RESEARCH REPORT
On the Relationship Between Coefficient Alpha and Composite Reliability
Robert A.Peterson and Yeolib Kim
The University of Texas at Austin
Cronbach’s coefficient alpha is the most widely ud estimator of the reliability of tests and scales.However,it has been criticized as being a lower bound and hence underestimating true reliability.A popular alternative to coefficient alpha is composite reliability,which is usually calculated in conjunction with structural equation modeling.A quantitative analysis of 2,524pairs of coefficient alpha and composite reliability values derived from empirical investigations revealed that although the average composite reliability value (.86)exceeded the average corresponding coefficient alpha value (.84),the difference was relatively inconquential for practical applications such as meta-analysis.Keywords:coefficient alpha,composite reliability,meta-analysis
Six decades ago,Cronbach’s (1951)minal article,“Coeffi-cient Alpha and the Internal Structure of Tests,”was published in Psychometrika .This article changed the manner in which test and scale reliabil
ity was conceptualized and measured.Coefficient alpha has since become a standard component of the toolkits of rearchers attempting to measure reliability,and the article has been cited nearly 17,000times in GoogleScholar.It is,without question,the most widely ud estimator of test and scale reliabil-ity in the social sciences.Indeed,the number of Google citations likely underestimates its u in practice.This is becau coefficient alpha has become an icon,and,analogous to other statistical or psychometric icons,it is often no longer associated with its creator.Stated somewhat differently,its attribution or citation history in publications has progresd from “Cronbach’s coefficient alpha”to “coefficient alpha”to “alpha.”
In addition to being widely applied,coefficient alpha has been widely investigated.Its measurement and statistical properties have extensively evaluated and discussions and arguments have both praid and criticized its u and interpretation.See,for example,Cortina (1993),Schmitt (1996),and Sijtsma (2009)for reprentative articles discussing the u and interpretation of coefficient alpha.Moreover,meta-analys (e.g.,Peterson,1994;Rodriguez &Maeda,2006)have empirically explored factors that affect the values of coefficient alpha under a variety of conditions.The magnitude of coefficient alpha has often been the focus of investigation,with conclusions generally being that “alpha under-estimates the true reliability of a measure that is not tau equiva-lent”(Osburn,
2000,p.344)or that “departure from classical tau-equivalence does lead to a small downward bias in alpha when
ud as a composite reliability measure”(Bacon,Sauer,&Young,1995,p.394).Cronbach (2004,p.402)himlf agreed with the asssment of a downward bias in coefficient alpha due to “a small mathematical detail that caus the alpha coefficient to run a trifle lower than the desired value.”More specifically,there is near universal agreement that unless there is (esntial)tau equivalency,coefficient alpha is a lower bound on true reliability (e.g.,Novick &Lewis,1967;Sijtsma,2009).
Becau coefficient alpha tends to be viewed as a lower bound on true reliability,numerous alternative estimators of true reliability have been proffered.The alternative estima-tors include what are usually termed the stratified alpha coef-ficient (Cronbach,Schonemann,&McKie,1965);Raju coeffi-cient,Angoff-Feldt coefficient,Feldt coefficient,Feldt-Gimer coefficient,lambda2coefficient,mazimized lambda4coeffi-cient,standardized alpha coefficient,maximal reliability coef-ficient (summarized in Osburn,2000);and the beta coefficient (Revelle &Zinbarg,2009;Zinbarg,Revelle,Yovel,&Li,2005),among others.
One category of proffered estimators consists of tho that are bad on structural equation modeling (e.g.,Bacon et al.,1995;Fornell &Larcker,1981;Graham,2006;Green &Yang,2009b;Hei &Bohrnstedt,1970;Jöreskog,1971;McDonald,1999;Raykov,1997;Yang &Green,2011).Given that the variance of obrved scale scores in classical test theory can be decompod as
天安门升旗攻略X 2ϭT 2ϩE 2,
where X reprents the obrved test or scale score variance,T reprents the true test or scale score variance,and E reprents
error variance,the true reliability of a test or scale ϭ⌻2/X 2
.Thus,the reliability of a test or scale is reprented by the ratio of its true score variance divided by its obrved score variance.As such,the variance components can be estimated using structural equation modeling.When true reliability is estimated using struc-tural equation modeling,the resulting estimate is typically referred to as composite reliability (CR).The claimed benefits of a struc-tural equation modeling approach include “better”(e.g.,typically
This article was published Online First November 5,2012.
Robert A.Peterson and Yeolib Kim,Office of the Vice President for Rearch,The University of Texas at Austin.
