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Congeneric and (Essentially) Tau-Equivalent Estimates of Score
Reliability: What They Are and How to Use Them
Educational and Psychological Measurement The online version of this article can be found at: can be found
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Educational and Psychological Measurement Educational and
Psychological Measurement
Congeneric and (Essentially)
Tau-Equivalent Estimates of
Score Reliability
What They Are and How to Use Them
James M. GrahamWestern Washington University Coefficient alpha, the most commonly used estimate of internal consistency, is oftenconsidered a lower bound estimate of reliability, though the extent of its underestimation isnot typically known. Many researchers are unaware that coefficient alpha is based on theessentially tau-equivalent measurement model. It is the violation of the assumptionsrequired by this measurement model that are often responsible for coefficient alpha’sunderestimation of reliability. This article presents a hierarchy of measurement models thatcan be used to estimate reliability and illustrates a procedure by which structural equationmodeling can be used to test the fit of these models to a set of data. Test and data characteris-tics that can influence the extent to which the assumption of tau-equivalence is violated arediscussed. Both heuristic and applied examples are used to augment the discussion.
Keywords: reliability; structural equation modeling; congeneric; tau-equivalent
Anumberofstudieshaveshownthatignoranceregardingfundamentalmeasure- ment issues has reached an endemic level (Vacha-Haase, Kogan, & Thompson, 2000; Whittington, 1998). Although many doctoral programs include exposure to sta-tistics and research design, measurement issues are often ignored in education andpsychology programs. As a result, issues such as reliability are often misconstrued(Aiken, West, Sechrest, & Reno, 1990; Pedhazur & Schmelkin, 1991). Althoughsome progress has been made in educating researchers about reliability, such as thedissemination of the fact that reliability is an artifact of the sample, not the test(Thompson & Vacha-Haase, 2000), many researchers still lack the basic knowledgenecessary to accurately estimate reliability.
The most commonly used measure of internal consistency, coefficient alpha, is based on the essentially tau-equivalent measurement model, a measurement modelthat requires a number of assumptions to be met for the estimate to accurately reflect Author’s Note: Please address correspondence to James M. Graham, Department of Psychology, Western
Washington University, 516 High Street, Bellingham, WA 98225-9089; e-mail:
2006 Sage Publications. All rights reserved. Not for commercial use or unauthorized distribution.
the data’s true reliability (Raykov, 1997a). Violation of these assumptions causes coef-ficient alpha to underestimate the true reliability of the data (Miller, 1995). In fact,there are a number of other measurement models, “parallel,” “tau-equivalent,” and“congeneric” (Feldt & Brennan, 1989, pp. 110-111; Lord & Novick, 1968, pp. 47-50),that can be used to estimate reliability.
This article describes these measurement models as they apply to classical test the- ory and reliability. The use of structural equation modeling (SEM) path diagrams toevaluate the fit of these models is described. A procedure for obtaining an accurateestimate of reliability based upon these findings is outlined. Finally, test and data char-acteristics that influence the extent of coefficient alpha’s underestimation of reliabilityare presented. Whereas previous work has explored the theoretical underpinnings ofthese concepts in greater depth and complexity (Miller, 1995; Raykov, 1997a, 1997b),it is my intention to describe these concepts in a more accessible format with ampleuse of both heuristic and applied examples.
Reliability in Structural Equation Modeling
Classical test theory (CTT) is based on the premise that the variance in observed scores (X) is due in part to true differences in the latent trait being measured (T) and inpart to error (E). This can be represented in the equation X = T + E. This basic equationcan be represented in an SEM path diagram in which one measured variable, X, andtwo latent variables, T and E, are shown in relation to one another. Unidirectional paths(with path coefficients set to 1) from the latent variables to the measured variable indi-cate that measured scores are an additive combination of the true and error scores.
Additionally, the latent variables are uncorrelated with one another.
In CTT, a reliability coefficient (ρ ) is the proportion of observed score (X) vari- ance accounted for by the true score (T) variance, as shown by the following equation(Miller, 1995): The present discussion focuses on one type of reliability, internal consistency, asopposed to other types of reliability, such as test-retest, alternate forms, or interrater.
The SEM model described above is not identified. As no parameters are given for the partitioning of the observed score variance into the two latent variables, there arean infinite number of ways in which the variance of X can be divided into T and E.
