Transform Ordinal Data from Questionnaire into Proper Interval Data

One way to transform ordinal level data into interval scale is to use some kind of Item Response model. A well-known example is the Rasch model, which extends the idea of the parallel test model from classical test theory to cope with binary-scored items through a generalized (with logit link) mixed-effect linear model (in some of the 'modern' software implementation), where the probability of endorsing a given item is a function of 'item difficulty' and 'person ability' (assuming there's no interaction between one's location on the latent trait being measured and item location on the same logit scale-which could be captured through an additional item discrimination parameter, or interaction with individual-specific characteristics-which is called differential item functioning). The underlying construct is assumed to be unidimensional, and the logic of the Rasch model is just that the respondent has a certain 'amount of the construct'-let's talk about subject's liability (his/her 'ability'), and call it θθ, as does any item that defines this construct (their 'difficulty'). What is of interest is the difference between respondent location and item location on the measurement scale, θθ. To give a concrete example, consider the following question: "I found it hard to focus on anything other than my anxiety" (yes/no). A person suffering from anxiety disorders is more likely to answer positively to this question compared to a random individual taken from the general population and having no past history of depression or anxiety-related disorder. An illustration of 29 item response curves derived from a large-scale US study that aims to build a calibrated item bank assessing anxiety-related disorders(1,2) is shown below. The sample size is N=766N=766; exploratory factor analysis confirmed the unidimensionality of the scale (with first eigenvalue largely above the second eigenvalue (by a 17-fold amount), and unreliable 2nd factor axis (eigenvalue juste above 1) as confirmed by parallel analysis), and this scale shows reliability index in the acceptable range, as assessed by Cronbach's alpha (α=0.971α=0.971, with 95% bootstrap CI [0.967;0.975][0.967;0.975]). Initially, five response categories were proposed (1 = 'Never', 2 = 'Rarely', 3 = 'Sometimes', 4 = 'Often', and 5 = 'Always') for each item. We will here only consider binary-scored responses.