Estimating Internal Consistency Using Bayesian Methods

Transcription

1 Journal of Modern Alied Statistical Methods Volume 0 Issue Article Estimating Internal Consistency Using Bayesian Methods Miguel A. Padilla Old Dominion University, maadill@odu.edu Guili Zhang East Carolina University, zhangg@ecu.edu Follow this and additional works at: htt://digitalcommons.wayne.edu/jmasm Part of the Alied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons Recommended Citation Padilla, Miguel A. and Zhang, Guili (20) "Estimating Internal Consistency Using Bayesian Methods," Journal of Modern Alied Statistical Methods: Vol. 0 : Iss., Article 25. DOI: /jmasm/ Available at: htt://digitalcommons.wayne.edu/jmasm/vol0/iss/25 This Regular Article is brought to you for free and oen access by the Oen Access Journals at DigitalCommons@WayneState. It has been acceted for inclusion in Journal of Modern Alied Statistical Methods by an authorized editor of DigitalCommons@WayneState.

2 Journal of Modern Alied Statistical Methods Coyright 20 JMASM, Inc. May 20, Vol. 0, No., //$95.00 Estimating Internal Consistency Using Bayesian Methods Miguel A. Padilla Old Dominion University Guili Zhang East Carolina University Bayesian internal consistency and its Bayesian credible interval (BCI) are develoed and Bayesian internal consistency and its ercentile and normal theory based BCIs were investigated in a simulation study. Results indicate that the Bayesian internal consistency is relatively unbiased under all investigated conditions and the ercentile based BCIs yielded better coverage erformance. Key words: Bayesian internal consistency, coefficient alha, confidence interval, Bayesian confidence interval, coverage robability. Introduction Psychological constructs are the building blocks of sychological/behavioral research. Indeed, one can easily argue that constructs are the foundation of these two sciences. A tyical way of measuring a construct is through a questionnaire containing items that are urorted to indirectly measure the construct of interest; thus, it becomes imortant that the items be consistent or reliable so that the questionnaire itself is consistent or reliable. Although there are several methods of measuring or estimating the reliability of a questionnaire, by far the most commonly used is coefficient alha. Coefficient alha has remained oular since its introduction in Cronbach s (95) article based on the work of Guttman and others in the 940s (Guttman, 945). Coefficient alha is a measure of internal consistency for a grou of items (i.e., questions) that are related in that they measure the same sychological/behavioral construct (Cortina, 993). There are three main reasons for coefficient alha s oularity. First, Miguel A. Padilla is an Assistant Professor of Quantitative Psychology in the Deartment of Psychology. His research interests are in alied statistics and sychometrics. him at: maadill@odu.edu. Guili Zhang is an Assistant Professor of Research Methodology in the Deartment of Curriculum and Instruction. Her research interests are in alied statistics. her at: ZHANGG@ecu.edu. coefficient alha is comutationally simle. The only required quantities for its comutation are the number of items, variance for each item and the total joint variance for all the items; quantities that can easily be extracted from the item covariance matrix. Second, coefficient alha can be comuted for continuous or binary data: this is a significant advantage when working with right/wrong, true/false, etc. items. Third, it only requires one test administration: Most other forms of reliability require at least two test administrations, which come at a cost of time and resources. For these reasons coefficient alha s ower to assess the sychometric roerty of the reliability of a measurement instrument is widely used and it has remained relatively unchanged for over 60 years. The advent of Bayesian methodology has brought about exciting and innovating ways of thinking about statistics and analyzing data. Bayesian methods have several advantages over traditional statistics, sometimes referred as frequentist statistics (Gelman, 4; Lee, 4), but two advantages stand out. First, researchers can now incororate rior knowledge or beliefs about a arameter by secifying a rior distribution for the arameter in the model; thus, the analysis is now comosed of data and rior knowledge and/or beliefs. By contrast, traditional or frequentist analyses are comosed only of data. Through this combination of data and rior knowledge, more can be learned about the henomenon under study and knowledge about the henomenon can be udated accordingly. Second, Bayesian methods rovide 277

