Faculty of Science Introduction Knowledge Structures Parameter Estimation Implementation in R Concluding Remarks Department of Psychology Outline Introduction Knowledge Structures Parameter Estimation Maximum Likelihood Estimation Minimum Discrepancy Method Minimum Discrepancy ML Estimation Parameter estimation in probabilistic knowledge structures Implementation in R Concluding Remarks J¨ urgen Heller & Florian Wickelmaier Psychoco 2011 1 | J¨ urgen Heller & Florian Wickelmaier Introduction Knowledge Structures Parameter Estimation Implementation in R Concluding Remarks Introduction Knowledge Structures Parameter Estimation Implementation in R Concluding Remarks . . . Numbers in Science . . . . . . Numbers in Psychology . . . “Anthropometry, or the art of measuring the physical and mental faculties of human beings, enables a shorthand description of any individual by measuring a small sample “When you can measure what you are of his dimensions and qualities. This will speaking about, and express it in numbers, sufficiently define his bodily proportions, you know something about it; but when his massiveness, strength, agility, keenness you cannot measure it, when you cannot of senses, energy, health, intellectual ca- express it in numbers, your knowledge is of pacity and mental character, and will con- a meager and unsatisfactory kind: it may stitute concise and exact numerical val- be the beginning of knowledge, but you are ues for verbose and disputable estimates.” scarcely, in your thoughts, advanced to the (Francis Galton, 1905) stage of science, whatever the matter may be.” (William Thomson Kelvin, 1889) 2 | J¨ urgen Heller & Florian Wickelmaier 3 | J¨ urgen Heller & Florian Wickelmaier
Introduction Knowledge Structures Parameter Estimation Implementation in R Concluding Remarks Introduction Knowledge Structures Parameter Estimation Implementation in R Concluding Remarks . . . Numbers in Psychology . . . Knowledge Structures (Doignon & Falmagne, 1985, 1999) Goals So, imagine that some committee of experts has carefully designed ◮ Characterizing the strengths and weaknesses in all parts of a an ‘Athletic Quotient’ or ‘A.Q.’ test, intended to measure athletic knowledge domain prowess. Suppose that three exceptional athletes have taken the ◮ Precise, non-numerical characterization of the state of test, say Michael Jordan, Tiger Woods and Pete Sampras. knowledge that is computationally tractable Conceivably, all three of them would get outstanding A.Q.’s. But ◮ Building upon results from discrete mathematics and exploiting these high scores equating them would completely misrepresent how the power of current computers essentially different from each other they are. One may be tempted ◮ Adaptive knowledge assessment to salvage the numerical representation and argue that the ◮ Efficiently identifying the current state of knowledge based on assessment, in this case, should be multidimensional. However, asking a minimal number of questions adding a few numerical dimensions capable of differentiating ◮ Adapting to the already given responses as experienced Jordan, Woods and Sampras would only be the first step in a teachers do in an oral examination sequence. Including Greg Louganis or Pele to the evaluated lot ◮ Personalization in technology-enhanced learning would require more dimensions, and there is no satisfactory end in ◮ Automatically select content that a person is ready to learn sight. (Falmagne et al., 2006, p. 63) 4 | J¨ urgen Heller & Florian Wickelmaier 5 | J¨ urgen Heller & Florian Wickelmaier Introduction Knowledge Structures Parameter Estimation Implementation in R Concluding Remarks Introduction Knowledge Structures Parameter Estimation Implementation in R Concluding Remarks Deterministic Theory Example Definitions Study on Fear Symptoms (Stouffer et al., 1950) ◮ A knowledge domain is identified with a set Q of ◮ U.S soldiers who have been under fire report different physical (dichotomous) items reactions to the dangers of battle ( N = 93) ◮ The knowledge state of a person is identified with the subset ◮ Knowledge domain Q = { a , b , c , d } (item “solved” when K ⊆ Q of problems in the domain Q the person is capable of options in parenthesis are chosen) solving a Violent pounding of the heart (sometimes, or often) ◮ A knowledge structure on the domain Q is a collection K of Feeling of weakness, or feeling faint (sometimes, or often) b subsets of Q that contains at least the empty set ∅ and the c Urinating in pants (sometimes, or often) set Q d Losing control of the bowels (once, sometimes, or often) ◮ The subsets K ∈ K are the knowledge states 6 | J¨ urgen Heller & Florian Wickelmaier 7 | J¨ urgen Heller & Florian Wickelmaier
