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Development of a descriptive fit statistic for the Rasch model.

The Rasch model (Rasch 1960/1980) has become a standard measurement model for the analysis and validation of educational and psychological tests and for the purpose of scaling examinees (Embretson, 2000; Hambleton, 1991; Rupp, 2006). This is due to the appealing properties of the model which are generally referred to as objective measurement (Karabatsos, 2000). These properties include parameter separability, existence of a common interval scale for both persons and items, unidimensionality of measurement, and existence of a sufficient statistic to estimate person and item parameters independently of each other (Baghaei, 2009; Fischer, 2006). Nevertheless, these features are not achieved automatically upon subjecting raw data to the Rasch model. The degree to which these essential properties of measurement are attained depends on the fit of data to the Rasch model.

The attractive properties of the Rasch model and the assumptions of the model--that is, unidimensionality, parallel item characteristic curves, and stochastic independence--allow for elegant methods of testing the model. Over the years, numerous global and local statistical tests have been proposed to check the conformity of data to the Rasch model principles (see Maydeu-Olivares, 2013).

Most of the fit tests available for the Rasch model rely on the principles of statistical hypothesis testing based on the null hypothesis in which the Rasch model holds, and an alternative hypothesis in which the Rasch model does not hold. The most widely used methods for testing the Rasch model are Andersen's likelihood ratio test (LR test) (Andersen, 1973), Martin-Lof's test (Martin-Lof, 1973), [Q.sub.1] test (Van den Wollenberg, 1982), and [R.sub.1] test (Glas, 1988). However, statistical tests for the Rasch model have certain disadvantages that affect their utility and reliability as model checks. The most serious problem of statistical significance testing is the sensitivity to sample size and the number of items. For instance, statistical tests might lead to statistically significant results even under practically negligible deviation from the Rasch model when sample size is large (high statistical power) or statistically nonsignificant results even under large deviations from the Rasch model when samples are small and thus have low statistical power (Gustafsson, 1980). Thus, sample size determination given type-I-risk [alpha], type-II-risk [beta], and model deviation [delta] for testing the Rasch model is needed (Kubinger, 2009). Nevertheless, only a few methods for sample size determination under restrictive assumptions are available (Draxler, 2010; Kubinger, 2011), while most of the statistical tests for the Rasch model do not permit sample size determination. Lastly, statistical tests for the Rasch model test perfect fit while an actual data set never fits a mathematical model perfectly.

Due to the above-mentioned limitations of statistical testing, descriptive fit statistics are used to provide a measure for the conformity of data to the Rasch model, which can be either global or local. Local fit statistics, such as the residual-based infit and outfit values (Wright, 1979), provide a measure for the conformity of portions of the data on both items and persons to the Rasch model specifications. Global descriptive fit statistics, i.e., measures for the overall conformity of data to the Rasch model, appear to be rarely used in practice.

Consequently, the goal of the present study is to develop global descriptive fit statistics for checking whether the Rasch model holds. Such measures are commonly used in structural equation modeling literature (see West, 2012) but are not used in IRT circles. A number of statistics are proposed and evaluated under different conditions of varying test length and sample size using a simulation study. The proposed statistics are based on the rationale of two statistical tests for the Rasch model, which are reviewed below.

Andersen's likelihood ratio test

The rationale behind Andersen's LR test (Andersen, 1973) is the invariance property of the Rasch model: if the Rasch model holds, equivalent item parameter estimates should be obtained from different subsamples of the data within sampling error. For example, item calibrations based on low scorers and high scorers should yield approximately the same parameter estimates. In addition, Andersen (1973) showed that if items have substantially different discriminations they would have different difficulty estimates. Therefore, the approach is sensitive to the violation of parallel item characteristic curves (ICC). The power of the LR test against the property of parallel ICC's was confirmed by Van den Wollenberg (1979). Likewise, Gustafsson (1980) demonstrated that the LR test has power against 2PL and 3PL models. Nevertheless, Suarez-Falcon and Glas (Suarez-Falcon, 2003) showed that the test has low power with respect to multidimensionality of the data.

