The Rasch Model has been applied in a variety of disciplines and for a wide range of topics such as development and validation of scale ( Noor Lide Abu Kassim 2007;Hula et. al. 2006; Doyle et. al. 2005; Tennant et. al 2004), psychology research (Fox & Jones, 1998), cognitive development(Grey & Fox, 1996; Bond, 1995, Akbariah, 2006), language learning (Taguchi 2005; Yuan, 2005, Ainol & Noor Lide, 2003) and health (Pallant & Tennant, 2007; Jackson et. al. 2002) to name a few. The advantages of using Rasch Model is due to the interval nature of the measure it provides and the theoretical independence of item difficulty and person ability scores from the particular samples used to estimate them
Rasch (1960) developed a probabilistic model in which each item difficulties(b) and person difficulty (θ) are simultaneously estimated. There are two version of the model which are the dichotomous(Rasch, 1960) and the polytomous(Andrish, 1979) model. The Rasch Model is a unidimensional model that has two main assertions: (1) that the easier the item is, the more likely it will be answered correctly or agreed upon; (2) The more able a person, the more likely they will correctly answer an item compared to a less able person. The ability parameter for person and difficulty parameter for item represent the position of person and items on a latent trait they share. Therefore, the analysis will results in a scale on which both persons and items are mapped onto the same construct in the same scale-free units, together with the standard error.
The equation for Rasch Model is conventionally written as:
Pni = e(θ n –bi)
1 + e(θ n –bi)
where Pni is the probability of person n on item i scoring a correct (x =1) response rather than an incorrect (x = 0) response, given person ability (θn) and item difficulty (bi). This probability is equal to the constant e, or natural log function (2.7183) rose to the difference between a person’s ability and an item’s difficulty (θ n–bi), and then divided by 1 plus this same value (Bond & Fox, 2001, p. 201). In short, the two parameters are used in the model to determine the probability of person nsucceeding on item i (Rasch, 1960).
Bond and Fox (2001) further explained that the model presumes that the probability of a certain respondents to give a right answer to particular items is a logistic function of the relative distance between the item difficulty parameter and the respondents’ ability parameter. They further states that the Rasch Model is based on a simple idea that all respondents are more likely to endorse easy items than difficult items. Hence, in Rasch analysis, items that receive lower ratings are more difficult to endorse than item that receive higher ratings. Parameter estimates for each item are expressed in logits(log-odd) probability units (Wright and Stone, 1979). A logit is the distance along the line of variables that increase the odds of observing the event by a factor of 2.718..This formula can be expressed as a logit model presented below:
ln Pni = θ n –bi
1 – Pni
where ln is the normal log, P is the probability of person n affirming item i; θ is the person’s ability, and b is the item difficulty. Fitting data to the Rasch Model thus places both item and person parameter estimates on the same log-odd unit(logit) scale, and it is this that gives the linear transformation of the raw score.
The model can be extended to the polytomous case and is known as the rating scale model(Andrich, 1978):
ln Pnij = θ n –bi – τj
1 – Pnij-1
where the additional τ represents the threshold(0.5 probability point) between adjacent categories.
Statistics indicating fit to the model test how far the observed data match that expected by the model. When the observed response pattern coincide either or does not deviate greatly from the expected response pattern, the items fit the measurement model and constitute a true Rasch scalae (van Alphen et. al. (1994). This is opposite to the practice in statistical modelling where models are developed to best represent the data.
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