TRI para escalas politômicas. Dr. Ricardo Primi Programa de Mestrado e Doutorado em Avaliação Psicológica Universidade São Francisco

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1 TRI para escalas politômicas Dr. Ricardo Primi Programa de Mestrado e Doutorado em Avaliação Psicológica Universidade São Francisco

2 Modelos Modelo Rasch-Andrich Rating Scale Model (respostas graduais) Rasch-Masters Partial Credit Model (créditos parciais) Características (Parâmetros) Generalização do modelo de Rasch para escalas likert Estima um índice de dificuldade por item e k-1 limiares (tresholds) gerais para as categorias (k = número de pontos na escala) Mais geral e flexível. Generalização do modelo de Rasch para itens politômicos Estima k-1 limires por item (k = número de pontos na escala) Samejima s Graded Response Model Generalização do modelo de 2 parâmetros para itens politômicos Estima um índice de discriminação e k-1 limiares (tresholds) as categorias (k = número de pontos na escala) Generalized Rating Scale Model ou Muraki s Modified Graded Response Model Generalização do modelo de 2 parâmetros para escalas likert Estima um índice de dificuldade por item e k-1 limiares (tresholds) gerais para as categorias (k = número de pontos na escala) Muraki s Generalized Partial Credit Model Generalização do modelo de 2 parâmetros para escalas likert Estima um índice de dificuldade por item e k-1 limiares (tresholds) gerais para as categorias (k = número de pontos na escala)

3 Lógica dos modelos: Curvas características das respostas P( x i P( x 1) exp( i ) 1, i ) (1) P( x 0) P( x 1) 1 exp( ) i For this dichotomous case we have two probability equations: 1 P ( x 0) ; and 1 exp( 1 ) exp( 1) P ( x 1). 1 exp( ) 1 Figure 4. Item characteristic curve for a dichotomous (2-category) item. Kennedy (2005). Constructing Measurement Models for MRCML Estimation: A Primer for Using the BEAR Scoring Engine. Berkeley: BEAR

4 Rasch-Masters Partial Credit Model For the polytomous case (Partial Credit Model; Masters, 1981), the following equation shows the probability that a person with a proficiency of will respond in category c rather than in any other category on item i, given item difficulty parameters i =( i1, i2, im ). exp c j0 k P( x i c, i ), (2) m exp ( ) k0 j0 ( ) where m is the number of steps (number of categories-1) for the item. For example, for a 3-category item (with 2 steps), ij ij P( x P( x P( x Note the convention s exp(0) m i i i exp 0) 1) 2) k k 0 j0 2 exp ( ij ) j0 exp( i1) ;and 2 k 1 exp( i1) exp(2 ( i1 i2 )) exp ( ) exp ( ij ) j0 exp( i1 i2 ) exp(2 ( i1 i2 )). 2 k 1 exp( i1) exp(2 ( i1 i2 )) 1 exp( i1) exp(2 ( i1 i2 )) exp ( ) ( 1 1 ; k 1 exp( i1) exp(2 ( i1 i2 )) exp ( ) k 0 j0 1 k 0 j0 2 k 0 j0 ij ij ij ij 1and 0 j0 ( ) 0; and that ) is the sum of the numerators for all categories. ij

5 Representação gráfica Rasch-Masters Partial Credit Model Figure 5. Category probability curves and ij values for a 3-category polytomous item.

6 PSYCHOMETRIKA-VOL 47, NO. 2. JUNE, 1982 A RASCH MODEL FOR PARTIAL CREDIT SCORING GEOFF N. MASTERS UNIVERSITY OF CHICAGO A unidimensional latent trait model for responses scored in two or more ordered categories is developed. This "Partial Credit" model is a member of the family of latent trait models which share the property of parameter separability and so permit "specifically objective" comparisons of persons and items. The model can be viewed as an extension of Andrich's Rating Scale model to situations in which ordered response alternatives are free to vary in number and structure from item to item. The difference between the parameters in this model and the "category boundaries" in Samejima's Graded Response model is demonstrated. An unconditional maximum likelihood procedure for estimating the model parameters is developed. Key words: latent trait, Rasch model, ordered categories, partial credit.

7 Intuição do modelo : Rasch-Andrich Rating Scale Model Figure 6. i, 1 and 2 representations for the polytomous case with 3 categories.

