A practical alternative to Classical Test Theory (CTT) construction methods is the Rasch model that belongs to the family of IRT models. IRT models generally formulate the probability of endorsing a given item as a function of the latent trait such as mindfulness [18,19]. A number of IRT models have been proposed, such as the Rasch model or the Birnbaum model (the former being a special case of the latter) [18,19]. Different IRT models rely on different measurement assumptions and hence different models are used for different purposes. The Rasch model allows the explicit testing of basic statistical properties that cannot be investigated by means of CTT [20,21]. For example, one can explicitly test by means of Rasch analysis, if it is legitimate to collapse single item scores into a sum score. Similarly, only if Rasch-type models corroborate the claim that measurement on an interval scaled level is achieved, can additive or multiplicative computations be justified [21-23]. From a mathematical point of view, the Rasch model tests whether the relationship between item responses and a latent trait such as mindfulness adheres to an s-formed curve, or ogive—similar to a cumulative normal curve (Figure 1). CTT models implicitly assume a linear relationship, although the underlying model is only rarely tested in an explicit manner. Moreover, linear models pose logical restrictions [24], and ogive-type forms are frequently more plausible, particularly with regard to psychological variables. This is because ogive-type models assume that an increase in extreme (high or low) levels of a respective latent variable has only a small impact on the general probability associated with endorsement of the respective item, whereas for intermediate levels of the same latent trait characteristic, an increase in the trait is associated with a substantial increase in the endorsement probability. The Rasch model may therefore be seen as a useful complementary approach to standard CTT.

It should be noted that other item response theory models, such as the Birnbaum model, do not fit the measurement axioms that allow for explicitly testing interval level measurement [22,25]. However, the conjoint measurement theory, as presented by Luce and Tukey [26], and formalized by Krantz and colleagues [27], lays out the conditions for psychological variables to be measured on an interval level. It has been shown that, while the Rasch model adheres to its axioms in many circumstances, other IRT models are not in line with the conjoint measurement theory [23]. Rasch models have been employed in a substantial number of psychometric studies [28-31]. However, in research related to mindfulness measurement, most questionnaires have been developed using CTT approaches. Two exceptions where IRT models have been used are the Developmental Mindfulness Survey (DMS) [32], which was developed using a Rasch model approach, and the psychometric investigation of the MAAS which has been validated using the Graded Response Model [33]. The authors of the DMS reported excellent fit values, although this scale has not been externally validated nor employed in other published studies. The MAAS is probably one of the most widely used instruments in the field, but IRT results suggest that the scale may measure the construct of mindlessness rather than mindfulness.

More psychometric research is needed to clarify contemporary research questions related both to the conceptualization and operationalization of mindfulness. Given the high interest in mindfulness research, existing instruments should be tested by means of methods complementary to CTT, such as the Rasch modeling, as this will facilitate a deeper understanding of mindfulness and its measurement. This article subjects the Freiburg Mindfulness Inventory to a Rasch analysis in order to tackle the questions related to its factorial structure from a different perspective than CTT. Due to the increasing application of MBI in clinical contexts, we will focus on the applicability of the FMI in clinical populations.

The sample consisted of a clinical population of N = 130 British patients (female = 112, male = 17, missing = 1) who were treated in a single private medical clinic for integrative medicine (Southampton, UK). The aim of the study was to investigate the impact of spiritual and exceptional experiences on health, thereby particularly focusing on the moderating role of mindfulness [34]. Seventy-one participants were diagnosed with chronic fatigue syndrome, 29 with irritable bowel syndrome, 9 with migraine, and 29 with other symptoms (not specified). Six hundred questionnaires were disseminated by post along with an introduction letter, a consent form and a pre-paid envelope for returning the survey. In total, 130 of 600 questionnaires were returned together with a signed consent form, leading to an overall return rate of 22%. Mean age was 46.9 years (SD = 14.7). Sixty-nine of the participants were married, 35 were single, 25 divorced, and 1 person was widowed. Seventy-nine had children, and 49 did not (2 responses missing). Twenty participants had no educational degree, 5 had high school degree, 16 undergraduate degree, 69 graduate, 5 held a PhD, and 15 did not report on educational degrees. Slightly more than half of the participants (69) reported following a spiritual or meditative practice such as Yoga regularly, as opposed to 61 who did not. All participants had to give informed consent before they were allowed to participate in the study. The study was approved by the relevant internal review board of the respective university.

The items of the FMI-14 are shown in Table 1. Response options range from 1 (almost never) to 4 (almost always). The empirical alpha of the FMI-14 in this sample was r = 0.88. The FMI captures trait mindfulness and has been shown to have good psychometric properties including a high internal consistency (alpha of 0.86 in an initial validation study), and it has been shown to correlate positively with health indicators [13-15]. Furthermore, the scale was able to differentiate between mindfulness practitioners and non-practitioners [12]. A recent confirmatory factor analysis showed good fit indices [9]. However, the confirmatory factor analysis has also suggested that there may be ambiguity with regard to the dimensional structure of the instrument as the authors found comparable evidence for both an one-factorial and a two-factorial solution [9].

