Effect of rating scale categories on reliability and fit statistics of the Malay Spiritual Well-Being Scale using Rasch analysis.
1Aqil Mohammad Daher, 2Sayed Ahmad Hassan, 3Than Winn, 4Mohd Ikhsan Selamat

1,4Population health and preventive medicine, faculty of medicine University Teknologi MARA, Sungai Buloh, Selangor Malaysia
2Department of Psychiatry, faculty of medicine University Teknologi MARA, Selayang, Selangor, Malaysia
3Department of Community Medicine, Mahsa University College, Kuala Lumpur, Malaysia

Abstract
Introduction
Handful literatures have explored the effect of rating categories on reliability based on classical test theory indices, fewer studies have employed the item response theory in examining reliability. We carried out this study to examine the effect of rating scale categories (RSC) on reliability and fit statistics of the Malay Spiritual Well-Being Scale using Rasch model.
Methods
The Malay Spiritual Well-being scale (SWBS) with the original six and newly introduced three and four rating categories (RSC) was distributed randomly among three different samples of 50 participants each in different community screening programs at in Selangor state , Malaysia.
Results
The mean age of respondent in the three samples ranged 36-39 year, majority was females in all samples, and Islam was the major religion of respondents. Malay race was predominating followed by Chinese then Indian. Most participants were employed and more than a half in all samples reported as being married. The original 6 rating categories (RSC) showed better targeting of 0.99 and smallest model error (0.24). The INFIT MNSQ and ZSTD was 1.1 and (–) 0.1) respectively. The six categories (RSC) achieved highest person and item and reliability of 0.86 and 0.85 respectively. These reliabilities yielded highest person (2.46) and item (2.38) separation indices compared to other categories.

Conclusion
The person and item reliability and the fit statistics seemed to be better with six categories (RSC) compared to four and three categories(RSCs). No recommendation to be were made regarding reducing number of rating scale categories of the Malay SWBS

Introduction

Ensuring validity and reliability of an instrument have has stimulated researchers to formulate various approaches that optimize the psychometric properties of instrument. Reliability of a questionnaire represents the reproducibility of scores across different test situation. Reliability quantifies the measurement error and can be defined as the percentage of explained variance to the total possible variance of scores. (1)
According to classical test theory (CTT), reliability has been expressed in terms of stability, equivalence, and consistency. Methods of assessing each aspect include test-retest reliability as a measure of stability, alternate or parallel form as a measure of equivalence and the internal consistency is usually quantified using Cronbach’s alpha and/or Kuder–Richardson-20 (KR-20)(2).
With the inception of the modern Item response theory (IRT), a new paradigm has emerged in assessing reliability. Rasch model is a statistical model which is closely related to IRT; it examines the observed data against a standard model and provides insight to the reliability, validity and fitness of that instrument to the target population. Rasch quantifies person ability and allocates it along a continuum of item difficulty to provide insight/guide about performance of the test. The more the information obtained about person ability and item difficulty is obtained the less measurement error and better reliability.
The main difference in concept of reliability in regards to CTT and IRT is the Standard Error (SE) used in the calculation of reliability coefficient. In the CTT the item difficulty estimate is, sample rated, computed as sample response to that item, assuming the ordinal raw score as linear when it is not. The SE of measurements is then derived from the average of sample ability, so the estimation based on average person ability not individual person ability. As such the error variance will be overestimated especially with inclusion of extreme persons who got less error variance (3).
Unlike the conventional CTT, Rasch provides two indicators of reliability; person and item reliability related to quantified person ability and item difficulty. Rasch models provide a direct estimate of the modeled error variance for each estimate of a person’s ability and an item’s difficulty measured on interval scale scores(4). Individual SEs are more useful than a sample or test average, and the extreme scores are usually excluded because their SEs are infinitely large and those measures therefore contain little information about those person’s or item’s location at the extreme of the ability continuum (5, 6).

Adding to that, person and item reliability are then transformed into separation index to overcome the restriction of range of reliability value of being 0-1, this separation index used to gauge reliability when reliability remained unchanged across repeated analysis of the same data.
The number of categories of the rating scale and its effect on reliability has been a subject of debate over the last decades. The earlier attempt to identify the effect of category number on reliability had revealed that there is no effect of the number of categories on the reliability (7). Studies showed unchanged test-retest reliability using rating categories with three, five, seven or nine response categories (8, 9) . Other researchers had also confirmed those finding [pic](10-13).
Contradictory to the above-mentioned findings, authors have concluded that increasing the number of response categories would improve the reliability; inter-rater reliability (14), test-retest reliability [pic](15-17). Findings from other published work confirmed that reliability increase with increasing response categories but not beyond six categories [pic](18-20). In regards to the optimum number of categories, authors argued that reliability is better with seven responses (14, 21) and others recommended a five instead (22, 23).