3 ESTIMATING INTERNAL CONSISTENCY USING BAYESIAN METHODS credible intervals (BCIs), the Bayesian analog to confidence intervals (CIs). However, credible intervals have a different interretation from confidence intervals. A confidence interval allows one to make statements, such as we are 95% confident that the interval catures the true oulation arameter. By contrast, a BCI allows one to say that we are 95% confident that the true oulation arameter lies between the bands of the credible interval, a simler and more owerful statement. This is, in fact, the interretation most researchers would like to make with confidence intervals. A Bayesian coefficient alha retains the simlicity and ower of the original coefficient alha, but it also has the advantages of Bayesian methodology. By incororating rior internal consistency information into the current estimation of coefficient alha, more can be learned about the internal consistency of a measurement instrument and knowledge about the instrument can be udated accordingly. Additionally, credible intervals are generated for the Bayesian coefficient alha. The bootstra is the common method for generating confidence intervals for coefficient alha; however, the bootstra confidence interval has the same interretation as the confidence interval from traditional statistical methods. With credible intervals direct statements can be made about where the true oulation coefficient alha lies, which is a clear advantage over the standard confidence interval. Prior Research on Coefficient Alha CIs As with all statistics, coefficient alha is a oulation arameter and must be estimated from samles; thus, it is subject to samling error that contributes to the variability around the true oulation arameter. Due to this, current statistical thinking and ractice oint to the need for roviding confidence intervals to sulement oint estimates and statistical tests (Duhachek, Coughlan & Iacobucci, 5; Duhachek & lacobucci, 4; Iacobucci & Duhachek, 3; Maydeu-Olivares, Coffman & Hartmann, 7). Many rofessional ublications are beginning to require authors to rovide CIs in addition to oint estimates, standard errors and test statistics. For examle, the American Psychological Association Task Force on Statistical Inference (Wilkinson, 999) emhasizes the obligation of researchers to rovide CIs for all rincial outcomes; however, generating CIs for coefficient alha has remained somewhat elusive and rarely imlemented in ractice. Confidence intervals for coefficient alha were first introduced by Kristof (963) and Feldt (965). These CIs assume that items are normally distributed and strictly arallel (Allen & Yen, 979; Crocker & Algina, 986), which imlies that the item covariance matrix is 2 comound symmetric; i.e., σ + σ I ( i= j) where σ are the item variances, and σ 2 are the item covariances, and I(.) is the indicator function. These confidence intervals, however, do not erform well when items are not strictly arallel (Barchard & Hakstian, 997). Given that the strictly arallel assumtion is unreasonable in alied research and that these CIs do not erform well when this assumtion is violated may be the reason why coefficient alha CIs are rarely imlemented in alied research (Duhachek & lacobucci, 4). An imrovement to the CIs roosed by Kristof (963) and Feldt (965) was introduced by van Zyl, Neudecker and Nel (0) who showed that the standard method of estimating coefficient alha is a maximum likelihood estimator (MLE) and derived its corresonding CIs. Although the coefficient alha MLE assumes that items are normally distributed, a major advantage is that it does not require the comound symmetry assumtion of the item covariance matrix. In a simulation study, Duhacket and Iacobucci (4) comared the erformance of the coefficient alha CIs for the method roosed by Feldt (965) and the MLE roosed by van Zyl, et al. (0) under a nonarallel measurement model. Their results indicate that the MLE method consistently outerformed the cometing methods across all simulation conditions, but because the MLE method assumes that items are normally distributed, when the assumtion is violated, the results can be untrustworthy. Normally distributed items are not a comletely realistic assumtion in sychological/behavioral research. Most items in measurement instruments are dichotomous 278