The Rasch model, part of the IRT model family, can be understood as a stochastic approximation to conjoint measurement theory that allows for explicitly testing the existence of interval metric measurement in a given data set [35]. The Rasch model assumes that the probability of a subject endorsing an item can be exclusively seen as a function of the distance between the item difficulty and the person ability on a linear scale [20]. Rasch models possess the feature of “specific objectivity” indicating that individual measures can be compared independently of the subset of relevant items (item invariance). Similarly, item difficulties are considered to be independent of the sample in which the questionnaire was administered if the Rasch model can be corroborated (person invariance).

More formally, within the dichotomous Rasch model, the probability p of person i solving item j, is in its most simple form a logistic function of the difference of ability θ (or attitude etc.) and item difficulty β on a linear scale. The response to the item is often denoted by X; endorsing a dichotomous item (“yes”) can be denoted by X = 1, whereas non-endorsing (“no”) can be denoted by X = 0:

Hence, the probability of endorsing a given item is a function of the ability of that person and the respective difficulty of the item. The probability of endorsing the item monotonically increases with the respective ability of the person increasing. Both item as well as person parameter values in the context of Rasch analysis are measured in “logit” units. For example, an item parameter is calculated as the natural logarithm of the odds-ratio (i.e., the probability of solving an item divided by the probability of not solving an item). A logit of 0 equals a probability of 50% of endorsing a given item, whereas a probability of 75% (90%) equals approx. a logit value of 1.1 (2.2). Probabilities smaller than 50% are identical with regard to the size of logit, but they are indicated by negative signs (e.g., logit of 25% is approx. −1.1 logits). The curve-linear relation between probability of endorsing an item and the associated amount of the person ability can be seen in Figure 1.

From the various Rasch models available Andrich's Rating Scale Model (RSM) was employed [36]. This model seemed ideal for our data set, as it allows a) for the analysis of a scale consisting of items with more than two answer options (i.e., polytomous items), and b) with the same number of answer options across all items. Additionally, in contrast to Master's Partial Credit Model [37], the RSM is appropriate for smaller samples sizes such as in the present study [38]. We used the software program Winsteps 3.92 for the statistical analyses [21,39]. Alpha was set at .01 for each test, if not explicitly stated otherwise. We followed the recommendations for Rasch analysis and model test recommended by Bond and Fox [21], and Tennant and Connaghan [40].

Substantial differences between the expected and empirically observed matrix values generally indicate a poor model fit. The INFIT and OUTFIT coefficients are commonly employed statistical parameters for describing the fit / misfit between empirically observed data and a theoretical model. The OUTFIT coefficient is defined as the mean squared deviations (MNSQ) from the model's expected values that can easily be affected by outliers (hence the name “OUT”FIT). The INFIT coefficient weights the MNSQ by the variance of observed values and is more robust towards outliers as it is more strongly determined by values within the interquartile distance (hence “IN”FIT). The INFIT coefficient is generally preferred for assessing the model fit. A value of 1 indicates an ideal fit whereas values greater than 1.5 are considered as indicators of poor fit [21]. The respective threshold of 1.5 suggests that the data would exhibit more than 50% variability than actually predicted by the model.

It is reasonable to assume for polytomous item formats that, for example, to endorse the answer options fairly often or almost always a higher level of ability is needed compared to the answers options rarely or occasionally. However, this hypothesis can and should be empirically tested. Rasch analysis provides the category thresholds coefficients τ that indicate the ability level at which the probability for two adjacent categories is 50%. If the relation between categories decoding higher ability levels and their respective τ form are well-ordered (i.e., transitive), then the polytomous format can be seen as established.

A common test for the presumed one-factoriality of the Rasch model is a principal component analysis based on the residuals after the extraction of the Rasch factor. If one-factoriality can be observed, then no further factors should be present. Since concise and universal criteria have yet to be established for determining whether a potential additional dimension should to be considered, results were interpreted according to the recommendations of Linacre [2010]. Therefore >60% of variance explained by the Rasch factor and <5% explained by the largest potential additional dimension was considered to indicate a good fit. Additionally, an eigenvalue <3 indicates that the potential second dimension has only marginal explanatory power.

From a psychometric point of view a good measurement instrument should be unaffected by specific sample characteristics such as age or gender. Lack of this property is called “Differential Item Functioning” (DIF). In non-technical terms, items should be equally difficult for all subgroups such as women or men, i.e., no DIF should be present. We included DIF analyses for diagnosis in our analysis as it is highly desirable that mindfulness instruments can be used irrespectively of such sample characteristics. Furthermore, as mindfulness instruments are frequently administered to individuals both with and without proficiency in mindfulness and systematic meditative or spiritual practice or lack thereof, it is important that the instrument is “fair” to both groups with regard to the difficulty level. Lack of DIF between two groups is indicative of both groups using the same skill for the instrument. We tested DIF for the variables “mindfulness practice” and “spiritual practice” (both binary criterions). Finally we included central demographic variables (age and education). Due to the gender imbalance (117 female vs. 12 male), we did not compute DIF for gender. To control for alpha error inflation, we applied Bonferroni correction (i.e., adjusting the alpha in each DIF variable by the amount of tests).

