Thus, in the ordinal model, in addition to the set of item intercept parameters (�� k) and item discrimination parameters (a k), a set of C-2 thresholds ��c (c = 2, �� ., C ? 1) with ��1 = 0 are estimated for the cumulative logits. As a result, �� selleck bio c ? �� k compares the relative frequency in categories c and lower (Yik �� c) with that in categories higher than c (Yik > c). With ��1 = 0, the item intercept parameters (�� k��s) represent the probability of response in categories two or higher versus the probability of Yik = 1.
Model II��Mixed-Effects Regression Model for NDSS Score Over Time Consider the following linear mixed model (LMM) for the continuous NDSS average score y ij for subject i at time point j, regressed on the time value T ij : yij=(��0+��0i)+(��1+��1i)Tij+?ij (4) where �� 0 is the overall population intercept, indicating population mean NDSS score at baseline; �� 0i is the intercept deviation for subject i; �� 1 is the overall population slope, that is, change in mean NDSS score per unit change in time; �� 1i is the slope deviation for subject i; and ? ij is an independent error term distributed normally with mean 0 and variance . The errors are independent conditional on both �� 0i and �� 1i. With two random subject-specific effects, the population distribution of intercept and slope deviations is assumed to be a bivariate normal N(0,�� ��), where S �� is the 2��2 variance�Ccovariance matrix given as: ����=[�Ҧ�02�Ҧ�0��1�Ҧ�0��1�Ҧ�12] This model indicates the linear effect of time both at the individual (�� 0i and �� 1i) and population (�� 0 and �� 1) levels.
For this study, the LMM in equation (4) was implemented in SAS PROC MIXED for the analysis of the averaged NDSS score over time. Model III��Longitudinal 2-PL IRT Model for NDSS Items Over Time When a total of m items are repeatedly measured across time for N subjects, the observed binary response to item k for subject i at time point j is denoted as Yijk. For the analysis of such data at the item level, Liu and Hedeker (2006) incorporated the random subject effect components (�� 0i and �� 1i in Equation 4), a crucial feature for longitudinal models, into the 2-PL Model (Equation 3) as logitijk=(��ijk��0k+��0i)+(��ijk��1k+��1i)Tij+��ijkak��ij, (5) Here, Xijk denotes the indicator variable for the kth item for subject i at time point j, Tij denotes the time value associated with the item response, �� 0k and �� 1k are the population intercept and linear trend for the kth item (both are fixed-effect parameters), and �� 0i and �� 1i are the same random subject effects as defined in Model II, reflecting individual deviations from the population intercepts and linear trends.
Similar to the 2-PL model in Equation (3), the item discrimination parameters a k correspond to the SDs (or factor loadings) for the items. These discrimination parameters are constrained to be invariant over Cilengitide time.