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To describe adaptive testing, six different steps can be identified (Hambleton, 1991): choice of an item response model, creation of an item bank, deciding on a starting point for testing, selection of subsequent test items, scoring/ability estimation and deciding on termination criteria for the test. The theories described in this paragraph can be used to ensure quality of the items in the item bank.
In (Dousma, 1995), an extensive description of test item analysis can be found. Such analysis focuses on level of difficulty, quality of alternatives and discriminative power of a test item described in the following formulas:
General level of difficulty: p=number of students answering item correctly/total number of students. Since this value does not take guessing into consideration, the level of difficulty corrected for guessing is defined as: p=p(1p)/(na1), where na represents the number of alternatives per item;
Attractiveness of an alternative: a=number of students choosing the alternative/total number of students. The a value of an item should not approximate or be equal to zero, and should always be smaller than p. If not, either the item is incorrect, or the course does not pay enough attention to how such a question should be answered. All avalues of an item should be approximately equal, so each alternative functions the way it is supposed to;
Quality of alternatives: Za=([Xa][X])/Sx, where [Xa] equals the mean score for students choosing the alternative and [X] equals the mean score of all students. Sx is the standard deviation of all testscores. The quality of a wrong alternative should be negative, and the quality of the right answer should be positive. This way, the item can distinguish good students from bad students;
The discriminating power of an item, which should be high so that it can distinguish good students from bad students. Here, only the simple calculation is provided: select two equally sized subgroups of students, one representing the highest overall scores, the other representing the lowest overall scores. Identify the number of students answering the item correctly in both groups. Divide both by the size of the group, and then subtract the number for the last group from the number for the first group. According to (Dousma, 1995), choosing 27% of the total group of students for both subgroups provides the best results.
The theory described in (Dousma, 1995) is therefore useable to ensure quality of testitems within an adaptive testing environment. The actual model used to decide on the right questions to ask a student means choosing a specific implementation of the Item Response Theory (IRT). This theory is based on the idea that a students answer to a question presents a certain probability of that student mastering the specific concept. Therefore, scoring and ability estimation as well as test termination is based on the results of a number of tests, representing a subset of all the tests available. In standard Artificial Intelligence literature, the use of Bayesian Belief Networks for such reasoning is advocated. The following two paragraphs describe the IRT and Bayesian Belief Network theories.
Item Response Theory
The Item Response Theory is aimed at providing information about the functional relation between an estimate of a students mastery level of a concept (his or her latent construct) and the likelihood that a given respondent will make a correct response to a given item (the Probability of Correct Response or PCR). In general, three models are identified based on using the difficulty parameter, the discrimination level and the guessing factor of a test item: the so called Rasch model using only the difficulty parameter, the 2parameters model using both the difficulty parameter and the discrimination parameter and the 3parameter model. According to Hambleton et al., this last model is most appropriate in adaptive testing (Hambleton, 1991).
Using IRT, each item on a test has its own characteristic curve (ICC), which represents the probability that the student will be able to provide an answer which will enhance the believe the student is a master. The mathematical expression for the threeparameter model is (Hambleton, 1991):
EMBED Unknown EMBED Unknown
Here, Pi(() is the probability that a student with ability ( answers item i correctly, depicted as an Sshaped curve with values between 0 and 1 over the ability scale. The difficulty parameter for item i is depicted as bi, the item discrimination parameter as ai and the pseudochancelevel as ci. This last parameter takes into account performance at the low end of the ability continuum, in terms of guessing. The number of items in the test is given by n, and the factor D is a scaling factor usually approximately 1.7. Using this formula, the Item Information Function (IIF) can be calculated. This is a representation of the amount of information provided by each item at any given level of latent construct. Information in IRT terminology describes how well the item can discriminate between individuals with different latent constructs. The general equation as well as the one used for the threeparameter model are given below.
EMBED Unknown EMBED Unknown
Equation a  Item Information Function
EMBED Unknown
Equation b  Threeparameter model IIF
Here, Qi(() is equal to 1Pi((). Using this information function, for each latent construct the item providing the most information can be chosen to increase probability of a student being either a master or a nonmaster. To propagate this knowledge throughout the course hierarchy, Bayesian Belief Networks can be used.
Bayesian Belief Networks
Many ITS systems use Bayesian network theory to update believe about a students knowledge. In the following two pictures, the situation is shown for two ways to present an environment where two questions are used to determine whether a student masters a subject or not.
