Geodesic
Theoretical basis of the iterative process of the joint assessment of difficulties in tasks and levels of training students
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 19 (2016) no. 1, pp. 107-123 
I. S. Shreifel; I. N. Eliseev. Theoretical basis of the iterative process of the joint assessment of difficulties in tasks and levels of training students. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 19 (2016) no. 1, pp. 107-123. http://geodesic.mathdoc.fr/item/SJVM_2016_19_1_a10/
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we study the iterative process of the joint numerical assessment of levels of training students and difficulties in tasks of diagnostic tools using the dichotomous response matrix $A$ of $N\times M$ size, with allowance for the contribution of tasks of different difficulty to the assessments obtained. It is shown that not for any matrix $A$ there exist infinite iterative sequences, and in the case of their existence, they do not always converge. A wide range of sufficient conditions for their convergence have been obtained, which are based on the following: 1) matrix $A$ contains at least three different columns; 2) if one places the columns of $A$ in non-decreasing order of column sums, then for any position of the vertical dividing line between the columns there exists a row, which has, at least, one unity to the left of the line, and at least one zero to the right of the line. It is established that the response matrix $A$ obtained as a result of testing reliability satisfies these two conditions. The properties of such matrices have been studied. In particular, the equivalence of the above conditions of primitiveness of the square matrix $B$ of order $M$ with the entries $b_{ij}=\sum^N_{\ell=1}(1-a_{\ell i})a_{\ell j}$ has been proved. Using the matrix analysis we have proved that the primitiveness of the matrix $B$ ensures the convergence of iterative sequences, as well as independence of their limits of the choice of the initial approximation. We have estimated the rate of convergence of these sequences and found their limits.

[1] Rasch G., Probabilistic Models for Some Intelligence and Attainment Tests, With a Foreword and Afterword by B. D. Wright, The Univ. of Chicago Press., Chicago–London, 1980

[2] Wright B. S., Masters G. N., Rating Scale Analysis: Rasch Measurement, MESA Press, Chicago, 1982

[3] Eliseev I. N., “Algoritmy iteratsionnykh protsedur vychisleniya otsenok urovnya podgotovki studentov”, Izvestiya vuzov. Elektromekhanika, 2013, no. 3, 83–90

[4] Eliseev I. N., Shrayfel' I. S., “Issledovanie sushchestvovaniya i edinstvennosti otsenok maksimal'nogo pravdopodobiya parametrov latentnykh peremennykh odnoparametricheskoy dikhotomicheskoy modeli Rasha”, Informatizatsiya obrazovaniya i nauki, 2011, no. 3(11), 117–129

[5] Eliseev I. N., Metody, algoritmy i programmnye kompleksy dlya rascheta kharakteristik diagnosticheskikh sredstv nezavisimoy otsenki kachestva obrazovaniya, Monografiya, 2 izd., pererab. i dop., Lik, Novocherkassk, 2013

[6] Eliseev I. N., Sharayfel' I. S., “Model' otsenivaniya latentnykh parametrov dikhotomicheskoy modeli Rasha”, Izvestiya vuzov. Severo-Kavkazskiy region. Tekhnicheskie nauki, 2011, no. 6, 37–46

[7] Eliseev I. N., Sharayfel' I. S., “O sostoyatel'nosti dikhotomicheskoy modeli Rasha”, Izvestiya vuzov. Severo-Kavkazskiy region. Tekhnicheskie nauki, 2012, no. 5, 127–136

[8] Eliseev I. N., “Teoreticheskie osnovy algoritma rascheta latentnykh peremennykh programmnym kompleksom RILP-1M”, Programmnye produkty i sistemy, 2011, no. 2, 67–71

[9] Eliseev I. N., “Teoreticheskie osnovy algoritma rascheta latentnykh peremennykh programmnym kompleksom RILP-2”, Izvestiya vuzov. Severo-Kavkazskiy region. Tekhnicheskie nauki, 2011, no. 3, 3–8

[10] Semov A. M., Semova M. A., Khlebnikov V. A., “Edinyy iteratsionnyy protsess sovmestnoy kolichestvennoy otsenki trudnostey zadaniy i urovney podgotovlennosti uchastnikov testirovaniya”, Tr. Tsentra testirovaniya, 2, Prometey, M., 1999, 54–60

[11] Khorn R., Dzhonson Ch., Matrichnyy analiz, Per. s angl., Mir, M., 1989 | MR

[12] Il'in V. A., Poznyak E. G., Osnovy matematicheskogo analiza, Chast' I, Uchebnik dlya vuzov, 7-e izd., FIZMATLIT, M., 2005