Applications of Stochastic Models on Test mis-scaling in Educational and Psychological Measurement

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1 Alcatos of Stochastc Models o Test ms-scalg Educatoal ad Psychologcal Measuremet Dr. R.Arumugam, M. Raja Assstat Professor,Deartmet of Maematcs, Peryar Maamma Isttute of Scece ad Techology, Thajavur Assstat Professor,Deartmet of Educato, Peryar Maamma Isttute of Scece ad Techology, Thajavur Abstract : I s aer stochastc models are develoed to study e rate of test ms-scalg educatoal ad sychologcal measuremet. The mea ad varace of geometrc ad egatve bomal models are gve to esure e umber of ms-scalg below a reshold. Numercal llustrato are also rovded. Keywords: Stochastc model, Ms-scalg ad Threshold. I. INTRODUCTION I e last decade, essay tems have bee cluded major educatoal assessmets, such as e Natoal Assessmet of Educatoal Progress (NAEP) ad e Thrd Iteratoal Maematcs ad Scece Study (TIMSS) (Alle, Carlso, & Zelea, 999; Mart & Kelly, 996). Meawhle, classroom teachers are advsed to use essay questos to comlemet multle-choce tems. Varous resoses geerated from essay tems demad a large amout of maower test gradg. Whle o graders la to mae mstaes, accdetal errors are lely to occur durg e huma oeratos (Wag, 993). The urose of s study s to exame e chace of test ms-scalg usg arorate models statstcs. The estmato of advertet gradg errors ca serve as a bass for qualty cotrol educatoal ad sychologcal measuremets. II. LITERATURE REVIEW Statstcal models have bee sought to ehace qualty cotrol varous rojects. I a dustral statstcs, qualty cotrol measures are adoted maly to esure e total umber of feror cdets below a reshold. Bssssell (970) revewed; It s ofte assumed at such evets follow e Posso Law. The assumtos of costat mea level ad deedece are ofte volated ractce (. 5). I educatoal ad sychologcal measuremets, test ms-scalg ca be treated as a secfc d of cdets. I a classroom settg, Lyma (998) oted at "Every teacher recogzes at grades are raer arbtrary ad subjectve" (. 07). I a large-scale assessmet, t s eve more dffcult to assume e same level of average erformace amog varous graders. Hece, e assumto of a costat mea erformace level s ofte volated large ad small-scale assessmets, whch maes e Posso model arorate for most real-lfe alcatos (Rasch, 980; Wag, 993). III. STOCHASTIC MODELS I a test scorg rocess, a cotrast ca be set to dfferetate outcomes of correct gradg ad ms-scalg. A evet w dchotomous outcomes s tycally modeled by a Beroull tral. For a welldesged test, e ossblty of ms-scalg () s ot hgh. Qualty cotrol measures, such as arragemet of schedules for short breag, ca be troduced e gradg rocess to esure at e DOI:0.3883/IJRTER DDO 33

2 Iteratoal Joural of Recet Treds Egeerg & Research (IJRTER) Volume 04, Issue 06; Jue - 08 [ISSN: ] umber of ms-scalg cases s o larger a a secfc level. I ractce, e gradg rocess may ee o utl occurrece of e ms-scalg. The, a brea sesso ca be scheduled to refresh e graders, ad us, hel cotrol e umber of ms-scalg below level. To facltate descrto of e stochastc model, oe may defe X to be e total umber of successes before e ms-scalg, ad to be e robablty of obtag exactly X successes. Accordgly, e total umber of trals deeds o e reshold level ad e umber of correctly-graded cases (X) before reachg e reshold. Sce e evet of ms-scalg haes by accdet, e umber of correctly-graded cases (X) may vary amog e graders. Gve a level of e reshold, e exected value of X ca be emloyed to schedule brea sessos before reachg a ms-scalg cdet o e tral. f (x) ( X ) ( X ) IV. GEOMETRIC STOCHASTIC PROCESS Uder a codto of zero tolerace, oe may wsh to schedule a brea erod for test graders before e frst occurrece of ms-scalg. Usg symbol s to rereset successful test gradg ad m to rereset e frst ms-scalg, oe may dect e stochastc rocess e followg cha of evets: sssm X tm e s Ths stochastc cha ca arse oly oe way,.e., e grader has successfully graded test questos o e frst X trals, ad eds u w e frst ms-scalg case o e ( X ) tral. Thus, e chaces of obtag exactly X successes ror to e frst falure s: ( ) ( )...( ) Where, s e robablty of ms-scalg each tral. Ths stochastc rocess ca be descrbed by a robablty fucto for dfferet values of X: x ( ) ; x 0,,,,3,......( ) Equato () defes a geometrc rocess "because e robabltes form a geometrc seres w a commo rato ". Sce e ossblty of ms-scalg each tral () s less a, e total robablty follows: ( ) ( )... ( ) V. MAIN RESULTS The exected umber of correctly graded s, X) () Sce a brea sesso s arraged before evet of e frst ms-scalg (.e., =), e watg tme for e frst ms-scalg s: X ) X ) X) (3) Sce, X ), X ) V( X All Rghts Reserved 34