EMBED ABCFlowCharter6.Document EMBED ABCFlowCharter6.Document
Figure STYLEREF 1 \n 0 SEQ Figure \* ARABIC \r 1 1  Two ways of representing a Bayesian Belief Network
If an instructor must provide the different probabilities based on experience, the second representation is clearly preferable from an efficiency point of view. This leaves the problem of deciding on P(MC1,C2), P(MC1,C2),P(MC1,C2) and P(MC1,C2).
For this, Bayesian nets can be used. Basically, this theory states:
EMBED Equation.3
Combining this with standard Bayesian probability theory, the following calculation describes the application of this rule to the example above:
Translating the rule to the situation in the example, it states: P(MC1,C2)=P(M,C1,C2)/(P(C1C2)P(C2))
Now, since C2 does not influence C1, the following is true: P(MC1,C2)=P(M,C1,C2)/(P(C1)P(C2))
And, reapplying the rules to P(M,C1,C2), the final result states: P(MC1,C2)=P(C1M)P(C2M)P(M)/(P(C1)P(C2))
Since for both P(C1) and P(C2) the following rule is true: P(C)=P(C,M)+P(C,M)
the final formula states: P(MC1,C2)=P(C1M)P(C2M)P(M)/(((P(C1M)P(M))+(P(C1M)P(M)))(((P(C2M)P(M))+(P(C2M)P(M)))
This is basically how Bayesian networks are used in adaptive testing: based on information about answers to questions belonging to a certain concept, the probability that a student is either a master or a nonmaster at that concept is calculated. One of the drawbacks of use Bayesian nets is known as the optimal factoring problem. With the transformations presented above as an example, the problem comes down to choosing the right order of events. If P(C1,M,C2) was transformed instead of P(C1,C2,M), at least four extra steps would be needed. For calculating belief in larger networks, several algorithms targeting such problems have been developed. In general, finding the optimal factoring is considered to be an NPhard problem, but a polynomial time algorithm exists for generating optimal factoring for treestructured belief networks such as the one used here (Ambrosio, 1991). This theory, and algorithms to use can be found in for example (Ambrosio, 1991).
Combining test analysis and Bayesian Belief Networks
When the rules of itemanalysis are applied to the situation where a Bayesian network is used to determine mastery, a basic adaptive testing system aimed at deciding on mastery can be described. In such a system, the item pool should be created by an expert and the entry level of a student should be provided by either the student or the instructor. Basically, the probability of a student being a master can be compared to two threshold values: if it is smaller than the lowest threshold nonmastery is decided, if it is larger than the highest mastery is decided. If no decision can be made, a new question will be asked until there is enough evidence to make a decision. Therefore, for each item the right answer, the number of alternatives, the number of students choosing a specific alternative and information about a student who has answered the question being a master or nonmaster is needed.
From the above description, the following conclusions can be drawn:
This approach requires extensive calibration of the system. Many values are dependent on knowledge about the number of students or the number of masters answering the specific item;
The system can become a learning system if it keeps refining its values based on student responses;
After a student has provided the system with a temporary entrylevel in the form of a concept, the system should decide which concepts are mastered by the student. However, the total number of questions should be limited, for a long list of questions will bore the student.
There are various systems implementing the theory described above. For this project, the ContentBalanced Adaptive Testing system (CBAT2) described in (Huang, 1996) seems useful. This approach extends the theory described by Frick and Welch who resolved the need of the traditional Weiss/Kingsburry IRT implementation for extensive calibration.
The ContentBalanced Adaptive Testing system
EMBED ABCFlowCharter6.Document
Figure STYLEREF 1 \n 0 SEQ Figure \* ARABIC 2  Curriculum hierarchy in CBAT2, (Huang, 1996)
The CBAT2 system, described in (Huang, 1996), aims at providing a solution for the following problems not addressed by the Frick and Welch approach: content balance, intelligent selection of test items, security so that selected test items do not form a pattern, possibility of associating questions with multiple content areas and the notion of twolevel assessment as will be described below. The Domain Knowledge component in this system consists of the directed acyclic graph of a curriculum hierarchy. As described before in this chapter, this graph associates content areas and questions in a course as is shown in REF _Ref411015811 \* MERGEFORMAT Figure 02.