4 PADILLA & ZHANG (yes/no, true/false, etc.) or Likert-tye items with several ordinal items: for these item tyes, normality is an unrealistic assumtion. From this ersective, Yuan and Bentler (2) extended the results of the coefficient alha MLE to a wider range of distributions, ointing out that it is robust to some violations of normality. However, they oint out that it is difficult to verify conditions to which the coefficient alha MLE is robust to item non-normality. Thus, if the conditions cannot be verified theoretically then they are even more difficult to verify in alied work. Yuan and Bentler (3) built on this by introducing what Maydeu-Olivares and colleagues (7) call asymtotically distribution-free (ADF) CIs for coefficient alha. In this study the authors comared the ADF, MLE, and bootstra coefficient alha CIs estimated from the Hokins Symtom Checklist (HSCL; Derogatis, Liman, Rickels, Uhlenhuth & Covi, 974). The results of Yuan and Bentler suggest that the ADF CIs are between the MLE and bootstra methods in terms of their accuracy. However, they oint out that the ADF CIs cannot describe the tail behavior of coefficient alha of the HSCL due to the small samle (n = 49); they suggest that with a larger samle size the ADF CIs could better describe the distribution of coefficient alha. Maydeu-Olivares, et al. (7) extended the work by Yuan, et al. (3) by simlifying the comutation of ADF CIs and investigating its erformance under several simulation conditions. Of articular interest was the comarison of the ADF CIs to the MLE CIs under various conditions of skewness and kurtosis. In general, they concluded that - with aroximately normal items - the MLE CIs erform well even with a samle size as small as. However, once the items begin to deviate from normality, the ADF CIs begin to outerform the MLE CIs. In articular, the ADF CIs outerform MLE CIs with as little a samle size of. When the samle size gets larger than the ADF CIs erform well regardless of the skewness and kurtosis investigated by the researchers. Recent research has thus been fruitful in investigating the roerties of coefficient alha CIs; however, these CIs are based on traditional frequentist statistics. As such, they have the traditional interretation of CIs and cannot be udated with rior information. The rimary urose of this study is to develo a Bayesian internal consistency estimate and to evaluate its erformances by investigating some of its roerties through simulation. Coefficient Alha Consider a measurement instrument containing items, y, y 2,, y, that indirectly measure a single dimension, attribute, or construct. A useful and common comutation in the sychological/behavioral sciences is the comosite Y = y + y y. This comosite is laced in statistical models such as ANOVA and regression when conducting research using the corresonding attribute as a variable. Therefore, it is imortant to know the reliability of the comosite and hence the construct being measured. A oular way to estimate the comosite reliability is through coefficient alha. Coefficient alha for the oulation is defined as ii σ i= αc =. () σij i= j= where and σ is the sum of all item variances ii i= σ ij is the sum of all item variances i= j= and covariances. For a samle of size n, oulation arameters are relaced by samle estimates to obtain a coefficient alha estimate as ˆii σ i= ˆ αc =. (2) ˆ σ ij i= j= Note that coefficient alha is being subscrited with c to distinguish it from the other forms that 279

5 ESTIMATING INTERNAL CONSISTENCY USING BAYESIAN METHODS will shortly be introduced. Recall that Zyl, et al. (0) showed that ˆc α is the MLE for α c. Coefficient alha has three interesting roerties imlied from the classical true score model (Allen & Yen, 979; Crocker & Algina, 986). First, when all items have equal true scores that relate equally to the observed scores along with equal measurement error variance, the items are said to be arallel. In this case the covariance matrix for the items has a comound 2 symmetric structure; i.e., σ + σ I ( i= j). Second, when the measurement error variances are not equal, the items are said to be tauequivalent. In both of these conditions coefficient alha is equal to the reliability of a measurement instrument. Lastly, when the true scores do not relate equally to the observed scores and measurement error variances are not equal, the items are congeneric. This last condition is the more general and in this case coefficient alha underestimates the reliability