The hierarchy of item difficulties was addressed as a further aspect for evaluating the validity of the scale. We hypothesized that items associated with the presence facet were easier to endorse than items related to the acceptance facet. This is based on the assertion of Kohls et al. [9] that the presence facet builds up earlier in the course of meditative proficiency than the acceptance facet.

Similar to the concept of reliability in classical test theory, Rasch measurement also provides reliability estimates. However, in the context of Rasch reliability, estimates are computed separately for item and persons parameters, whereas in CTT only item reliability is considered. Rasch reliability is calculated as the ratio of explained to total variance, with the resulting value ranging from 0 to 1 and can be interpreted in a similar vein to internal consistency (Cronbach's alpha) in CTT where values of r > 0.7 are deemed acceptable, and values r > 0.8 are considered to be good [21].

The one-factorial model with 14 items of the FMI was tested as it is probably the most widely used form of the FMI [12-14]. As previous research provided evidence for both the one-factorial as well as a two-factorial solution, we initially examined the dimensionality. The Rasch factor explained 47% of variance (expected variance explained: 46%) with an eigenvalue of 12.5. The second greatest factor explained 11% of variance with an eigenvalue of 3.0. Hence, clear evidence of a second factor seems to be present. Items 1, 2, and 3 exhibited positive loadings higher than .40 on the second factor, and for another four items negative loadings higher than 0.40 (10, 4, 9, 11) on the same second factor were observed. This second factor can tentatively be labeled “present moment attention”. The Rasch factor seemed to describe a latent factor embracing an emotionally neutral, non-judgmental stance. This suggests that a one-factorial solution does not fit with the data. As all items showed good fit values except for item 13 (INFIT MNSQ = 2.23; see Table 2) we decided to exclude this item from further analyses. One obvious reason for the misfit of item 13 is that it is the only negatively coded item, which is also placed at the end of the scale.

A Rasch model for each of the two proposed subfacets of the FMI, presence (items 1,2,3,5,7,10) and acceptance (items 4,6,8,9,11,12,14) was then tested separately, as originally suggested by Kohls et al. [9]. For the presence facet, one-factoriality and local independence could be confirmed, as the Rasch factor accounted for 72% (expected: 72%) with an eigenvalue of 15.2. The second potential factor accounted for only 9% of the variance (eigenvalue = 1.9) and was therefore not considered to be of practical relevance. Item fit indices were good with a maximum INFIT MNSQ of 1.3, and the overall INFIT MNSQ was very high (0.99). The item answer category thresholds were found to be ordered and evenly distributed. DIF was not present for age when we compared two groups on the basis of median split halves. DIF was also not found for diagnosis, education, mindfulness practice, and spiritual practice (such as prayer or meditation in general). The reliability was acceptable (0.78; separability: 1.89), although the scale was probably too easy for the sample (mean person parameter 0.53). Strong floor and ceiling effects were present as demonstrated by the higher dispersion of person parameters (−3.45 to 3.64) compared to item parameters (−0.75 to 0.81), hence limiting the discriminative capabilities of the instrument. However, taken together the presence facet exhibited good adherence to the Rasch model although reliability could be improved.

Similar results were obtained for the acceptance facet. One-factoriality could also be confirmed. The Rasch factor accounted for 61% of the variance (eigenvalue 10.9). The second strongest factor accounted for only 9% of the variance (eigenvalue = 1.6). The item fit indices were found to be good with a maximum INFIT mnsq of 1.2 (item 8); the mean INFIT mnsq was very high (0.99). The difficulty of the response options were ascending in the expected order (order of difficulty of response options: rarely – occasionally – fairly often – almost always). No DIF was found to be present for age (median split), diagnoses, education, mindfulness practice, or spiritual practice. Rasch reliability was good with r = 0.81 (separability 2.04) as was the targeting (i.e., mean person parameter of −0.01 was very close to mean item difficulty, anchored at 0.00). However, strong floor and ceiling effects remained as the person parameter distribution was much more dispersed (−3.99 to 3.98) than the item parameter distribution (−0.46 to 0.30). The two factors correlated substantially (0.68; p < 0.001); a value similar to that found by Kohls et al. [9].

To determine whether the hypothesized hierarchy of item difficulties could be confirmed in the data, a one-factorial Rasch model without item 13 was computed. Again, this model failed to adhere to the assumption of one-dimensionality. Despite this shortcoming, it may allow for a hierarchy of item difficulty to be derived. Ranking items by their relative difficulty may prove heuristically useful to deepening understanding of the construct of mindfulness. The mean difficulty for the presence items was −0.32 whereas the mean difficulty for the acceptance items was 0.24 (see Table 2). This substantial difference of approximately a 0.5 logits suggests that, as expected, presence items are easier to endorse than acceptance items.