
<span class="n">sim_success_probabilities</span> <span class="o">=</span> <span class="n">expit</span><span class="p">(</span><span class="n">sim_ability</span><span class="p">[</span><span class="n">sim_subject</span><span class="p">]</span> <span class="o">-</span>
<span class="n">sim_difficulty</span><span class="p">[</span><span class="n">sim_item</span><span class="p">])</span>
<span class="c1"># simulate grades</span>
<span class="n">sim_grade</span> <span class="o">=</span> <span class="n">binomial</span><span class="p">(</span><span class="n">n</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
<span class="n">p</span><span class="o">=</span><span class="n">sim_success_probabilities</span><span class="p">,</span>
<span class="n">size</span><span class="o">=</span><span class="n">sim_input</span><span class="p">[</span><span class="s1">'R'</span><span class="p">])</span>
<span class="c1"># Dictionary of data to give to STAN</span>
<span class="n">sim_data</span> <span class="o">=</span> <span class="p">{</span>
<span class="s1">'grade'</span><span class="p">:</span> <span class="n">sim_grade</span><span class="p">,</span>
<span class="s1">'subject'</span><span class="p">:</span> <span class="n">sim_subject</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span>
<span class="s1">'item'</span><span class="p">:</span> <span class="n">sim_item</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span>
<span class="p">}</span>
<span class="n">sim_data</span><span class="o">.</span><span class="n">update</span><span class="p">({</span><span class="n">i</span><span class="p">:</span> <span class="n">binary_sim_input</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">'R'</span><span class="p">,</span> <span class="s1">'I'</span><span class="p">,</span> <span class="s1">'S'</span><span class="p">]})</span>
<span class="n">recoverables</span> <span class="o">=</span> <span class="p">{</span>
<span class="s1">'ability'</span><span class="p">:</span> <span class="n">sim_ability</span><span class="p">,</span>
<span class="s1">'difficulty'</span><span class="p">:</span> <span class="n">sim_difficulty</span><span class="p">,</span>
<span class="p">}</span>
<span class="n">recoverables</span><span class="o">.</span><span class="n">update</span><span class="p">({</span><span class="n">i</span><span class="p">:</span> <span class="n">binary_sim_input</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">'sigma_ability'</span><span class="p">,</span>
<span class="s1">'mu_difficulty'</span><span class="p">,</span>
<span class="s1">'sigma_difficulty'</span><span class="p">]})</span>
<span class="k">return</span> <span class="n">sim_data</span><span class="p">,</span> <span class="n">recoverables</span>
<span class="c1"># define some input data</span>
<span class="n">binary_sim_input</span> <span class="o">=</span> <span class="p">{</span><span class="s1">'R'</span><span class="p">:</span> <span class="mi">10000</span><span class="p">,</span>
<span class="s1">'I'</span><span class="p">:</span> <span class="mi">15</span><span class="p">,</span>
<span class="s1">'S'</span><span class="p">:</span> <span class="mi">15</span><span class="p">,</span>
<span class="s1">'sigma_ability'</span><span class="p">:</span> <span class="mi">1</span><span class="p">,</span>
<span class="s1">'sigma_difficulty'</span><span class="p">:</span> <span class="mi">2</span><span class="p">,</span>
<span class="s1">'mu_difficulty'</span><span class="p">:</span> <span class="o">-</span><span class="mi">1</span><span class="p">}</span>
<span class="n">binary_sim_data</span><span class="p">,</span> <span class="n">binary_sim_recoverables</span> <span class="o">=</span> <span class="n">generate_binary_irt_data</span><span class="p">(</span><span class="n">binary_sim_input</span><span class="p">)</span>
<span class="c1"># print results</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'Here is our randomly generated data:'</span><span class="p">)</span>
<span class="n">pprint</span><span class="p">(</span><span class="n">binary_sim_data</span><span class="p">)</span> <span class="c1"># pprint makes the dictionary print nicely</span>
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<pre>Here is our randomly generated data:
{'I': 15,
'R': 10000,
'S': 15,
'grade': array([1, 1, 1, ..., 1, 1, 1]),
'item': array([10, 2, 3, ..., 5, 13, 6]),
'subject': array([10, 6, 12, ..., 11, 4, 15])}
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<p>Now that we have this simulated data, we can fit our model and check how well it recovers the values that we put into the simulator.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># fit model</span>
<span class="n">binary_fit</span> <span class="o">=</span> <span class="n">binary_irt_model</span><span class="o">.</span><span class="n">sampling</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">binary_sim_data</span><span class="p">,</span> <span class="nb">iter</span><span class="o">=</span><span class="mi">2000</span><span class="p">,</span> <span class="n">chains</span><span class="o">=</span><span class="mi">4</span><span class="p">)</span>
<span class="c1"># extract parameters</span>
<span class="n">binary_params</span> <span class="o">=</span> <span class="n">binary_fit</span><span class="o">.</span><span class="n">extract</span><span class="p">()</span>
<span class="c1"># print fit summary</span>
<span class="nb">print</span><span class="p">(</span><span class="n">binary_fit</span><span class="p">)</span>
<span class="c1"># draw summary plot</span>
<span class="n">f</span> <span class="o">=</span> <span class="n">binary_fit</span><span class="o">.</span><span class="n">plot</span><span class="p">()</span>
<span class="n">f</span><span class="o">.</span><span class="n">set_size_inches</span><span class="p">(</span><span class="mi">18</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
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<pre>Inference for Stan model: anon_model_67a0fead3b0bed682fd9788c315cf7f2.