of a measurement instrument. It is from these three conditions that the conclusion ˆc α ρ xx is made, where ρ xx is the reliability coefficient of a measurement instrument. Bayesian Internal Consistency The cornerstone of Bayesian methodology is Bayes theorem. Through Bayes theorem all unknown arameters are considered random variables. Due to this, rior distributions must be initially defined, which is a way for researchers to exress rior beliefs or available information before data are involved in the statistical analysis. Using the observed data y and rior distribution (θ), a osterior distribution π(θ y) of the arameters θ can be constructed. The osterior distribution can be fully exressed through Bayes theorem as ( θ y) ( θ) ( ) ( ) L π ( θ y) = L θ y θ L θ y θ dθ Θ ( ) ( ) (3) where L(θ y) is the data likelihood function and L ( y θ) ( θ) d θ Θ is a normalizing constant. One can directly see that the osterior is comosed of both actual data and rior beliefs or knowledge about the arameter. After the osterior π(θ y) is constructed it can be summarized by the mean and SD (or SE) along with other summarizing quantities. For examle, the mean and variance can be comuted as and E ( ) π ( ) θ y = θ θ y dθ (4) Θ 2 ( θ ) = E( ) π( ) var Θ θ θ d (5) where the SD var ( θ ) ( ) = y is also the SE for E θ y. At times, the median (or th ercentile) comuted as P( θ y m) = P( θ y m) (6) 2 is of interest as it is less influenced by extreme values. For the Bayesian coefficient alha (Balha), first start with the multivariate normal distribution. The osterior of a multivariate normal can be described by ( μσ, y) L( μσ, y) ( μσ, ) L( μσ, y) ( μσ ) ( Σ) π =. (7) On the far right of (6), note that the rior for the mean is directly deendent on the rior covariance, in addition, this indicates that a different rior is secified for the covariance matrix and mean vector. By choosing the following conjugate riors for both the covariance matrix and mean vector ( d ) Σ Λ (8) ~ W 0, 280

6 PADILLA & ZHANG μ Σ ~ N μ0, Σ, (9) n0 Anderson (984) and Schafer (997) showed that the osterior distribution for the covariance matrix and mean vector is Σ y ~ W n+ d 0, nn 0 ( n ) S + Λ+ ( y μ )( y μ ) n+ n (0) μ ( Σ, y) ~ N ( ny+ n0μ0), Σ n+ n0 n+ n0 () where W () denotes an inverted Wishart distribution and d 0, Λ, μ 0, and n 0 are hyerarameters chosen by the analyst, and y and S are the mean vector and covariance matrix estimated from the data. Thus, the osterior of the multivariate normal is described by two distributions which jointly are called the normalinverse Wishart distribution. Note that a rior needs be secified for μ and Σ. If no rior is available a generic noninformative rior such as ( θ ) can be used. In this case the osterior is comletely defined by the data. This arameterization fully describes the osterior and it can now be directly comuted. The coefficient alha osterior can be difficult to obtain directly. However, by simulating t =, 2,..., T values from (9) and (0) as Σ y and μ ( Σ, ) y, the estimation of the coefficient alha osterior distribution can be obtained as α = i= c σ i= j= ii σ ij (2) where σ ii () and σ t are elements of ij Σ y. A Bayesian coefficient alha (Balha) can then obtained as b ( α ) α = E y. (3) An alternative Bayesian coefficient alha (BalhaM) can be obtained through P ( αc αb, m) P( αc αb, m) y = y /2. Bayesian credible intervals can then obtained by the lower α/2 and uer α/2 ercentiles of the samle, where α is the tye I error rate. One can also obtain a normal theory based credible interval as α b ± Zα /2SD. Other summary measures can also be comuted as indicated above. Methodology Simulation A Monte Carlo simulation design was utilized to investigate the roerties of Bayesian coefficient alhas. First, the number of items was investigated: 5, 0, 5 and 20 and it was found that coefficient alha increases as a function of the number of items, however, it is constrained to one. Although it is ossible for tests and/or surveys to have more than 20 items, going beyond 20 items reaches a oint of diminishing returns in terms of investigating coefficient alha. Second, the mean item correlation ( r ) was investigated: 0.73, 0.223, and The mean item correlation is defined as r = 2 i< j r ij c ( ) (4) These mean items correlations were investigated because they generate coefficient alhas that range from 0. to 0.90, a sufficient range to investigate the roerties of the Balha. Third, samle size was also exlored:,,,, 2 and. As is the case for the number of items, going beyond a samle 28