Accordingly, to conduct Andersen's LR test (Andersen, 1973), the sample is divided into g score-level subsamples and the conditional likelihood function for each subsample and the entire sample is computed. If the Rasch model holds, the likelihood of the complete data should be approximated by the product of the likelihoods of the subsamples (Suarez-Falcon, 2003). Andersen (1973) showed that -2 times logarithm of the difference between the maximum likelihood (ML) of the whole sample and sum of likelihoods of the subsamples is asymptotically chi-square distributed with (g-1) (k-1) degrees of freedom, where g and k are the number of score groups and the number of items, respectively.

Fischer-Scheiblechner's S test

In Fischer-Scheiblechner's (Fischer & Scheiblechner, 1970) approach, the sample is divided into two subsamples and the item parameters are estimated in each of the subsamples. The difference between the item parameters across the subsamples is tested with the usual z-test of difference:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[??].sub.i1] and [[??].sub.i2] are the difficulty estimates of item i in subsamples 1 and 2, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are variances of the estimates of [[??].sub.i1] and [[??].sub.i2]. [S.sub.i] is an approximate standard normal deviate and [S.sup.2.sub.i] is approximately chi-squared distributed. Overall, S statistic is calculated by summing up chi-squared terms, i.e., [S.sup.2.sub.i] values for all item parameter pairs. According to Fischer and Scheiblechner (1970), S is chi-squared distributed with k-1 degrees of freedom, k being the number of items. However, van den Wollenberg (1979) states that the summation of [S.sup.2.sub.i] values is possible only if chi-squared terms are independent. This condition is violated as the covariance between item parameters must be negative due to the norming condition of item parameters in each subsample. Since the covariance matrix does not enter the equation of S, it cannot be chi-squared distributed with k or k-1 degrees of freedom.

Present study

The goal of the present study is to develop global fit statistics for checking the dichotomous Rasch model. In general, for descriptive fit statistics to be useful they should meet several conditions. First, in the absence of differential item functioning (DIF),- i.e., where items have different parameter estimates based on different subsamples of the same location on the latent trait- the fit statistic should be near a constant value. Second, in the presence of DIF, the fit statistic should quantify the extent of DIF. In particular, it should become larger depending on the number of DIF items and the magnitude of DIF. Lastly, when quantifying DIF, the fit statistic should not be affected by sample size. More specifically, in the absence of DIF, the fit measure should be near a constant value independent of the sample size while, in the presence of DIF, the value should only quantify DIF without being affected by sample size. In order to develop fit measures for testing the Rasch model, the study investigated properties of various measures to evaluate if those requirements are fulfilled.

METHOD

Proposed fit measures for testing the Rasch model

We propose some descriptive fit measures based on the principle of stability of item parameter across subsamples, which will then be examined with a simulation study.

Root-mean-square deviation (RMSD). RMSD is the square root of the mean square difference between item parameters estimated in two subgroups after bringing them onto a common scale:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[??].sub.i1] is the estimated item parameter in the first subgroup (e.g., examinees with low scores), [[??].sub.i2] is the estimated item parameter in the second subgroup (e.g., examinees with high scores), and k is the number of items. Following the rationale of the Andersen's LR test, if the Rasch model holds in the population, equivalent item parameter estimates should be obtained, apart from sampling error, which means the RMSD should be close to zero.

Standardized root-mean-square deviation (SRMSD). SRMSD is the RMSD divided by the pooled standard deviation ([SD.sub.pooled]) of item parameters for both subgroups:

SRMSD = RMSD/[SD.sub.pooled]

The pooled standard deviation is given by:

[SD.sub.pooled] = SD([[beta].sub.i1]) + SD([[beta].sub.i2])/2

Likewise, if the Rasch model holds, the RMSD should be near zero.

Normalized root-mean-square deviation (NRMSD). The NRMSD is the RMSD divided by the range of estimated item parameters in both subgroups:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the maximum of the item parameters in both subgroups and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the minimum of the item parameters in both subgroups. Again, if the Rasch model holds, the SRMSD should be near zero.