The original spiritual Well-Being scale- English version under investigation is a widely used self administered scale that measures the spiritual Well-Being. Up to date no published article has examined the suitability of the rating scale whether in the original setting and language or when it has been translated, particularly in Malay Language. The objective of this paper is to empirically investigate the potential effect of number of response categories on the reliability and fit statistics of the Malay version SWBS using Rasch analysis.

Materials and method

Instrument
The Malay version of the SWBS has been translated from original English version according to Malay language according to the WHO recommendation by WHO (24), in which two independent translators carried out the forward translationors and another two backward translators. came up with This resulted in a Malay-English translated version and a consensus reached about the final version of the Malay version of SWBS. The original SWBS was developed by Palotzian and C. Ellison (1983). It has 20 items which are anchored to a 6 point Likert-like scale from strongly agree to strongly disagree.

Statistical analysis
Data were entered and analyzed using Winstep Rasch analysis software.
Rasch analysis provides a customary summary statistics of fit indices like INFIT MNSQ (mean square), INFIT ZSTD(Standardized mean square) ; and reliability indices including; person reliability, item reliability; and person and item separation indices achieved by the instrument and the target sample. Fit statistics indicate how accurately or predictably data fit the model, reliability is the likelihood that the same order of items and persons will be replicated in a different sample. Separation index is, calculated as the number of SE of spread among the items/persons, the ability to define a distinct groups of items difficulty and person abilities.

Sample /setting • The current work is a part of the validation and reliability testing study of the Malay SWBS, in which pre-testing was a continuous process to ensure a valid and reliable instrument. The final linguistically checked version of the Malay SWBS with original 6 response categories was administered to a sample of 50 respondents who were participating in an ongoing community screening program that covers most area of Selangor state in Malaysia. Interestingly all participants were duly verified to be Malaysians who are able to read and write in Malay language as a requirement to participate in this study. Ethical approval was obtained from relevant institutional ethics committee.

Statistical analysis
Data were entered and analyzed using Winstep Rasch analysis software.
Rasch analysis provides a customary summary statistics of fit indices like INFIT MNSQ (mean square), INFIT ZSTD(Standardized mean square) ; and reliability indices including; person reliability, item reliability; and person and item separation indices achieved by the instrument and the target sample. Fit statistics indicate how accurately or predictably data fit the model, reliability is the likelihood that the same order of items and persons will be replicated in a different sample. Separation index is, calculated as the number of SE of spread among the items/persons, the ability to define a distinct groups of items difficulty and person abilities.

Results • The summary of category structure and Category Probability plot of the original 6 categories rating scale showed the need to collapse the categories as the step calibration (Rasch-Andrich thresholds) was not increasing monotonically (category disordering) and the category measure showed poor progression(Figure 1).

• Different iteration were obtained to identify which categorization would perform better, the better rating scale combination was decided by category probability plot and Item and Person Infit MNSQ and ZSTD. Figures 2 and 3 show the category probability plot of the best possible post-hoc categorization along with table 1 that shows the summary statistics of the original rating scale and post-hoc analysis.

Table: 1
| categories |Original 123456 |Post-hoc 111234 |Post-hoc 111223 |
| |Persons |Items |Persons |Items |Persons |Items |
|STATISTICS |Mean |SD |Mean |SD |Mean |SD |
|SEPARATION |2.46 |2.38 |2.86 |2.70 |2.76 |2.77 |
|SE |0.11 |0.9 |0.14 |0.13 |0.22 |0.20 |

• We found that four and three categories as shown in table 1 were the best among other iterations. • It is observable that the 4 category targeting was very good compared to the original 6 categories and yielded highest person reliability and separation. It was noticeable that the mean person INFIT MNSQ of the four categories achieved the ideal value of 1 and the person mean INFIT ZSTD was unchanged between different categorization. The smallest model error was achieved by the 4 categories. The mean item INFIT MNSQ was 1 for the four and three categories achieved the ideal value of 1 and the mean item INFIT ZSTD was) for the 6 and 3 categories. Item and person reliability and separation indices were not dramatically changed and all were in good range.

