Fluent Essays

Please see attached - need it in 8 hours. There are 2 parts. APA citation is a must for all parts. PART 1 Read the attached article and answer the following questions –(MINIMUM 300 WORDS) • Why are ordinal (or Likert) scales typically the form of data collected when studying human behavior within the fields of education, management, psychology, and related social and behavioral sciences? • Describe the differences found when using a normal-curve (NC) transformation versus using Likert scales for the psychometric analysis of diversity measures with entrepreneurial teams. • In the authors evaluation of three different reliability estimates (standard deviation, error variance, and unidimensionality), which was found to the preferred method and which was the least preferred? Why? • Be sure to use scholarly references to back your assertions and cite! PART 2 Responses needed for these 2 discussions (minimum 150 words for each discussion) Discussion 1 Likert scales are very insignificant when it comes to studying human behavior in various fields like education, behavioral sciences, and psychology for different reasons. The scale is ordinal primarily because of its association with discrete values like 1,2 and 3, just to mention a few (Wu & Leung, 2017). The data collected in this case constitute continuous values in a given interval. It is important to mention that considering the Likert scale, interval value is never accepted. In the context of studying human behavior is critical to use the ordinal or Likert scale since the associated data resonate with the nature of humans engagements. The attitudes and related opinions are highly critical sources of data that require an ordinal scale to collect. It follows that the objectivity of acquiring viable and feasibility information based on the discrete values is very high and significantly possible. Most of the respondents dealing with the Likert scale are capable of comprehending the implications of this psychometric scale holistically. Data collected from using the scale is very pragmatic and assists immensely in addressing views and opinions (Wu & Leung, 2017). The researchers apply this firm of scale to understand the underlying views and opinions from associated parties effectively. The Likert scale is a very important aspect of explicitly understanding the perspectives that human behavior can adopt. It remains critical to affirm that the scale is essential in acquiring data that is generally relevant. The scale types such as the Guttman scale, Thurstone scale, and Bogardus scale are essential and efficient in ensuring that the ultimate goal of collecting discrete data is met. Various questions are provided when using the Likert scale that serves a major role in promoting authenticity and reliability in various dimensions. The differences that exist between using normal-curve (NC) transformation versus using Likert scales are very vivid and highly pragmatic since they are based on the provided facts. The NC transformation scale is associated with data that are closer or nearer to the underlying values in a situation where the normal curve assumption is upheld (Deng et al., 2015). Another key difference between the nominal curve transformation and the Likert scale is that the associated transformed data is usually less skewed considering their average value or on average. This is a very significant component and factor to consider since it can get utilized in situations where there is a need to measure various diversities. It follows that transformed data is relevant and essential when it comes to measuring different diversities, thereby giving the most reliable and effective outcome. In evaluating the three different reliability estimates that are standard deviation, error variance, and unidimensionality, the most effective and preferred method is the standard deviation. It Is important to mention that standard deviation provides an opportunity to engage in accurate and reliable estimations. The three diversity measures are supposed to embrace effectiveness in the application when it comes to statistical analysis. In this context, adopting an option that provides amicable results and operational excellence in various dimensions remains a very subjective step. Standard deviation is an alternative that is most preferred since it yields a highly efficient and accurate parameter estimate as oppose to other reliability estimates (Deng et al., 2015). On the other hand, the unidimensionality reliability estimates are