We may have possible ordered categories. Considering the response variable it can take on M R, the value of (one of the ordered categories) with probability m = Prob(R = m) φm If we have a rating scale with the categories (1) not proficient, (2) proficient, and (3) exceeds proficient (with respect to academic achievement), we can represent these outcomes as = Prob(R = 1) = Prob(“not proficient”) φ1 = Prob(R = 2) = Prob(“proficient”) φ2 = Prob(R = 3) = Prob(“exceeds proficient”) φ3 In these models, we work with cumulative probabilities, since we want to capture the ordinal nature of the response options (outcome variable). We consider cumulative probabilities as = Prob(R = + + + m) ≤ φ1 φ2 … φm In our example, = φ1 = + φ1 φ2 = + + = 1 φ1 φ2 φ3 Also note that the third equation is redundant, because if you know the first two, the third is known. Generally, we only need -1 cumulative probabilities are estimated. M The cumulative logit function can be used to represent the idea of cumulative probabilities = A logistic regression can be formulated = + ηmi θm βX In this model, there is an intercept for each category (called thresholds) and a common m θm slope There are a couple of assumptions in the use of this model, including the assumption that β. affects the odds ratios for each category in the same way. X m Level-1 Model To continue the example above with the 3-level performance scale, the level-1 model would be = + X1ij η1ij θ1j β1j = + X1ij η2ij θ2j β2j
We may have M possible ordered categories. Considering the response variable R, it can take on the value of m (one of the ordered categories) with probability φm = Prob(R = m) If we have a rating scale with the categories (1) not proficient, (2) proficient, and (3) exceeds proficient (with respect to academic achievement), we can represent these outcomes as φ1 = Prob(R = 1) = Prob(“not proficient”) φ2 = Prob(R = 2) = Prob(“proficient”) φ3 = Prob(R = 3) = Prob(“exceeds proficient”) In these models, we work with cumulative probabilities, since we want to capture the ordinal nature of the response options (outcome variable). We consider cumulative probabilities as = Prob(R ≤ m) = φ1 + φ2 + … + φm In our example, = φ1 = φ1 + φ2 = φ1 + φ2 + φ3 = 1 Also note that the third equation is redundant, because if you know the first two, the third is known. Generally, we only need M -1 cumulative probabilities are estimated. The cumulative logit function can be used to represent the idea of cumulative probabilities = A logistic regression can be formulated ηmi = θm + βX In this model, there is an intercept θm for each category m (called thresholds) and a common slope β. There are a couple of assumptions in the use of this model, including the assumption that X affects the odds ratios for each category m in the same way. Level-1 Model To continue the example above with the 3-level performance scale, the level-1 model would be η1ij = θ1j + β1j X1ij η2ij = θ2j + β2j X1ij
10.HGLM.docx