Item response modelling for estimating symptom severity
The concept of the IRT model is shown in Figure 1. The score for each of the 33 items is an ordinal variable in commonly accepted clinical terms: 0 = normal, 1 = slight, 2 = mild, 3 = moderate and 4 = severe. For each item, a graded-response logit model was used for describing the probability of a subject’s score for each item13:
\(P\left(Y_{\text{ij}}\geq k\right)=\frac{e^{a_{j}\left(S_{i}-b_{\text{jk}}\right)}}{1+e^{a_{j}\left(S_{i}-b_{\text{jk}}\right)}}\)Equation 1
\(P\left(Y_{\text{ij}}=k\right)=P\left(Y_{\text{ij}}\geq k\right)-P\left(Y_{\text{ij}}\geq k+1\right)\)Equation 2
Equation 1 describes P (Yijk ) as the probability that the score of subject i for item j(Yij ) is at least k, where Si is the severity for subject i ;aj is called the discrimination parameter for item j , reflecting the ability of the item to differentiate the severity among the patients; and bjk is called difficulty parameter of score k for item j , representing the severity at which there is a 50% probability of obtaining a score ≥k for that item. The probability that the score of subjecti for item j (Yij ) is k can then be derived in Equation 2.
As such, the 33 item-level graded-probability models described by Equations 1 and 2, one for each item, collectively estimate a severity level for each patient at a given point in time, mirroring the patient’s sum of scores (Figure 1). The graphical representation of Equation 1 and Equation 2 are called Item Characteristic Curve (ICC) and Category Characteristic Curve (CCC), respectively; they can be visualized in Figure 1.
The difficulty and discrimination parameters were determined by fitting Equations 1 and 2 to the item scores of the entire dataset; effectively, the severity in the same patient at different visits were estimated independently without correlation. Baseline severity values were assumed to follow a standard normal distribution with a mean of zero and a variance of one. The severity values in subsequent visits were anchored to the baseline, with an estimated shift in their means and variances. This way all the IRT model parameters were identifiable14. The distribution of the estimated severity values was plotted over time to explore the disease progression.