Background

Cronbach's α coefficient (hereinafter referred to as Alpha [1]) is widely used as an index of scoring reliability and is often reported in social and behavioral studies [2, 3]. However, very few authors report the confidence intervals (CIs) of Alpha in their papers, although this has been suggested by many researchers [4–7].

Conception of research questions

The American Psychological Association Task Force on Statistical Inference suggests, "ways provide some effect-size estimate when reporting a p value" [4]. The task force also noted that "interpreting the size of observed effects requires an assessment of the reliability of the scores" because score unreliability weakens effect size [5]. Confidence intervals around Alpha are advocated for both absolute and relative decisions made by researchers [6, 7]. Alexander et al., in their critical review of research methods regarding quality improvement research, also stated that 74.1% of the studied articles discussed effect size [8]. These reports underscore the need for research on the disclosure of Alpha and 95% CIs in quality-of-life studies.

Most research on quality of life uses observed scores obtained from surveys and questionnaires to make conclusions and inferences regarding quality-of-life outcomes. False responses on patient questionnaires can be caused by aberrant answers, guessing, inattentiveness (which may result from fatigue), misunderstanding or response misfit, according to item response theory [9, 10]. These issues can result in false alarms (i.e., false positive and false negative errors) coming from medical diagnoses and leading to make vain efforts in actions taking. It is of interest to discuss the possibility of replacing observed response scores with expected ones to make decisions and inferences by applying probability theory to studies of quality-of-life outcomes.

Objectives

This article is organized as follows. First, a set of EORTC QLQ-C30 component items developed to assess the quality of life of cancer patients [11] was given to 115 patients who were diagnosed with nasopharyngeal carcinoma. Second, we present a brief introduction of the Rasch [12] model, which we performed to determine expected response scores by the probability function of item difficulty and person ability. Third, Alphas for 95% CIs were generated by a SPSS syntax procedures for reliability 95% CI estimation referring to both observed and expected response scores on EORTC QLQ-C30 by manipulating different person numbers. Fourth, graphical representations of skewness analyses with 95% CIs were applied to justify differences between cancer stages across items, by which the misfitted items on the EORTC QLQ-C30 were examined in further detail with both observed and expected response scores.

Methods

Research data resource

A total of 115 patients newly diagnosed with nasopharyngeal carcinoma were recruited to answer the first 28 questions on the Taiwanese version of the EORTC QLQ-C30 questionnaire, which requires answers on a scale of 1 to 4 (1 = Not at all, 2 = A little, 3 = Quite a bit, and 4 = Very much). The two composite questions requiring answers on a 1-7 scale were excluded from the study ("How would you rate your overall health during the past week?" and "How would you rate your overall quality of life during the past week?"). All examinees were treated in one of three Taiwanese medical centers and had a follow-up evaluation a minimum of three months later without complete remission. Accordingly, the 115-person × 28-item observed-response matrix was created. This study was approved by the institutional review boards of the hospitals.

Data analysis using the Rasch model

We assumed that the 28-item EORTC QLQ-C30 fits Rasch model's specification and forms a unidimensional construct. If any item could not fit the Rasch model expectation, patients were deemed to have exhibited unexpected behaviors, such as aberrant responses, guessing or inattentiveness, which may have led to responses outside of the model's expectations (as we assumed all 28 items constructed a one-dimensional latent trait).

When the data fit the model's expectations, the infit and outfit mean square error (MNSQ) statistics had an expected value of unity on the items. The values of the MNSQ statistics show the amount of distortion of the measurement system. Values less than unity (or over-fit) showed that the items were too predictable (i.e., there was redundancy). Values greater than unity (or under-fit) indicated unpredictability (i.e., noise). The MNSQ statistics were chi-square statistics divided by their degrees of freedom. Items with infit MNSQ beyond the range of 0.5-1.5 were usually regarded as misfitting or poor-fitting [13, 14].

Excel module for data simulation

For our approach to succeed, an Excel-VBA(visual basic for application) module according to the formulas of Cronbach's α (see Formula 1) and skewness (see Formula 2) was programmed.

α = N N − 1 ( 1 − ∑ i = 1 N σ Y i i 2 σ x 2 ) ,
(1)

Cronbach's α is defined in Formula 1, where N is the number of items, σ x 2 is the variance of the observed total test scores for the sample of examinees and σ Y i 2 is the variance of item i for the sample of examinees.