Correspondence concerning this article should be addresd to Robert A.Peterson,Office of the Vice President for Rearch,The University of Texas at Austin,Austin,TX 78712.E-mail:rap@mail.utexas.edu
Journal of Applied Psychology ©2012American Psychological Association 2013,Vol.98,No.1,194–1980021-9010/13/$12.00DOI:10.1037/a0030767
194
嘀咕的意思larger)estimates of true reliability than possible through coeffi-cient alpha becau construct loadings or weights are allowed to vary,whereas the loadings or weights for coefficient alpha are constrained to be equal.Conquently,structural equation model-ing has the ability to empirically asss and overcome some of the limiting assumptions of coefficient ,Raykov,2001).
Purpo
怎么做肠粉Despite the claimed advantages of a composite reliability coef-ficient over coefficient ,Gree
英语不客气n&Yang,2009b;Raykov, 2001),to date there have been no empirical comparisons of cor-responding coefficient alpha and composite reliability values.Al-though composite reliability coefficients have been shown to be larger than coefficient alpha analytically,and through simulations, there have been no comparisons of the two coefficients bad on practical applications using nonartificial data.Therefore,the re-arch question addresd here using a meta-analysis approach is twofold:Does composite reliability unequivocally produce a“bet-ter”(i.e.,larger)estimate of true reliability than coefficient alpha under identical rearch conditions?If so,then what methodolog-ical characteristics,if any,systematically relate to possible differ-ences in estimates of true reliability produced by composite reli-ability and coefficient alpha?
If coefficient alpha systematically underestimates true reliability relative to composite reliability,then using coefficient alpha to correct validity correlation coefficients for attenuation would lead to overstating the magnitude of relationships.This could especially be a problem when conducting a meta-analysis in which concerted efforts are made to correct effect ,correlations)for attenuation to make them amenable to meaningful aggregation.In brief,the purpo of the prent study was to asss the compa-rability of coefficient alpha and composite reliability values de-rived under applied rearch conditions.
Method
举世瞩目
The specific objective of the prent study was to compare values of coefficient alpha and composite reliability derived from the same data analys under applied conditions.Conquently, the methodology ud followed that of Peterson and Brown(2005) in their comparison of corresponding beta coefficients and corre-lation coefficients drawn from regression analys of various be-havioral data ts.As Peterson and Brown(2005)noted,“Al-though such a comparison could be made using synthetic data, synthetic data often do not capture the nuances and relationships that exist in actual data”(p.177).The same logic follows here. Following an initial arch of journals in the EBSCO,JSTOR, and GALE databas using such terms as coefficient alpha and composite reliability,24journals were lected from which cor-responding pairs of coefficient alpha and composite reliability measure values were harvested.The journals encompasd ,Journal of Applied Psychology),, Journal of Marketing),,Journal of Operations Management),and ,Computers and Education). Every article published in the journals over the period1996 through2011was personally examined to determine whether the results of an empirical study were reported and,if so,whether the article reported corresponding coefficient alpha and composite reliability values.Beca
u the metrics ,coefficient alpha and composite reliability values)were not necessarily men-tioned in article titles,abstracts,or meta-tags,computer-bad data collection approaches typically ud in meta-analys were not appropriate.Conquently,a very labor-intensive examination process was undertaken to harvest the metrics of interest.(To illustrate the effort required,Shook,Ketchen,Hult,&Kacmar, 2004,estimated that only3%of the articles in management re-porting reliability coefficients reported both coefficient alpha and composite reliability.)
A total of2,524pairs of coefficient alpha values and composite reliability values were obtained from327articles containing381 parate studies in24journals(the median number of pairs per journal was67).The values were bad on respons from 155,308individuals.Analogous to Peterson and Brown(2005),all identified pairs of coefficient alpha and composite reliability val-ues were harvested,even tho that might be considered outliers or reflect reporting-or publication-related errors.In addition to the values,data were obtained regarding the sample sizes underlying the studies and various methodological characteristics of the un-derlying tests or scales.The characteristics included the number of items comprising the test or scale who reliability was being evaluated,the number of categories in the items,the type of sample providing the data(college students,consumers,business-people),the item format(only endpoint anchors,numerical values on ite
m categories,verbal values on item categories),and whether there was an odd or even number of item categories.Other data collected included the mode of scale administration(lf-report vs. interviewer),test or scale ,whether focud on a study participant[respondent-centered],or an independent stimu-lus[stimulus-centered]),and the nature of the construct being measured(whether it rved as a dependent or an independent variable in the analysis).Becau the data consisted of pairs of reported values,data issues common to meta-analys including the file-drawer problem and reporting errors were not deemed problematic(e below).To facilitate comparisons,all coefficient alpha and composite reliability values were rounded to two deci-mal places prior to analysis.