Therefore, it is impossible to estimate the reliability of X unless the test consists ofmore than one item. In multiple-item tests, each item has its own true and error scores;again, without further specifications, there are still an infinite numbers of ways to par-tition the variance of items. To estimate the reliability of these items, we must identifythe model by making further assumptions.
Estimates of reliability within CTT assume that all observed variables measure a single latent true variable. Many researchers erroneously believe that reliability provides a 2006 Sage Publications. All rights reserved. Not for commercial use or unauthorized distribution.
Educational and Psychological Measurement Structural Equation Modeling (SEM) Path Diagram
of Unidimensional Composite True Variable
measure of test unidimensionality. In actuality, reliability assumes that unidimen-sionality exists (Miller, 1995). Failure to meet the assumption of unidimensionalitywill result in an inaccurate and often misleading estimate of reliability.
The assumption of unidimensionality for a four-item test is represented as an SEM path diagram in Figure 1. Here the variables are shown with subscripts in the text, forexample, X , but without subscripts in the figures, for example, X1. These refer to the same variables. The model in Figure 1 shows a single latent variable (T) as beingresponsible for part of the observed score variances of each individual test item (X , X , etc.). Additionally, each item has a unique error term associated with that item. In themodel shown in Figure 2, there are no numbers on the paths from the latent true vari-able to the individual item observed variables. The relationship between the compositetrue score and item true scores can be defined a number of ways and will be dealt withat a later point.
Estimating Reliability Within SEM
As previously noted, reliability is the proportion of true to observed score variance.
To provide an estimate of reliability in SEM, therefore, it is necessary to create estimates 2006 Sage Publications. All rights reserved. Not for commercial use or unauthorized distribution.
Basic Reliability Structural Equation Modeling (SEM) Path Diagram
of both the true and observed score variances. Using SEM, a measure’s total observedscore variance can be made available by creating a composite observed variable (X).
This variable is created by adding the variances of the individual observed variables(X , X , etc.) while taking into account the shared variance of the individual observed variables (Miller, 1995; Raykov, 1997a). This process, represented in SEM terms, isshown in Figure 2. Whereas the composite variable X is represented as a circle, thedirection of the arrows show that X is a direct result of the sum of the individualobserved item variances, taking into account the variance shared between items.
The creation of an estimate of the true score variance has already been discussed; it is the variance of the latent variable labeled T in Figures 1 and 2. To calculate a reliabil-ity estimate with this information, one has only to apply Equation 1. Alternatively, anestimate of reliability can be obtained from the model shown in Figure 2, by squaringthe implied correlation between the composite latent true variable (T) and the compos-ite observed variable (X) to arrive at the percentage of the total observed variance thatis accounted for by the “true” variable.
Measurement Models
To identify the measurement model used to estimate the composite true variable, it is necessary to make further assumptions above and beyond unidimensionality. There 2006 Sage Publications. All rights reserved. Not for commercial use or unauthorized distribution.
Educational and Psychological Measurement are a number of measurement models that may be useful in estimating the reliability oftest items, each of which requires that the data used meets different requirements.
The parallel model. The parallel model is the most restrictive measurement model for use in defining the composite true score. In addition to requiring that all test itemsmeasure a single latent variable (unidimensionality), the parallel model assumes thatall test items are exactly equivalent to one another. All items must measure the samelatent variable, on the same scale, with the same degree of precision, and with the sameamount of error (Raykov, 1997a, 1997b). All item true scores are assumed to be equalto one another, and all error scores are likewise equal across items. When applied tothe CTT equation, each item k for individual i can be shown as The parallel model can be used to identify the SEM path diagram shown in Figure 2.
To do so, each of the paths from the composite true variable to the individual item vari-ables are set to 1, signifying that each measured variable measures the same latent vari-able with the same degree of precision and the same scale. Additionally, the individualitem error variances are constrained to be equal to one another; in Amos (Arbuckle,2003), this is accomplished by setting the variance of the error terms to a letter (param-eters with the same letter are constrained to equality).
The tau-equivalent model. The tau-equivalent model is identical to the more restrictive parallel model, save that individual item error variances are freed to differfrom one another. This implies that individual items measure the same latent variableon the same scale with the same degree of precision, but with possibly differentamounts of error (Raykov, 1997a, 1997b). All variance unique to a specific item istherefore assumed to be the result of error. The tau-equivalent model implies thatalthough all item true scores are equal, each item has unique error terms: The SEM path diagram for the tau-equivalent model is identical to the parallel modelpath diagram, save that error variances are no longer constrained to equality.