• The six categories rating scale was collapsed into 4 categories (Strongly agree, Agree, Disagree Agree and, Strongly Disgree) in one version and into 3 categories in another version (agree, neither agree nor disagree, and disagree). Subsequently, the modified versions of Malay SWBS with FOUR and THREE categories were administered randomly into TWO samples of 50 respondents at other sessions of community screening program.

• Table 2 shows the characteristics of the study samples, the mean age was almost comparable in the three samples. In terms of gender, females were predominating in the three samples with higher proportion reported in the third sample (70.8%). Muslims were the majority in the first and third samples while Christian were predominating the second sample. In regards to educational level, the third sample reported a higher proportion of primary and secondary education (27.7% 44.7 % respectively) compared to other samples, while the second and third samples reported higher proportion of university degrees 75.6 % and 46.9 % respectively. Malay participants were majority in the first and third samples followed by Chinese while Chinese were majority in the second sample followed by Malay. Employed participants were highly represented in all samples compared to other working status groups. More than a half in all samples reported as being married and less than half (43.8 % ) and around third 28.9% reported single status in the first and second samples respectively

Table 2 descriptive statistics of the study samples
| | |6 cat |4 cat |3 cat |
|Age Mean(SD) | |36(11) |37(9) |39(12) |
|Gender n(%) |Male |19(38.8 ) |18(40.0) |14(29.2) |
| |Female |30(61.2 ) |27(60.0) |34(70.8) |
|Religion n(%) |Muslim |21(42.9) |11(24.4) |31(64.6) |
| |Buddhist |13(26.5) |8(17.8) |10(20.8) |
| |Hindu |12(24.5) |5(11.1) |4(8.3) |
| |Christian |3(6.1) |15(33.3) |3(6.3) |
| |Others |0(0) |6(13.3) |0(.0) |
|Educational Level n(%) |No formal education |0(0) |0(0) |3(6.4) |
| |Primary |1(2) |0(0) |13(27.7) |
| |Secondary |15(30.6) |5(11.1) |21(44.7) |
| |Diploma |10(20.4) |6(13.3) |8(17.0) |
| |University Degree |23(46.9) |34(75.6) |2(4.3) |
|Race n(%) |Malay |21(42.0) |11(24.4) |31(64.6) |
| |Chinese |16(32.0) |16(35.6) |10(20.8) |
| |Indian |13(26.0) |10(22.2) |7(14.6) |
| |Others |0(0) |8(17.8) |0(.0) |
|Occupation n(%) |Unemployed |1(2.1) |1(2.2) |8(17.0) |
| |Employed |45(93.8) |34(75.6) |18(38.3) |
| |Pensioner |1(2.1) |0(0) |3(6.4) |
| |Student |0(0) |7(15.6) |3(6.4) |
| |Housewife |1(2.1) |3(6.7) |15(31.9) |
|Marital Status n(%) |Single |21(43.8) |13(28.9) |7(14.6) |
| |Married |25(52.1) |31(68.9) |39(81.3) |
| |Separated/Divorced |2(4.2 ) |1(2.2) |0(0.0) |
| |Widow |0(0) |0(0) |2(4.2) |
| | | | | |

Table 3: Summary statistics of

| categories |Original 123456 |112344 |112233 |
| |Persons |Items |Persons |Items |Persons |Items |
|STATISTICS |Mean |SD |Mean |SD |Mean |SD |
|SEPARATION |2.46 |2.38 |1.98 |1.84 |1.75 |2.23 |
|SE |0.11 |0.9 |0.18 |0.13 |0.19 |0.27 |

Table 3 depicts the fits statistics and reliability indices for the three categorization of the rating scale. From the table above we observe that the original 6 categories achieved a good targeting of the less than one error and achieved the smallest model error compared to the newly introduced 4 and 3 categories.
Although the difference was not sizeable, the 3 categories achieved the smallest mean person INFIT MNSQ of 1.03 with the smallest SD of 0.38.
The three categories registered the ideal mean person INFIT ZSTD of ZERO and the others of -0.1. The SD of person INFIT ZSTD reduced slightly toward the smallest number of categories.
The three categories registered the lowest mean and SD of the item INFIT MNSQ compared to other categories. On the other hand 6 categories registered the lowest mean item INFIT ZSTD of ZERO. The SD of the item measure has increased slightly with four categories and sizably with three categories to reflect more variation in the estimate of item difficulty.
In terms of reliability and separation index, it is observable that the 6 categories yielded the highest person and item reliability compared to others. Similarly, the separation indices were higher for the 6 categories.