Chi square to degree of freedom ratio [X.sup.2]/df. The chi square to degree of freedom ratio is commonly applied in the framework of structural equation modeling (SEM) to assess model fit (see West, 2012). The rationale is that the expected value of the [X.sup.2] for a correct model equals the degree of freedom. Thus, if the Rasch model holds, [X.sup.2/df] should be close to one. The current study investigated [X.sup.2]/df for both the Andersen's LR test and the Fischer and Scheiblechner's S statistic.

Root mean square error of approximation (RMSEA) The RMSEA (Steiger, 1980) is a widely used fit measure in structural equation modeling:

RMSEA = [square root of (max ([chi square] - df,0)/df(n - 1))]

When the chi-square is less than the degree of freedom, the RMSEA is set to zero. In the current study, the RMSEA based on both the Andersen's LR test and the Fischer and Scheiblechner's S statistic is investigated. If the Rasch model holds, the RMSEA should be near zero.

Simulation study

In order to investigate the properties of the proposed measures, simulations based on two general conditions were carried out: (1) without differential item functioning or null hypothesis conditions and (2) with differential item functioning or alternative hypothesis conditions. In both conditions, data were simulated with n = 100, 200, 300, 400, 500, 600, 700, 800, 900, and 1,000 examinees in combination with k = 10, 20, 30, 40, and 50 items. In the alternative hypothesis conditions, data were simulated with eight DIF items. The magnitude of DIF was 0.6 or 1/10 of the range of the simulated item parameters.

The item parameters were set as equally spaced within the interval [-3, 3], which corresponds to the whole spectrum of item difficulties that arise in practice. Meanwhile, the person parameters of examinees were randomly drawn from N (0, 1.5), again corresponding to the values of person parameters that are likely to occur in practice. Moreover, simulations were conducted in R (R Core Team, 2015) using the eRm package (Mair, 2015).

In order to compute the proposed fit statistics, data sets were divided into high scorers and low scorers, based on the mean of the raw scores. Next, the item parameters were estimated separately in the two subsamples. Lastly, the item parameters where brought on to a common scale.

In each condition, the fit statistic in question was computed for 10,000 replications. In addition, for each fit statistic, we computed mean, standard deviation as well as minimum and maximum over all replications.

RESULTS

Null hypothesis condition

First, the null hypothesis condition was investigated, that is, those without differential item functioning. As for the RMSD, results revealed that this fit statistic is highly dependent on sample size; the larger the sample size, the lower the fit statistic. For example, while the mean RMSD is 0.61 with N = 100 examinees and k = 10 items, the mean RMSD drops to 0.18 with N = 1,000. Since the SRMSD and the NRMSD are based on the RMSD, these measures to a certain extent are also dependent on the sample size. For instance, while the mean SRMSD is 1.47 with N = 100 examinees and k = 10 items, it drops to 0.41 with N = 1,000 examinees. Likewise, when the mean NRMSD is 0.09 with N = 100 examinees with k = 10 items, it drops to 0.03 with N = 1,000 examinees (1).

As expected, the mean of [chi square]/df is around 1.00 for both the ratio based on the test statistic of Andersen's LR test and the S statistic. Moreover, this fit statistic does not depend on the sample size or test length. For example, the Andersen [chi square]/df with N = 100 and k = 10 items is 1.01, and with N = 1,000, and k=10 is 0.99. But [chi square]/df based on the S statistic very slightly changes with test length.

Regarding the RMSEA, the mean value of this fit statistic based on the Andersen's LR test and the S statistic is around zero as expected. However, RMSEA seems to be somewhat dependent on sample size, when sample size is lower than N = 400 in the case of k = 10 and lower than 300 in the case of k > 10. For instance, the Andersen's RMSEA for N = 100 and k = 10 is 0.03, while this value drops to 0.01 for N = 400. These properties of the investigated fit statistics seem to hold for k > 10. The results of the null hypothesis condition with eight DIF items are shown in Figure 1.

In sum, the results suggest that RMSD, SRMSD, and NRMSD are not suitable as fit statistics because they are highly dependent on sample size in the absence of DIF. For this reason, only [chi square]/df and RMSEA will be discussed in the alternative hypothesis condition.