Formula 2 shows the equation for skewness:

S k e w n e s s = n ( n − 1 ) ( n − 2 ) ∑ ( x i − x ¯ s ) 3 ,
(2)

where n is the number of responses, x i is the value of individual responses and x ¯ and s denote the mean and standard deviation of the total responses, respectively.

p n i x = exp ( ∑ j = 0 x ( θ n - δ ij ) ∑ k = 0 m i exp ( ∑ j = 0 k ( θ n - δ ij ) ,
(3)

In Formula 3, based on the Rasch rating scale model [15] shown in Additional file 1, the probability of a person n in a specific category j of item i is yielded by person ability θ and item difficulty δ on an m-point scaled questionnaire.

The expected response scores would be generated by Rasch model's probability function of item difficulties and person measures (see Additional file 1). We anticipate that the person expected scores could produce item MNSQ statistics between 0.5 and 1.5 when item difficulties are anchored according to the original 115 (person) × 28 (item) observed response data set.

The author-programmed A Excel-VBA module for computing test Cronbach's α, item/person skewness coefficient and person expected response score generation is shown (in Additional file 2) according to the item difficulties (Table 1) yielded by WINSTEPS [16] using the Rasch rating scale model [15] from the original 115 (person) × 28 (item) observed matrix.

Table 1 The EORTC QLQ-C30 questionnaire, sorted in descending order of difficulty (logit units) according to the patient responses in this study (questions 29 and 30, which were not reported, are excluded)
Full size table

Procedures for data simulation

1. Alpha 95% CIs by both observed and expected scores

The present study built on the data that were collected from the 115 patients answering the aforementioned 28 items on the EORTC QLQ-C30. As mentioned above, we investigated 95% CIs from the perspective of Alpha by examining different person numbers, such as 115, double(230) and trible(445) with a SPSS syntax procedures (see Additional file 3) referring to both observed and expected scores on EORTC QLQ-C30 to examine any difference between scenarios.

2. Generating expected response data sets by simulation

The random response, for instance with a dichotomous scale, less than the probability P n i = exp ( θ n − δ i ) 1 + exp ( θ n − δ i ) , an item response of 1 would be assigned for that examinee on the item. On the other hand, had the random number been greater than the response probability, an item response of 0 would be recorded. This is a standard item response generation method as used by previously published papers [17–19]. We can easily extend it to a polytomy-like EORTC QLQ-C30 to generate the expected item response data. The Rasch rating scale model [15] probability generation program is shown in Additional file 1 and was applied to yield a data set of expected response scores constructed by the 1-4 scaling scheme.

3. Item-by-item skewness analysis by a bootstrapping procedure

To compare the quality of life on the 28-item EORTC QLQ-C30 questionnaire, we applied item-by-item comparisons [20, 21] of the patient attitudes according to cancer stage. For this purpose, skewness analyses were performed by a bootstrapping method [22–24], sampling with replacement of patient responses at each item, each time selecting 500 responses with 100 replications in ascending order to filter out the median skewness and its 95% CI.

The correlation coefficients between item difficulties (in logit (log odds) units calibrated by WINSTEPS), observed and expected response summation scores across items and the skewness yielded by both observed and expected responses on items are reported. We will examine whether the skewness of expected responses can earn a higher correlation to other counterparts than the skewness of observed scores. A graphical representation of the comparison of patient attitudes toward quality of life across the entirety of the questionnaire will be presented.

Statistical analyses

Statistical analyses and plotting were performed by SPSS software for Windows (Version 15, SPSS, Chicago, IL).

Results

Item difficulties of EORTC QLQ-C30

The most difficult item was question 5 (Do you need help with eating, dressing, washing yourself or using the toilet?), as shown in Table 1. The easiest question was 18 (Were you tired?). Expected response scores, as anticipated, presented an acceptable infit MNSQ more often than observed scores, which displayed two items outside of the range 0.5-1.5. From the observed score perspective, question 5 had an infit MNSQ of 1.6, and question 28 had an infit MNSQ of 1.74, indicating some aberrant responses on those two questions that negatively affected the inferences and conclusions that were made.