7 ESTIMATING INTERNAL CONSISTENCY USING BAYESIAN METHODS size of reaches a oint of diminishing returns in terms of investigating coefficient alha (Duhachek & lacobucci, 4); however, these are samle sizes tyically found in sychological/behavioral research. Table resents coefficient alha as a function of mean item correlation and number of items and shows a reasonable range of coefficient alha that may be found in sychological/behavioral research. Table : Poulation Coefficient Alha for Items by Mean Item Correlations Items Mean Item Correlations Multivariate normal data were generated with mean vector zero and correlation matrix R of dimensions defined by the number of items in the simulation. R was chosen to have homogenous off-diagonal elements that generated the corresonding mean item correlation. For each condition of the simulation study,000 relications were obtained. In each relication, Balha was comuted along with the SE and 95% BCIs. Relative bias for Balha was comuted as: ˆ αb α ˆ αb =. (5) α The average of the estimated SE was comuted as SE = B i= SE ( ˆ αbi, ) B (6) where B is the number of relications. Lastly, two forms of BCI intervals were comuted. The first BCI was obtained by the lower α/2 and uer α/2 ercentiles of the samle. The second BCI was obtained as ˆ α /2 ( ˆ b ± Zα SE αb). The coverage robability of the 95% BCIs were comuted as the roortion of times the BCI contains the oulation arameter α c. Coverage can be judged by forming confidence intervals around the coverage. Coverage should not fall aroximately two standard errors (SEs) outside the nominal coverage robabilities () (Burton, Altman, Royston & Holder, 6). The standard error is defined as ( ) SE ( ) = (7) B where B is the number of simulations in the study. For the current study, =.95 with SE ( ) = and the CI is [ ].936,.964. Thus, coverage that falls outside this CI is considered unaccetable. For this study, Balha and corresonding 95% BCIs were estimated from a total of,000 simulations from the osterior distribution. In addition, the rior for Balha was set to be noninformative. A noninformative rior essentially lets the data essentially seak for themselves. Results Relative bias for Balha and corresonding standard errors are reorted in Table 2. First, Balha and BalhaM always tend to slightly underestimate the oulation coefficient alha; however, both Balha and BalhaM are relatively unbiased under all investigated conditions. Second, Balha and BalhaM estimates get closer to the oulation coefficient alha as samle size increases. Third, Balha and BalhaM estimates get closer to the oulation coefficient alha as the number of items increases. In addition, Balha and BalhaM estimates get better as the mean item correlation increases. Lastly, BalhaM is consistently closer to the oulation coefficient than Balha although the difference is nominal. 282

8 PADILLA & ZHANG In terms of the standard error (SE), a few things should be ointed out. First, the SEs are smaller as the mean item correlation increases. Second, standard errors imrove as samle size increases as should be exected. For samles sizes from to, the SE difference is nominal when the number of items is between 5 and 0. When the number of items is between 5 and 20, the SE difference is nominal regardless of the samle size. Third, the SEs imrove as the mean item correlation increase although the difference can be considered nominal; in most of these conditions, increasing the number of osterior samles should imrove the estimation of the SEs. Table 2: Balha and BlahaM Relative Bias with Standard Errors* Number of Items Samle Size Mean Item Correlation , (.89) , (.034) , (.0792) , (.0835) , (.0699) -.073, (.058) , (.0673) -.098, (.0577) -.09, -.04 (.0557) -.028, (.0477) , (.042) -.009, (.0357) , (.0) -.004, (.0424) , (.034) -.067, (.0467) -.003, (.0387) , (.0285) , (.0860) , (.0666) , (.0468) -.02, (.0546) -.037, (.0423) , (.0295) 0 -.0, (.0426) -.045, (.0370) , (.0335) , (.0287) , (.0235) , (.099) , (.0324) , (.0256) , (.077) , (.0295) , (.023) , (.06) , (.0672) , (.056) -.052, -.05 (.0348) -.036, (.040) -.004, (.034) , (.02) , (.032) , (.0275) , (.0246) , (.0209) , (.067) , (.042) , (.0245) , (.084) , (.025) , (.029) , (.067) -.007, (.03) , (.0580) -.0, (.0335) -.023, (.0440) , (.0254) -.029, (.0292) , (.068) , -.00 (.0260) -.004, (.094) , (.030) , (.029) , (.065) , (.00) , (.094) , (.075) , (.045) , (.032) -.003, (.0096) -.004, -.00 (.0088) *Note: The first number is Balha followed by BalhaM. Numbers in arentheses are standard errors. 283

