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README.md

Moodle Adaptive Test Activity

The Adaptive Test activity enables a teacher to create tests that efficiently measure the takers' abilities. Adaptive tests are comprised of questions selected from the question bank that are tagged with a score of their difficulty. The questions are chosen to match the estimated ability level of the current test-taker. If the test-taker succeeds on a question, a more challenging question is presented next. If the test-taker answers a question incorrectly, a less-challenging question is presented next. This technique will develop into a sequence of questions converging on the test-taker's effective ability level. The test stops when the test-taker's ability is determined to the required accuracy.

The Adaptive Test activity uses the "Practical Adaptive Testing CAT Algorithm" by B.D. Wright published in Rasch Measurement Transactions, 1988, 2:2 p.24 and discussed in John Linacre's "Computer-Adaptive Testing: A Methodology Whose Time Has Come." MESA Memorandum No. 69 (2000).

This Moodle activity module was created as a collaborative effort between Middlebury College and Remote Learner. The latest code, documentation, and bug-tracker can be found at https://github.com/middlebury/moodle-mod_adaptivequiz.

The Question Bank

To begin with, questions to be used with this activity are added or imported into Moodle's question bank. Only questions that can automatically be graded may be used. As well, questions should not award partial credit. The questions can be placed in one or more categories.

This activity is best suited to determining an ability measure along a unidimensional scale. While the scale can be very broad, the questions must all provide a measure of ability or aptitude on the same scale. In a placement test for example, questions low on the scale that novices are able to answer correctly should also be answerable by experts, while questions higher on the scale should only be answerable by experts or a lucky guess. Questions that do not discriminate between takers of different abilities on will make the test ineffective and may provide inconclusive results.

Take for example a language placement test. Low-difficulty vocabulary and reading-comprehension questions would likely be answerable by all but the most novice test-takers. Likewise, high-difficulty questions involving advanced gramatical constructs and nuanced reading-comprehension would be likely only be correctly answered by advanced, high-level test-takers. Such questions would all be good candidates for usage in an Adaptive Test. In contrast, a question like "Is 25¥ a good price for a sandwich?" would not measure language ability but rather local knowledge and would be as likely to be answered correctly by a novice speaker who has recently been to China as it would be answered incorrectly by an advanced speaker who comes from Taiwan -- where a different currency is used. Such questions should not be included in the question-pool.

Questions must be tagged tagged with a 'difficulty score' using the format 'adpq_n' where n is a positive integer, e.g. 'adpq_1' or 'adpq_57'. The range of the scale is arbitrary (e.g. 1-10, 0-99, 1-1000), but should have enough levels to distinguish between question difficulties.

The Testing Process

The Adaptive Test activity is configured with a fixed starting level. The test will begin by presenting the test-taker with a random question from that starting level. As described in Linacre (2000), it often makes sense to have the starting level be in the lower part of the difficulty range so that most test-takers get to answer at least one of the first few questions correctly, helping their moral.

After the test-taker submits their answer, the system calculates the target question difficulty it will select next. If the last question was answered correctly, the next question will be harder; if the last question was answered incorrectly, the next question will be easier. The system also calculates a measure of the test-taker's ability and the standard error for that measure. A next random question at or near the target difficulty is selected and presented to the user.

This process of alternating harder questions following correct answers and easier questions following wrong answers continues until one of the stopping conditions is met. The possible stopping conditions are as follows:

  • There are no remaining easier questions to ask after a wrong answer.
  • There are no remaining harder questions to ask after a correct answer.
  • The standard error in the measure has become precise enough to stop.
  • The maximum number of questions has been exceeded.

Graph of an attempt

Test Parameters and Operation

The primary parameters for tuning the operation of the test are:

  • The starting level
  • The minimum number of questions
  • The maximum number of questions
  • The standard error to stop

Relationship between maximum number of questions and Standard Error

As discussed in Wright (1988), the formula for calculating the standard error is given by:

Standard Error (± logits) = sqrt((R+W)/(R*W))

where R is the number of right answers and W is the number of wrong answers. This value is on a logit scale, so we can apply the inverse-logit function to convert it to an percentage scale:

