An Empirically . . . What We Do First Justification for . . . What If Use min for . . . How Success in a Task Towards a Second . . . Depends on the Skills Level: How to Describe Not- . . . Resulting Equation Two Uncertainty-Based Solving the Resulting . . . Conclusion Justifications of a Home Page Semi-Heuristic Rasch Model Title Page ◭◭ ◮◮ Joe Lorkowski 1 , Olga Kosheleva 2 , and Vladik Kreinovich 1 ◭ ◮ Departments of 1 Computer Science and 2 Teacher Education Page 1 of 13 University of Texas at El Paso 500 W. University Go Back El Paso, Texas 79968, USA lorkowski@computer.org, olgak@utep.edu, vladik@utep.edu Full Screen Close Quit
An Empirically . . . What We Do 1. An Empirically Successful Rasch Model First Justification for . . . • For each level of student skills, the student is usually: What If Use min for . . . Towards a Second . . . – very successful in solving simple problems, How to Describe Not- . . . – not yet successful in solving problems which are – Resulting Equation to this student – too complex, and Solving the Resulting . . . – reasonably successful in solving problems which are Conclusion of the right complexity. Home Page • To design adequate tests, it is desirable to understand Title Page how a success s in a task depends: ◭◭ ◮◮ – on the student’s skill level ℓ and ◭ ◮ – on the problem’s complexity c . Page 2 of 13 1 • Empirical Rasch model predicts s = 1 + exp( c − ℓ ) . Go Back • Practitioners, however, are somewhat reluctant to use Full Screen this formula, since it lacks a deeper justification. Close Quit
An Empirically . . . What We Do 2. What We Do First Justification for . . . • In this talk, we provide two possible justifications for What If Use min for . . . the Rasch model. Towards a Second . . . How to Describe Not- . . . • The first is a simple fuzzy-based justification which Resulting Equation provides a good intuitive explanation for this model. Solving the Resulting . . . • This will hopefully enhance its use in teaching practice. Conclusion Home Page • The second is a somewhat more sophisticated explana- tion which is: Title Page – less intuitive but ◭◭ ◮◮ – provides a quantitative justification. ◭ ◮ Page 3 of 13 Go Back Full Screen Close Quit
An Empirically . . . What We Do 3. First Justification for the Rasch Model First Justification for . . . • Let us fix c and consider the dependence s = g ( ℓ ). What If Use min for . . . Towards a Second . . . • When we change ℓ slightly, to ℓ + ∆ ℓ , the success also How to Describe Not- . . . changes slightly: g ( ℓ + ∆ ℓ ) ≈ g ( ℓ ). Resulting Equation • Thus, once we know g ( ℓ ), it is convenient to store not Solving the Resulting . . . g ( ℓ +∆ ℓ ), but the difference g ( ℓ +∆ ℓ ) − g ( ℓ ) ≈ dg dℓ · ∆ ℓ. Conclusion Home Page • Here, dg dℓ depends on s = g ( ℓ ): dg dℓ = f ( s ) = f ( g ( ℓ )). Title Page ◭◭ ◮◮ • In the absence of skills, when ℓ ≈ −∞ and s ≈ 0, adding a little skills does not help much, so f ( s ) ≈ 0. ◭ ◮ • For almost perfect skills ℓ ≈ + ∞ and s ≈ 1, similarly Page 4 of 13 f ( s ) ≈ 0. Go Back • So, f ( s ) is big when s is big ( s ≫ 0) but not too big Full Screen (1 − s ≫ 0). Close Quit
An Empirically . . . What We Do 4. First Justification for the Rasch Model (cont-d) First Justification for . . . • Rule: f ( s ) is big when: What If Use min for . . . Towards a Second . . . • s is big ( s ≫ 0) but How to Describe Not- . . . • not too big (1 − s ≫ 0). Resulting Equation • Here, “but” means “and”, the simplest “and” is the Solving the Resulting . . . product. Conclusion Home Page • The simplest membership function for “big” is µ big ( s ) = s . Title Page ◭◭ ◮◮ • Thus, the degree to which f ( s ) is big is equal to ◭ ◮ s · (1 − s ) : f ( s ) = s · (1 − s ) . Page 5 of 13 • The equation dg dℓ = g · (1 − g ) leads exactly to Rasch’s Go Back 1 Full Screen model g ( ℓ ) = 1 + exp( c − ℓ ) for some c . Close Quit
An Empirically . . . What We Do 5. What If Use min for “and”? First Justification for . . . • What if we use a different “and”-operation, for exam- What If Use min for . . . ple, min( a, b )? Towards a Second . . . How to Describe Not- . . . • Let us show that in this case, we also get a meaningful Resulting Equation model. Solving the Resulting . . . • Indeed, in this case, the corresponding equation takes Conclusion the form dg dℓ = min( g, 1 − g ) . Home Page • Its solution is: Title Page ◭◭ ◮◮ • g ( ℓ ) = C − · exp( ℓ ) when s = g ( ℓ ) ≤ 0 . 5, and • g ( ℓ ) = 1 − C + · exp( − ℓ ) when s = g ( ℓ ) ≥ 0 . 5. ◭ ◮ • In particular, for C − = 0 . 5, we get a cdf of the Laplace Page 6 of 13 distribution ρ ( x ) = 1 2 · exp( −| x | ) . Go Back Full Screen • This distribution is used in many applications – e.g., to modify the data in large databases to promote privacy. Close Quit