The essentially tau-equivalent model. The essentially tau-equivalent model is, as its name implies, essentially the same as the tau-equivalent model. Essential tau-equivalence assumes that each item measures the same latent variable, on the samescale, but with possibly different degrees of precision (Raykov, 1997a). Again, as withthe tau-equivalent model, the essentially tau-equivalent model allows for possiblydifferent error variances.
The difference between item precision and scale is an important distinction to make. Whereas tau-equivalence assumes that item true scores are equal across items,the essentially tau-equivalent model allows each item true score to differ by an additive 2006 Sage Publications. All rights reserved. Not for commercial use or unauthorized distribution.
constant unique to each pair of variables (Miller, 1995; Raykov, 1997a). Mathemati-cally, this assumption can be represented as shown: These equations reflect the fact that, although items’true scores are being measured on the same “scale” (similar variances), they may differ in terms of “precision” (differ-ent means). A “precise” measure would be one in which the measured values fordifferent items would be closely grouped together, whereas the measured values ofdifferent items would be widely spread out in an “imprecise” measure. For example,consider a test designed to measure the latent variable depression in which each item ismeasured on a 5-point Likert-like scale, from strongly disagree to strongly agree.
Responses to the items “I feel sad sometimes” and “I almost always feel sad” are likelyto share similar distributions, though perhaps with different modes. This might be dueto the fact that, though both questions measure the same latent variable on the samescale, the second question is worded more strongly than the first. As long as the vari-ances of these questions are similar across respondents, they are both measuringdepression in the same scale, whereas their precision in measuring depression differs.
The inclusion of an additive constant affects only an item’s mean, not its variance or covariances with other items. As reliability is a variance-accounted-for statistic, itis unaffected by differing means. Therefore, for the purposes of estimating reliability,the SEM path diagram for the essentially tau-equivalent model is identical to that ofthe tau-equivalent model.
It should be noted that coefficient alpha, the most widely used estimate of internal consistency, is an estimate of reliability based on the essentially tau-equivalent model.
Using SEM procedures with the essentially tau-equivalent measurement model willresult in a reliability estimate that is equal to Cronbach’s alpha. Because it is based onthe essentially tau-equivalent model, Coefficient alpha assumes that all items measurethe same latent trait on the same scale, with the only variance unique to an item beingcomprised wholly of error.
The congeneric model. The congeneric model is the least restrictive, most general model of use for reliability estimation. The congeneric model assumes that each indi-vidual item measures the same latent variable, with possibly different scales, withpossibly different degrees of precision, and with possibly different amounts of error(Raykov, 1997a). Whereas the essentially tau-equivalent model allows item truescores to differ by only an additive constant, the congeneric model assumes a linearrelationship between item true scores, allowing for both an additive and a multiplica-tive constant between each pair of item true scores, as shown below: To identify the model shown in Figure 2 with the congeneric model, the path from the latent true variable to one of the measured items is set to 1, whereas the other paths 2006 Sage Publications. All rights reserved. Not for commercial use or unauthorized distribution.
Educational and Psychological Measurement from the true variable to the items are left free to be estimated. This indicates that thetrue scores of the other items are expressed in terms of the true score of the fixed item.
Any of the measured items can be chosen as the “scaling” variable, with no effect onthe outcome of the model.
The Hierarchical Nature of Measurement Models
Each of the previously discussed measurement models is part of a nested hierarchy.
The congeneric model shown in Equation 5 is the most general, least restrictive modelfor use in reliability estimation. If all multiplicative constants in the congeneric modelare set to 1 (inferring that item true scores are measured on the same scale, or have thesame standard deviation), we arrive at the essentially tau-equivalent, or coefficientalpha, model shown in Equation 4. If all additive constants are then set to 0 (inferringthat not only do item true scores have the same variance, but they are measured withthe same degree of precision, or have the same mean), we arrive at the tau-equivalentmodel shown in Equation 3. Finally, if all error variances are set to equal one another,we arrive at the parallel model shown in Equation 2, where all observed and latentvariables are equivalent across items.
Estimating Reliability With the Hierarchical Model
Coefficient alpha is considered a lower bound estimate of reliability, and the extent of coefficient alpha’s underestimation of the reliability cannot be typically known.