least preferred following the nature of estimates that are associated with the same. This reliability estimate compared to the other counterpart options is the least and less reliable. Discussion 2 An ordinal scale is a scale (of measurement) that uses labels to classify cases into ordered classes. Ordinal scale tests are often used in social science settings because of how they can measure behavioral attitudes by asking people to respond to a series of statements topics, in terms of the extent to which they agree or disagree that would tap into the test taker’s intellectual and emotional components of attitudes. Ordinal scales tests, such as the Likert tests, are easy to interpret once the data has been received, and it allows test-takers to provide opinions based on the statement’s description. Likert scales also offer anonymity on self- administered questions to reduce social pressure and desirability bias (McLeod, 2019). The limitation of the tests is the questions in place can have the tendency for the test takers to present themselves in a generally favorable fashion. An example would be if the test is measured based on discrimination; test-takers would lie because they do not want to be seen as bullies or racists. Lovler and Miller (2020) describe Normal Curves (NC) as a theoretical distribution that exists in our imagination as perfect and symmetrical, with the scores concentrated on the middle than the tails. The psychometric analysis article describes those 117 individuals who took the diversity measurement tests, which is about 13 items to measure diversity. Researchers used 4 waves to distribute the tests, only to use the first wave tests because there were not many changes to the individuals’ answers from the other wave distribution. Researchers found that the sample skewness and the distribution curve provided a smaller critical value (Deng, et al., 2014). The Likert scale provided more marginal and well-fitted data by the one-factor model, which is easier to interpret to measure human characteristics. The article describes that the reliability estimates following the transformed data tend to be smaller than the Likert data (Deng, et al., 2014). Based on the descriptions provided, since NC’s tend to be on a hypothetical side and provide “perfect” scores, using the Likert scale can be realistic and provides reliability scores. The test takers can manipulate the Likert tests, but it provides a more measurement ability to convert or transform test scores in a meaningful unit (Lovler and Miller, 2020). See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/273586003 Psychometric Properties of Measures of Team Diversity With Likert Data Article in Educational and Psychological Measurement · May 2014 DOI: 10.1177/0013164414541275 CITATION 1 READS 68 3 authors, including: Some of the authors of this publication are also working on these related projects: Asymptotics View project Estimating equation View project L. Deng Beihang University (BUAA) 6 PUBLICATIONS 38 CITATIONS SEE PROFILE Ke-Hai Yuan University of Notre Dame 140 PUBLICATIONS 3,974 CITATIONS SEE PROFILE All content following this page was uploaded by Ke-Hai Yuan on 12 June 2015. 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Marcoulides and Ke-Hai Yuan Psychometric Properties of Measures of Team Diversity With Likert Data Published by: http://www.sagepublications.com at: can be foundEducational and Psychological MeasurementAdditional services and information for http://epm.sagepub.com/cgi/alertsEmail Alerts: http://epm.sagepub.com/subscriptionsSubscriptions: http://www.sagepub.com/journalsReprints.navReprints: http://www.sagepub.com/journalsPermissions.navPermissions: What is This? - Jul 4, 2014OnlineFirst Version of Record >> by guest on July 5, 2014epm.sagepub.comDownloaded from by guest on July 5, 2014epm.sagepub.comDownloaded from http://epm.sagepub.com/ http://epm.sagepub.com/content/early/2014/07/03/0013164414541275 http://www.sagepublications.com http://epm.sagepub.com/cgi/alerts http://epm.sagepub.com/subscriptions http://www.sagepub.com/journalsReprints.nav http://www.sagepub.com/journalsPermissions.nav http://epm.sagepub.com/content/early/2014/07/03/0013164414541275.full.pdf http://online.sagepub.com/site/sphelp/vorhelp.xhtml http://epm.sagepub.com/ http://epm.sagepub.com/ Article Educational and Psychological Measurement 1–23 � The Author(s) 2014 Reprints and permissions: sagepub.com/journalsPermissions.nav DOI: 10.1177/0013164414541275 epm.sagepub.com Psychometric Properties of