Parameter Recovery

Written by Andersen Chen

1. Introduction

Providing students with good study recommendations begins with understanding what they already know. At Knewton, we’re combining existing work on psychometrics and large scale machine learning to perform proficiency estimation more accurately and at a larger scale than ever before.

As Alejandro Companioni [discussed in a previous post], Item Response Theory (IRT) is the core model around which our proficiency estimation is built. IRT proposes that there are latent features of students (proficiency) and questions (difficult, discrimination, and guessability) which explain the accuracy of responses we see on assessments. However, most IRT analyses assume that students bring the same level of proficiency to every question asked of them. This might be fine for a 2-hour standardized test, but we are interested in helping students over an entire course. If we’re doing our job, students should be improving their proficiency as time goes on!

This summer, I made our IRT models more sensitive to gains (and losses) in student proficiencies over time. I’ll leave the details of that model to another post, but as a teaser, I’ll show you our first figure. In this figure, the $x$-axis represents time as measured by the total number of questions answered. Dot represent the question which was answered at that time. Red dots represent incorrectly answered questions while green dots represent correctly answered questions. The ${y}$-value of the dot is its difficulty. The green line tracks our simulated student proficiency, and the blue line tracks our recovered student proficiency paramter given the questions that they have answered.

In this post, I will discuss three methods which we used to evalute the performance of our algorithms and discuss their relative strengths and weaknesses. We’ll focus on the mean-squared error, the log-likelihood, and the Kullback-Liebler divergence.

A visual representation of our temporal IRT model

2. Evaluating results

To tackle the problems I faced, I explored statistical inference algorithms on probabilistic graphical models. To judge my results, I simulated classes of students answering sets of questions, and saw how accurately my algorithms recovered the parameters of the students and questions.

One method of quantifying the accuracy of estimates is the mean-squared error (MSE). It takes the mean of the square of the differences between the estimated parameters and their actual values. In symbols, if ${\theta^{\text{act}}}$ is the vector of model parameters used to generate our data, and ${\theta^{\text{rec}}}$ are the parameters our method recovered, then

$\displaystyle \text{MSE} = \frac{1}{N} sum_{i=1}^N \left(\theta^{\text{act}}_i — \theta_i^{\text{rec}}\right)².$

While the MSE is a good indicator for accuracy in many places, it has problems when models have multiple solutions at different scales. Let’s see why this is through an example.

Suppose we have a class of students answering a set of questions. Suppose the questions are actually very hard and that the students happen to be very proficient at the material. Just looking at this set of data, however, we have no way of knowing that these students are actually very proficient. We can only assume the most likely scenario — the students have average proficiences and are answering the questions competently. From the data alone, we can only hope to discern the values of item parameters relative to the proficiency of students and vice-versa. We cannot hope to know their values absolutely. So there are many equally valid interpretations at different scales of the same data. Because the MSE looks at the difference between the two parameters, it will penalize parameters that are scaled differently, even though those parameters might be an equally valid solution given the data!

Let’s look at Figure 2. We see that the ordering is basically preserved, which we could measure with Pearson’s ${r}$. but the recovered parameters are not on the same scale. If that were the case, then the line of best fit would have slope one and ${y}$-intercept ${0}$. However, in this case, the slope is closer to ${2/3}$. In statistical terms, the variance of the recovered parameters is much less than the variance of the actual parameters.

Recovered student proficiencies plotted against actual student proficiencies from a simulation.

The log-likelihood gives us a more meaningful method of measuring accuracy. The log-likelihood tells us the log-probability of our recovered parameters given our data. At instantiation, our algorithms should have low log-likelihoods — the parameters that we guess first are random and don’t fit our data well. Our algorithms should iterate toward higher log-likelihoods, hopefully converging at the set of parameters with the highest log-likelihood. This is the philosophy behind the Expectation-Maximization algorithm. But the log-likelihood is susceptible to tail events. For instance, if a question is in reality extremely hard but, through random chance, a few students with average proficiency answer the question correctly, then maximizing the log-likelihood will lead to marking these very hard questions as easier than they actually are. This, of course, could be solved with more data from large numbers of extremely proficient students, but this data is often hard to come by. Instead, we introduce another way of measuring model fit: the Kullback-Leibler (KL) divergence. Suppose we have probability distributions ${P}$ and ${P’}$, generated by our original and recovered parameters, respectively. That is,

$\displaystyle P(\text{datum}) = \frac{1}{# \text{ of data}} Pr(\text{datum} | \theta^{\text{act}} )$

and

$\displaystyle P’(\text{datum}) = \frac{1}{# \text{ of data}} Pr(\text{datum} | \theta^{\text{rec}} )$

In our case, a datum is one student’s response to one question and the ${theta}$ paramters are the student proficiency and question discrimination, difficult, and guessability discussed in Alex’s post. Then we can define

$\displaystyle D_{KL} (P||P’) = \displaystyle sum_i \lnleft({\frac{P(i)}{P’(i)}}\right) P(i).$

If ${P = P’}$, then ${D_{KL}(P||P’) = 0}$. Moreover, Gibbs’ inequality implies that ${D_{KL}(P||P’) geq 0}$. If the original and recovered parameters generate similar probabilities, then the KL Divergence should be small. Note that since the KL divergence involves the ratio of the likelihoods, it is less susceptible to the influence of tail events than the log-likelihood alone.

The log-likelihood and KL divergence both use likelihoods to measure fit, which means that they only care about fit of the parameters to the data, and not the exact convergence to the original. So they often prove to be reliable measures of fit to judge our algorithms on. For instance, even though the MSE of our recovered parameters is large, Figure 2 shows us that our algorithm has likely converged (since the log-likelihood and KL divergences are not changing much) and give us a reliable measure of model fit.

The log-likelihood and KL divergence of our recovered parameters through a run of our algorithm.