One common reason for coefficient alpha’s underestimation of reliability is the viola-tion of the essentially tau-equivalent model. For example, if data characteristics indi-cate that the test items measure the same latent variable in different scales, coefficientalpha (based on the essentially tau-equivalent model) would underestimate the reli-ability, which would be better estimated using the congeneric model. Using the appro-priate model given the characteristics of the data can provide a much more accurateestimation of reliability. As the measurement models used in estimating reliability arehierarchical, the fit of the data can be tested to each model in a step-by-step manner,working from least restrictive/parsimonious to most restrictive/parsimonious. Thisallows the assumptions of the reliability estimates to be tested and the best possiblemodel chosen.
The process of determining which measurement model to use to estimate reliability is a simple one. First, the fit of the congeneric model is tested, and fit statistics areobtained. Next, the fit of the essentially tau-equivalent/tau-equivalent model is tested,and the resulting fit statistics are compared to those obtained from the congenericmodel. If the decrease in fit is large enough to be considered meaningful (by whatevermeans the individual researcher chooses—fit statistics, χ2 change, etc.), the conge-neric model is used to estimate reliability. If the fit statistics are similar, the fit of thetau-equivalent model is compared to the fit of the parallel model. If the difference in fitbetween the tau-equivalent and parallel models is meaningful, the tau-equivalentmodel is used. If the differences are not meaningful, the parallel model is used.
2006 Sage Publications. All rights reserved. Not for commercial use or unauthorized distribution.
The fit of the congeneric model will always be better than or equal to all other models, as the congeneric model is the least restrictive. For the sake of parsimony,however, the most restrictive feasible model should always be used. The method ofdetermining whether difference in fit between two competing models is meaningful islargely a matter of what fit statistics a given researcher prefers. For the followingexamples, the goodness-of-fit index (GFI; Jöreskog & Sörbom, 1984), the compara-tive fit index (CFI; Bentler, 1990), and the root mean square error of approximation(RMSEA; Steiger & Lind, 1980) are used in conjunction with change in χ2 statistics toevaluate model fit.
It should be noted that the χ2 goodness-of-fit test statistic utilizes traditional statisti- cal significance testing procedures and is therefore highly subject to the size of thesample being used. As stated by Bentler and Bonett (1980), In very large samples virtually all models that one might consider would have to berejected as statistically untenable. . . . This procedure cannot be justified, since the chi-square value . . . can be made small simply by reducing the sample size. (p. 591) Although the use of the χ2 statistic alone for the purpose of determining model fit isquestionable, χ2 statistics can be of use in comparing the fits of nested models in whichthe sample size is held constant across models (Thompson, 2004).
Another important consideration in the examples that follow is the method of esti- mation used. Coefficient alpha and the majority of commonly used statistical proce-dures use ordinary, or unweighted, least squares (OLS), a method of estimation thatmaximizes explained variance while minimizing unexplained, or error, variance.
Maximum likelihood, another method of estimation, attempts to maximize the fit ofthe data to a given model. As each method of estimation serves a different function,both are used in the following examples. Maximum likelihood is used to initially testthe fit of the data to the different models, and OLS is used to provide a reliability esti-mate comparable to (or, in the case of the tau-equivalent model, exactly equal to)coefficient alpha.
Heuristic example. As a demonstration of how coefficient alpha underestimates score reliability when the assumption of essential tau-equivalence is violated, con-sider the following example. This example uses a fictional five-item measure with 60individuals. To create the items, a true score was first created for each individual byarbitrarily entering numbers from a computer number pad. These true scores rangedfrom 1 to 9, with a mean of 5.15 and a standard deviation of 2.11. Error terms were thencreated for items to vary within individuals and to be completely uncorrelated withone another. An additive constant was then created for each item so that the constantsvaried from item to item but were constant across participants. The additive constantswere arbitrarily set to 1 for x , 4 for x , 9 for x , 5 for x , and 3 for x It should be noted that the inclusion of the additive constants does not impact the results of these analy-ses, as reliability is dependent on score variance and measures of dispersion are notimpacted by additive constants. In regards to reliability, essential tau-equivalence and 2006 Sage Publications. All rights reserved. Not for commercial use or unauthorized distribution.