In this hierarchy, questions associated with the lower levels require knowledge of specific concepts and skills. Questions at higher levels require general knowledge. The system uses a twolevels approach towards both content balancing and assessment at different levels. This is based on the fact that mastery at a higher level means mastery at all lower levels which are prerequisite for the higher level. This has at least three consequences:
By repeating this approach, the knowledge level at each of the components involved can be tested;
The test list will not become too large;
Test from lower levels can be used at higher levels.
If for instance the system starts evaluating a students knowledge at the course level, the system assesses knowledge at both the course level and the module level. If the system evaluates a student at module level, it assesses a student at both module and concept level. This approach is depicted in REF _Ref411015937 \* MERGEFORMAT Figure 03.
EMBED ABCFlowCharter6.Document
Figure STYLEREF 1 \n 0 SEQ Figure \* ARABIC 3  Curriculum hierarchy for a course and module test, (Huang, 1996)
According to Huang, the discriminatory level is usually difficult to calibrate and its meaning is difficult to be understood by test designers, it is therefore defined as a constant with value 1.2. The guessing factor is simply defined as one divided by the number of alternatives. To calculate the difficulty level of a question, a more complex formula is used:
EMBED Unknown
Here, initi is the initial difficulty of the question, Ri and Wi are the number of times item i was answered correctly or incorrectly respectively. The constant 20 is used as a normalization factor. The system has a learning ability since (i is defined as:
EMBED Unknown
where n is the number of answers for item i in the past, (j was the temporary proficiency of the student who gave the jth answer for Qj; kj=0 if the jth answer is correct, and kj=2 if the jth answer is incorrect. The function f converts a ( value to a difficulty value and is currently linear. The test designer assigns the initial difficulty, which has a value from 0 to 1. Using this approach, it is no longer necessary to calibrate the system using empirical data. As the question is used in tests, the difficulty level converges to (i/(Ri+Wi), thereby minimizing the influence of the initial difficulty.
Each student starts with an initial proficiency, and the first question is selected based on this proficiency. Based on empirical data, the system calculates new temporary proficiencies along with the confidence degree for the proficiency. The question selection procedure first decides on the right component, and than selects a question from this component. The choice for the component is based on weights of the components involved. However, if proficiency is already decided on one of the components, this component is not included in the candidate components. Selection of a question belonging to the component is based on the amount of information that a question may provide for the students assessment as described by the IIF. A set of questions with the highest score become candidate questions, among which a question is randomly selected. This is done to prevent the test from becoming deterministic.
To decide on test termination, the likelihood ratio LR described by Frick and Welch can be used (Collins, 1996). This approach is Bayesian in nature, and is described as:
EMBED Unknown
Here, s=1 if item i is answered correctly and s=0 otherwise, f=1 if item i is answered incorrectly and f=0 otherwise; n is the number of items in the item pool, i is the event that a right answer is provided, M is the event of a student being a master and N the event of a student being a nonmaster; Pom and Pon represent the students prior master probability and nonmaster probability respectively.
The test then terminates when two conditions are met:
LR has passed an upper confidence criterion or LR has passed a lower confidence criterion;
every child components has at least a predefined minimum number of questions selected for the test.
After termination, the temporary proficiency of each component becomes its proficiency. Based on this proficiency mastery or nonmastery can be decided.
Comparing CBAT2 to theory previously described, three assumptions in the system may need further development. First, by not including a discriminatory level in the algorithm, previously described problems might occur in the testing environment. If the goal is to measure the students ability to answer difficult questions, the algorithm is useful. However, if the goal is to decide on mastery or nonmastery, a discriminatory level should be included to be able to choose which question can best be used to decide on this.
Second, the guessing factor is highly dependent on the quality of the alternatives. The initial value could be set as is done in the CBAT2 system, but as empirical results become available, the guessing factor should take the quality of alternatives into account. This is also mentioned in (Hambleton, 1991), where instead of guessing factor the term pseudochancelevel parameter is used. The pseudochancelevel parameter described the portion of lowestscoring students answering a question right, and is in general lower than the guessing factor used in CBAT2.
Finally, in CBAT2, the preliminary proficiency of a student is used in the difficulty accumulator. However, this seems a bit peculiar: the preliminary proficiency is only an estimate of the final proficiency which is the result of testing the student. Practically, the preliminary proficiency doesnt tell much about the real student state, especially since no probability measure is used. In this case, it seems more sensible to use the final proficiency to update all questions asked. This is more complex, but seems to be more correct.
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