Alpha values for different person numbers

In Table 2 we can see that expected scores earn higher Alpha and its 95% CIs across person numbers than the observed responses. The more person number can produce the narrower 95% CI band.

Table 2 Comparisons of test reliability between observed and expected scores by numbers of person.
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Skewness analysis with 95% CI on misfitting items

As shown in Table 1, questions 5 and 28 exhibited greater infit MNSQ errors, indicating some non-randomized, or, say, unexpected aberrant responses on these two items. The skewness and 95% CI of these two misfitted items, calculated from the two data sets of observed and expected scores, are shown on both sides of Figure 1. Significant differences were seen in stage I by the 95% CI far from others of stage II, III and VI on item 5, for which skewness only in stage II displayed a lower value (3.18) by observed scores than by expected scores, (4.32). On item 28, observed raw scores showed a significant difference between stages I and II, but the expected scores did not. When using Rasch measures to generate expected scores for prediction, slightly better results might be obtained when the measure estimates are based on adjusted data. A productive adjustment makes observations in extreme categories slightly less extreme[16].

Figure 1
figure 1

Skewness analyses of questions 5 and 28 of the EORTC QLQ-C30. *, statistically significant difference in the cancer stage groups by the band of 95% CI not overlaied (the dot denotes point estimation of skewness coefficient; the border line in red represents the value of skewness coefficient equal to zero).

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Overall skewness analysis with 95% CI across items

In Figure 2 we can see that there were 6 items on stages showing significant differences in quality-of-life attitude by the expected response scores, but 11 items on stages gave significant differences by the observed response scores, indicating different inferences would have been made if we had chosen different data sources (i.e., observed and expected scores in Figure 2).

Figure 2
figure 2

Skewness analyses for EORTC QLQ-C30 by an item-by-item approach of cancer stages. *, statistically significant difference in the cancer stage groups by the band of 95% CI not overlaied (the dot denotes point estimation of skewness coefficient; the border line in red represents the value of skewness coefficient equal to zero).

Full size image

As shown in Table 3, the skewness for expected response scores had a higher correlation coefficient (≥ 0.90) than the skewness with observed scores (some ≤ 0.90), as compared to other counterparts, such as Rasch transformed item difficulties (which were derived from original observed response scores), observed and expected summation response scores. This indicates that the expected scores would be more meaningful than the observed scores for making inferences.

Table 3 Relationship between items among difficulty (logit), skewness, observed scores and expected scores (item number = 28).
Full size table

Discussion

The reliability of a test in a given population is defined as the ratio of true score variance to observed score variance [25]. In empirical research, true scores cannot be directly determined. One of the most frequently used methods of estimating internal consistency reliability is Cronbach's coefficient α, as shown in Formula 1, which is frequently cited (at least 5,590 times, including approximately 325 social science citations, per year) [2]. Our data indicate that CIs for Cronbach's α should be more widely calculated and reported for published data. The SPSS syntax procedure of intervals for reliability coefficients is included in Additional file 3, although it can be estimated in various ways [7]. We hope that in the future, authors will report Cronbach's α routinely and note what estimation methods they use. This would help clarify their score reliabilities and how these are affected by sampling error variance.

Findings

We utilized the Rasch model, using the term "ability" for a latent trait that underlies the questionnaire responses of the participants, to determine the probability of each examinee answering a given question accurately [26]. The expected response scores could then be derived from the original observed data matrix, which we used to obtain a more accurate Alpha and its intervals. Furthermore, the graphical representation of item-by-item skewness analyses using expected response scores was plotted to make more appropriate inferences and conclusions of difference detection among groups of interest, particularly on misfit responses.

Strengths of the study

Most data analysis of quality-of-life questionnaires is based on classic test theory (CTT). In recent years, CTT has been gradually replaced by item response theory (IRT) [10, 14, 27, 28]. Whereas CTT concerns the accuracy of observed scores, IRT concerns the accuracy of ability scores [29]. In this study, we applied IRT-based Rasch analysis to generate expected response scores and confidence intervals, approaching a more accurate understanding of our patient population and enabling proper treatment decisions.