Results
Table1contains characteristics of the respective distributions of the coefficient alpha and composite reliability values obtained from the meta-analysis.On average,bad on a pairwi compar-ison of the two values,composite reliability values exceeded coefficient alpha values by about.018units,or2.1%,with a standard deviation of.047.Although this difference was statisti-Table1
Characteristics of Coefficient Alpha and Composite
Reliability(CR)体育保健
Characteristic␣distribution CR distribution Range0.46–0.990.54–0.99 Mean.84.86 Median.86.88 Standard deviation.08.07 SkewnessϪ.80Ϫ1.04 Kurtosis.62 1.24
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COEFFICIENT ALPHA AND COMPOSITE RELIABILITY
cally significant (p Ͻ.001),significance was due more to the relatively large sample size and repeated measure analysis than a substantive difference.More specifically,there was no difference between coefficient alpha and composite reliability values in 27%of the obrvations;50%of the differences in values were within Ϯ.01units;and approximately 61%of the values within a pair were within Ϯ2%of each other.Simultaneously,though,in 59%of the obrvations,composite reliability values were larger than coefficient alpha values;coefficient alpha values were larger than composite reliability values in 15%of the obrvations.The correlation between coefficient alpha and composite reliability values was .80.Figure 1contains a scatter diagram of the pairs of coefficient alpha and composite reliability values.
猪大肠做法
Table 2contains a summary of the results of one-way repeated measures analys of variance,with coefficient alpha and compos-ite reliability being the dependent variables and rearch method-ology characteristics being the independent variables.(Subsample sizes do not always equal the total sample size for a particular characteristic due to missing data.)Although the vast majority of the differences between coefficient alpha and composite reliability values for particular methodological characteristics were statisti-cally significant,this was due more to the relatively large number of paired values analyzed than the magnitude of the differences.The important thing to note from Table 2is that the relationship between coefficient alpha and composite reliability mean values was relatively consistent across the methodological characteristics.An alternative way to examine the relationships between coef-ficient alpha and composite reliability is to explore a subt of the obrvations in which coefficient alpha values were larger than composite reliability values.This was done by creating a category of obrvations wherein coefficient alpha values were at least .05units larger than the corresponding composite reliability values (in other words,the differences in values were greater than 1SD ).There were 134such obrvations.Comparison of the rearch methodology characteristics,respectively,associated with the 134obrvations,and the remaining 2,390obrvations revealed only one distinguishing difference.Coefficient alpha values were more likely to be larger than composite reliability values when stimulus-centered measurement was undertaken.
To asss the possibility of publication or availability bias,we conducted two ancillary analys.First,we obtained independent samples of coefficient alpha and composite reliability values from articles in the arched journals reporting one of the values,but not both.Specifically,153coefficient alpha values and 153composite reliability values were independently harvested from the 24
jour-
Figure 1.Scatterplot of coefficient alpha and composite reliability
values.
Table 2
Coefficient Alpha and Composite Reliability Values for Rearch Characteristics
Characteristics N Mean ␣Mean CR Sample size Ͻ100ءء910.820.88100–199ءء6970.840.85200–299ءء
6910.850.87300or more ءء1,0450.850.86Type of sample
College students ءء3500.860.89Consumers ءء
6960.860.87Businesspersons ءء1,3600.830.85Mixed ءء
1060.870.88Number of item categories Not given 1610.850.854120.770.785ءء8520.830.856ء230.850.857ءء
1,4440.850.878or more ء
310.870.89Number of items Not given 760.850.852ءء2470.800.833ءء9000.830.854ءء6390.850.875ءء3180.870.886ءء1800.880.897ءء750.870.908280.880.899190.890.9210
100.920.9411or more 320.890.89Item format
Only endpoints labeled ءء3900.850.87Numerical values on categories ءء5310.840.86Verbal values on categories ءء1,2390.840.86Cannot tell ءء3640.840.85Nature of scale Odd number ءء2,2930.840.86Even number 500.830.84Cannot tell
1810.840.84Administrative mode Self ءء
2,3270.840.86Interviewer ءء1790.860.88Not given ء
180.900.86Scale orientation
Respondent-centered ءء1,2950.850.87Stimulus-centered ءء1,0300.840.85Both ءء
1990.850.87Nature of construct Dependent ءء5150.850.86Independent ءء
1,2620.840.85Cannot tell/both ءء
747
0.86
0.87
朋可以组什么词Note .CR ϭcomposite reliability.ء
Mean difference significant at p Ͻ.01.ءء
Mean difference significant at
p Ͻ.001.