4 chains, each with iter=2000; warmup=1000; thin=1;
post-warmup draws per chain=1000, total post-warmup draws=4000.
mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
ability[0] 0.16 7.6e-3 0.21 -0.23 0.02 0.16 0.29 0.6 755 1.01
ability[1] -1.26 7.7e-3 0.21 -1.67 -1.39 -1.27 -1.13 -0.85 728 1.01
ability[2] -0.64 7.6e-3 0.21 -1.03 -0.78 -0.64 -0.52 -0.23 724 1.0
ability[3] 1.46 7.6e-3 0.22 1.05 1.31 1.45 1.6 1.9 815 1.0
ability[4] -0.21 7.8e-3 0.21 -0.61 -0.35 -0.21 -0.08 0.21 705 1.01
ability[5] -0.46 7.7e-3 0.21 -0.86 -0.6 -0.46 -0.33 -0.04 715 1.0
ability[6] -0.57 7.6e-3 0.21 -0.98 -0.71 -0.57 -0.44 -0.15 754 1.0
ability[7] 0.72 7.8e-3 0.21 0.31 0.58 0.71 0.86 1.15 752 1.0
ability[8] 0.4 7.7e-3 0.21-3.9e-3 0.26 0.4 0.54 0.85 766 1.0
ability[9] -0.11 7.6e-3 0.21 -0.52 -0.25 -0.12 0.02 0.32 754 1.0
ability[10] -0.01 7.9e-3 0.21 -0.42 -0.15 -0.02 0.12 0.42 718 1.01
ability[11] -0.1 7.8e-3 0.21 -0.49 -0.24 -0.1 0.03 0.33 730 1.01
ability[12] -0.31 7.6e-3 0.21 -0.71 -0.45 -0.31 -0.18 0.12 767 1.0
ability[13] 0.19 7.7e-3 0.21 -0.21 0.05 0.19 0.32 0.61 720 1.0
ability[14] 0.88 7.8e-3 0.21 0.48 0.73 0.87 1.02 1.32 755 1.01
difficulty[0] 1.81 0.03 0.56 0.64 1.46 1.8 2.17 2.92 476 1.01
difficulty[1] -2.3 0.03 0.59 -3.51 -2.67 -2.31 -1.92 -1.13 526 1.01
difficulty[2] -2.21 0.03 0.58 -3.45 -2.58 -2.21 -1.84 -1.07 502 1.01
difficulty[3] -1.02 0.03 0.57 -2.22 -1.37 -1.02 -0.65 0.1 484 1.01
difficulty[4] 0.5 0.03 0.56 -0.65 0.16 0.49 0.87 1.62 469 1.01
difficulty[5] -0.41 0.03 0.56 -1.59 -0.77 -0.41 -0.05 0.71 475 1.01
difficulty[6] 1.54 0.03 0.56 0.38 1.19 1.54 1.91 2.67 485 1.01
difficulty[7] 1.48 0.03 0.56 0.3 1.12 1.48 1.84 2.59 475 1.01
difficulty[8] -1.58 0.03 0.57 -2.74 -1.94 -1.58 -1.21 -0.44 488 1.0
difficulty[9] -0.41 0.03 0.56 -1.58 -0.76 -0.42 -0.05 0.72 470 1.01
difficulty[10] 0.48 0.03 0.56 -0.68 0.13 0.48 0.84 1.61 475 1.01
difficulty[11] 2.04 0.03 0.57 0.86 1.69 2.04 2.41 3.16 474 1.01
difficulty[12] -3.32 0.03 0.62 -4.58 -3.71 -3.3 -2.92 -2.13 568 1.0
difficulty[13] 4.01 0.03 0.58 2.8 3.64 4.01 4.37 5.2 505 1.01
difficulty[14] -0.44 0.03 0.57 -1.62 -0.79 -0.45 -0.08 0.68 477 1.01
mu_difficulty -0.83 0.03 0.59 -1.98 -1.19 -0.85 -0.47 0.44 484 1.01
sigma_difficulty 2.19 0.01 0.49 1.48 1.85 2.11 2.43 3.38 1331 1.0
sigma_ability 0.75 4.0e-3 0.17 0.5 0.63 0.73 0.84 1.14 1773 1.0
lp__ -4364 0.11 4.09 -4373 -4367 -4364 -4361 -4357 1363 1.0
Samples were drawn using NUTS at Sun Jan 22 09:25:37 2017.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at
convergence, Rhat=1).
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