Educational and Psychological Measurement Variance/Covariance Matrix for Heuristic Data
tau-equivalence are essentially the same thing. Item scores (x , x , . . . , x ) were cre- ated by applying Equation 4. This procedure created a data set that perfectly meets theassumption of essential tau-equivalence.
To test the impact of the violation of the assumption of tau equivalence on reli- ability, two additional variables were created. These variables (x and x ) were cre- ated using the same error scores and initial true scores as Item x ; however, the true scores for x and x differed from the true scores for Items x through x by a multipli- cative constant. These items were created using Equation 5, where b = 5 and b = 10.
This resulted in items that were congeneric to the original five items but violated tau-equivalence. The variance/covariance matrix for items x through x is presented in The following analyses were conducted using Amos (Arbuckle, 2003), an SEM software program with a graphical, user-friendly interface. Initially, Items x through x were subjected to a reliability analysis using the tau-equivalent measurement model. Although the correlation between X and T was not explicit in the model, Amosallows one to select “standardized estimates” and “all implied moments” as outputoptions. This produces a correlation matrix between all latent and observed variablesincluded in the model. The correlation between X and T using OLS as a method of esti-mation was squared to provide a reliability estimate. As shown in Table 2, this resultedin a tau-equivalent reliability of .91. When this same set of item scores was subjectedto a reliability analysis using a congeneric measurement model, it also resulted in areliability estimate of .91. Because the items scores are perfectly tau-equivalent to oneanother, and because the tau-equivalent model is a special case of the congenericmodel, the reliability estimates obtained by either method are identical.
Because the tau-equivalent and congeneric measurement models are nested mod- els, the difference in fit of these two models can be obtained by using maximum likeli-hood as the method of estimation and looking both at the differences in fit indices andat the change in χ2, which is obtained by simply subtracting the χ2 and degrees of free-dom of the congeneric model from the tau-equivalent model. The fit indices shown inTable 2 show that both the tau-equivalent and congeneric models provide an excellentfit to the data, with CFIs and GFIs greater than .9 and RMSEA values less than .08. In 2006 Sage Publications. All rights reserved. Not for commercial use or unauthorized distribution.
Model Fit and Reliability Estimates for Heuristic Data
Note: GFI = goodness-of-fit index; CFI = comparative fit index; RMSEA = root mean square error ofapproximation.
fact, given that the data were constructed to fit the model, the fit indices indicatenear-perfect fit. Additionally, the difference in χ2 between these two models is neitherlarge nor statistically significant. Both models fit the data equally well; therefore, onemight choose to select the more restrictive, parsimonious tau-equivalent estimate overthe congeneric estimate.
Next, Item x was replaced by Item x and the same sets of analyses were run again.
Item x is identical to Item x , save that the true score of x differs from the true score of x by a multiplicative constant of 5. As seen in Table 2, the tau-equivalent measure of reliability (.76) is substantially lower than the congeneric measure of reliability (.97).
The fit indices suggest that the data fits the congeneric model rather well but has a poorfit with the tau-equivalent model. Additionally, the χ2 difference in fit between thesemodels is both large and statistically significant; therefore, the congeneric modelappears to be the best fit for the data. Had one simply used Cronbach’s alpha withouttesting whether the data met the tau-equivalence assumption, one would have underes-timated the reliability of the item scores.
Finally, Table 2 also shows the results of the reliability analyses using x instead of x . The true score of x differs by a multiplicative constant of 10 from the true score of Item x . Again, the fit indices and the χ2 difference test agree that the data best fit the congeneric model. As shown, were one to use Cronbach’s alpha, one might errone-ously assume that a near-perfectly reliable congeneric measure is made up of almosthalf error variance.
Factors Affecting Coefficient Alpha’s Underestimation of Reliability
As demonstrated, coefficient alpha underestimates the reliability of test scores when the test violates the assumption of tau-equivalence. Specifically, the larger the 2006 Sage Publications. All rights reserved. Not for commercial use or unauthorized distribution.
Educational and Psychological Measurement violation of tau-equivalence that occurs, the more coefficient alpha underestimatesscore reliability. Both the present example and previous work (Raykov, 1997b) havedemonstrated that the presence of even a single item that is not tau-equivalent to theother items can have a dramatic impact on the accuracy of coefficient alpha; however,the impact that violating the assumption of tau-equivalence can have is also dependenton a number of other factors.