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PETERSON AND KIM
nals in the meta-analysis.The average coefficient alpha value was .83;the average composite reliability value was.85.Second,we harvested22pairs of coefficient alpha and composite reliability values from manuscripts submitted for publication consideration but not accepted(from the first autho
r’s reviewer files).In this t, the mean coefficient alpha value was.87;the mean composite reliability value was.88.Thus,although the data were somewhat limited,the ancillary analys suggest that the results of the meta-analysis are reasonably general.
Discussion and Conclusions
The answer to the first question underlying in this rearch,“... does composite reliability unequivocally produce a‘better’(i.e., larger)estimate of true reliability than coefficient alpha under identical rearch conditions?”is a qualified“yes,but.”The an-swer to the cond question underlying this rearch,“...what methodological characteristics,if any,systematically relate to pos-sible differences in estimates of true reliability produced by com-posite reliability and coefficient alpha?”is a qualified“virtually none.”
Estimates of true reliability produced by composite reliability were,on average,larger than tho produced by coefficient alpha. This was expected given the formulaic differences in calculating the two coefficients(coefficient alpha is a constrained version of composite reliability).For the entire sample of obrvations,the mean composite reliability value was.86,whereas the mean co-efficient alpha value was.84;the mean difference calculated on individual pairs of the two values was.018.Fro
m a technical perspective,the.018difference could be interpreted as reprent-ing the extent to which the assumptions underlying the calculation of coefficient alpha have not been met.The magnitude of the difference was relatively consistent across the various method-ological characteristics studied.
The empirically derived means and mean difference obrved in this study generally corroborate results reported in methodologi-cally oriented comparisons of coefficient alpha and composite reliability.For example,Green and Yang(2009b)calculated23 pairs of coefficient alpha and(linear)composite reliability values under a variety of artificial data conditions(e their Table1). Their mean coefficient alpha value was.598and their mean (linear)composite reliability value was.605;the mean difference was.007.They also reported a coefficient alpha value of.778and a(linear)composite reliability value of.819(difference of.041) for an eight-item scale administered to828individuals.Ferketich (1990)prented an example comparing coefficient alpha and composite reliability bad on a10-item scale administered to590 individuals.The reported coefficient alpha value was.8446, whereas the reported composite reliability value was.8649;the difference was.0203.Raykov(1997)calculated a coefficient alpha value of.896and a composite reliability value of.912(difference of.016)for an eight-item scale administered to165individuals. Bacon et al.(1995)reported a coefficient alpha value of.596and a composite reliability value of.651(difference of.0
55)for a simulated data t.The overall similarity of the differences be-tween the two reliability coefficients across the various studies suggests that there is little practical difference between them. Even though composite reliability values obrved in the prent study were,on average,consistently“better,”that is,larger,than corresponding coefficient alpha values,in general the differences were not practically meaningful such that u of coefficient alpha to correct for attenuation would not be expected to systematically “inflate”the value of a validity ,mean␣1/2ϭ.917; mean CR1/2ϭ.927)relative to composite reliability when applied
in the standard attenuation correction formula.Moreover,the finding that about61%of the values in a pair were withinϮ2%of each other suggests that the differences in values were generally of little practical ,Lee&Frisbie,1999).Thus, although composite reliability values were typically larger than corresponding coefficient alpha values,claims that coefficient alpha grossly underestimates true reliability as compared with composite reliability need to be tempered.
Accordingly,it is instructive to note that although composite reliability values were typically larger than coefficient alpha val-ues,in15%of the pairs of values coefficient alpha was larger than composite reliability;in about6%of the pairs,coefficient alpha values were significantly larger than corresponding composite reliability values.The percentages support the conclusions of Bentler(20
09),Green and Yang(2009a),and Raykov(2001)that, under certain conditions,such as when there are correlated errors among the items in a test or scale,values of coefficient alpha may exceed values of composite reliability.Unfortunately,it was not possible to determine the specific reasons for the patterns of values from the prent data.Possible reasons include rounding differences,inappropriate structural models ud when estimating composite reliability,reporting errors in the values harvested, computational differences,or certain types of correlated data er-rors.Likewi,it was not possible to discern the reasons for tho instances in which composite reliability values significantly ex-ceeded coefficient alpha values.Such differences may also be due to rounding or reporting errors,certain data configurations,or even misapplication of the two reliability ,when rela-tionships are formative rather than reflective,reliability coeffi-cients are not meaningful).More rearch is required to identify conditions leading to values of coefficient alpha being larger than values of composite reliability.
The results of the prent study suggest that,at a minimum, coefficient alpha and composite reliability values might be ud interchangeably when correcting validity coefficients or effect sizes in meta-analys with few practical conquences.Although coefficient alpha values may generally be lower bounds on true reliability,their u in practice should not be deleterious to knowl-edge development.
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Received January3,2012
Revision received October1,2012
Accepted October2,2012Ⅲ
198PETERSON AND KIM