Measures of Team Diversity With Likert Data Lifang Deng1, George A. Marcoulides2, and Ke-Hai Yuan3 Abstract Certain diversity among team members is beneficial to the growth of an organization. Multiple measures have been proposed to quantify diversity, although little is known about their psychometric properties. This article proposes several methods to evalu- ate the unidimensionality and reliability of three measures of diversity. To approxi- mate the interval scale required by the measures of diversity, a transformation on the Likert-item scores is proposed. Ridge maximum likelihood is used to deal with the issue of small sample size, and methods for evaluating the significance of the difference of two reliability estimates with correlated samples are also developed. Results with a real data set on entrepreneurial teams indicate that different measures of diversity may correspond to significantly different estimates of reliability. Results also indicate that diversity measures obtained with the transformed data tend to be more unidi- mensional than their counterparts obtained from Likert data. However, diversity mea- sures obtained from Likert data tend to yield greater reliability estimates. Among the three examined measures of diversity, the standard deviation is found to yield greater and more efficient reliability estimates than the others and is thus recommended. Keywords unidimensionality, reliability, normal-curve transformation, ridge structural equation modeling 1 Beihang University, Beijing, China 2 University of California, Santa Barbara, CA, USA 3University of Notre Dame, Notre Dame, IN, USA Corresponding Author: Ke-Hai Yuan, University of Notre Dame, 123a Haggar Hall, Notre Dame, IN 46556, USA. Email: [email protected] by guest on July 5, 2014epm.sagepub.comDownloaded from http://epm.sagepub.com/ Introduction The compositional diversity of team members within an organization has been shown to affect the performance and growth of the organization (Harrison & Klein, 2007; Van Knippenberg, De Dreu, & Homan, 2004). Among various kinds of diver- sity (e.g., demographic, informational, experiential, or personality attributes), not all have been determined to be beneficial to the growth of an organization. Some researchers have indicated that it is merely the differences between team members in terms of their skill level, knowledge, and perspectives that are needed to foster crea- tivity and innovation (e.g., Guzzo & Shea, 1992). However, the findings in the extant literature have not been consistent (Jackson, Joshi, & Erhardt, 2003; Stewart, 2006; Van Knippenberg et al., 2004; Webber & Donahue, 2001) and indicate that our understanding of diversity and its role is still relatively limited. To facilitate a better understanding of diversity, Harrison and Klein (2007) classified diversity into three distinctive types: separation, variety, and disparity. Separation is for the difference in position or opinion among team members; variety is used to describe diversity in expertise, knowledge, or experience; and disparity refers to inequality in status or resources held among team members. Such a classification allows researchers to identify different roles of different types of diversity. A variety of measures have also been proposed to quantify different types of diversity among individuals within a team. According to Harrison and Klein (2007), separation should be measured by either the standard deviation or the average of Euclidean distances, variety should be measured by the so-called Blau’s (1977) index or entropy (Teachman, 1980), and disparity should be measured by the coefficient of variation or the ratio of the average of the absolute differences over the mean. Harrison and Klein (2007) also discussed the type of scales required by each of these diversity measures and emphasized that measuring separation requires the observed data on team members to be at the interval scale, whereas measuring disparity requires the observed data to be at the ratio scale. However, in the study of human behavior within the fields of education, management, psychology, and related social and behavioral sciences, it is extremely difficult to obtain data at the ratio or even interval scales. What are typically obtained are data collected from a survey using questionnaires that are commonly only ordinal or Likert type. Nevertheless, research- ers still regularly apply procedures that require interval-scale data to ordinal data. For example, factor analysis is commonly applied to Likert