The Rasch model is the formal measurement model required to construct quantitative measurement from dichotomous or ordinal data [11 ,14 (link),15 (link)]. It is used whenever a set of items are intended to be summed together to give a total score. The pattern of responses from such data is checked against the model expectations, which is a parametric probabilistic form of Guttman Scaling [16 ].
Thus the process of Rasch analysis is concerned with testing to see if the data accord to model expectations, satisfy the various assumptions of the model, and other key measurement issues such as the absence of differential item functioning [17 (link)]. For example, the assumption of local independence can be characterised as comprising two elements, response dependency and trait dependency [18 (link)]. The former is where items are linked in some way, such as a series of walking items reflecting increasing distances. The latter is multidimensionality. Both these are tested by analysis of the residuals where the former is judged to be absent when residual correlations are below 0.3, and the latter to be unidimensional where patterns of items in the residuals (as identified by a Principal Component Analysis - PCA) are shown to give similar person estimates [19 (link)]. Response dependency can be accommodated by grouping locally dependent sets of items into 'testlets' [20 (link)]. Where testlets of different lengths are constructed the item residual standard deviation may be inflated.
Another assumption is that of the stochastic ordering of items, testing the probabilistic Guttman pattern. This is confirmed by a series of fit statistics, where Chi-Square based statistics are shown to be non-significant (i.e. no deviation from model expectation) after adjustment for multiple testing [21 (link)]. Summary residual statistics, under conditions of perfect fit, are expected to have a mean of zero and standard deviation of one, whereas in practice the latter should be below 1.4, except where testlets have been used to accommodate local dependency issues, when the standard deviation becomes inflated [22 ]. Individual item residuals are expected to be within the range ± 2.5. Differential Item Functioning (DIF) is deemed absent when there is no significant difference in the residuals (via ANOVA) across key contextual groups, such as age or gender. For analysis of DIF three age groups were used: persons under and up to 38 years (N = 116), 39 to 46 years (N = 99) and persons 46 years or older (N = 104). These groups were based upon distribution to obtain similar numbers within groups to support an ANOVA analysis of the residuals.
Reliability is reported as a Person Separation Index, similar to Cronbach's alpha when data are normally distributed. As both items and persons are calibrated on the same metric, where data fit the Rasch model it is possible to examine the targeting of the items in the scale. A properly targeted instrument would have a mean population value of zero logits, which is also where the items of the scale are centred. Also, when data fit the model, a raw score-interval scale transformation becomes available. This means that the ordinal score, achieved by simply summing the items together, can be transformed into an interval scale latent estimate for use in parametric statistics, and for calculating change scores. This is available because under the Rasch model the raw score is a sufficient statistic for the estimate of the person ability, and the property of specific objectivity (parameter separation) fulfils the requirements to satisfy the axioms of conjoint measurement to provide interval scaling [23 (link)-26 (link)]. In summary the process of Rasch analysis tests the viability of sets of items to be used as valid and reliable additive scale, including aspects of invariance across groups, and compliance with the requirements for constructing interval scale measurement. Further details of the process are given elsewhere [27 (link)-29 (link)].
The sample size of 638 is sufficient for both a factor analysis of 22 items, and to give a high degree of precision (i.e. item location estimates within 0.3 logit with 99% confidence) for the Rasch analysis [30 ].
The study was approved by The Regional Ethical Review Board in Gothenburg and conduced in compliance with the Helsinki declaration. All subjects included in the study signed a written informed consent allowing their data to be used for research purposes.
The Rasch software used was RUMM2030 [31 ]. CFA and EFA in MPlus6 [32 ] and all other analysis in SPSS Version 18 [33 ].
Thus the process of Rasch analysis is concerned with testing to see if the data accord to model expectations, satisfy the various assumptions of the model, and other key measurement issues such as the absence of differential item functioning [17 (link)]. For example, the assumption of local independence can be characterised as comprising two elements, response dependency and trait dependency [18 (link)]. The former is where items are linked in some way, such as a series of walking items reflecting increasing distances. The latter is multidimensionality. Both these are tested by analysis of the residuals where the former is judged to be absent when residual correlations are below 0.3, and the latter to be unidimensional where patterns of items in the residuals (as identified by a Principal Component Analysis - PCA) are shown to give similar person estimates [19 (link)]. Response dependency can be accommodated by grouping locally dependent sets of items into 'testlets' [20 (link)]. Where testlets of different lengths are constructed the item residual standard deviation may be inflated.
Another assumption is that of the stochastic ordering of items, testing the probabilistic Guttman pattern. This is confirmed by a series of fit statistics, where Chi-Square based statistics are shown to be non-significant (i.e. no deviation from model expectation) after adjustment for multiple testing [21 (link)]. Summary residual statistics, under conditions of perfect fit, are expected to have a mean of zero and standard deviation of one, whereas in practice the latter should be below 1.4, except where testlets have been used to accommodate local dependency issues, when the standard deviation becomes inflated [22 ]. Individual item residuals are expected to be within the range ± 2.5. Differential Item Functioning (DIF) is deemed absent when there is no significant difference in the residuals (via ANOVA) across key contextual groups, such as age or gender. For analysis of DIF three age groups were used: persons under and up to 38 years (N = 116), 39 to 46 years (N = 99) and persons 46 years or older (N = 104). These groups were based upon distribution to obtain similar numbers within groups to support an ANOVA analysis of the residuals.
Reliability is reported as a Person Separation Index, similar to Cronbach's alpha when data are normally distributed. As both items and persons are calibrated on the same metric, where data fit the Rasch model it is possible to examine the targeting of the items in the scale. A properly targeted instrument would have a mean population value of zero logits, which is also where the items of the scale are centred. Also, when data fit the model, a raw score-interval scale transformation becomes available. This means that the ordinal score, achieved by simply summing the items together, can be transformed into an interval scale latent estimate for use in parametric statistics, and for calculating change scores. This is available because under the Rasch model the raw score is a sufficient statistic for the estimate of the person ability, and the property of specific objectivity (parameter separation) fulfils the requirements to satisfy the axioms of conjoint measurement to provide interval scaling [23 (link)-26 (link)]. In summary the process of Rasch analysis tests the viability of sets of items to be used as valid and reliable additive scale, including aspects of invariance across groups, and compliance with the requirements for constructing interval scale measurement. Further details of the process are given elsewhere [27 (link)-29 (link)].
The sample size of 638 is sufficient for both a factor analysis of 22 items, and to give a high degree of precision (i.e. item location estimates within 0.3 logit with 99% confidence) for the Rasch analysis [30 ].
The study was approved by The Regional Ethical Review Board in Gothenburg and conduced in compliance with the Helsinki declaration. All subjects included in the study signed a written informed consent allowing their data to be used for research purposes.
The Rasch software used was RUMM2030 [31 ]. CFA and EFA in MPlus6 [32 ] and all other analysis in SPSS Version 18 [33 ].
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