Item Response Models items are developed to measure a single ability or trait, and these items are administered to P J test takers. In this model, the items are scored as correct (1) or incorrect (0). The Rasch model, a one-parameter item response model, is specified to estimate the log-odds of a correct response, as a function of person ability and item difficulty. j p The typical formulation of the Rasch (1 parameter) model is a function of person ability and j θ item difficulty i b. Working with logits (log-odds) through the HGLM framework, we reorganize the model Notice that is now In the logit link function notation of HLM, we have the log-odds link βi –bj. as a function of the difference between person ability and item difficulty = ηjp π0j – πqj To define the scale in logits, we set persons or items to have a mean of 0. Then, the probability that examinee correctly answers item is j p This is the same formula we saw earlier in the case of regressing binomial outcomes, where we employ the logit link function (the log-odds of c success), , and want to turn that log-odds into ηij a probability. In a standard Rasch IRT model, person abilities and item difficulties are fixed. In the HLM formulation of the Rasch model, person abilities vary randomly over a population of test takers. Level-1 Model The sampling model is Bernoulli (a binomial sampling model where there is = 1 trial). = 1 m Yij with probability We use the logit link function φij. The level-1 model is then is person ability j π0j is a indicator variable that = 1 if response for person is to item 0 otherwise Xpij i j p, is the difference in log-odds of a correct response between item and a reference item p πpj (omitted due to redundancy) for person j
Item Response Models P items are developed to measure a single ability or trait, and these items are administered to J test takers. In this model, the items are scored as correct (1) or incorrect (0). The Rasch model, a one-parameter item response model, is specified to estimate the log-odds of a correct response, as a function of person j ability and item p difficulty. The typical formulation of the Rasch (1 parameter) model is a function of person j ability θ and item i difficulty b. Working with logits (log-odds) through the HGLM framework, we reorganize the model Notice that βi is now –bj. In the logit link function notation of HLM, we have the log-odds link as a function of the difference between person ability and item difficulty ηjp = π0j – πqj To define the scale in logits, we set persons or items to have a mean of 0. Then, the probability that examinee j correctly answers item p is This is the same formula we saw earlier in the case of regressing binomial outcomes, where we employ the logit link function (the log-odds of c success), ηij , and want to turn that log-odds into a probability. In a standard Rasch IRT model, person abilities and item difficulties are fixed. In the HLM formulation of the Rasch model, person abilities vary randomly over a population of test takers. Level-1 Model The sampling model is Bernoulli (a binomial sampling model where there is m = 1 trial). Yij = 1 with probability φij. We use the logit link function The level-1 model is then π0j is person j ability Xpij is a indicator variable that = 1 if response i for person j is to item p, 0 otherwise πpj is the difference in log-odds of a correct response between item p and a reference item (omitted due to redundancy) for person j
Item-Response-Models.docx