a useful study tool - but why in exams? it reflects the proper meaning of knowledge

1. Confidence-based assessment in the 1st year medical end-of-year exam Tony Gardner-Medwin Physiology, UCL a useful study tool - but why in exams? it reflects the proper meaning of knowledge conventional marking disadvantages able students how did the students do in the exam? conf-asst was a more reliable measure of student ability it saves on the number of questions required • a useful study tool - but why in exams? • it reflects the proper meaning of knowledge • conventional marking disadvantages able students • how did the students do in the exam? • conf-asst was a more reliable measure of student ability • it saves on the number of questions required

2. increasing nescience What is Knowledge? Knowledge depends on degree of belief, or confidence: knowledge uncertainty ignorance misconception delusion Knowledge depends on degree of belief, or confidence: • knowledge • uncertainty • ignorance • misconception • delusion =0 -log2(confidence*) for truth of a =1 true proposition >>1 Measurement of knowledge requires the eliciting of confidence (or *subjective probability) for the truth of correct statements. This requires a proper scheme of incentives

3. 100% 80% C=3 60% Subjective 40% Expectation of Score C=2 20% C=1 0% 0.5 0.75 1 Subjective Probability LAPT confidence-based scoring scheme Confidence Level 1 2 3 Score if Correct 1 2 3 Score if incorrect 0 -2 -6 P(correct) < 67% >67% >80% Odds < 2:1 >2:1 >4:1

4. conventional marking disadvantages able students Suppose 4 students go for the same answer options in an exam: 75 , 25  Ai is confident of all his answers Bo is very hesitant about all her answers Cy is realistic (expects 75%), but can’t distinguish reliable & uncertain answers Di is confident of 50 answers (90% ) and uncertain of the others (60% ) Clearly:Di > Cy > Bo, Ai Di has extra insight - about her knowledge, or maybe about subtleties in questions How can she use this insight? Conventional scoring: Her only option is to omit uncertain answers: % correct:Ai = Bo = Cy = 75%, Di = 45% negative marking score (±1):Ai = Bo = Cy = 50%, Di = 40% Confidence-based scoring: She can moderate her confidence: Ai enters all at C=3, Bo at C=1: Ai = Bo = 25% Cy enters all at C=2: Cy = 33% Di splits answers C=3, C=1: Di = 48% [adjust] 33%  

5. NB over-confidence can actually get you places !! • Summary aims • reward the ability to distinguish reliable and uncertain answers (whatever the reason for uncertainty) • penalise confident errors more than errors from uncertainty • What people sometimes think is the aim! • to penalise a general over-confidence or under-confidence • - probably helped by practice & feedback, but not an exam issue

6. 100% 5%, 95% 90% percentiles 80% % correct 70% 60% 50% i-c exF exM i-c exF exM i-c exF exM @ C=1 @ C=2 @ C=3 How well did students discriminate? exam: 500 T/F Qs, in 2 sessions, each 2hrs 331 students: 190 F, 141 M

7. y = x1.67 equality (only expected for a pure mix of certain knowledge and total guesses) scores if uncertainty is homogeneous and correctly reported theoretical scores for homogeneous uncertainty, based on an information theoretic measure 100% A. a 80% b 60% c confidence-based score 40% d 20% 0% 0% 20% 40% 60% 80% 100% (50% correct) conventional scaled score (“simple” score)

8. Breakdown of credit and variance due to uncertainty Simple scores (scaled conventional scores) - 65% of the variance came from answers at C=1, but only 18% of the credit. Confidence-based scores: these give less weight to uncertain answers; uncertainty variance is then more in proportion to credit, and was reduced by 46% (relative to the variation of student marks)

9. Exam marks are determined by: 1. the student’s knowledge and skills in the subject area 2. the level of difficulty of the questions 3. chance factors in the way questions relate to details of the student’s knowledge 4. chance factors in the way uncertainties are resolved (luck) (1) = “signal” (its measurement is the object of the exam) (3,4) = “noise” (random factors obscuring the “signal”) Confidence-based marks improve the “signal-to-noise ratio” The most convincing test of this is to compare marks on one set of questions with marks for the same student on a different set . A good correlation means we are measuring something about the student, not just “noise”

10. No. Confidence scores are better than simple scores at predicting even the conventional scores on a different set of questions. This can only be because they are a statistically more efficient measure of knowledge. The correlation, across students, between scores on one set of questions and another is higher for confidence than for simple scores. But perhaps they are just measuring ability to handle confidence ?

11. How should one handle students with poor calibration? Significantly overconfident:2 students (1%) e.g. 50% correct @C=1, 59%@C=2, 73%@C=3 Significantly underconfident:41 students (14%) e.g. 83% correct @C=1, 89%@C=2, 99%@C=3 Maybe one shouldn’t penalise such students Adjusted confidence-based score: Mark the set of answers at each C level as if they were entered at the C level that gives the highest score. mean benefit = 1.5% ± 2.1% (median 0.6%)

12. y = x1.67 a equality (only expected for a pure mix of certain knowledge and total guesses) b c scores if uncertainty is homogeneous and correctly reported d theoretical scores for homogeneous uncertainty, based on an information theoretic measure 100% A. 80% 60% confidence-based score 40% 20% 0% 0% 20% 40% 60% 80% 100% simple scaled score (50% correct) (100% correct)

13. simpleconfconf (adj) Signal / noise variance ratio: 2.8 5.3 4.3 Savings in no. of Qs required: - 48% 35%

14. SUMMARY CONCLUSIONS • Adjusted confidence scores seem the best scores to use (they don’t discriminate on the basis of the calibration of a person’s confidence judgements, and are also the best predictors of performance on a separate set of questions). • Reliable discrimination of student knowledge can be achieved with one third fewer questions, compared with conventional scoring. • Confidence scoring is not only fundamentally more fair (rewarding students who can correctly identify which answers are uncertain) but it is more efficient at measuring performance. • www.ucl.ac.uk/~cusplap

16. Principles that students seem readily to understand :- • both under- and over- confidence are impediments to learning • confident errors are far worse than acknowledged ignorance and are a wake-up call (-6!) to pay attention to explanations • expressing uncertainty when you are uncertain is a good thing • thinking about the basis and reliability of answers can help tie bits of knowledge together (to form “understanding”) • checking an answer and rereading the question are worthwhile • sound confidence judgement is a valued intellectual skill in every context, and one they can improve