Standard Error (± %) = ((1 / ( 1 + e^( -1 * sqrt((R+W)/(R*W)) ) ) ) - 0.5) * 100

Looking at the Standard Error function, it is important to note that it depends only on the difference between the number of right and wrong answers and the total number of answers, not on any other features such as which answers were right and which answers were wrong. For a given number of questions asked, the Standard Error will be smallest when half the answers are right and half are wrong. From this, we can deduce the minimum standard error possible to achieve for any number of questions asked:

  • 10 questions (5 right, 5 wrong) → Minimum Standard Error = ± 15.30%
  • 20 questions (10 right, 10 wrong) → Minimum Standard Error = ± 11.00%
  • 30 questions (15 right, 15 wrong) → Minimum Standard Error = ± 9.03%
  • 40 questions (20 right, 20 wrong) → Minimum Standard Error = ± 7.84%
  • 50 questions (25 right, 25 wrong) → Minimum Standard Error = ± 7.02%
  • 60 questions (30 right, 30 wrong) → Minimum Standard Error = ± 6.42%
  • 70 questions (35 right, 35 wrong) → Minimum Standard Error = ± 5.95%
  • 80 questions (40 right, 40 wrong) → Minimum Standard Error = ± 5.57%
  • 90 questions (45 right, 45 wrong) → Minimum Standard Error = ± 5.25%
  • 100 questions (50 right, 50 wrong) → Minimum Standard Error = ± 4.98%
  • 110 questions (55 right, 55 wrong) → Minimum Standard Error = ± 4.75%
  • 120 questions (60 right, 60 wrong) → Minimum Standard Error = ± 4.55%
  • 130 questions (65 right, 65 wrong) → Minimum Standard Error = ± 4.37%
  • 140 questions (70 right, 70 wrong) → Minimum Standard Error = ± 4.22%
  • 150 questions (75 right, 75 wrong) → Minimum Standard Error = ± 4.07%
  • 160 questions (80 right, 80 wrong) → Minimum Standard Error = ± 3.94%
  • 170 questions (85 right, 85 wrong) → Minimum Standard Error = ± 3.83%
  • 180 questions (90 right, 90 wrong) → Minimum Standard Error = ± 3.72%
  • 190 questions (95 right, 95 wrong) → Minimum Standard Error = ± 3.62%
  • 200 questions (100 right, 100 wrong) → Minimum Standard Error = ± 3.53%

What this listing indicates is that for a test configured with a maximum of 50 questions and a "standard error to stop" of 7%, the maximum number of questions will always be encountered first and stop the test. Conversely, if you are looking for a standard error of 5% or better, the test must ask at least 100 questions.

Note that these are best-case scenarios for the number of questions asked. If a test-taker answers a lopsided run of questions right or wrong the test will require more questions to reach a target standard of error.

Minimum number of questions

For most purposes this value can be set to 1 since the standard of error to stop will generally set a base-line for the number of questions required. This could be configured to be greater than the minimum number of questions needed to achieve the standard of error to stop if you wish to ensure that all test-takers answer additional questions.

Starting level

As mentioned above, this usually will be set in the lower part of the difficulty range (about 1/3 of the way up from the bottom) so that most test takers will be able answer one of the first two questions correctly and get a moral boost from their correct answers. If the starting level is too high, low-ability users would be asked several questions they can't answer before the test begins asking them questions at a level they can answer.

Scoring

As discussed in Wright (1988), the formula for calculating the ability measure is given by:

Ability Measure = H/L + ln(R/W)

where H is the sum of all question difficulties answered, L is the number of questions answered, R is the number of right answers, and W is the number of wrong answers.

Note that this measure is not affected by the order of answers, just the total difficulty and number of right and wrong answers. This measure is dependent on the test algorithm presenting alternating easier/harder questions as the user answers wrong/right and may not be applicable to other algorithms. In practice, this means that the ability measure should not greatly affected by a small number of spurrious right or wrong answers.

As discussed in Linacre (2000), the ability measure of the test taker aligns with the question-difficulty at which the test-taker has a 50% probability of answering a question correctly.

For example, given a test with levels 1-10 and a test-taker that answered every question 5 and below correctly and every question 6 and up wrong, the test-taker's ability measure would fall close to 5.5.

Remember that the ability measure does have error associated with it. Be sure to take the standard error ammount into account when acting on the score.

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