All other things being considered equal, tests with a greater number of items are less vulnerable to underestimation when tau-equivalence is violated than tests withonly a small number of items (Raykov, 1997b). This is due to the fact that, when asingle item violates tau-equivalence, the proportion of true score variance that is con-generic to the other item true scores is smaller when one has a greater number of itemsthan when one has fewer items.
Because the tau-equivalence model assumes that items are measured on the same scale, examining item standard deviations may be of some utility. If the standard devi-ations of item scores composing a test are vastly different from one another, they arelikely to be being measured on different scales. Such a comparison might be made byconstructing confidence intervals about item standard deviations, and visually exam-ining them for equivalence. If the items standard deviations were not equivalent, onemight be alerted that the data may be failing the assumption of tau-equivalence. Itshould be noted that the standard deviation of an item could be impacted by both vari-ance in the true score and variance in the error term associated with the item. An exam-ination of the standard deviations of item true scores (as opposed to item observedscores) would give a better estimate of the degree to which tau-equivalence is violated.
Because item true scores are not typically known, calculating the standard deviationsof the item true scores is not likely to be feasible.
Finally, tests that use multiple response formats across items are more likely to vio- late the assumption of tau-equivalence than those that do not. For example, the itemtrue scores from several true-false items are likely to be measured on a different scalethan the true scores of items that are scored on a 6-point Likert-type scale. The use of dif-ferent response formats typically indicates that the items are being measured on differentscales, and is likely to result in different true-score standard deviations from item to item.
Applied example. Whereas the heuristic example demonstrated how coefficient alpha underestimates reliability when the assumptions of tau-equivalence are violated,an example using actual data may also be instructive in providing an example of how aconfluence of the previously discussed factors can impact the accuracy of coefficientalpha. The following data are from a study conducted by Graham and Conoley (2006)using the Dyadic Adjustment Scale (DAS; Spanier, 1976). The DAS is a commonlyused measure of relationship quality, designed for use with cohabiting couples. TheDAS is a 32-item self-report measure in a variety of response formats whose sum resultsin a number from 0 to 151, with a higher number denoting greater relationship quality.
Whereas the total score from the DAS is most often used in applied research, the testdeveloper originally divided the scale into four subscales. One of these subscales, Affec-tive Expression, consists of 4 items inquiring about levels of agreement between 2006 Sage Publications. All rights reserved. Not for commercial use or unauthorized distribution.
Variance/Covariance Matrix for Dyadic Adjustment Scale (DAS)
Husband Data From Graham and Conoley (2006)
spouses on demonstrations of affection and sex relations (both on 6-point Likert-typescales) and about whether the couple has disagreed recently as a result of being too tiredfor sex and not showing love (both dichotomously scored yes–no). A recent reliabilitygeneralization meta-analysis of the DAS indicated a mean Affective Expressionsubscale score reliability of .71 across studies (Graham, Liu, & Jeziorski, in press).
Table 3 presents the variance/covariance matrix of the 4 affective expression items of 60husbands from the study conducted by Graham and Conoley.
Initially, one might notice that the number of items comprising this subscale is relatively small; this makes the alpha reliability of scores from these items more vul-nerable to violations of essential tau-equivalence. One might next notice that two ofthe items are scored dichotomously, whereas the other two items are scored on a6-point Likert-type scale. This use of different response formats makes violation ofthe assumption of tau-equivalence more likely. Next, one might notice that the vari-ances of the two dichotomously-measured items appear to be substantially lower thanthe two items scored on a 6-point scale. Confidence intervals constructed about thestandard deviations of Items 4 (.95 ± .15) and 6 (.99 ± .15) do not overlap with theconfidence intervals constructed about the standard deviations of Items 29 (.47 ± .07)and 30 (.42 ± .06).
In all instances, this measure gives the appearance of violating the assumption of essential tau-equivalence. As such, it would be expected that coefficient alpha wouldprovide a lower estimate of reliability than a congeneric measure. Using both SPSSand the tau-equivalent SEM model described above, these data have a Cronbach’salpha of .72. Following the procedures for estimating congeneric reliability, an esti-mate of .83 is obtained. The data fit the congeneric model (GFI = .925; CFI = .865;RMSEA = .288) better than the tau-equivalent model (GFI = .761; CFI = .463;RMSEA = .363). Additionally, the congeneric model had a statistically significantlybetter fit than the tau-equivalent model, ∆χ2(3) = 32.1, p < .001. Across studies, theAffective Expression subscale of the DAS has been shown to consistently result inscores with lower reliability than scores on the other subscales (Graham et al., in press).