data for item selection or scale development (Raykov & Marcoulides, 2011). Although such a practice may still yield interpretable results, a better method is to factor analyze the polychoric correlation matrix (Babakus, Ferguson, & Jöreskog, 1987). Given that the observed values in Likert data used to determine the above-mentioned diversity measures are somewhat arbitrary, we propose to transform them to avoid the arbitrariness. The transformation is based on threshold values under the normal curve, parallel to those used in estimating polychoric correlations (Olsson, 1979). We can call it the normal- curve (NC) transformation. Although the transformed data are still limited in number of values, we argue that they are closer to the conditions required by diversity 2 Educational and Psychological Measurement by guest on July 5, 2014epm.sagepub.comDownloaded from http://epm.sagepub.com/ measures than the commonly used Likert data. To see the effect of the transforma- tion, we will study the psychometric properties of several diversity measures when applied to both Likert and transformed data. Unidimensionality and reliability are probably the two most basic psychometric properties one has to consider for any scale or instrument. Unidimensionality implies that the statistical dependence among the items can be accounted for by a single underlying latent trait, and reliability informs about the degree to which the observed individual differences are indicative of true individual differences on the latent dimension of interest. Measures of diversity, especially those for measuring separa- tion with Likert data, are also subject to such properties if they aim to properly cap- ture any underlying trait. In particular, when measurements in a scale are not unidimensional, the empirical meaning of the scale will be different from the mean- ing assigned to it, which will create interpretational confounding (e.g., Anderson & Gerbing, 1982; Bagozzi, 1980; Burt, 1973, 1976). Reliability is equally important because, for measurements with a low reliability index, the observed values of the obtained measurements can be mostly due to random errors. Additionally, because the value of the determined reliability index sets a bound on validity (Allen & Yen, 1979), a high reliability (index) is a necessary condition for high validity (Raykov & Marcoulides, 2011). We hope that by studying the unidimensionality and reliability of different measures of diversity, the inconsistent findings obtained to date on the roles of diversity can be better understood. The methodological development presented in this article was motivated by the need to study the psychometric properties of diversity measures based on 13 Likert items administered to entrepreneurial teams. Because the number of teams plays the role of sample size, which is not sufficiently large, a method to deal with the issue of small sample sizes was also needed especially when using factor analysis to evaluate the unidimensionality of the diversity measures. For such a purpose, we make use of the ridge maximum likelihood (ML) method originally developed in Yuan and Chan (2008). This method has been shown to yield more accurate parameter estimates than the normal-distribution-based maximum likelihood (NML) even for normally distrib- uted data. We also develop methods for evaluating the significance of the difference of two reliability estimates with correlated samples. This enables us to determine whether different measures of diversity correspond to significantly different reliabil- ity estimates. If different diversity measures yield significantly different reliability estimates, then it is better to use the one that corresponds to the greatest reliability. In the next section, the methodological components for studying the unidimen- sionality and reliability of different measures of diversity are given, including the formulations of different diversity measures, the NC transformation, ridge ML, and standard error (SE) for difference of reliability estimates. A real data set with Likert scale and its analysis are presented in the following section. We conclude with a dis- cussion and recommendations. It is important to note that our focus is on the psycho- metric properties (unidimensionality and reliability) of different measures of Deng et al. 3 by guest on July 5, 2014epm.sagepub.comDownloaded from http://epm.sagepub.com/ diversity, not on interrater reliability issues (for further details on interrater reliabil- ity, see Algina, 