Discussion
Reliability is complex concept where there is no single measurement/parameter that quantifies this psychometric property. To our knowledge this is the first study which examines the effect of number response categories on reliability of the Malay SWBS using Rasch model. Of the advantages of Rasch model measurement, it allows estimation of/measuring the source of error arising from person and items, whereas other method of testing reliability report statistics about the performance of the respondents only.
The observed difference of/in the Infit MNSQ of the persons was not substantial/sizable between the three groups and it shows a good fit as it approaches 1(25-27). The minimal changes of person infit ZSTD did not show significantly better fit of the four and three categories compared to 6 and didn’t cause a major noise to the fit statistics. The variation in the estimate of person ability is comparable between 6 and 4 categories and still lower than 3 categories.

The highest possibility that person will behave in similar way when they are subjected to items of similar difficulties was conferred by the 6 categories (0.86), We expected that 6 categories would perform better to separate the respondent to endorse the items into more than three strata which is considered good . Similarly Item reliability reduced with four categories compared to 6 categories and it can be said that items give less reliable information than the persons in this sample. In addition, the item separation was better for the 6 categories (2.38) and three categories (2.23) compared to 4 categories (1.84), i.e. item endorsement (difficulty) by respondents were separated into THREE levels Compared to TWO.

The findings may be explained by the fact that reducing the number of categories means reducing endorsement spectrum which yield score with little variance which will affect the separation index that is needed to be as high as possible (28-30).

The rating categories which yielded high reliability and separation index were found to be those with many response categories .These findings are in line with previous studies which showed that increasing number of categories would increase reliability[pic](14, 17-21).
Our results contradict the findings reported by Zhu where the smaller number of categories improved the reliability (31). Also in opposite to other finding that reliability were not affected or better with smaller number of response categories[pic](11-13), and Garner who emphasized that 20 categories are necessary needed for better reliability(32).

Limitation
Although Rasch has been perceived to handle small sized data, the study might be limited by the small sample size which will increase the SE and hence reduce reliability. However the reliability indices were high enough with such small sample. Our intention is to present real life data rather than relying on simulation of large data base.
Secondly, using different groups of respondents to answer the three different rating scale categorization might affect the comparison. It would be better if the SAME persons answer the 3 different categorizations, but practically it was not feasible.

Conclusion
The person and item reliability and, to lesser extent, the fit statistics seemed to be better with six categories compared to four and three categories. No recommendation to be is made regarding reducing number of rating scale categories of the Malay SWBS.

Acknowledgment

The authors wish to express gratitude to the Research Management Institute of the Universiti Teknologi MARA for providing a research grants (600-RMI/ST/DANA 5/3/ Dst (82/2011) that enables researcher to accomplish this humble work as a part of PhD project at UiTM.
The authors thank Dr Zarilah AbdulAziz for her great help in analyzing the data. The authors also thank Mohd Zali Mohd Nor from Universiti Putra Malaysia whom comments and review helped finalizing this manuscripts.

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-----------------------
Figure 1: Category Probability plot -6 categories (123456)

SUMMARY OF CATEGORY STRUCTURE. Model="R"
+------------------------------------------------------------------
|CATEGORY OBSERVED|OBSVD SAMPLE|INFIT OUTFIT||STRUCTURE|CATEGORY|
|LABEL SCORE COUNT %|AVRGE EXPECT| MNSQ MNSQ||CALIBRATN| MEASURE|
|-------------------+------------+------------++---------+--------+
| 1 1 19 2| .78 -.11| 2.03 3.04|| NONE |( -2.39)| 1
| 2 2 45 5| .28* .08| 1.31 1.36|| -.88 | -1.18 | 2
| 3 3 99 10| .25* .33| .92 .93|| -.59 | -.42 | 3
| 4 4 353 36| .55 .66| .75 .70|| -.78 | .38 | 4
| 5 5 114 12| .90 1.09| 1.21 .85|| 1.99 | 1.27 | 5
| 6 6 344 35| 1.79 1.67| .74 .78|| .26 |( 2.36)| 6
|-------------------+------------+------------++---------+--------+
|MISSING 6 1| .05 | || | |
+------------------------------------------------------------------
OBSERVED AVERAGE is mean of measures in category. It is not a parameter estimate.

