26 The reliability and construct validity of student perceptions of teaching quality 2.3.3 The item response theory model Item response theory (IRT) was used to model students’ responses to the Impact! questionnaire items at the different measurement moments. All items scored (k = 15) are associated with a unidimensional latent variable theorized as “teaching quality”. The model used was what is called the generalized partial credit model (GPCM, Muraki, 1992). A response to item k in category m (m = 0, 1, or 2) pertaining to a teacher j, by a student i, at time point t, is denoted by 1992). A response to item k in categ ry m (m i, at time point t, is denoted by . The probabil respectively, and is the quality of teacher j 2.3.4 The Generalizability Theory model the quality of teacher j as perceived by student i random effects, that is, . (2) Table 2.1 Parameter Interpretation ijtk U m = exp( ) ( | ) 1 exp( ) k ijt km ijtk ijt k ijt kh h m p U m h a q d q a q d - = = + - å k a km d ijt q | | ijt j t i j jt it j q q w t d e = + + + + | main effect teacher main effect time main effect students nested in interaction teachers & time i j jt it j e . The probability of the response is given by 1992). A response to item k in category m (m = 0, 1, or 2) pertaining to a teacher j i, at time point t, is denoted by . The probability of the response is given by (1) respectively, and is the quality of teacher j as perceived by student i at time point t. 2.3.4 The Generalizability Theory model the quality of teacher j as perceived by student i at time point t random effects, that is, . (2) The interpretation and the distribution of these random effects are given in Table 2.1. Table 2.1 Generalizability theory model: parameters, interpretations and distributions Parameter Interpretation Distribution ijtk U m = exp( ) ( | ) 1 exp( ) k ijt km ijtk ijt k ijt kh h m p U m h a q d q a q d - = = + - å k a km d ijt q ijt q | | ijt j t i j jt it j q q w t d e = + + + + 2 2 | | 2 | main effect teacher (0, ) main effect time (0, ) main effect students nested in teachers (0, ) interaction teachers & time (0, ) j j t t i j i j jt jt it j N N N N q s w s t s d s e 2 2 | interaction students & time nested in teachers (0, ) it j N s (1) where The model used was what is called the generalized partial credit model (GPCM, Muraki, 1992). A r s nse to tem k in category m (m = 0, 1, or 2) pertaining to a teacher j, by a student i, at time point t, is denoted by . The probability of the response is given by (1) where and are the item discrimination and an item location parameter of item k, respectively, and is the quality of teacher j as perceived by student i at time point t. 2.3 4 The G neraliz b lity Theory m del In the present context, the generalizability theory (GT) model is a linear model that decomposes the quality of teacher j as perceived by student i at time point t, that is, , into a number of random effects, that is, . (2) The interpretation and the distribution of these random effects are given in Table 2.1. Table 2.1 Generalizability theory model: parameters, interpretations and distributions Parameter Interpretation Distribution ijtk U m = exp( ) ( | ) 1 exp( ) k ijt km ijtk ijt k ijt kh h m p U m h a q d q a q d - = = + - å k a km d ijt q ijt q | | ijt j t i j jt it j q q w t d e = + + + + 2 2 | | 2 | main effect teacher (0, ) main effect time (0, ) main effect students nested in teachers (0, ) interaction tea hers & time (0, ) j j t t i j i j jt jt it j N N N N q s w s t s d s e 2 2 | interaction students & time nest d in teachers (0, ) it j N s and 1992). A response to item k in category m (m = 0, 1, or 2) pertaining to a teacher j i, at time point t, is denoted by . The probability of the response is given by (1) where are the item discrimination and an item location paramet r of item respectively, and is the quality of teacher j as perceived by student i at time point t. 2.3.4 The Generalizability Theory model the quality of teacher j as perceived by student i at time point t random effects, that is, . (2) The interpretation and the distribution of these random effects are given in Table 2.1. Table 2.1 Generalizability theory model: parameters, interpretations and distributions Parameter Interpretation Distribution ijtk U m = exp( ) ( | ) 1 exp( ) k ijt km ijtk ijt k ijt kh h m p U m h a q d q a q d - = = + - å k a km d ijt q ijt q | | ijt j t i j jt it j q q w t d e = + + + + 2 2 | | 2 | i effect teacher (0, ) main effect time (0, ) main effect students nested in teachers (0, ) interaction teachers & time (0, ) j j t t i j i j jt jt it j N N N N q s w s t s d s e 2 2 | interaction students & time nested in teachers (0, ) it j N s are the item discrimination and an item location parameter of item k, respectively, and 1992). A response to item k in category m (m i, at time point t (1) respectively, and is the quality of teacher j as perceived by student i 2.3.4 The Generalizability Theory model the quality of teacher j as perceived by student i at time point t random effects, that is, . (2) Table 2.1 Generalizability theory model: parameters, interpretations and distributions Parameter Interpretation Distribution ijtk U m = exp( ) ( | ) 1 exp( ) k ijt km ijtk ijt k ijt kh h m p U m h a q d q a q d - = = + - å k a km d ijt q ijt q | | ijt j t i j jt it j q q w t d e = + + + + 2 2 | | 2 main eff ct t acher (0, ) main eff ct time (0, ) main effect students nested i te hers (0, ) interactio teacher & tim (0, ) j j t t i j i j jt jt N N N N q s w s t s d s 2 is the quality of teacher j as perceived by student i at time point t. 3.4 The generalizability the ry del In the present context, the generalizability theory (GT) model is a linear model that decomposes the quality of teacher j as perceived by student i at time point t, at is, j, by a stude t k, i at time point t. t, that is, , into a numb r of Distribution ijt q 2 2 2 (0, ) (0, ) (0, ) i j jt N N N N 2 2 | it j N s , into a number of random effects, that is, 1992). A response to item k in category m (m = 0, 1, or 2) pertaining to a teacher j i, at time point t (1) respectively, and is the quality of teacher j as perceived by student i 2.3.4 The Generalizability Theory model the quality of teacher j as perceived b student i at time point t random effects, th t is, . (2) Tabl 2.1 Generalizability theory model: parameters, interpretations and distributions Parameter Interpret tion Distribution ijtk U m = exp( ) ( | ) 1 exp( ) k ijt km ijtk ijt k ijt kh h m p U m h a q d q a q d - = = + - å k a km d ijt q ijt q | | ijt j t i j jt it j q w t d e = + + + + 2 main effect teacher (0, ) main effect time (0, ) j j t t N N q s w s 2 . (2) The interpretation and the distribution of these random effects are given in Table 2.1.
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