1978; Schuster & Smith, 2002; Shrout & Fleiss, 1979). Methodology This section first introduces the three diversity measures that will be used in the anal- ysis of the real data. Then, the NC transformation is described. Ridge ML for factor analysis is reviewed next. Formulas for SE of the difference of two reliability esti- mates are developed at the end of this section. These measures and techniques will be used to analyze the real data in the subsequent section. Diversity Measures Let xijk be the score of person k on item j within team i, k = 1, 2, . . . , ni; j = 1, 2, . . . , p; i = 1, 2, . . . , N: Three measures of diversity derived from xijk will be studied. These are the average of absolute distances (aad) among team members, aadij = 2 ni(ni � 1) Xni�1 k1 = 1 Xni k2 = k1 + 1 jxijk1 � xijk2j; ð1Þ the average of absolute deviations from the mean (aadm) of team members, aadmij = 1 ni � 1 Xni k = 1 jxijk � �xijj, ð2Þ where �xij = Pni k = 1 xijk=ni; and the standard deviation (sd) among team members, sdij = ½ 1 ni � 1 Xni k = 1 (xijk � �xij)2�1=2: ð3Þ Two measures for separation were recommended by Harrison and Klein (2007). One is the standard deviation in which the denominator is ni instead of ni � 1: Another is the square root of the average of the squared Euclidean distances (xijk1 � xijk2 ) 2, in which k1 = k2 is not distinguished from k1 6¼ k2: According to Biemann and Kearney (2010), these measures may contain substantial bias due to including terms that are obviously 0 or without correcting for the loss of degrees of freedom. The diversity measure in (1) only includes the absolute distances for differ- ent team members, and degrees of freedom loss are accounted for in (2) and (3). Parallel to the average Euclidean distance (aed) in Harrison and Klein (2007) or Biemann and Kearney (2010), we define aedij as aedij = 2 ni(ni � 1) Xni�1 k1 = 1 Xni k2 = k1 + 1 (xijk1 � xijk2 ) 2 #1=2 : 4 Educational and Psychological Measurement by guest on July 5, 2014epm.sagepub.comDownloaded from http://epm.sagepub.com/ Because aedij is proportional to sdij (see Hays, 1981) and any results of reliability and unidimensionality analysis of aedij would be identical to those of sdij, we do not separately examine aedij in this article. As we were not able to locate any references in the literature in which the aadmij in (2) or an index that is proportional to aadmij has been proposed to measure diver- sity, aadmij can be regarded as a new measure. The psychometric properties of the three measures, aad, aadm, and sd, will be examined through real data analysis in the following section. Quantities in the form of the average of absolute distances (e.g., aad) are not pre- sented as stand-alone measures in either Harrison and Klein (2007) or Biemann and Kearney (2010). Instead, they are divided by the team mean score for measuring dis- parity. Another measure for disparity recommended by Harrison and Klein (2007) is the coefficient of variation. Since these measures require the observed data to possess the properties of ratio scale, they may not be applicable to Likert data and will not be studied in this article. Similarly, variety will not be measured through Likert data and neither do we study Blau’s index or the entropy. Normal-Curve Transformation Since all three measures of diversity (aad, aadm, sd) are obtained by arithmetic oper- ations, they are ideally applicable to data that are of interval scale (Harrison & Klein, 2007). However, as indicated previously, measurements in the social and behavioral sciences are typically Likert or ordinal scale. To approximate interval scales, we pro- pose a transformation to Likert data in this subsection. With a total of Nt = PN i = 1ni individual observations and c categories for a given item, let q̂l be the proportion of observations 1 for category l: Following the conven- tion of polychoric correlations (Olsson, 1979), we may assume that, for each observed xijk , there is an underlying continuous variable zijk ;N (0, 1) such that xijk = l whenever zijk belongs to the interval (hl�1, hl�, where h0 \h1 \ � � � \hc are threshold values to be estimated. This implies that the probability for xijk = l is given by ql = F(hl ) � F(hl�1), l = 1, 2, . . . , c, where F(�) is the cumulative distribution function of z;N (0, 1), with h0 = � ‘ and hc = ‘: Thus, the marginal maximum likelihood estimate of hl is given by ĥl = F �1( Xl t = 1 q̂t ), l = 1, 2, . . . , c � 1: ð4Þ Based on this underlying NC assumption, we propose a transformation to the Likert