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The two curves start to diverge at an interval of about days, and this pattern is consistent for all subtests. This explained why the average interval was much longer for this group. Although these students may constitute a distinct population, they do not differ much from students in other countries with respect to overall initial score. Hierarchical Linear Models The descriptive information is consistent with the hypothesis that the test is sensitive to changes in English ability as a function of interval.
The main motivation for conducting HLM analyses was to quantify the relationship between interval and gains in a way that would not be distorted by potential differences in student populations across countries and to obtain a valid test of statistical significance of this relationship. Our HLM is designed to achieve both of these goals.
We first present what we call the base model and then present what we call the alternative model, which we use to address a particular potential source of bias in the base model. The effect of interest is the coefficient on interval. All other effects in the model are included only to improve the estimate of the effect of interval and to get an appropriate measure of uncertainty about this effect. We now describe these other effects and why they should achieve these goals.
We include dummy variables in the model for individual countries to prevent country differences from biasing our estimated relationship between interval and gains. As noted previously, there were large differences in the distributions of interval across countries, especially at the higher end of the interval distribution. These differences could have led to bias in the estimated relationship between interval and score gains if students from different countries had systematically different average values of interval because of unobserved test taker characteristics that were related to score gains.
It seemed reasonable to assume that such characteristics might exist, so we needed to treat differences among countries as a potential confounding factor and find a way to prevent this potential confounding factor from biasing the estimated coefficient on interval. Treating countries as dummy variables avoids this potential confounding, as it ensures that the effect of interval is estimated using only variation in interval among students but within countries i. Differences in the average value of interval across countries do not therefore contribute to the estimated effect of interval, and therefore, differences among students in different countries cannot be a source of bias.
We include random effects in the model for testing groups. By definition, students in the same testing group had the same interval; thus, it was not possible to include dummy variables for testing groups because that would have left no variation with which to estimate the effect of interval.
However, it was possible to include random effects for testing groups.
Doing so is beneficial and important. As described earlier, students belonging to the same testing group were likely to share similar instructional or testing experiences. These factors could have caused students who shared a testing group to have systematically higher or lower gains than would have been predicted based on only the interval and the country for that group.
The testing group random effects are introduced to account for unobserved differences in the average gains in different testing groups that remain after accounting for both interval and country effects. Accounting for this variation through the random effects leads to a more accurate estimate of the effect of interval and an accurate standard error for this estimated effect.
Our model assumes a linear relationship between interval and gains within countries. It is reasonable to question whether a nonlinear relationship would be more appropriate. The linear specification is also supported by a graphical analysis.
Figure 4 provides a scatterplot of total gains versus interval where both quantities are centered by their respective country means. This represents the relationship between total gains and interval within countries, aligning closely to how Model 1 identifies the effect of interval through the use of country dummy variables.
The fact that these two curves nearly coincide supports the assumption that the relationship between gains and interval within country is well approximated by the linear specification of Model 1. Figure 4 Open in figure viewer PowerPoint Scatterplot of total score gain versus interval where both are centered around their respective country means.
The horizontal gray line at 0 is provided for reference. Our model also assumes a common linear relationship across countries. It is reasonable to question whether allowing the slope relating interval to gains varies across countries. We tested such a model against our simpler alternative, and again, our model was preferred by standard model comparison criteria.
Definitions of the JQC for A2 and B2 levels were accomplished through whole-panel discussion, using the B1 descriptions as a starting point. These JQC descriptions served as the frame of reference for the standard-setting judgments; that is, panelists were asked to consider the test questions in relation to these definitions. A modified Angoff approach was implemented following the procedures of Tannenbaum and Wylie , which included three rounds of judgments informed by feedback and discussion between rounds.
Prior to judgments made on the first section Listening Comprehension , the panelists were trained in the process and then given opportunity to practice making their judgments.
At this point, they were asked to sign a training evaluation form confirming their understanding and readiness to proceed, which all did. In Round 1, for each test question, panelists were asked to judge the percentage of just qualified candidates for the A2 and B2 levels who would answer the question correctly. They used the following judgment scale expressed as percentages : 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, The panelists were instructed to focus only on the alignment between the English skills demanded by the question and the English skills possessed by JQCs, and not to factor random guessing into their judgments.
For each test question, they made judgments for each of the two CEFR levels A2 and B2 before moving to the next question.
After making judgments, panelists received feedback on individual and group judgments. The average was then rounded to the next highest whole number; it is this whole number that represents the recommended cut score. Similarly, the highest and lowest cut scores presented to the panelists are first rounded to the next highest whole number before being presented to the panelists as feedback.
Panelists were then asked to share their judgment rationales. As part of the feedback and discussion, p values percentage of test takers who answered each question correctly , were shared. In addition, p values were calculated for candidates scoring at or above the 75th percentile on that particular section i. The partitioning, for example, enabled panelists to see any instances where a question was not discriminating, or where a question was found to be particularly challenging or easy for test takers at the different ability levels.
After discussion, panelists made Round 2 judgments. In Round 2, judgments were made again at the question level; panelists were asked to take into account the feedback and discussion from Round 1, and were instructed that they could make changes to their ratings for any question s , for either A2 or B2 levels, or both.
The Round 2 judgments were compiled, and feedback similar to that presented in Round 1 was provided. In addition, impact data from the October test administration were presented; panelists discussed the percentage of test takers who would be classified into each of the levels currently recommended percent below A2, percent above B2, and the percent between the two current recommendations for cut scores for A2 and B2, which includes students who would be classified 5 at the A2 and B1 levels.
At the end of the Round 2 feedback and discussion, panelists were given instructions to make Round 3 judgments. In Round 3, panelists were asked to consider the cut scores for the overall section e. Specifically, panelists were asked to review the JQC definitions of all three levels and to decide on the recommended cut score for B1, taking into account the Round 2 cut score recommendations and discussions regarding the A2 and B2 levels.
They were instructed that for Round 3, they should indicate their final cut score recommendation at the section level for A2 and B2, then locate the B1 cut score, using the A1 and B2 cut scores as references. The transition to a section-level judgment places emphasis on the overall constructs of interest i. This modification had been used in previous linking studies e. At the conclusion of Round 3 judgments for each section, the process was repeated for the next test section, starting with the general discussion of what the section measured and a discussion of minimum skills needed to reach each of the targeted CEFR levels JQC definitions , followed by three rounds of judgments and feedback.
After final Round 3 judgments were compiled for all three sections, the results of the standard setting were presented to the panel and final evaluations were completed. The tables summarize the results of the standard setting for Levels A2 and B2 for Rounds 1 and 2, and for Levels A2, B1, and B2 for the final round of judgments.
The results are presented in raw scores, which is the metric that the panelists used.