From Figure 1 and 2 we can see that the 95%CIs for an one-samnple case far from the skewness value of zero is deemed that a statistically significant difference in an attitude toward positive (with a negatively skewed distribution if higher scores represent positive) or toward negative (with a positively skewed distribution if lower scores represent negative) is found (p < .05). However, for the two-sample case it should be cautious about interpreting the graphical reports when examining the attitude difference between groups by non-overlapping 95% confidence intervals (CIs). According to the criterion of 95% CIs derived from two independent samples(e.g., A and B), there is a distance beyond 2*SE_A + 2*SE_B between those two means with more than 95% confidence to regard as a significant difference emerged between two groups. That is said that (1) bootstrapping approach is especially suited to data following a non-normal distribution; (2) the strict criterion of 95% CIs derived from two independent samples shows that MB - MA > 2 × SE_A + 2 × SE_B > 2 × sqrt(SE_A^2 + SE_B^2) which holds a high confidence to regard as a significant difference emerged in two groups by checking non-overlapping 95% confidence intervals (CIs).

Limitations of the study

We simulated data from 115 patients newly diagnosed with nasopharyngeal carcinoma who responded to the 28 questions on the Taiwanese version of the EORTC QLQ-C30 questionnaire that requires answers on a scale of 1 to 4. Some bias might have occurred as a result of the Taiwanese translation of the questionnaire. If so, the item-calibrated difficulties, including step threshold difficulties, would be different among patients from different countries, which may lead to a variety of expected response scores by the Rasch model. The Alpha and 95% CIs shown in Table 2, according to sample size, would also be different in relation to different types of response outliers [9], different test forms, different numbers of questions, misspecifications and confounding independent variables in a single reliability generalization analysis [28].

The Cronbach's alpha has been frequently reported in CTT for a long time though Rasch person separation reliability was reported better than Cronbach's alpha as a test indication in recent years [28]. We do not decline the use of WINSTEPS command by SIFILE = data file name to simulate a data file equivalent to the raw data to obtain Rasch person separation reliability 95% CIs when demonstrating the use of Cronbach's alpha 95% CIs in Additional file 3.

In this study we assumed that the 28-item EORTC QLQ-C30 fits Rasch model's specification and forms a unidimensional construct. However, to our knowledge till now, it has not been either previously tested or reported using a Rasch analysis rather than sample and item dependent classical test models. For the reason of space limitation for this manuscript, we have not explored the dimensionality and invariance of the EORTC QLQ-C30 like the General Health Questionnaire (GHQ)-12 did [30] in detail through the Rasch model in this paper. A brief report regarding dimension and invariance of items has been enclosed in Additional file 4.

Applications

There are concerns that the expected response scores derived from observed raw scores may also yield unreliable results. The frequent counts of total items (similar to the item difficulties in this study) endorsed by examinees would not be overly distorted if we assume that the errors made by them were randomized across the questions. The specific misfit items observed by a particular group (e.g., stage II cancer patients on item 28 in Figure 2) could be well adjusted by expected response scores generated by the Rasch model's probability theory. In addition, the shortcoming of missing data handled in CTT could be overcome through the derived measures estimated by Rasch modeling.

The SPSS syntax procedures in Additional file 3, which has been revised referring to the previously published article [7], can help authors easily report Alpha intervals in studies involving quality-of-life investigation and surveillance. Furthermore, the skewness coefficient and its 95% CI, referring to expected response scores, can be used to verify whether the assessed attitude is neutral (approaching a normal distribution), in absolute agreement (with a negatively skewed distribution if higher scores represent agreement) or in disagreement (with a positively skewed distribution if lower scores represent disagreement). Use of this coefficient merits further study in other kinds of quality-of-life surveys. One major obstacle is that users may need some training to interpret the visual representation of the skewness analysis.

Conclusion

Intervals for reliability coefficients can be estimated in various ways. We demonstrated bootstrap and SPSS syntax procedures in this study. There is no particular reliability coefficient or CI estimation method that authors are expected to universally invoke [31, 32]. After all, each of the factors in a quality-of-life assessment was different in our study; however, they did not independently vary. We suggest that authors routinely report CIs in quality-of-life publications, to clarify the statistical estimates of the effect of sampling error variance on score reliability. These requirements may also facilitate the understanding that scores in a questionnaire are not completely associated with invariant reliability. Furthermore, using expected response scores is encouraged. Graphical skewness representation with 95% CI is also encouraged to depict patient quality of life in future studies.