xijk by yijk = (ĥl�1 + ĥl )=2 if xijk = l, l = 1, 2, . . . , c: ð5Þ Deng et al. 5 by guest on July 5, 2014epm.sagepub.comDownloaded from http://epm.sagepub.com/ Notice that there are only c � 1 finite values of ĥl in (4), and we cannot use ĥ0 = � ‘ or ĥc = ‘ because they will result in yijk = � ‘ when xijk = 1 or yijk = ‘ when xijk = c: We further propose to use ĥ0 = F �1(:5=Nt ) and ĥc = F �1(1 � :5=Nt ): ð6Þ The proposed values in (6) are equivalent to assigning a value of .5 to cells with zero number of observations in the analysis of contingency tables, because we can think of an extra category xijk = 0 below xijk = 1 and another extra category xijk = c + 1 above xijk = c, and both had zero number of observations. The proposed values in (6) are also similar to the so-called continuity correction in applying the central limit theo- rem to categorical data (Feller, 1945), where a step of .5 is used when jumping from 1 to the next whole number. Notice that the correction in (6) is for yijk to avoid being �‘ or ‘ whenever xijk = 1 or c: If the nominal number of categories is c but only c � 1 or fewer number of categories are observed, we may simply treat the unobserved categories in the middle as having probability of zero by just applying the correction to the end points of xijk: We need to note that the transformed yijk do not possess the property of interval scales, although they avoid the arbitrary nature of Likert data that assign consecutive whole numbers to ordered categories. Closely related to polychoric correlation, the rationale of the transformation in (5) depends heavily on the assumption of a normal curve underlying the observed frequencies. If the NC assumption holds, the yijk obtained by the NC transformation determined by equations (4), (5), and (6) is sim- ply the middle point of the interval zijk belongs, and thus represents the best predic- tion of the true value of zijk in the sense of smallest absolute mean difference. Applying each of the three diversity measures, aadij, aadmij, and sdij, to the trans- formed yijk yields three more measures of diversity. In the next section, their reliabil- ity and unidimensionality are examined, and the results are contrasted with those obtained based on Likert data. Ridge Maximum Likelihood for Factor Analysis With Small Sample Sizes As indicated in the previous section, the number of teams, N , plays the role of sample size when evaluating the psychometric properties of the diversity measures aad, aadm, and sd. Since it can be expensive to have a large N , we use ridge ML for factor analysis of the diversity measures in (1) to (3) when studying their unidimensionality. Unless all the ni are sufficiently large, the diversity measures in (1) to (3) cannot be regarded as normally distributed. As such, we expect ridge ML to work better than NML when factor analyzing the diversity measures. Let S be a sample covariance matrix of size p, and we are interested in modeling S = E(S) by a confirmatory factor model S(u) = LFL 0 + C, ð7Þ 6 Educational and Psychological Measurement by guest on July 5, 2014epm.sagepub.comDownloaded from http://epm.sagepub.com/ where L is a factor loading matrix, F is a factor correlation matrix, and C is a diago- nal matrix of measurement errors/uniquenesses. The widely used NML procedure for covariance structure analysis is to minimize FML(S, S(u)) = tr½SS�1(u)�� log jSS�1(u)j� p for parameter estimation. Let a . 0 be a small number and Sa = S + aI, with I being the identity matrix. The ridge ML developed in Yuan and Chan (2008) is to estimate ua by minimizing FML(Sa, S(ua)), and let the estimates be denoted by ûa: The corresponding estimates û for u are obtained by subtracting a from each of the elements of ûa corresponding to the diagonal elements of C, leaving the other elements of ûa unchanged. Standard errors of û are obtained by a sandwich-type covariance matrix, which accounts for the unknown underlying population distri- bution of the involved diversity measure. As for overall model evaluation, Yuan and Chan (2008) showed that, unless a = 0, TML = (N � 1)FML(Sa, S(ua)) does not asymptotically follow the nominal chi-square distribution x2df even if data are normally distributed. They developed a rescaled statistic TRML and an adjusted sta- tistic TAML: Parallel to the development for NML in Satorra and Bentler (1994), TRML asymptotically follows a distribution whose mean equals df , and TAML asymptotically follows a distribution whose mean and variance equal those of the approximating