9 ESTIMATING INTERNAL CONSISTENCY USING BAYESIAN METHODS The Bayesian credible intervals are dislayed in Table 3 and are more interesting. In general, most of the credible intervals fall within the accetable range of [.936,.964] based on,000 relications. In addition, the ercentile based BCIs are consistently closer to 0.95 than the normal theory based BCIs. With 5 items, only two BCIs were not within the accetable range. When the number of items shifts to 0, seven BCIs were not within the accetable range, but most of the unaccetable BCIs are normal theory based. As the number of items increases, more BCIs begin to fall outside the accetable range, but once again, most of the unaccetable BCIs are normal theory based. However, the unaccetable BCIs occur when the numbers of items are between5 to 20 and are aired with the smaller samle sizes. Secifically, when the numbers of items are 5, unaccetable BCIs occur at a samle size of. Also, when the numbers of items are 20, unaccetable BCIs occur at samle sizes of to. In both cases, more normal theory BCIs become unaccetable as the item mean correlation increases. However, the ercentile based BCIs tend to remain more stable and closer to Conclusion The building blocks of sychological/behavioral research are sychological constructs, which are tyically indirectly measured through items on questionnaires. It is crucial to have items that are consistent or reliable in order for research results to be trustworthy and useful. A oular method for estimating a form of reliability is internal consistency via coefficient alha (Cronbach, 95; Guttman, 945). However, coefficient alha has remained unchanged for over 60 years. Many rofessional ublications are encouraging and/or mandating researchers to sulement their arameter estimates with CIs. Although CIs for coefficient alha have recently enjoyed fruitful research (Barchard & Hakstian, 997; Duhachek & lacobucci, 4; Feldt, 965; Kristof, 963; Maydeu-Olivares, et al., 7; van Zyl, et al., 0; Yuan & Bentler, 2; Yuan, et al., 3), they are rarely imlemented in alied research. In addition, all current coefficient alha CIs are frequentist based and, as such, they have the traditional, less desirable CI interretation and cannot use rior information to stabilize inferences or udate information. This study develoed a Bayesian coefficient alha (Balha or BalhaM) and its corresonding BCIs. The results from the Monte Carlo investigations indicate that Balha and BalhaM are relatively unbiased under all investigated conditions of the simulation. However, Balha and BalhaM have the added advantage of having the BCIs, which have the interretation researchers really want to make with CIs. Again, BCIs allow one to make the following simler and more owerful statement: results show 95% confidence that the true oulation arameter lies between the bands of the credible interval. In terms of coverage, the ercentile based BCIs erformed better than the normal theory based BCIs. In articular, the normal theory BCIs begin to erform oorly when the mean item correlation is r =.30, and the condition worsens as the number of items increases. However, increasing the samle size offsets these conditions. In fact, having a samle size of 2 or more aears to rovide rotection against this breakdown of the normal theory BCIs. Conversely, the ercentile based BCIs remain more consistent, but begin to become unaccetable with the smaller samle sizes and when the number of items is between 5 and 20. However, they remain accetable as long as the samle size is at least. Thus, ercentile based BCIs are recommended over the normal theory BCIs. In general, this suggests that as the number of items increases a larger samle size is required to rovide stable inferences. This is not a surrising result. In traditional frequentist statistics, this would be the only otion. However, in Bayesian methodology there are two otential additional otions to stabilize inferences. First, the number of osterior samles can be increased. This would increase the recession of the estimates. Second, a rior can be secified, which will stabilize inferences that, in turn, will rovide better coverage. It should be noted that the urose of this study was to demonstrate how a Bayesian internal consistency can be estimated under the basic assumtions made of reliability (Allen & 284