8 Rasch-Andrich Rating Scale Model When using the partial credit model we generally parameterize the difficulty of achieving a score of j on item i and represent it with ij. That is, ij is the proficiency level required to expect an equal chance of responding in category j or in category j-1 on item i. Alternatively, we might think of the average of the ij 's as an overall item difficulty, and the step difficulties as each step's deviance from the average. In looking at item difficulties in this way we are saying that each ij can be formulated as i + ij, where ij is the deviance from the average item difficulty for item i at step j. Note that in this case the last tau parameteris equal to the negative sum of the others so that the sum of all the tau parameters equals zero, m 1 im ik k1. A graphical representation of this formulation for an item with two steps (and therefore three categories) is shown in Figure 6. The rating scale model is a special case of the partial credit model in which the tau parameters for step j are the same for every item. That is, 11 = 21 = 31..., = 22 = 32..., etc. In this formulation, our measurement model becomes exp j0 k P( x i c i, ), m exp [ ( )] c k 0 j0 [ ( )] where i =( i, 1, 2,... m-1 ). Again, the final tau parameter, m, is not estimated because it is constrained to make the sum of all the tau parametersequal to zero. i i j j

9 PS CHOMETR Kk~VOL. 43, NO. 4. DECEMBER, 1978 A RATING FORMULATION FOR ORDERED RESPONSE CATEGORIES DAVID ANDRICH THE UNIVERSITY OF WESTERN AUSTRALIA A rating response mechanism for ordered categories, which is related to the traditional threshold formulation but distinctively different from it, is formulated. In addition to the subject and item parameters two other sets of parameters, which can be interpreted in terms of thresholds on a latent continuum and discriminations at the thresholds, are obtained. These parameters are identified with the category coefficients and the scoring function of the Rasch model for polychotomous responses in which the latent trait is assumed uni-dimensional. In the case where the threshold discriminations are equal, the scoring of successive categories by the familiar assignment of successive integers is justified. In the case where distances between thresholds are also equal, a simple pattern of category coefficients is shown 1o follow. Key words: logistic latent trait, Rasch model, thresholds, ordered categories.

10 Modo alternativo de se expressar o modelo: Thurstonian threshold

11 Samejima s Graded Response Model SGRM The graded response model assumes that the response categories can be ordered along a continuum. Samejima s model estimates a set of boundary location parameters for each item. The following equations are for the homogenous case of the graded response model, which assumes that within each item the discriminations of the options are equal (i.e., there is a single discrimination for each item), but it allows discriminations to vary across items. The boundary response function (BRF) is defined as P * i g j exp Ai j big 1expAi j bi g, where A i is the product of the item discrimination parameter (a i ) with D, bi g is the boundary location parameter for boundary g, * * 0 j 1and m j P (C.2) P 1 0, (C.3) where g = m 1 and m is the number of response categories. Then the category response function (CRF) is defined as * P P P. (C.4) i j i j i j g g g1 Guyer, R., & Thompson, N.A., (2011). User s Manual for Xcalibre 4.1. St. Paul MN: Assessment Systems Corporation.

12 Embretson & Reise (2000)

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14 Generalized Rating Scale Model ou Muraki s Modified Graded Response Model e Muraki s Generalized Partial Credit Model GRSM The generalized rating scale model is a generalization of Samejima s model in which one set of boundary location parameters are estimated for each domain. The GRSM estimates a single item location parameter which adjusts the common boundary parameters based on the items location. The boundary response function (BRF) is defined as P * i g j exp Ai j bi c g, 1expAi j bi c g and the option response functions are then computed from Equation C-4. (C.5) GPCM and the RPCM The partial credit models are most a In the partial credit models, the probability of responding by selecting a particular response option, g, is computed directly from P i gg1, g j g exp exp m g0 h h0 g0 Ai j big, Ai j bi g (C-6) where bi g is the boundary (or step ) location parameter. For the RPCM, A i = 1.0 for all items.

15 Fitting a Polytomous Item Response Model to Likert-Type Data Eiji Muraki Educational Testing Service This study examined the application of the MML-EM algorithm to the parameter estimation problems of the normal ogive and logistic polytomous response models for Likert-type items. A rating-scale model was developed based on Samejima s (1969) graded response model. The graded response model includes a separate slope parameter for each item and an item response parameter. In the rating-scale model, the item response parameter is resolved into two parameters: the item location parameter, and the category threshold parameter characterizing the boundary between response categories. For a Likert-type questionnaire, where a single scale is employed to elicit different responses to the items, this item response model is expected to be more useful for analysis because the item parameters can be estimated separately from the threshold parameters associated with the points on a single Likert scale. The advantages of this type of model are shown by analyzing simulated data and data from the General Social Surveys. Index terms: EM algorithm, General Social Surveys, graded response model, item response model, Likert scale, marginal maximum likelihood, polytomous item response model, rating-scale model.

16 Comparação dos modelos (Embretson & Reise, 2000)

17 Comparação dos modelos (Embretson & Reise, 2000)

18 Modelos l Rating scale (resposta graduada) distâncias iguais l Partial credit (créditos parciais) distâncias livres

19 CCI para itens Likert: Modelo de crédito Parcial e Resposta Graduada

20 Dificuldade no MRG

21 Item 1

22 Item 2

23 Item 3

24 Item 4

25 Item 5

26 Item 6

27 Item 7 e 8

28 Estratégia de análise l Análise de itens Análise da estrutura das escalas graduadas l Estrutura dos tresholds l Average response measure r item total Infit Outfit j = 0 to m E = m j=0 jp j Var(E) = P j ( j E) 2 Outfit = m N j=0 j=1 (O E) 2 Var(E) N Precisão da escala Item map

29 Alguns textos com guidelines interessantes l l

30 Exemplo l IDTP