distribution. Since the details of ridge ML have already been described in Yuan and Chan (2008), no further elaboration is given here. Our pur- pose is to apply ridge ML to evaluate the unidimensionality of each of the three measures of diversity in (1) to (3) and to determine whether the corresponding sample covariance matrix can be reasonably fitted by a one-factor model. Following the recommendation of Yuan and Chan (2008), a = p=N is used in applying the ridge ML. In order to fully justify applying a factor analysis to each of the diversity measures, we do not need to assume that each of aadij, aadmij, or sdij is identically distributed across i = 1, 2, . . . , N: The development in Lee and Shi (1998) implies that the vector di = (aadi1, aadi2, . . . , aadip) 0 does not need to have the same population covariance as i varies. Since for both reliability and unidimensionality the analysis is based on the sample covariance matrix S of the corresponding diversity measures with the assump- tion E(S) = S, our study of the psychometric properties of di is for the population rep- resented by the sample di, i = 1, 2, . . . , N: We will further discuss this point in the concluding section. Standard Error for Difference of Two Reliability Estimates With Correlated Samples Among the many available estimates of reliability for equally weighted composite scores, coefficient alpha is most widely used in practice even though it can over- or underestimate the population reliability (Raykov, 1997). Another popular estimate is coefficient omega defined through the factor loadings and error variances by fitting Deng et al. 7 by guest on July 5, 2014epm.sagepub.comDownloaded from http://epm.sagepub.com/ the sample covariance matrix to a one-factor model (McDonald, 1999). Both are applicable when evaluating the reliability of the different diversity measures. Our interest is whether different diversity measures will yield significantly different relia- bility estimates. Thus, we need to have an estimate of the SE of the difference of two estimates of alpha or omega. When the two estimates are independent, the variance of the difference of the two estimates is simply the summation of the variances of the two estimates of alpha or omega. However, with respect to the three diversity measures, the variance or SE of the difference of two estimates of alpha or omega depends on their correlation. Since the SE for the difference of two reliability esti- mates with correlated samples will facilitate comparison of reliabilities in other con- texts, and the literature to date does not contain such a development, we provide more details for obtaining consistent SEs of the difference of two estimates of alpha and omega, respectively. We also present the necessary notation and formulas for calculating the SEs. The complete details leading to the calculation formulas are given in Appendices A and B. Let S = (sjk ) be a sample covariance matrix of size p, and s = vech(S) be the vector by stacking the elements in the lower-triangular part of S: Then, with p� = p(p + 1)=2, s is a vector of p�31, and the sample coefficient alpha is given by â = g(s) = p p � 1 (1 � Xp j = 1 sjj= Xp j = 1 Xp k = 1 sjk ) = p p � 1 (1 � a0s b0s ), where a is a p�31 vector whose elements are 1 corresponding to sjj and 0 elsewhere; and b is also a p�31 vector whose elements are 1 corresponding to sjj and 2 corre- sponding to sjk when j 6¼ k: For example, at p = 3, s = (s11, s21, s31, s22, s32, s33)0, a = (1, 0, 0, 1, 0, 1)0, and b = (1, 2, 2, 1, 2, 1)0: We need to have the Jacobian matrix or the matrix of derivatives of g(s) with respect to the elements of s, and it is given by _g(s) = � p p � 1 ½ 1 b0s a � a0s (b0s) 2 b�: With s1 = vech(S1) and s2 = vech(S2) from two correlated samples, standard error for â2 � â1 = g(s2) � g(s1) also involves the variance-covariance matrices of s1 and s2: Denote these by G11 = Var( ffiffiffi n p s1), G22 = Var( ffiffiffi n p s2), and G12 = Cov( ffiffiffi n p s1, ffiffiffi n p s2), where n = N � 1: These are consistently estimated by their sample counterparts, with details given in Appendix A. With the introduced notation, the result given in Appendix B implies that ffiffiffi n p ½(â2 � a2) � (â1 � a1)� is asymptotically normally dis- tributed with mean zero and variance consistently estimated by t̂ 2 a = _g 0(s1)Ĝ11 _g(s1) + _g 0(s2)Ĝ22 _g(s2) � 2 _g0(s1)Ĝ12 _g(s2): ð8Þ It follows from (8) that the SE of (â2 � â1) is consistently estimated by t̂a= ffiffiffi n p , which will be used in the next …