# The Accurate Measurement of Students' Learning in E-Learning Environments
## 2 Background
### Fixed-Parameter Calibration  
In educational measurement, IRT has been widely used in placing persons' abilities from different calibrations on a common metric or in equating [21]. In practice, the equating allows us to compare persons' abilities from different test forms. For example, graduate record examinations (GREs) use multiple test forms for various reasons including item security, and equating allows us to place examinees' abilities from different forms on a common metric for comparison. This equating procedure is essential in comparing the performance of persons over repeated measures. Three common approaches in IRT have been used to capture academic growth in the literature: the linear transformation of separate calibrations, concurrent calibration, and fixed-parameter calibration (FPC). In the specific context of MOOCs, [PERSON] and [PERSON] [7] proposed to use equating for fair and equitable measurement of student learning in MOOCs. In their work, they used the linear transformation of separate calibrations. In this study, we propose to use FPC to establish a common metric for the measurement of learning in MOOCs because FPC allows us to establish a common metric more easily without the process of score transformation, or the FPC method establishes a common metric in a way that fits well in MOOCs settings. The key question of equating in education measurement is how to place the new and old items on a common scale for comparison. Two different ways are possible for equating: independent calibration with linking and FPC [8]. In the independent calibration, we first calibrate (or estimate item parameters) the new and old item together and then place the calibrated parameters for the new items on the existing scale using the linear transformation that can be found by linking items. On the contrary, FPC does not require linking items, which makes FPC more suitable for the measurement of learning in MOOCs. When calibrating IRT using FPC, the item parameters of a subset of items, called operational item parameters, are fixed at their previously calibrated values, and only the item parameters of new items are calibrated. As a result, the new items are placed on the same metric as the previously calibrated items [8; 22]. Figure 3 illustrates FPC. In the figure, the item parameters of Item 2 and Item 3 in IRT Calibration 2 are fixed to their previously calibrated values in IRT Calibration 1, and only the parameters of Person 3, Item 4, and Item 5 are estimated in IRT Calibration 2. The fact that the item parameters of Item 2 and Item 3 are exactly the same in the two calibrations indicates that the two calibrations are on a common metric. FPC is simpler than separate calibrations because it does not require any transformation, but simply requires fixing some of the item parameters to their previously calibrated values during calibration. Furthermore, FPC is well suited for establishing a common metric across multiple administrations of a MOOC because those multiple administrations are highly likely to share a large number of items, which can be used as operational items in FPC. In educational measurement, FPC has been widely used in online calibration used in computerized adaptive testing (CAT) to calibrate new items in the tests on the same metric with previously calibrated items [23; 24; 25]. In addition to the common metric, FPC provides an additional advantage in the measurement of learning in MOOCs. Because of the nonmandatory and self-paced nature of MOOCs, typical MOOCs data contain a large number of missing responses to items [26]. In general, IRT is known to be robust to missing data [27]. As a result, IRT scores are expected to be more robust to missing data than the percentage or raw total scores. of all 55 items at their previously calibrated values. In general, the operational items in FPC are some subset of items.