# An Evaluation of Healthy Eating Scale for Patients with Pre-Diabetes Using Rasch Analysis
## 2 Materials and Methods
### Introduction to Rasch Model and Data Analysis  
The specialized software RUMM2030 [38] was used for Rasch analysis. RUMM2030 is one of many software packages that can carry out Rasch analysis. The estimation algorithm used in RUMM2030 is the pairwise conditional estimation procedure which generalizes the equation for one pair of items in which the person parameter is eliminated to all pairs of items taken simultaneously [39]. Of numerous software packages that can be used to perform Rasch analysis [40], the software packages that use the same algorithm as RUMM2030 include Ganz and R (using the \"pairwise\" or the \"sirt\" package). A list of software packages to perform Rasch analysis can be accessed on the Institute for Objective Measurement [40] webpage. The Rasch model was developed by [PERSON] and is an approach to item analysis where item difficulty and individual ability may be estimated [41]. The original Rasch model was formulated for dichotomous items. This model uses sum scores from ordinal responses to calculate estimates to represent person ability or achievement, and item locations [42]. There are polytomous extensions of the Rasch model [43], which can be used where scores represent categories of increasing level of a trait and are applicable to the use of Likert scales. The polytomous Rasch model was developed by [PERSON] (1978) [44]. In this study, instead of a generalized partial credit model (GPCM), we used a partial credit model (PCM) which derives from multiple-choice tests. A partial credit model (PCM) can be explained in educational contexts, where an outcome is not always either a correct or incorrect answer, where it is desirable to identify responses that lie in between a completely correct and incorrect answer [45]. Since our scale was in the form of an MCQ without a definite true or false answer but indicated a positive or negative endorsement of items, we applied a PCM by imposing the equality constraint of item discriminations. The PCM assumes that item slopes are equal across items [46]. In our 15-item questionnaire relating to food habits, each item with four categories, we assumed each item contributed equally for item discrimination. The GPCM extends the PCM by including a discrimination term [47] and implements a varying slope parameter [48]. The GPCM allows non-equal item discriminations which are yet to be explored in our current tool. However, we aim to apply GPCM in future studies with a large sample. The equations for the two versions of the Rasch model are shown below [37]. Dichotomous:  
\[P\{X_{ni}=x\}=\frac{e^{x(\beta_{n}-\delta i)}}{1+e^{x(\beta_{n}-\delta i)}}\]  
Polytomous:  
\[P\{X_{ni}=x\}=\frac{e^{-\tau 1i-\tau 2 ix\ \cdot\ \cdot\ \cdot\ \cdot\ -\tau xi+x(\beta_{n}-\delta i)}}{ \sum_{xi=0}^{ni}e^{-\tau 1i-\tau 2 ix\ \cdot\ \cdot\ \cdot\ \cdot\ -\tau xi+x(\beta_{n}-\delta i)}}\]  
where \(\beta_{n}\) is the location of person \(n\) and \(\delta_{i}\) is the location of item \(i\). \(\tau_{xi}\), \(x=1\), \(2\), \(\ldots\), \(mi\) are thresholds which partitioned the latent continuum of item \(i\) into \(m_{i}+1\) ordered categories. \(X\) is the response value that qualifies the expression by \(\beta_{n}\)-\(\delta_{i}\). Rasch analysis is considered the gold standard with which the validity of scales can be evaluated [49]. Scales that comprise ordinal level items, such as a Likert-style response scale, have historically been developed using classical test theory, however, this approach has limitations due to inconsistent or unknown differences between the levels on the scale [43]. The model makes several hypotheses/requirements to ensure the instrument has Rasch properties. Unidimensionality is a fundamental requirement and indicates that only one construct is being measured [50,51]. It is defined as the absence of any meaningful pattern in the residuals [49]. Another requirement is local independence, which indicates the scores are related to each other through the construct only.