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
Final estimation of fixed effects: (Unit-specific model) Fixed Effect Coefficient Standard error t-ratio Approx. d.f. p-value For INTRCPT1, ψ0 INTRCPT2, γ00 1.551164 0.036655 42.318 7639 <0.001 For IND1 slope, ψ1 INTRCPT2, γ10 0.530704 0.047733 11.118 38002 <0.001 For IND2 slope, ψ2 INTRCPT2, γ20 0.281744 0.046264 6.090 38002 <0.001 For IND3 ψ3 INTRCPT2, γ30 -0.082213 0.044577 -1.844 38002 0.065 For IND4 slope, ψ4 INTRCPT2, γ40 1.055976 0.051815 20.380 38002 <0.001 For IND5 slope, ψ5 INTRCPT2, γ50 -0.476906 0.043173 -11.046 38002 <0.001 A comparison to Rasch estimates of item location using a full ordinal partial-credit model (Winsteps): A Model for Impact (e.g., gender difference in trait level) ψ0j = γ00 + γ01(sex)j +u0j ψ1j = γ10 ψ2j = γ20 ψ3j = γ30 ψ4j = γ40 ψ5j = γ50
In this model, ψ0j is common to all items in level-1, as it is the location of the reference item to identify the scale; thus it is the overall difference between males and females on all items – the difference in ability between male and female respondents. Final estimation of fixed effects: (Unit-specific model) Fixed Effect Coefficient Standard error t-ratio Approx. d.f. p-value For INTRCPT1, ψ0 INTRCPT2, γ00 1.644895 0.043713 37.629 7638 <0.001 FEMALE, γ01 -0.181614 0.045773 -3.968 7638 <0.001 For IND1 slope, ψ1 INTRCPT2, γ10 0.530804 0.047740 11.119 38002 <0.001 For IND2 ψ2 INTRCPT2, γ20 0.281827 0.046271 6.091 38002 <0.001 For IND3 slope, ψ3 INTRCPT2, γ30 -0.082237 0.044585 -1.844 38002 0.065 For IND4 slope, ψ4 INTRCPT2, γ40 1.056250 0.051822 20.382 38002 <0.001 For IND5 slope, ψ5 INTRCPT2, γ50 -0.477160 0.043183 -11.050 38002 <0.001 A Model for DIF Final estimation of fixed effects: (Unit-specific model) Fixed Effect Coefficient Standard error t-ratio Approx. d.f. p-value For INTRCPT1, ψ0 INTRCPT2, γ00 1.652380 0.053370 30.961 7638 <0.001 FEMALE, γ01 -0.195052 0.073457 -2.655 7638 0.008 For IND1 slope, ψ1 INTRCPT2, γ10 0.535299 0.069964 7.651 37997 <0.001
FEMALE, γ11 -0.008013 0.095727 -0.084 37997 0.933 For IND2 slope, ψ2 INTRCPT2, γ20 0.350346 0.068304 5.129 37997 <0.001 FEMALE, γ21 -0.126609 0.092908 -1.363 37997 0.173 For IND3 slope, ψ3 INTRCPT2, γ30 -0.172108 0.064635 -2.663 0.008 FEMALE, γ31 0.171673 0.089290 1.923 37997 0.055 For IND4 slope, ψ4 INTRCPT2, γ40 1.006111 0.075437 13.337 <0.001 FEMALE, γ41 0.095013 0.103819 0.915 37997 0.360 For IND5 slope, ψ5 INTRCPT2, γ50 -0.459143 0.063078 -7.279 <0.001 FEMALE, γ51 -0.034561 0.086560 -0.399 37997 0.690