Alternative hypothesis condition

In the alternative hypothesis conditions, results revealed that the higher the proportion of DIF items to the entire number of items, the higher the value of the fit statistics. That is, the magnitudes of the fit statistics depend on the number of investigated items. The more items investigated holding the number of DIF items constant, the lower the fit measure. In the following section, we will discuss the conditions with eight DIF items in more detail.

The results revealed that [chi square]/df based on the LR test and the S statistic are both dependent on the sample size, while these statistics were found to be independent of sample size in the null hypothesis condition. Results showed that the RMSEA based on the LR test and the S statistic is not affected by sample size as long as the sample size is at least N = 200. For example, the mean [chi square]/df value based on the LR test for N = 100 and k=10 with eight DIF items is 1.86, while this value increases to 9.45 when sample size is N = 1,000. On the other hand, the mean RMSEA value based on the LR test for N = 100 and k=10 is 0.08 and slightly increases to 0.09 when sample size is N = 1,000.

In sum, the RMSEA based on the Andersen's LR test and the S statistic are the only fit statistics that are not dependent on the sample size. Strictly speaking, RMSEA seems not to be reliable when n = 100, but with N [greater than or equal to] 200 RMSEA only quantifies the magnitude of DIF without considering the sample size. However, this fit statistic is not very sensitive to misfit, as the values do not change noticeably between null and alternative hypotheses. The results of the alternative hypothesis condition with eight DIF items are depicted in Figure 2.

Results showed that the value of [chi square]/df is a function of both the sample size and the ratio of DIF items to the entire number of items. When eight out of 10 items are DIF, i.e., 80% of the items, the value of [chi square]/df for a sample size of 100 is 1.86 and increases with sample size. However, the same value when eight out of 20 items are DIF, i.e., 40% of the items, is 1.41 and increases with sample size. When eight out of 50 items are DIF, i.e., 16% of the items, the value is 1.15 for a sample size of 100. Therefore, the interpretation of [chi square]/df depends on the amount of DIF we are ready to accept in the data. If we consider 100 as the smallest acceptable sample size for conducting Rasch model analysis and 0% as the smallest tolerable magnitude of DIF in the data we need [chi square]/df values a lot smaller than 1.15. Therefore, a maximum value of 1.03 (the largest value for [chi square]/df in the null hypothesis condition where there was no DIF) for this value should indicate perfect fit to the Racsh model. However, note that this value has different standard errors for different test lengths in the null hypothesis condition which allows for more generous cut-off values (2).

DISCUSSION

In this study, an attempt was made to develop descriptive measures of fit for the dichotomous Rasch model. Accordingly, a number of fit statistics based on the property of parameter invariance of the Rasch model were evaluated in a simulation study. Furthermore, the simulation studies were carried out under the specific conditions of test length and sample size.

Most of the available global model fit measures are based on statistical hypothesis testing. Such fit assessment procedures are sensitive to large sample sizes since statistical power increases. Furthermore, such methods evaluate perfect fit of the data to the Rasch model. In this study a descriptive method, namely, Andersen's [chi square]/df, is suggested to evaluate the overall fit of data to the Rasch model. The proposed method in this study is not based on statistical null hypothesis testing and is independent of sample size. Based on simulation studies, cut-off values for the statistic for different test lengths are suggested. The statistic is a complement to the available fit statistics based on null hypothesis testing and not a replacement.

Results showed that while all the fit statistics are more or less independent of test length in the null hypothesis condition, three of them--RMSD, SRMSD, and NRMSD--are dependent on sample size. The means of these statistics vary substantially across sample sizes, and therefore do not meet the requirements we specified above for efficient fit values. Meanwhile, the other four measures--Andersen [chi square]/df, S statistic [chi square]/df, Andersen RMSEA, and S statistic RMSEA--are independent of sample size in the null hypothesis condition. In this condition, the mean values for Andersen [chi square]/df and S statistic [chi square]/df are near one, and for Andersen RMSEA and S statistic RMSEA, they are near zero across all sample sizes. As a result, the S statistic [chi square]/df seems to be dependent on the test length to some degree, as the value for a test length of 10 is around 1.10 but the value approaches one as test length increases. However, the problem with Andersen RMSEA and S statistic RMSEA values is that these measures, although being robust against sample size and test length, are insensitive to model violations. In the H1 condition, where the Rasch model does not hold, these values are around .10 (k=10) and .06 and .04 when k= 30 and k=40, respectively. This indicates that there is not much difference in these values in the H0 and [H.sub.1] conditions, which limits their utility as indicators of model violation.