CATEGORY PROBABILITIES: MODES - Structure measures at intersections
P ++-----------+-----------+-----------+-----------+-----------++
R 1.0 + +
O | |
B | |
A |111 44|
B .8 + 1111 444 +
I | 111 44 |
L | 111 44 |
I | 11 4 |
T .6 + 111 44 +
Y | 11 22222222222 4 | .5 + 112222 222 44 +
O | 22211 22 4 |
F .4 + 22 11 224 + | 222 111 4422 |
R | 222 11 4 2 |
E | 222 11 44 22 |
S .2 + 2222 111 44 333333 22 +
P |2222 ****333 333***333 |
O | 33**** 1111 222*333|
N | 333333****44 111111 222|
S .0 +*******************444444 111111111111+
E ++-----------+-----------+-----------+-----------+-----------++ -3 -2 -1 0 1 2 Person [MINUS] Item MEASURE

Category Probability plot- 3 categories (111223)

SUMMARY OF CATEGORY STRUCTURE. Model="R"
+------------------------------------------------------------------
|CATEGORY OBSERVED|OBSVD SAMPLE|INFIT OUTFIT||STRUCTURE|CATEGORY|
|LABEL SCORE COUNT %|AVRGE EXPECT| MNSQ MNSQ||CALIBRATN| MEASURE|
|-------------------+------------+------------++---------+--------+
| 1 1 163 17| -.68 -1.00| 1.32 1.47|| NONE |( -2.59)| 1
| 2 2 467 48| .04 .27| .84 .73|| -1.45 | .00 | 4
| 3 3 344 35| 2.30 2.14| .81 .84|| 1.45 |( 2.59)| 6
|-------------------+------------+------------++---------+--------+
|MISSING 6 1| -1.52 | || | |
+------------------------------------------------------------------

CATEGORY PROBABILITIES: MODES - Structure measures at intersections
P ++---------+---------+---------+---------+---------+---------++
R 1.0 + +
O | |
B | |
A | |
B .8 +1111 3333+
I | 11 33 |
L | 11 22222 33 |
I | 111 22222 22222 333 |
T .6 + 11 222 222 33 +
Y | 11 22 22 33 | .5 + 122 223 +
O | 22211 33222 |
F .4 + 22 11 33 22 + | 22 11 33 22 |
R | 222 111 333 222 |
E | 22 11 33 22 |
S .2 + 222 111 333 222 +
P |2 33*11 2|
O | 3333 1111 |
N | 33333333 11111111 |
S .0 +3333333333333333 1111111111111111+
E ++---------+---------+---------+---------+---------+---------++ -3 -2 -1 0 1 2 3 Person [MINUS] Item MEASURE

Figure2: Category Probability plot -4 categories(111234)

SUMMARY OF CATEGORY STRUCTURE. Model="R"
+------------------------------------------------------------------
|CATEGORY OBSERVED|OBSVD SAMPLE|INFIT OUTFIT||STRUCTURE|CATEGORY|
|LABEL SCORE COUNT %|AVRGE EXPECT| MNSQ MNSQ||CALIBRATN| MEASURE|
|-------------------+------------+------------++---------+--------+
| 1 1 163 17| -.47 -.69| 1.44 1.60|| NONE |( -2.35)| 1
| 2 2 353 36| -.22 -.12| .77 .77|| -1.19 | -.39 | 4
| 3 3 114 12| .33 .57| 1.07 .90|| 1.34 | .72 | 5
| 4 4 344 35| 1.42 1.33| .81 .83|| -.16 |( 1.86)| 6
|-------------------+------------+------------++---------+--------+
|MISSING 6 1| -1.03 | || | |
+------------------------------------------------------------------
OBSERVED AVERAGE is mean of measures in category. It is not a parameter estimate.

CATEGORY PROBABILITIES: MODES - Structure measures at intersections
P ++-----------+-----------+-----------+-----------+-----------++
R 1.0 + +
O | |
B | |
A |111 44|
B .8 + 1111 444 +
I | 111 44 |
L | 111 44 |
I | 11 4 |
T .6 + 111 44 +
Y | 11 22222222222 4 | .5 + 112222 222 44 +
O | 22211 22 4 |
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q õçÙçõÎõçõçÉÅÉÅÉÅÉÅÀ¸±ª£¸ª±ª¸ª˜?…?…~wpfphœL>F .4 + 22 11 224 + | 222 111 4422 |
R | 222 11 4 2 |
E | 222 11 44 22 |
S .2 + 2222 111 44 333333 22 +
P |2222 ****333 333***333 |
O | 33**** 1111 222*333|
N | 333333****44 111111 222|
S .0 +*******************444444 111111111111+
E ++-----------+-----------+-----------+-----------+-----------++ -3 -2 -1 0 1 2 Person [MINUS] Item MEASURE