Introduction Knowledge Structures Parameter Estimation Implementation in R Concluding Remarks Introduction Knowledge Structures Parameter Estimation Implementation in R Concluding Remarks Example Example Study on Fear Symptoms (Stouffer et al., 1950) Study on Fear Symptoms (Stouffer et al., 1950) ◮ Absolute frequencies N R of response patterns ◮ Hasse-Diagram of response patterns { a , b , c , d } item a b c d N R 1 0 0 0 40 { a , b , c } { a , b , d } { a , c , d } 0 0 0 0 7 0 1 0 0 2 { a , b } { a , d } 1 0 0 1 3 1 1 0 0 23 { b } { a } 1 0 1 1 1 1 1 0 1 9 1 1 1 0 1 ∅ 1 1 1 1 7 84 42 9 20 8 | J¨ urgen Heller & Florian Wickelmaier 9 | J¨ urgen Heller & Florian Wickelmaier Introduction Knowledge Structures Parameter Estimation Implementation in R Concluding Remarks Introduction Knowledge Structures Parameter Estimation Implementation in R Concluding Remarks Example Probabilistic Knowledge Structures Study on Fear Symptoms (Stouffer et al., 1950) Rationale ◮ If there are response errors then knowledge states K ⊆ Q and ◮ Hasse-Diagram of response patterns (excluding { a , b , c } ) response patterns R ⊆ Q have to be dissociated { a , b , c , d } Definition (Falmagne & Doignon, 1988a, 1988b) ◮ A probabilistic knowledge structure is defined by specifying { a , b , d } { a , c , d } a ◮ a knowledge structure K on a knowledge domain Q (i.e. a collection K ⊆ 2 Q with ∅ , Q ∈ K ) { a , b } { a , d } b d ◮ a marginal distribution P K ( K ) on the knowledge states K ∈ K ◮ the conditional probabilities P ( R | K ) to observe response { b } { a } pattern R given knowledge state K c The probability of the response pattern R ∈ R = 2 Q is predicted by � ∅ P R ( R ) = P ( R | K ) · P K ( K ) K ∈K 10 | J¨ urgen Heller & Florian Wickelmaier 11 | J¨ urgen Heller & Florian Wickelmaier
Introduction Knowledge Structures Parameter Estimation Implementation in R Concluding Remarks Introduction Knowledge Structures Parameter Estimation Implementation in R Concluding Remarks Local stochastic independence Theory Assumptions Probabilistic Knowledge Structure on Q = { a , b , c , d } ◮ Given the knowledge state K of a person . 2 ◮ the responses are stochastically independent over problems 0 . 3 0 β a ◮ the response to each problem q only depends on the abcd β b . 2 probabilities β c . 2 β d of a careless error β q 0 . 2 of a lucky guess η q 0 abc 0 ◮ The probability of the response pattern R given the knowledge acd 0 . 3 . 2 abd η a state K reads η b 0 . 2 η c bc η d � � � � 0 P ( R | K ) = β q · (1 − β q ) · η q · (1 − η q ) bd q ∈ K \ R q ∈ K ∩ R q ∈ R \ K q ∈ Q \ ( R ∪ K ) . 2 0 ∅ 12 | J¨ urgen Heller & Florian Wickelmaier 13 | J¨ urgen Heller & Florian Wickelmaier Introduction Knowledge Structures Parameter Estimation Implementation in R Concluding Remarks Introduction Knowledge Structures Parameter Estimation Implementation in R Concluding Remarks Data Maximum Likelihood Estimation Observed frequencies N R of response patterns R ⊆ Q = { a , b , c , d } EM Algorithm ◮ Formulate the likelihood as if we have available the absolute frequencies M RK of subjects who are in state K and produce pattern R (complete data) instead of the absolute frequencies abcd bcd N R of the response patterns R ∈ R (incomplete data) acd abd abc cd bd bc ad ac “Expectation” “Maximization” ab d ( t +1) , ˆ Estimate ˆ η ( t +1) , ˆ π ( t +1) Compute β c ( t ) , ˆ b E ( M RK ) = N R · P ( K | R , ˆ η ( t ) , ˆ π ( t ) ) based on m RK = E ( M RK ) β a {} 0 10 20 30 14 | J¨ urgen Heller & Florian Wickelmaier 15 | J¨ urgen Heller & Florian Wickelmaier
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