Hence, the practical measure seems to be Andersen's [chi square]/df as it is near one in the [H.sub.0] condition across all sample sizes and test lengths and noticeably deviates from one in the [H.sub.1] condition. The standard deviation of this measure, however, varies across different test lengths, which restricts building a single confidence interval for use in applied settings. Therefore, we need to devise different cut-off values depending on the test length. Using the mean standard errors across all sample sizes the one-sided 68% confidence intervals in Table 1 can be built as cut-off values for Andersen [chi square]/df for different test lengths.

The reason for building 68% confidence intervals instead of 95% was to lower the chances of false acceptance of the Rasch model based on the suggested fit measure. If Andersen [chi square]/df exceeds these values for different test lengths, the Rasch model should be rejected, whereas if it falls below these values, the Rasch model holds. For using this fit statistic no especial software is needed. If the Rasch model package computes Andersen's LR test then the statistic can easily be computed by dividing the chi square value by its associated degrees of freedom.

The fit statistic developed in this study should be sensitive to the violation of parallel item characteristic curves because if the discriminating powers of items are different, they would have substantially different difficulty estimates based on low and high scoring groups (Andersen, 1973).

Future research should further investigate the cut-off values across other test lengths proposed in this study. It was established that the fit statistic specifically targets DIF and violation of parallel item response functions. Accordingly, future research should study its sensitivity against other types of model violation, such as multidimensionality and violation of conditional independence. A corollary of the test, with sensitivity to the unidimensionality assumption, can be developed based on the rationale of the suggested test here (Martin-Lof, 1973). This entails dividing the test into two subsets of easy and hard items, and estimating persons' abilities from the two subsets. Then alternative fit indices in line with those developed in this study can be devised. Such a fit index provides information with respect to the violation of the unidimensionality assumption. If a test taps a single dimension, different subsets of items should yield equivalent ability parameters apart from random deviations. Lastly, the fit statistic developed in this study is limited to dichotomous items and future research should focus on developing a similar measure for polytomous items.

Purya Baghaei

Islamic Azad University, Iran

Takuya Yanagida

University of Applied Sciences, Austria

Moritz Heene

Ludwig-Maximilians University, Germany

Author info: Correspondence should be sent to: Dr. Purya Baghaei, English Department, Mashhad Branch, Islamic Azad University, Mashhad, Iran. pbaghaei@mshdiau.ac.ir

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(1) A table which depicts the mean of all proposed statistics in the null hypothesis condition (when there is no DIF) for different sample sizes and test lengths can be obtained from the authors.

(2) A table which depicts the mean of [chi square]/df and RMSEA for the Andersen' LR test and the S statistic in the alternative hypothesis condition (with eight DIF items) for different sample sizes and test lengths can be obtained from the authors.

Caption: FIGURE 1 Mean LR test [chi square]/df statistic for N = 100, 200, 300, 400, 500, 600, 700, 800, 900, and 1,000 for k = 10, 20, 30, 40, and 50 items under the null hypothesis condition (left panel) and alternative hypothesis condition (right panel) with 8 DIF items

Caption: FIGURE 2. Mean LR test RMSEA Statistic for N = 100, 200, 300, 400, 500, 600, 700, 800, 900, and 1,000 for k = 10, 20, 30, 40, and 50 Items under the Null Hypothesis Condition (left panel) & Alternative Hypothesis Condition (right panel) with 8 DIF Items.

TABLE 1 Suggested Cut-off Values for Andersen's [chi square]/df
Value for Different Test Lengths

             k                    Andersen's [X.sup.2]/df

             10                         1.45
             20                         1.32
             30                         1.26
             40                         1.23
             50                         1.20

k = number of items


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Author:Baghaei, Purya; Yanagida, Takuya; Heene, Moritz
Publication:North American Journal of Psychology
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Date:Mar 1, 2017
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