An Empirically . . . What We Do 6. Towards a Second Justification First Justification for . . . • The success s depends on how much the skills level ℓ What If Use min for . . . exceeds the complexity c of the task: s = h ( ℓ − c ). Towards a Second . . . How to Describe Not- . . . • For each c , we can use the value h ( ℓ − c ) to gauge the Resulting Equation students’ skills. Solving the Resulting . . . • For different c , we get different scales for measuring Conclusion skills. Home Page • This is similar to having different scales in physics: Title Page – a change in a measuring unit leads to x ′ = a · x ; ◭◭ ◮◮ e.g., 2 m = 100 · 2 cm; ◭ ◮ – a change in a starting point leads to x ′ = x + b ; Page 7 of 13 e.g., 20 ◦ C = (20 + 273) ◦ K. Go Back • In physics, re-scaling is usually linear, but here, 0 → 0, Full Screen 1 → 1, so we need a non-linear re-scaling. Close Quit
An Empirically . . . What We Do 7. How to Describe Not-Necessarily-Linear Re- First Justification for . . . Scalings What If Use min for . . . • If we first apply one reasonable re-scaling, and after Towards a Second . . . that another one, we still get a reasonable re-scaling. How to Describe Not- . . . Resulting Equation • For example, we can first change meters to centimeters, Solving the Resulting . . . and then replace centimeters with inches. Conclusion • Then, the resulting re-scaling from meters to inches is Home Page still a linear transformation. Title Page • In mathematical terms, this means that the class of ◭◭ ◮◮ reasonable e-scalings is closed under composition. ◭ ◮ • Also, if we have a re-scaling, e.g., from C to F, then Page 8 of 13 the “inverse” re-scaling from F to C is also reasonable. Go Back • In precise terms, this means that the class of all reason- able re-scalings is invariant under taking the inversion. Full Screen Close Quit
An Empirically . . . What We Do 8. How to Describe Re-Scalings (cont-d) First Justification for . . . • Thus, we can say that reasonable re-scalings form a What If Use min for . . . transformation group. Towards a Second . . . How to Describe Not- . . . • Our goal is computations. Resulting Equation • In a computer, we can only store finitely many param- Solving the Resulting . . . eters. Conclusion Home Page • Thus, each re-scaling must be determined by finitely many parameters. Title Page • Such groups are called finite-dimensional . ◭◭ ◮◮ • So, we need to describe all finite-dimensional transfor- ◭ ◮ mation groups that contain all linear transformations. Page 9 of 13 • It is known that all functions from these groups are Go Back fractionally-linear f ( s ) = a · s + b c · s + d. Full Screen Close Quit
An Empirically . . . What We Do 9. Resulting Equation First Justification for . . . • We consider a transformation s ′ = f ( s ) between What If Use min for . . . s = h ( ℓ − c ) and s ′ = h ( ℓ − c ′ ) . Towards a Second . . . How to Describe Not- . . . • We showed that this transformation is fractionally- Resulting Equation linear f ( s ) = a · s + b c · s + d. Solving the Resulting . . . Conclusion • When s = 0, we should have s ′ = 0, hence b = 0. Home Page • We can now divide both numerator and denominator Title Page A · s by d , then f ( s ) = C · s + 1 . ◭◭ ◮◮ • When s = 1, we should have s ′ = 1, so A = C + 1, and ◭ ◮ f ( s ) = (1 + C ) · s C · s + 1 . Page 10 of 13 • For c ′ = 0, we thus get Go Back h ( ℓ − c ) = (1 + C ( c )) · h ( ℓ ) Full Screen C ( c ) · h ( ℓ ) + 1 . Close Quit
An Empirically . . . What We Do 10. Solving the Resulting Equation Explains the First Justification for . . . Rasch Model What If Use min for . . . • We know that Towards a Second . . . h ( ℓ − c ) = (1 + C ( c )) · h ( ℓ ) How to Describe Not- . . . C ( c ) · h ( ℓ ) + 1 . Resulting Equation Solving the Resulting . . . • Differentiating both sides w.r.t. c and taking c = 0, we Conclusion get a differential equation whose general solution is Home Page 1 Title Page h ( ℓ ) = 1 + exp( k · ( c − ℓ )) . ◭◭ ◮◮ • By changing measuring units for ℓ and c to k times ◭ ◮ smaller ones, we get the Rasch model Page 11 of 13 1 h ( ℓ ) = 1 + exp( c − ℓ ) . Go Back Full Screen Close Quit
Recommend
More recommend
