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 (Yij ≥ k ) 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.