3 Iteratoal Joural of Recet Treds Egeerg & Research (IJRTER) Volume 04, Issue 06; Jue - 08 [ISSN: ] Table : The relatosh betwee P ad x) x) Table : The relatosh betwee P ad V(x) V(x) For a gve test, f e chace of ms-scalg () s small, e e watg tme for a brea sesso ca be loger. I a extreme case, multle-choce (MCQ) tests are graded by a mache whch has equal to zero each tral ad e watg tme ca be fte. Therefore, ere s o eed for a brea uless e gradg mache s broe dow. VI. A NEGATIVE BINOMIAL MODEL I a more comlcated codto, a test s graded by a total umber of graders. For ay grader, w reshold deed o e qualty cotrol requremet, ad may tae a value larger a oe. To mae sure e ms-scalg evet occurrg o e ( X ) tral, e revous ( ) ms-scalg evets must already hae e recedg ( X ) trals. The robablty of ( ) ms-scalg cases o e frst X ) trals s gve by ( X X ( ) The robablty of ms-scalg o tral X ) s. ( X 0,,, All Rghts Reserved 35

4 Iteratoal Joural of Recet Treds Egeerg & Research (IJRTER) Volume 04, Issue 06; Jue - 08 [ISSN: ] Equato (4) follows a egatve bomal dstrbuto (Feller, 957). The robablty geerato fucto for e egatve bomal dstrbuto s ( ) t. Because e overall qualty cotrol s based o cumulatve erformace of e graders, e stochastc rocess volves deedet radom varables,, X X. Thus, e qualty cotrol reshold hges o e dstrbuto of X X,... Lucly, e robablty geerato fucto P X ( t) X follows ( ) t ( ) t (5) Where,.... Based o uqueess of e robablty geeratg fucto (Port, 994), must have a egatve bomal dstrbuto w arameters ad. X) The exected value for ( ) ad V( X ca be resultg from e egatve bomal dstrbuto, ad e result s (see Casella & Berger, 990). So, e watg tme before occurrece of e ( ) X ) q X ) (6) (7) Table 3 : The relatosh amod, ad x) x) ms-scalg s:. Table 4 : The relatosh amod,, q ad V(x) q V(x) All Rghts Reserved 36

5 Iteratoal Joural of Recet Treds Egeerg & Research (IJRTER) Volume 04, Issue 06; Jue - 08 [ISSN: ] VII. DISCUSSION I a comarso betwee (3) ad (6), oe may ote at e geometrc rocess ca be treated as a secal case ( = ) of e egatve bomal dstrbuto. Based o e results (6), e watg tme for a brea erod ca be loger f e overall tolerace level s hgher ad e chace of ms-scalg () s small. I summary, artly due to dffereces e otato choce, e well-establshed geometrc ad egatve bomal dstrbutos have yet to be used models of test ms-scalg. I oer felds, Johso ad Kotz (969) have oted at "e egatve bomal dstrbuto s frequetly used as a substtute for e Posso dstrbuto whe t s doubtful wheer e strct requremets, artcularly deedece, for a Posso dstrbuto wll be satsfed" (. 35). Thus, e geometrc ad egatve bomal models rovde alteratve choces at are more flexble a e Posso model educatoal ad sychologcal measuremets. VIII. CONCLUSION The geometrc rocess s develoed from a sgle-grader scearo uder a olcy of zero tolerace for test ms-scalg. Hece, e result equato (3) may be more alcable a local settg whch a teacher has bee assged to grade tests for a etre class. The egatve bomal rocess, o e oer had, seems arorate for state or atoal assessmet at volves more a oe test grader. I bo cases, e watg tme for test ms-scalg has bee derved from e corresodg stochastc rocesses. The results ca be emloyed to schedule brea erods to esure e error of msscalg below a reshold. REFERENCES I. Alle, N. L., Carlso, J. E., & Zelea, C. A. (999). The NAEP 996 techcal reort.washgto, DC: Natoal Ceter for Educato Statstcs. II. Bssell, A. F. (970). Aalyss of data based o cdet couts. The Statstca, 9 (3), III. Casella, G ad Berger, R. L. (990). Statstcal ferece. Pacfc Grove, CA: Broos. Draer, N. R., & Lawrece, IV. W. E. (970). Probablty: A troductory course. Chcago, IL: Marham. Feller, W. (957). A troducto to robablty eory ad ts alcatos (d ed.). New Yor, NY: Joh Wley & Sos. V. Lyma, H. B. (998). Test scores ad what ey mea. Bosto, MA: Ally & Baco. VI. Rasch, G. (980). Probablstc models for some tellgece ad attamet tests.chcago, IL: Uversty of Chcago Press. VII. Wag, J. (993). Smle ad herarchcal model for test msgradg. Educatoal ad Psychologcal Measuremet, 53, All Rghts Reserved 37