51 3 The probability of the response is given by: generalized partial credit model (GPCM, Muraki, 1992). A response to item k in category 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: ( !"#$ = | !"#) = %&'()*!+"#$,-!%) /0∑ %&'(2*!+"#$,-!&) & , (1) where $ and $) are the item discrimination and an item location parameter for item k respectively, and !"# is the quality of teacher j as perceived by student i at time point t. 3.3.7 The generalizability theory model decomposes the quality of teacher j as perceived by student i at time point t, that is, !"# number of random effects, that is: !"# = " + # + !|" + "# + !#|" . (2) distribution is common in IRT modelling. , (1) where 43 theorized to be the quality of a teacher’s teaching. The model used is what is called the generalized partial credit model (GPCM, Muraki, 1992). A response to item k in category m (m 0, 1, or 2) pertaining to a teacher j, by a student i, at time point t, is denoted by !"#$ = . The probability of the response is given by: ( !"#$ = | !"#) = %&'()*!+"#$,-!%) /0∑ %&'(2*!+"#$,-!&) & , (1) $ and $) are the item discrimination and an item location parameter for item k, respectively, and !"# is the quality of teacher j as perceived by student i at time point t. 3.3.7 The generalizability theory model In the present context, the generalizability theory (Brennan, 2001) model is a linear model that decomposes the quality of teacher j as perceived by student i at time point t, that is, !"#, into a number of random effects, that is: !"# = " + # + !|" + "# + !#|" . (2) The interpretation and the distribution of these random effects are given in Table 3.2. Note that the variance attributed to teachers is set equal to 1.0. This is done to identify the model. All other variance components are relative to the teacher variance and, therefore, this model identification does not influence the outcomes of the evaluation of hypotheses. Identifying the model by assuming that one of the latent variable distributions has a standard normal distribution is common in IRT modelling. and 43 theorized to be the quality of a teacher’s teaching. The model used is what is called the generalized partial credit model (GPCM, Muraki, 1992). A response to item k in category m (m = 0, 1, or 2) pertaining to a teacher j, by a student i, at time point t, is denoted by !"#$ = . The probability of the response is given by: !"#) = %&'()*!+"#$,-!%) /0∑ %&'(2*!+"#$,-!&) & , (1) he $ $) ar the item discrimination and an item location parameter for item k, r !"# is the quality of teacher j as perceived by student i at time point t. 3.3.7 T e eneralizability theory model In the present context, the generalizability theory (Brennan, 2001) model is a linear model that deco poses the quality of teacher j as perceived by student i at time point t, that is, !"#, into a nu ber of random effects, that is: !"# = " + # + !|" + "# + !#|" . (2) The interpretation and the distribution of these random effects are given in Table 3.2. Note that the variance attributed to teachers is set equal to 1.0. This is done to identify the model. All other variance components are relative to the teacher variance and, therefore, this model identification does not influence the outcomes of the evaluation of hypotheses. Identifying the model by assuming that one of the latent variable distributions has a standard normal distribution is common in IRT modelling. are the it iscrimi d n it m location parameter for item k, respectively, and generalized partial credit model (GPCM, Muraki, 1992). A response to item k in category = 0, 1, or 2) pertaining to a teacher j, by a student i, at time point t, is denoted by !"#$ The probability of the response is given by: ( !"#$ = | !"#) = %&'()*!+"#$,-!%) /0∑ %&'(2*!+"#$,-!&) where $ and $) r spectively, and !"# is the quality of teacher j as perceived by student i at time point t. 3.3.7 The generalizability theory model decomposes the quality of teacher j as perceived by student i at time point t, that is, !"# number of random ffect , that is: !"# = " + # + !|" + "# + !#|" d stribution is common in IRT modelling. is the quality of teacher j as perceived by student i at time point t. 3.3.7 The eneralizability theory model In t present context, th generalizability theory (Brennan, 2001) model is a lin ar m del that decomposes the quality of teach r j as perceived by student i at time point t, that is, 43 to the Impact! to item k in category m (m i, at time point t, is denoted by !"#$ = . ,-!%) +"#$,-!&) , (1) k, j as perceived by student i at time point t. i at time point t, that is, !"#, into a " . (2) nt a number of random effects, that is: m = 0, 1, or 2) pert ining o a teacher j, by a student i, at time point , s denot d b !"#$ = The probability of the response is given by: ( !"#$ = | !"#) = %&'()*!+"#$,-!%) /0∑ %&'(2*!+"#$,-!&) & , (1) where $ and $) respectively, and !"# is the quality of teacher j as perceived by student i at time point t. 3.3.7 The generalizability theory model deco s e quality of teacher j as perceived by student i at t me point t, that is, !"# n r of random effects, that is: !"# " + # + !|" + "# + !#|" . (2) distribution is common in IRT modelling. . (2) The interpretation and the distribution of these random effects are given in Table 3.2. Note that the variance attributed to teachers is set equal to 1.0. This is done to identify the model. All other varianc compone ts are relativ to the teacher variance and, therefore, this model identif ication does not influence the outcomes of the evaluation of hypotheses. Identifying the model by assuming that one f the latent variable distributions has a standard normal distribution is common in IRT modelling. Table 3.2 Generalizability theory model: parameters, interpretation and distributions Parameter Interpretation Distribution Table 3.2 Generali lity theory model: parameters, interpretation and distributions Parameter Interp etati stribution " main effect teacher ~ (0,1) # growth coefficient ~ (0, # 4 ) "# interaction teachers and time points ~ (0, " 4 # !|" effect students nested within teachers ~ 50, ! 4 |"6 5 6 7# residual students and time points ~ 90, 5 6 4 7#: !"# residual ~ 50, ! 4 "#6 main effect teacher gr coefficient interactio t ers and time points effect stude ts ested within teachers residual and time points sidual
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