# The Accurate Measurement of Students' Learning in E-Learning Environments
## 2 Background
### Item Response Theory  
IRT is a system of measurement models that has been widely used in educational measurement [17]. In general, measurement models are statistical models that establish the correspondence between latent variables (e.g., students' ability) and their manifestations (e.g., students' responses to items). The ultimate interest in measurement models is to make an inference about unobservable latent variables using observed variables. In essence, IRT is the logistic regression of observed item responses on the person characteristics (i.e., ability) and item characteristics (e.g., difficulty and discrimination). For example, the two-parameter logistic (2 PL) model, which is one of the popular IRT models, can be expressed as the following equation [18]:  
\[P[Y_{i}=1]=\frac{exp[a_{i}(\theta_{j}-b_{i})]}{1+exp[a_{i}(\theta_{j}-b_{i})]},\]  
where \(P[Y_{i}=1]\) represents the probability of getting an item \(i\) correct, \(\theta_{j}\) represents the ability of a person \(j\), \(b_{i}\) represents the difficulty of an item \(i\), and \(a_{i}\) represents the discrimination of an item \(i\). The equation for the 2 PL model can be plotted to describe the relationship between \(P[Y_{i}=1]\) and ability \(\theta_{j}\), which is often called an item characteristic curve (ICC). Figure 1 presents the ICCs for three different sets of item parameters. From the comparison between Line 1 and Line 3, it can be easily seen that the difficult parameter \(b\) shifts (or translates) the ICC along the \(x\) (or ability) axis. Line 3 represents a more difficult item because, given a fixed ability (say 2.5), the probability of a correct response is much lower in Line 3 than in Line 1. From the comparison between Line 1 and Line 2, it can be also easily seen that the discrimination parameter \(a\) is essentially a regression coefficient representing the strength (or quality) of the relationship. Around an ability of zero, Line 2 has more discrimination power than does Line 1 in that Line 2 shows greater differences in the predicted probability. In the equation for the 2 PL model, the distance between \(\theta_{j}\) and \(b_{i}\) determines the probability of correct responses. For example, an easy item administered to a person with a high ability (i.e., \(\theta_{j}-b_{i}>>\)) produces a high probability of getting the item correct. For more details about IRT, readers are referred to the specialized literature [17; 19]. In IRT, each student is assumed to have an underlying ability in a specific domain. The goal of IRT calibration (or estimation) is to locate each person's ability on a continuous latent variable. The strength of IRT mainly comes from the fact that IRT also locates the difficulty of each item on the same latent variable on which each person's ability is located. That is, IRT locates both the persons' abilities and the items' difficulties on the same latent continuum and decides relative distances among the persons' abilities and the items' difficulties on the continuum. In locating persons and items on a latent continuum (or estimating persons' and items' parameters), each IRT estimation (or calibration) needs to define its own metric because latent variables representing students' abilities do not have any inherent metric. For example, in IRT, the metric of a latent variable can be defined by fixing the variance of the latent variable to one. Figure 2 illustrates two separate IRTcalibrations having their own metrics. In the first calibration (IRT Calibration 1), three items (Item 1, Item 2, and Item 3) and two persons (Person 1 and Person 2) are located on the latent continuum. A larger value in the continuum represents higher-ability persons or harder items. For example, Person 2 has a higher ability than Person 1, and Item 3 is more difficult than Item 2. The second calibration (IRT Calibration 2) locates different sets of items (Item 2, Item 3, and Item 4) and persons (Person 3 and Person 4) on its own metric.