It appears that this underestimation may be in part due to the fact that these scores vio-late the assumption of tau-equivalence. In this example, Cronbach’s alpha underesti-mated the reliability of these scores by at least 10%, because the subscale better fitsa congeneric model! 2006 Sage Publications. All rights reserved. Not for commercial use or unauthorized distribution.
Educational and Psychological Measurement Discussion
The use of coefficient alpha to estimate reliability is often taken for granted, with little thought put into understanding the assumptions required for the obtained reli-ability estimate to be accurate. As a result, the reliability of published data is oftenneedlessly underestimated. Nearly all classical general linear procedures require thatcertain assumptions be met (normality, homogeneity of variance, etc.), yet these pro-cedures are routinely applied even when the basic assumptions have not been met(Wilcox, 1998). Wilkinson and the American Psychological Association (APA) TaskForce on Statistical Inference (1999) addressed this by stating simply that “you shouldtake efforts to assure that the underlying assumptions for the analysis are reasonablegiven the data” (p. 598). Although this statement may seem overly simplistic, the factremains that many students and researchers in education and psychology are unawareof many of the assumptions required by a given statistical procedure, are unaware ofhow to test those assumptions, or are unaware of acceptable alternatives should thoseassumptions not be met.
A survey of graduate programs in psychology reported that only 27% of programs reported that most or all of their students can appropriately apply methods of reliabil-ity measurement to their own research (Aiken et al., 1990). Aiken and colleagues(1990) concluded that this deficiency in understanding basic measurement concepts“opens the door to a proliferation of poorly constructed ad hoc measures, potentiallyimpeding future progress in all areas of the field” (p. 730). The assumption of essen-tially tau-equivalence in reliability is rarely discussed out of measurement and SEMcircles, and even the measurement literature rarely (if ever) considers the assumptionsof essentially tau-equivalence when reporting reliability. The case of tau-equivalencein the field of measurement is particularly perplexing when one considers other proce-dures commonly used in the development of measures. Whereas the exploratory andconfirmatory factor analytic procedures used in determining measure item composi-tion assume the more general case of congeneric items, the statistic most often used toestimate the reliability of the resultant item groupings assumes tau-equivalence. Inshort, the factor models and measurement models do not match! The procedures discussed here can be easily replicated using Amos or any other commonly available SEM software package. The use of SEM techniques does, how-ever, require large sample sizes. Although this is likely to limit this technique’s every-day use, it is amenable to the vast majority of psychometric studies which typically uselarge sample sizes. Certainly, examining whether one’s data meets the assumptions oftau-equivalence should be an important step in the initial development of any measure,as should the calculation of a congeneric estimate of reliability should the data fail thatassumption.
The present discussion is not intended to advocate for the use of a congeneric mea- sure of reliability over coefficient alpha in all cases. Measurement, like other areas ofscience, rewards parsimony. Because the tau-equivalent model estimates fewerparameters than the congeneric model, there are more opportunities to falsify the tau-equivalent model. The more falsifiable a model is, the more rigorous and persuasive it 2006 Sage Publications. All rights reserved. Not for commercial use or unauthorized distribution.
is considered if it is not subsequently falsified (Mulaik et al., 1989). Therefore, if bothmodels fit reasonably well, a tau-equivalent estimate of reliability is preferable over acongeneric estimate. If the congeneric model fits appreciably better, a congeneric esti-mate of reliability is preferable.
It is hoped that the present discussion provides an accessible framework for under- standing and testing the tau-equivalence assumption of Cronbach’s alpha and for cal-culating a congeneric estimate of reliability should that assumption fail. Testing the fitof the various measurement models in a hierarchical fashion is a simple matter giventhe current availability and ease of use of SEM software packages. Researchers areencouraged to always test whether their data are essentially tau-equivalent whendeveloping a measure and to use the appropriate measurement model to estimate reli-ability. Researchers with sufficiently large samples are also encouraged to utilize theseprocedures, even if the psychometric properties of their data are not the primary focus.
This is particularly important if a small number of items compose the measure, if itemsutilize different formats, or if items have largely different standard deviations. The use ofthese procedures can help ensure that the assumptions required by commonly used reli-ability estimates are not violated and that an accurate estimate of reliability is obtained.
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