10 PADILLA & ZHANG Table 3: Balha and BalahM Bayesian Credible Interval Coverage Number of Items Samle Size Mean Item Correlation , , , , , , , , , , , , , ,.948.9, , , , , , , ,.96.96, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,.968 *Note: The first number is the ercentile BCIs followed by the normal theory based BCIs. Unaccetable coverage is bolded; accetable coverage is within [.936,.964]. Yen, 979; Crocker & Algina, 986), thus, study rovides a sringboard from where future research on Bayesian coefficient alha can be conducted. However, like any simulation study, this research is limited by the tye and number of conditions investigated. In this study, only homogenous items were investigated. Additionally, items were continuous and normally distributed. Further research is required to investigate the robustness of a Bayesian coefficient alha to violations of the basic reliability assumtions and to establish its roerties under binary or ordinal items. 285

11 ESTIMATING INTERNAL CONSISTENCY USING BAYESIAN METHODS As noted by Duhachek and Iacobucci (4) and Maydeu-Olivares, et al. (7), reorting only a oint estimate of coefficient alha is no longer sufficient. With inferential techniques reorting the SE and CIs rovide more information as to the size and stability of the oint estimate; in this case the oint estimate is coefficient alha. Within this context, a Bayesian internal consistency estimate may rovide an attractive alternative to current coefficient alha CIs because it rovides researchers with BCIs that can be interreted in a way researchers want and can make use of rior information to stabilize inferences. References Allen, M. J., & Yen, W. M. (979). Introduction to measurement theory. Monterey, CA: Brooks-Cole Publishing Comany. Anderson, T. W. (984). An introduction to multivariate statistical analysis (2 nd Ed.). New York: Wiley. Barchard, K. A., & Hakstian, A. R. (997). The robustness of confidence intervals for coefficient alha under violation of the assumtion of essential arallelism. Multivariate Behavioral Research, 32(2), Burton, A., Altman, D. G., Royston, P., & Holder, R. L. (6). The design of simulation studies in medical statistics. Statistics in Medicine, 25(24), Cortina, J. M. (993). What is coefficient alha? An examination of theory and alications. Journal of Alied Psychology, 78(), Crocker, L. M., & Algina, J. (986). Introduction to classical and modern test theory. New York: Holt, Rinehart and Winston. Cronbach, L. (95). Coefficient alha and the internal structure of tests. Psychometrika, 6(3), Derogatis, L. R., Liman, R. S., Rickels, K., Uhlenhuth, E. H., & Covi, L. (974). The Hokins Symtom Checklist (HSCL): A selfreort symtom inventory. Behavioral Science, 9(), -5. Duhachek, A., Coughlan, A. T., & Iacobucci, D. (5). Results on the standard error of the coefficient alha index of reliability. Marketing Science, 24(2), Duhachek, A., & lacobucci, D. (4). Alha s atandard error (ASE): An accurate and recise confidence interval estimate. Journal of Alied Psychology, 89(5), Feldt, L. (965). The aroximate samling distribution of Kuder-Richardson reliability coefficient twenty. Psychometrika, 30(3), Gelman, A. (4). Bayesian data analysis (2 nd Ed.). Boca Raton, Fla.: Chaman & Hall/CRC. Guttman, L. (945). A basis for analyzing test-retest reliability. Psychometrika, 0(4), Iacobucci, D., & Duhachek, A. (3). Advancing Alha: Measuring reliability with confidence. Journal of Consumer Psychology, 3(4), Kristof, W. (963). The statistical theory of steed-u reliability coefficients when a test has been divided into several equivalent arts. Psychometrika, 28(3), Lee, P. M. (4). Bayesian statistics : An introduction (3 rd Ed.). London: Arnold. Maydeu-Olivares, A., Coffman, D. L., & Hartmann, W. M. (7). Asymtotically distribution-free (ADF) interval estimation of coefficient alha. Psychological Methods, 2(2), Schafer, J. L. (997). Analysis of incomlete multivariate data. New York: Chaman & Hall/CRC. van Zyl, J., Neudecker, H., & Nel, D. (0). On the distribution of the maximum likelihood estimator of Cronbach s alha. Psychometrika, 65(3), Wilkinson, L. (999). Statistical methods in sychology journals: Guidelines and exlanations. American Psychologist, 54(8), Yuan, K.-H., & Bentler, P. (2). On robusiness of the normal-theory based asymtotic distributions of three reliability coefficient estimates. Psychometrika, 67(2), Yuan, K.-H., Guarnaccia, C. A., & Haysli, B. (3). A study of the distribution of samle coefficient alha with The Hokins Symtom Checklist: Bootstra versus asymtotics. Educational and Psychological Measurement, 63(),